ABSTRACT A PROCEDURE FOR.MEASURING THE SEPARATE ‘EFFECTS OFIMAN-CONTROLLED INPUTS AND WEATHER ON YIELDS-- APPLIED T0 GRAIN SORGHUM YIELDS by 00 if Fred H. Abel There were two principal objectives in this study. The first was to estimate how changes in inputs have affected yield, and the second was to determine the effect of specifying alternative models. A single equation model was developed. The parameters were estimated by least squares regression analysis. The dependent variable was yield of grain sorghum per acre. There were 645 observations; observations on 129 counties in each of the agricultural census years, 1939-1959. Three kinds of independent variables were included--man-controlled input variables, dummy (0, 1) variables, and weather variables. The seven man-controlled input variables were: (1) Percent of grain sorghum acreage irrigated, (2) dollars spent on gas and oil per acre of cropland harvested, (3) pounds of fertilizer nutrients applied per acre of grain sorghum, (4) ratio of acres fallowed to acres of cropland har- vested, (5) average acres of grain sorghum per farm harvesting grain sorghum, (6) number of tractors per acre of cropland harvested, and (7) per acre value of land (to measure the interaction effects of land with technology). 2 Fred H. Abel Two sets of dummy (0, 1) variables were included-27 variables to represent the crop reporting districts and 4 variables to represent years. Four gets of weather variables were included: (1) Preseason pre- cipitation, (2) season precipitation, (3) season temperature, and (4) season interaction (temperature times precipitation). Three forms of geaggn weather variables were considered in detail: (a) A weather variable for each week of the growing season for each weather factor, (b) a polynomial of seventh degree for each weather factor, and (c) a season total variable for each weather factor. Estimates of the effect on average yield of changes in the level of the independent variables were obtained from the "complete" equation. This equation contained the seven man-controlled input variables, the 27 dummy variables for crop reporting districts, the four dummy variables for years, the preseason precipitation variable, the 23 season precipitation variables, the 23 season temperature variables, and the 23 season interaction variables. On the basis of this equation, it was estimated that of the 1,146 pound per acre increase in yield between 1939 and 1959, 27.4 percent was explained by changes in the level of the explicit man-controlled inputs, 46.1 percent by changes in the level of implicit man-controlled inputs, and 26.5 percent by changes in weather. Of the increase due to changes in explicit man-controlled inputs, almost all is due to changes in two inputs-- fertilizer, irrigation, and their interaction with land (value of land). Changes in weather during the growing season accounted for 85.4 percent of the total weather effects. Shifts in the location of production, 1939 to 1959, caused average yield to increase 50 pounds. Three hundred and eight other equations were estimated to estimate the effects of specifying alternative models. The fig for the "complete" equation was .855. When polynomial weather variables were substituted for 3 Fred H. Abel the weekly variable, F2 was .821. When season total weather variables were substituted for the weekly variables, §2 was .786. Omitting any set (man- controlled inputs, years, crop reporting district, season precipitation, season temperature, or season interaction) of variables from the "complete" equation caused R2 to decrease significantly. In almost all cases the magnitude of the coefficients remaining in the equation was affected. In some cases the level of significance and sign were also affected. A PROCEDURE FOR MEASURING THE SEPARATE EFFECTS OF MAN-CONTROLLED INPUTS AND WEATHER ON YIELDS-- APPLIED TO GRAIN SORGHUM YIELDS \\ By («5” Fred H§>Abel A THESIS Submitted to Michigigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Economics 1967 ‘ ACKNOWLEDGMENTS I would like to express my deep appreciation to Dr. Robert L. Gustafson for his many helpful suggestions, patience, and thoughtful advice during my course of study and preparation of this dissertation. I would also like to thank Dr. Lester Manderscheid for his advice, suggestions, and encouragement during my stay at Michigan State and during the preparation of this dissertation. Appreciation also goes to the personnel of the Farm Production Economics Division, Economic Research Service, United States Department of Agriculture, for giving me assistance and encouragement during the final stages of preparing this dissertation. Thanks are also due to Mrs. Kathy West for the many hours she spent on preparing the data, to Miss Marguerite Miller and.Mrs. Laura Flanders for their many hours spent in getting the computer to work for me, and to Mrs. Claudia Sitch for typing and retyping the many early drafts of this dissertation. My deep appreciation and gratitude to my wife for her patience, understanding, and encouragement. To my wife, and my sons Paul, Steven, and Mark, who make the whole thing worthwhile. ii TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES Chapter I. INTRODUCTION Need for Study Understanding the physical relationship Predicting crop yields Explaining changes in yields Using Aggregate Data Objectives Review of Literature Grain sorghum History of sorghum in the United States Grain sorghum botany Grain sorghum culture Weather studies Actual weather variable techniques Yield index techniques Functions used to explain changes in yields II. THE MODEL The Model The Dependent Variable The Independent Variables Man-controlled input variables Acres of grain sorghum harvested per farm harvesting grain sorghum Number of acres of cropland harvested per tractor or number of tractors per acre of cropland harvested Dollars spent on gas and oil per acre of cropland harvested Acres of grain sorghum irrigated as percentage of total acres of grain sorghum harvested Per acre value of land iii Page ii viii xii \OOQQm-PWNNH H 27 27 27 28 PROCEDURE Conclusion THE MODEL.-—Continued Man-hours of labor per acre Acres cultivated summer fallow Ratio: Acres fallowed to acres of cropland harvested- Pounds of commercial plant nutrients applied per acre of grain sorghum Dummy variables Years Crop reporting districts e Growing seasons Weather variables Weekly estimates technique Weather polynomial technique Seasonal total technique Conclusion The Unit of Observation, the Observations and the Data-—- The unit of observation The observations The data Man-controlled inputs and yields Dummy variables Weather variables Change in Data Procedure Used to Select Equations (Submodels) to Estimate Factors affecting changes in yields Consequences of leaving variables out of the equation-— Man-controlled inputs Year variables Crop reporting districts Growing seasons Preseason precipitation Polynomial weather variables weekly weather variables Location variables Procedure Used to Present Results Chapter IV Chapter V Chapter VI Chapter VII iv Chapter Page IV. THE "COMPLETE" EQUATION 61 The Coefficients 62 Man-controlled input variables 62 Percent irrigated 62 Acres per farm 62 Tractors per acre 62 Dollars spent on gas and oil 63 Value of land 63 Ratio acres fallowed to acres of cropland harvested--- 63 Fertilizer 63 Years 64 Crop reporting districts 65 Preseason precipitation 66 Season precipitation 66 Season temperature 67 Season interaction 67 Explaining the Change in Yield 67 Explaining the Changes in Yield Over Time 71 The change in yields, 1939-l944 74 The change in yields, 1944-1949 74 The change in yields, 1949-1954 75 The change in yields, 1954-1959 75 The change in yields, 1939-1959 76 Explaining Cross-sectional Differences in Yields 76 Effect of Shift in Acres on Average Yields 80 Independence of residuals 81 Comparison of Alternative Models 83 Conclusion 86 V. MAN-CONTROLLED INPUTS 87 Man-controlled Input Models 87 Simple Correlations 88 Percent irrigated 89 Acres per farm 91 Acres per tractor and tractors per acre 92 Dollars spent on gas and oil 93 Value of land 93 Labor 94 Acres fallowed and ratio: Acres fallowed to acres of cropland harvested 94 Fertilizer 95 Effects on Coefficients 95 Acres of grain sorghum harvested per farm harvesting grain sorghum 95 Dollars spent on gas and oil 98 Acres of cropland harvested per tractor 100 Number of tractors per acre of cropland harvested---—--—- 101 V Chapter V. MAN-CONTROLLED INPUTS.--Continued Man-hours of labor Percent acres of grain sorghum irrigated Value of land Acres fallowed Ratio: Acres fallowed to acres of cropland harvested-—- Fertilizer Conclusion VI. LOCATION AND YEARS Location Variables Year Variables VII. WEATHER VARIABLES weather Coefficients in Alternative Models Preseason precipitation Weekly weather variables Polynomial weather variables Polynomial precipitation variables Polynomial temperature variables Polynomial interaction variables Temperature and precipitation polynomial variables--- Season total weather variables Season total precipitation variables Season total temperature variables Weekly Estimates from Alternative Sets of Weather Variables Conclusions The Polynomial Models Precipitation polynomials Temperature polynomials Interaction polynomials Conclusion VIII. SUMMARY AND CONCLUSIONS Results Other Conclusions Model Flexibility Functional form Time series-cross section data Aggregation of data Type of crop considered Number and kind of weather variables vi 173 Page BIBLIOGRAPHY 175 APPENDIX A: THE DATA: PROBLEMS, SOURCES AND METHODS OF ESTIMATION-- 188 The Dependent Variable 188 The Independent Variables 188 The Basic Data 233 APPENDIX B: RESULTS OF REGRESSION ANALYSIS 234 APPENDIX C: TABLE OF POWERS 305 vii Table 1.1 o 3-1 0 3-20 3-30 3’50 3-60 3-70 4‘10 4‘20 4'30 4-6. 5-10 5'20 5-30 LIST OF TABLES Production of Feed grains in U. 8., 1956-63 Acres Harvested and Production of Grain Sorghum in U. S. and in 129 Counties Included in Study Average Yield of Grain Sorghum List of Submodels Containing Only Man-controlled Inputs as Independent Variables List of Sets of Equations Omitting Some Man-controlled Input Variables List of Equations Containing A Complete Set of MCIV ---------- The "Nearly Complete" Equations List of Submodels Containing Weather Polynomials of Varying Degrees Changes in Yields and Levels of Factors Between Years ------ Effects of Changes in Level of Factors on Changes in Yields-- Deviation in Average (Over Time) Level of Man—controlled inputs for Crop Reporting Districts Relative to Level in District 19 Yields of Crop Reporting Districts Relative to Crop Reporting District 19 and Factors Explaining the Differences-------- Effect of Shifts in Location of Production Between Selected Years fig for Equations Using Corrected Data Rz's for Models Containing Only Man-controlled Input Variables as Independent Variables Simple Correlations of the Man-controlled Input variables--- Basic Data Concerning Technology Variables Included in the Analysis viii Page 43 51 52 52 53 56 72 73 78 79 81 84 88 9O 96 5-90 5-10. 5-11. 5-12. 5-13 0 6-1. 6-2A o 6-2B. 6-20 . 7-40 7-5 0 Coefficients for Acres of Grain Sorghum Per Farm Variable--- Coefficients for Dollars Spent on Gas and Oil Coefficients for Tractors Per Acre Effects of Multicollinearity of Years with Labor on the Estimated Labor Coefficient Effect of Multicollinearity of Years with Labor on the Estimated Coefficients for Years Total Man WOrk Units Per Acre of Grain Sorghum Coefficients for Percent Irrigated Coefficients for Value of Land Variable Coefficients for Ratio: Acres Fallowed to Acres of Cropland Harvested Coefficients for the Fertilizer Variable a? for Pairs of Equations Differing Only by the Presence of Either Growing Seasons or Crop Reporting Districts--—-—- Coefficients for Crop Reporting Districts--Equations Containing Weekly weather Variables Coefficients for Crop Reporting Districts--Equations Containing Season Total Weather Variables Coefficients for Crop Reporting Districts--Equations Containing Seventh Degree Polynomial weather Variables---- Coefficient for Year Variables Coefficients for Preseason Precipitation Variables for "Nearly Complete" Equations Coefficients for Weekly Precipitation Variables from the "Nearly Complete" Equations Coefficients for Weekly Temperature Variables from the "Nearly Complete" Equations Coefficients for weekly Interaction Variables Average Temperature and Precipitation for Each Week in Growing Season ix Page 97 99 101 102 104 105 107 110 113 115 118 119 121 123 128 131 133 134 135 137 Table 7-6 0 7-7 0 7-8. 7-9. 7-10. 7-11. 7-12 0 7-13. 7-14. 7-15 0 7-16. 7-17 0 7-18 0 7-19. A-l o A-2. [1‘3 0 Net Effects of One Additional Inch of Precipitation-By weeks for Several Alternative Equations Net Effects of One Additional Degree of Temperature-By weeks for Several Alternative Equations Coefficients for Polynomial Precipitation Variables Coefficients for Polynomial Temperatures Coefficients for Polynomial Interaction Variables Coefficients for Season Total Weather Variables weekly Estimates of Effects of a Unit Increase in Precipitation or Temperature from Models Including weekly or Seventh Degree Polynomial Variables Coefficients for Precipitation and Temperature Polynomial Variables, Equation 301 1'12 for Models Containing Precipitation Polynomials of Varying Degrees Precipitation Polynomial, Coefficients from Selected Equations- —2 R for Models Containing Temperature Polynomials of Varying Degrees Temperature Polynomial, Coefficients from Selected Equations--— R2 for Equations Containing Interaction Polynomials of Varying Degrees Interaction Polynomial, Coefficients from Selected Equations--- Sources of Data on Acres of Grain Sorghum Harvested and Production of Grain Sorghum Sources of Data on Cropland Harvested and Tractor Numbers ------ Sources of Data on Acres of Grain Sorghum Harvested That Had Been Irrigated Data on Acres Irrigated by States, and Values Used to Estimate Acres of Grain Sorghum Irrigated in 1944 Sources of Data on Dollars Spent on Gas and Oil Page 138 140 143 145 147 149 152 153 159 160 162 163 165 166 188 189 189 190 191 Table A-6. A’? o A-8. A-9 o A-lo c A-11. A-12 o A—13. A-l4. A-15. A‘lé o A‘l? o A-18 c A-19 o A-ZOO B, Part8 1.41. C, Table of Powers. Index of Average Prices Paid by Farmers for Motor Supplies for Years Used in Study Data for U. S. on Dollars Spent on Gas and Oil, 1939, 1944, and 1949 Sources of Data for Value of Land and Buildings Per Acre-—----— Consumer Price Index for Years Used in Study Percent Buildings are of Land and Buildings, for Years and States Used in Study Data on Total Man WOrk Units Per Acre of Grain Sorghum, Farm Production Regions and Years Used in Study Sources of Data on Acres Fallow and Acres Idle and Fallow ------ Data on Acres Fallow and Acres Fallow and Idle for Ten Great Plains States, 1939 and 1944 Fertilizer Data for 1959, for Economic Subregions and State Parts Used in Study Fertilizer Data for 1954 for States Used in Study Fertilizer Data for 1949 by Farm Production Regions List of Counties Included in the Study with Data Related to Each County, Observation Number and County Number Key -------- Key to Crop Reporting Districts Used in Study Key for Growing Seasons List of Weather Stations from which.weather Data were Obtained, by County and Kind of Weather Data xi Page 191 192 192 193 193 194 195 195 197 197 198 200— 209 210 211 213- 232 242- 304 305 LIST OF FIGURES Figure Page 7-1. Estimated Effects of Precipitation-—For A One-Inch Increase in Weekly Total-A1ternative Techniques 154 7-2. Estimated Effects of Temperature-For A One-Degree Increase in weekly Tota1--A1ternative Techniques 155 / ‘ . CHAPTER I INTRODUCTION This study is concerned with developing a physical "production function" for grain sorghum. A single equation model is used and esti- mates of the parameters are obtained by regression analysis. An attempt is made to measure simultaneously the influence of weather, man- controlled inputs, and location of production on per acre yields of grain sorghum. The effects of omitting a variable or set of variables on the ability of the model to explain yields and on the coefficients of the variables remaining in the submodel are considered. Also alterna— tive forms of the weather variables and some of the man—controlled input variables are considered. The objectives of this study and relevant background information are presented in this chapter. The model and a detailed description of each of the variables included in the model are the subjects of the second chapter. The third chapter contains a discussion of problems and procedures. The results of the analysis are presented in the fourth through the seventh chapters. The eighth and final chapter contains the summary and conclusions. Detailed lists and discussions of the sources of data and the results (coefficients and indicated level of significance) are presented in the appendix. Need for Study There are three principal needs for physical production function studies. They are listed and briefly discussed below. v.4 p. 2 Understanding the Physical Relationship Botanists, agronomists, plant physiologists, horticulturists, and other plant scientists have a continuing interest in determining the relationship of environmental conditions and levels of man-controlled inputs to yields. Other groups that can use information about these relationships (1) Farmers, so they can make "correct" production decisions; (2) Agricultural supply firms, so they can anticipate demand for their products; (3) Agricultural policy makers, so they can estimate the effect of policy alternatives; (4) Agricultural marketing firms, so they can estimate supply; and (5) Agricultural economists, so they can determine optimum resource use 0 Predicting Crop Yields Producers, purchasers of agricultural crops, as well as persons concerned with agricultural policy and/or national planning have a con- tinuing interest in obtaining good projections of yields. This interest is so strong that the Crop Reporting Board of the United States Department of Agriculture makes monthly estimates during the growing season of the prospective yields of many crops. Knowledge of the relationship of location, weather, and man-controlled inputs to yields would facilitate this estimating procedure. Knowledge of these relationships would also aid in making long- run predictions of yields. This could be done by assuming "average" or "normal" weather and predicting changes or possible changes in the level 3 of the man-controlled inputs. The projected level of the man-controlled inputs for some years in the future could then be "plugged" into the model with average weather to estimate yields in that year. Such predic- tions are relevant for answering many questions concerning our ability to feed a rapidly expanding population or to feed the world. Explaining Changes in Yields The large changes in yields of certain crops in recent years has led to a desire to (1) determine the factors causing the change in yield and (2) measure the effect of each factor. The factors can be grouped as (1) man-controlled factors and (2) environmental factors. I It is important that the relationships of man-controlled inputs and environment to output be known so: (1) Activity analysis at all levels of aggregation can use "good" input-output coefficients; (2) the behavior of farmers and their supply response can be understood; (3) "correct" production recommendations to farmers and to agricultural industries can be made; (4) producers can make "correct" profit-maximizing decisions; and (5) agricultural policy that best meets the short- and/or long-run objectives of society and/or agriculture can be made. An example of current and major importance is the need to determine how much of the agricultural surplus was the result of changes in the level of man-controlled inputs and how much the result of "good" weather. The determination of this could have a major influence on agricultural policy. A great many studies have been conducted in an attempt to determine the influence of man-controlled inputs (MCI) and/or weather on yields. The lack of success in measuring the effect of weather and indeed the need 4 for a technique to do this are attested to by the large number of alterna- tive techniques developed in recent years. Of the three "needs" discussed above (understanding physical rela- tionships, predicting yields, and explaining yield changes), this study is primarily concerned with the last one. It is concerned with explain- ing yield changes and with developing a technique to explain yield changes. Usin e ate Data At an early stage of this study, a choice existed as to whether a model should be developed using experiment station data or using aggre- gate farm data. The principal advantage of using experiment station data is that very detailed information exists concerning such factors as: Date of planting, soil type, variety of seed, seedbed preparation, ferti- lizers applied, date of irrigation and amounts of water applied, chemicals applied, plant population, date of harvesting, etc. Also weather data are obtained at a location very near the plots, minimizing the problem of obtaining relevant weather data. This choice was rejected in spite of its advantages because it was decided that a model that explained experiment station yields was of little value save the implication that it would also be useful with aggre- gate data. Whether the model would give meaningful results when aggregate data were used would still have to be determined. It was decided that it would be better to determine if a model could be constructed that would give meaningful and useful results using available aggregate data. It was rejected also because of the desire to explain the change in aggregate yields. 5 Objectives There are two major objectives of this study. The first is to estimate how changes in inputs have affected changes in the per acre yield of grain sorghum. The second is to estimate the effect of alternative model specifications. Two minor objectives concerning alternative models are: (1) What are the effects on R2 and on the coefficients in the model of dropping certain variables or sets of variables; and (2) what are the effects of alternative ways of representing or measuring the factors. The objectives above include answering the following questions. (1) Can a model using time series-cross sectional data by agri- cultural census years and counties and containing as independent variables (a) man-controlled inputs, (b) years, (c) location, (d) preseason precipitation, (e) weekly values during the growing season for precipitation, (f) weekly values during the growing season for temperature, and (g) weekly values during the growing season for precipitation multiplied by temperature (interaction) explain the observed change in yield of grain sorghum? (a) How much of the change in the yield can be explained by changes in the man-controlled inputs? Changes in weather? Changes in man-controlled inputs not included explicitly in the model (years)? Shifts in the location of production? (b) How much of the differences in yields between locations is explained by variables associated with location but not included explicitly in the model? 6 (2) What are the effects on R2 and on the coefficients in a sub- model of dropping variables or sets of variables from the complete model? (a) (b) (e) (d) (e) (1‘) How well does a submodel containing only man-controlled inputs compare to the complete model? How well does a submodel containing only weekly weather variables compare to the complete model? How well does a submodel containing only years and loca- tions compare to the complete model? How does dropping the man-controlled input variables affect the coefficients for the weather variables? The years variables? The location variables? How does dropping each man-controlled input variable affect the coefficients for other man-controlled input variables? How does dropping the weather variables, the location variables, or the years variables affect the coefficients of the other variables remaining in the submodel? (3) What is the effect onR2 and on the coefficients in a submodel of substitution of variables? (a) (b) How does substituting season total weather variables for the weekly weather variables affect the R2 and the coeffi- cients of the other variables in the submodel? What is the effect onR2 and on the coefficient of other variables in the submodel of substituting for the weekly weather variables weather polynomials of degrees one, two, three, four, five, six, or seven? (4) (5) 7 (c) What is the effect of substituting average planting date variables for crOp reporting district variables onR2 and the coefficients of other variables in the submodel? Based on the alternative submodels estimated, what are the advantages and disadvantages of various submodels? Can the effect of weather on per acre yields be better estimated by dividing the relevant growing season into weeks and obtaining an estimate of the effect of the weather in each week: (a) Using weekly weather variables, or (b) using polynomial weather variables? Review of Literature Grain Sorghum Grain sorghum was used for this study because of the great increase in per acre yields realized in the last twenty years. Also grain sorghum was an important grain crop in the United States in 1963 and is increasing in importance (Table l-l). Table l-l.--Production of feed grains in the United States, 1956—633/ Year 3 Corn for grain: Oats f Barley' f Sorghum grain : 1,000 bushels l956-----: 3,075,336 1,151,398 376,661 204,881 1957-----: 3,045,355 1,289,880 442,761 567,506 1958----: 3,356,205 1,401,410 477,368 581,012 l959-----: 3,824,598 1,052,059 422,383 555,211 l960----: 3,908,070 1,155,312 431,309 619,867 1961 ...... : 3,625,530 1,011,398 395,669 479,751 1962-----: 3,636,673 1,020,371 436,448 509,685 1963----' 4,091,685 979,400 405,577 587,909 a/ Supplement for 1963 to Grain and Feed Statistics, USDA, ERS, Economic and Statistical Analysis Division, Statistical Bulletin No. 159, March 1964 and Agricultural Statistics, 1965, USDA. . . -o 8 Grain sorghum is also important as a world food grain where it ranks third, being exceeded only by rice and wheat. Most of the sorghum grain produced in the United States is used as animal feed, but about 75 percent of the world crop is consumed by humans (130).l/ About 90 percent of the 1958 world crop was grown in China, India, Manchuria, and French West Africa. It is also grown in many other areas including Asia Minor, Iran, Turkestan, Pakistan, Korea, Japan, Australia, Southern Europe, Central America, and South America (100). Sorghum grain is very similar to corn in nutrient content, contain- ing about 12 percent protein, 3 percent fat, and 70 percent carbohy- drates (130). Besides the use for food or feed, grain sorghum also has many industrial uses. The starch can be used for adhesives, sizing for paper and fabrics, and as drilling mud for the petroleum industry. Grits obtained from the endosperm can be used in brewing. The seed coat con- tains wax similar to cornauba wax that is used in making carbon paper, sealing wax, electrical insulation, and other products. Dextrose sugar, oil, and syrup, byhproducts of the wet milling industry, are used in foods. The sugar and syrup are used in canned fruit, and the oil is suitable for salad oil (130, 132). The recent development of hybrid varieties which can be grown in areas where previous varieties could not and which produce higher yields than previous varieties makes it likely that grain sorghum.will become even more important. l/ The numbers in parentheses refer to the publications listed in the Bibliography. 9 History of Sorghum in the United States Grain sorghum was introduced into the United States in the last half of the 19th century (133, 176). The first sorghum grown was tall like corn and was harvested by hand (130). In the 1910's dwarf varieties were introduced. These varieties were affected less by extreme weather and were rapidly adopted. In the 1920's double dwarf varieties were developed and again because of their advantages rapidly replaced other varieties (66, 170). The double dwarf varieties were small enough that they could be harvested with a combine. These standard varieties were continually improved by systematic breeding and selection to give higher yields and to be more resistant to insects, diseases, and extreme weather (43, 130, 170). Hybrid grain sorghum had been studied for many years before a technique for large-scale production of hybrid seed was discovered in 1954. The first commercial seed field was planted in 1955 and hybrid varieties were grown on a large scale for the first time in 1956 (l, 158). By 1960, 70 percent of all grain sorghum acreage was planted to hybrids (1). In 1958 there were more than 500 varieties of grain sorghum grown in the United States (130). It is certain with the increased development of hybrid varieties that there are even more varieties grown now. Grain Sorghum Botany Sorghum generally is divided into these two main classes: Forage types and grain types.;/ All sorghum varieties produce grain and almost all varieties can be used for forage. However, there are great varietal 1/ Although Broomcorn, Sudan grass, and Johnson grass belong to the sorghum genus, they are generally considered as separate crops because of their specialized uses (130). 10 differences. Those varieties that do well for forage produce poor yields of grain and grain varieties produce poor yields and possibly poor quality forage. There are dual purpose varieties that produce reasonable yields of both, but they do not do as well for either purpose as the specialized varieties. The grain of sorghum is small and the number of seeds per pound ranges from 12,000 to 35,000 seeds. This compares to about 14,000 seeds per pound for wheat (167). Grain sorghum weighs about 56 pounds per bushel. The rate of germination is poor with field germination being about 60 percent when the seed germinates 90 percent in the laboratory (167). Sorghum is not sensitive to soil types and can tolerate considerable quantities of alkali or salts (133, 167). The amount of moisture necessary to produce a crop does depend on soil type. Very low yields or even crop failure may be expected if the precipitation is less than 12 inches on sandy soil or 14 inches on heavier soils. In moist seasons, highest yields are obtained on the heavier soil (100). In most years 21 to 25 inches of water are needed for high yields (82, 130). Grain sorghum can tolerate too much moisture (flooding) better than many other crops (133). The timeliness of precipitation is also important. It has been shown that at the time of flowering there is a great increase in trans- piration without any change in the environment (3). It has been demonstrated that grain sorghum can utilize moisture from a depth of 90 inches (133). Grain sorghum can withstand greater extremes of heat than most other crops (133). However, yields are influenced by heat. If the temperature is high during the time that the crop is producing seed there will be higher rates of transpiration and less storage of sugar, starches, and other products of photosynthesis (160). The most favorable mean temperature 11 for the growth of sorghum is about 80° F. (130). The timing of tempera- ture is also important. High temperature at the time the plant heads and flowers is particularly harmful. Better yields are obtained if the plant comes to head after the period of greatest heat is past (176). New varieties have been developed that have changed the sensitivity of sorghum to temperature. At one time sorghum.could only be grown where the frost-free season was at least 160 days and with a mean July tempera- ture of at least 75° F. Now sorghum can be grown with a frost-free season as short as 130 days and a mean July temperature of near 70° F. (100, 174). Most of the grain sorghum.varieties grown in the United States grow to maturity in 90 to 120 days (130). The actual range is from 85 days to 140 days (100). The length of time to grow to maturity (to mature) is primarily a function of variety, but it is also a function of temperature and length of day. Sorghum is a "short day" species, which means that flower initia- tion is hastened if the days are short and is delayed if the days are long (130). For example, a deviation of one hour from an average day length of 12 hours will alter the growing period by about 10 days (100). The fact that the length of time for sorghum to mature is a func- tion of temperature is illustrated by the fact that the period from planting to pollination is twice as long at an average temperature of 68° E.as it is at 86° F. (100). All varieties are not affected to the same extent by day length and temperature. Thus, two varieties that may mature in the same length of time at one location may differ at another location where temperature and/or day length are different (130). 0 AL! . 12 Grain sorghum can be harvested when the moisture content of the heads is 25 to 30 percent. However, because the grain does not dry well in the bin, the moisture content should be less than 13 percent or else the grain must be artificially dried. Much of the recent interest in growing grain sorghum in the Corn Belt States is because of the avail- ability of farm grain dryers (133). Sorghum is subject to four general groups of diseases: (1) Those that reduce stands by rotting the seeds or killing the seedlings; (2) those that attack the leaves; (3) those that attack the heads; and (4) those that cause root or stalk rot. The most severe losses are generally caused by the root or stalk rot. Control of these diseases lies in the use of resistant varieties, seed treatment and/or crop rotation (167). Sorghum is also subject to insect attacks. The more common ones are chinch bugs, corn ear worm, corn leaf aphids, sorghum.midge, and grasshoppers. Ordinarily, injury to sorghum from insects is not very great or widespread (167). Grain Sorghum Culture Sorghum should be planted in a well prepared seedbed. In the more humid areas it is planted in the top of the seedbed but in drier areas planting is done in the bottom of a furrow (130). Sorghum may be planted in southern Texas as early as February 15 or as late as September 1 (100). The further north the growing area, the shorter is the range of possible planting dates. The general rule is not to plant until the soil is warm (60° F.) and to plant so that the crop will head after the hottest part of the growing season has passed (176). 13 Plant population is an important factor affecting yields (100). The desired plant population depends on whether the crop will be irrigated, planted in rows, or drilled. Given the desired plant population, the amount of seed to plant depends on germination rate, size of kernel, ability of variety to tiller, and hardness of endosperm of the kernel (167). Irrigation is an important cultural practice. It has been observed that irrigated sorghum yields two to five times as much as dry land sorghuml/ (43, 78). There are two major methods of irrigating sorghum. One is to apply 10 to 12 inches of water previous to planting and then no further irrigation. The second is to apply water when and as much as is needed (130). The importance of fertilization as a cultural practice depends on whether the crop is irrigated or not. Under dry land conditions sorghum shows little or no response to fertilizers (31, 67, 100, 130, 133). However, there is a great response to fertilizer, particularly nitrogen, if sorghum is irrigated (100, 130, 133). Since 1945, almost the entire crop of grain sorghum has been harvested with combines. The proportion of the crop harvested with com- bines had increased to 100 percent from about 10 percent in 1940 (130). Fallowing is another important cultural practice. The yields of sorghum on land fallowed the previous year are 50 to 90 percent greater than on similar land not fallowed (174, 167). The total production per acre from the two crop years of a three-year grain-grain-fallow rotation is about the same as from three years of continuous cropping (130). 1/ This is partly due to the fact that generally irrigated sorghum is fertilized while dry land sorghum is not. .. 14 weather Studies A large number of weather-crop yield studies have been made. Most have been concerned with understanding the physiology of the plant and/or with the prediction of yields and/or production. As this study is primar- ily concerned with the need to explain change in yields, only those techniques and studies related to this will be discussed. It will be noted if any technique used to explain yield changes also provides information pertinent to the other two "needs" discussed above. The techniques used to explain yield changes take weather effects into account in two ways. One way is to use actual weather variables and/or some transformation of the weather variables. The second way is to use "weather" variables derived from production figures. This tech- nique assumes the unexplained variation in production is all due to weather. Actual Weather Variable Techniques This section contains a review of techniques that contain actual weather variables, such as precipitation or temperature, or some trans— formation of actual weather variables. One of the principal advantages of these techniques is that the information obtained contributes to a better understanding of plant physiology and can be used in predicting yields. Other advantages are that it is possible to determine if a particular weather factor limits production and if and how much yields in the future can be changed by controlling or influencing weather. The final advantage that will be listed is that these techniques can be used on any crop and with any kind of units of observation. 15 One of the principal arguments against these techniques is that no matter how many weather variables are used and no matter how they are transformed it is inconceivable that all the effects of weather could be measured. This argument is indisputable but it remains to be determined if the major portion of the effects of weather can be measured by a few correctly specified weather variables. One of the principal disadvantages of this technique is that in general several variables are needed. This is a particularly severe disadvantage if the number of observations is small. Another disadvantage is that the time and effort needed to collect the data is quite large if the number of observations is large. Four different methods of including weather variables to explain yields will be discussed. The first method is the use of actual weather variables. The number of actual weather variables that could be used is almost limitless. Some of the actual weather variables used in weather- crop studies (but not necessarily for the purpose of explaining the effects of weather on yields) are: Annual precipitation, seasonal pre- cipitation, preseason precipitation, soil moisture, temperature, humidity, light, evaporation, wind velocity, and soil temperature. Only a limited number of studies have been conducted where actual weather variables have been used to explain yields. Of these some of the most important are those by Dr. Louis M. Thompson of Iowa State University. In his studies of wheat (156), grain sorghum (159): soybeans (159), and corn (159, 160), he used monthly totals of precipitation and monthly average temperatures for the principal months of the growing season as independent variables. Regression models containing these weather variables and a trend variable to capture the effects of changes in tech- nology were estimated. 16 Another method of using the weather variables but allowing for nonlinear effects of the weather variables on yields is to consider in addition to the direct weather variables these same variables raised to some power. The most common practice is to consider in addition to the linear term a quadratic term. The studies by Thompson illustrate this technique (155, 158, 159, 160). A third method of using weather variables to explain changes in crop yields which takes into account the effects of distribution as well as amount is to fit a polynomial in time to a set of weather data repre- senting consecutive short time periods within the growing season. To elaborate, the growing season (or year) is divided into a number of comparatively short time periods (such as weeks or two-day periods). The information on a particular weather variable within each of the time periods and the position of the particular time period in the sequence are the basic information used. The weather information for each period is weighted by the position of the period in the sequence and then summed to form a "new" variable. The number of variables needed and thus the number of different weightings needed depends on the degree of the poly- nomial to be fitted.;/ This technique has, to the writer's knowledge, never been used explicitly for the purpose of explaining the effect of weather on yields so that other factors affecting yields could be investigated. The tech- nique itself was introduced by R. A. Fisher in 1924 (52). It was used again and somewhat clarified by Floyd E. Davis, J. E. Pallesen and some 1/ This process is discussed in greater detail in the following chap- ter. Also, Dr. Fred H. Sanderson gives a very comprehensive treatment of this subject in chapter nine of his book, Methods of Crop Forecast- ‘igg, Harvard Economic Studies, Vol. 93, 1954. '.... --'v... u . Q.‘ -h ‘- 17 of their colleagues in the early 1940's (36, 37, 120). All of these investigators used a computational procedure called "orthogonal poly- nomials" to estimate the coefficients of their regression equations. In 1943, W. A. Hendrick and J. C. Scholl (65) demonstrated that the same objectives could be attained with usual regression analysis if the data were appropriately transformed into new variables.;/ They went on to compare the results obtained by using monthly data and weekly data. They concluded, "The weekly data do not enable one to estimate the average state yields more accurately, but they facilitate the measurement of seasonal changes in weather effects." Two recent studies using polynomials to capture the effects of weather were conducted by E. Huge and R. O'dell (136, 137). Their first study was on corn and the second on soybeans. Both studies use data obtained from experimental plots. Plots were selected upon which most of the technology had been constant. The corn yields had to be adjusted for the effects of changing from nonhybrid to hybrid varieties. Adjusted yields were used as the dependent variable in the analysis. The effects of changing soybean varieties were accounted for by the inclusion of a trend variable in the analysis. Other than for changes in varieties grown, technology was not different on the plots considered in the study. The last method used to include actual weather variables in a model to be discussed is the index method. In this method, actual weather variables are combined to form a single weather variable (an index of weather). This variable is then included in the model. The major weak- ness of all the indices considered here is their failure to take into account the effects of the distribution of the weather factors. 1/ This transformation is explained in more detail in the following chapter. 18 Many indices derived from weather variables are discussed in detail in a recent paper by Bernard Oury (119). A few of the indices discussed there are listed below to illustrate the nature of these indices. Thornthwaite developed a moisture index that was expected to ex- press the relative humidity or aridity during a period in a given location: Moisture index = precipitation — potentialfevaportranspiration potential evaportranspiration Lang suggested the following index: Index = precipitation = 2 temperature T where precipitation is measured in millimeters and temperature in degrees centigrade. These units of measurement for precipitation and tempera- ture are the same for all indices considered here. De Martonne modified this to avoid the problem of negative values by adding 10 to temperature, i.e., = _£__ I T+10 Kgppen suggested the following three alternatives: I 2 SP 5T+120 I : 2P T+33 _ P I - ___ T+7 Angstrgm considered this index: I = P 1.07T The indices were designed to use annual values of the weather varia- bles. However, they can be modified to use data based on shorter time periods. For example, De Martonne's index can be written, I : _§__ x number of periods in a year T+10 19 where P and T are averages of the periods. If P and T were averages of the monthly totals, the index I = Tim would have to be scaled upward by a factor of 12 (119). A variation of the index method is involved in the moisture stress concept developed by O. T. Demmead and R. H. Shaw (44). A plant is said to have experienced a moisture stress day if for a day the water needed by the plant was not available. Although the concept is simple the actual determination of both water need and water availability is extremely complex. Interested readers are referred to the original article (44). The moisture stress concept was used by Robert F. Dale in a recent study (35). The variable actually used was the number of nonstress days during the growing season. The results were quite good. However, the determination of nonstress days demands at present special empirical investigations. To be useful in aggregate models it is necessary that the number of nonstress days be determinable from regularly obtained weather data such as precipitation and temperature. Yield Index Techniques These techniques derive measures of the effects of weather on crop yields by considering how plant yields have varied on plots where tech- nology has remained "constant." The major reason that these techniques are considered is well stated by Robert F. Dale as follows, "After all, the plant experiences and integrates the same weather recorded only in part by our instruments as well as the complex plant-soil-weather inter- action and the side effects of insects and disease" (35). One of the principal advantages of these techniques is that only one variable for weather is needed in the model. Another advantage is 20 that if the index is "correct" the influence of all weather factors and their interactions are taken into account. Some of the disadvantages are: (1) No information is obtained that will allow the determination of which, if any, weather factors are limiting production; (2) No information is obtained that will aid in understanding the relationship of plants to environmental factors; (3) No information is obtained that will aid in either short- or long-run predictions; (4) Data of the kind used to date (experimental control plots and variety test plots) are not available in sufficient quantity (if at all) to derive indices for most crops; and (5) There is no reason to believe that weather at the test plot location(s) is typical or representative of the State or region. The yield indices are obtained by using data from experimental control plots or variety test plots. A linear trend is fitted to the data to remove the effects of factors which have changed consistently over time such as soil fertility. The index value is the ratio of actual yield to the trend yield. This method was used by Glenn L. Johnson in his study of burley tobacco in 1952 (84). Dale Hathaway used it again in 1954 in his study of the dry bean industry in Michigan (60). James L. Stalling, a student of G. L. Johnson, used this method to obtain indices for some major crops ‘by States, regions, and for the United States (145). In all of these studies control plots for yield experiments were used. . u .. I \p 1". Q \‘c .- A. ‘v a .- ~..—' ‘, 21 This method was modified somewhat by Lawrence Shaw who used data from variety yield test plots (143). Technology is not held constant on these plots as it is for the experimental control plots. Thus the problems of separating the effects of weather and technology are much greater. How- ever, these trial plots are located throughout the State on farms and so they are much more likely to be representative of the State. Functions Used to Explain Changes in Yield Functions reviewed will be limited to those that are for a particu- lar crop and that include weather and man-controlled inputs as variables. Principal studies containing the characteristics listed above can be classified into two groups. The first group are those that include a number of MCIV and a single variable (usually an index) to represent ‘weather. The second group are those that have a number of weather variables and a single variable (usually a time trend) to represent the MCIV. Good examples of studies using several non-weather variables and a single weather variable are those by D. Gale Johnson and Robert L. Gustafson (83), by Ludwig Auer (6), and by Shaw and Durost (143). In their study, Johnson and Gustafson used the following non-weather variables: Fertilizer, mechanization, variety index or degree of hybridi- zation, sunmler fallow, labor, value of land per acre, total cropland harvested, and irrigation. The only weather variable was average annual precipitation. Functions were estimated for wheat and corn. The value assigned to each variable was the change in the average level of the variable between two selected time periods. The time periods in the case of wheat were for the base period 1928-41, excluding 1933-36, and for comparison period 1945-54. “ 1. 1|» 22 Auer in his study also concentrated on MCI (6). However, his func- tions were based on time series data (1939-1960). He estimated functions by crops and by States. A total of 180 functions were estimated. The MCIV were an index of variety, pounds of fertilizer applied per acre, crop acreage, and a trend variable (to represent technology). The "weather" variable was a yield index calculated from data on experimental and test plots. The study of corn yield by Shaw and Durost (143) is similar to the studies by Gustafson and Auer as a single variable is used to capture the effects of weather and several MCIV are included. A yield index was constructed to represent weather for crop reporting districts in Ohio, Indiana, Illinois, Iowa, and Missouri. However, these were aggregated to obtain an index for the Corn Belt as a whole. The analysis was of a time series of yields. Examples of studies emphasizing weather variables are those by Thompson (156—160) and Studnes (150). In these studies technology was taken into account by the inclusion of a trend variable. Several weather variables (monthly totals or monthly averages of specific weather factors such as rainfall or temperature) were also included. Studnes' study differs from Thompson's in that a longer time series was considered and he attempted (after the results of the regression on yield) to decompose the trend term (technology) into its component parts using other data. It appeared to the author that the studies by Johnson and Gustafson, Auer, and Shaw and Durost did not take weather into account adequately and the studies by Thompson and Studnes did not take technology into account adequately. It was in the desire to remedy these inadequacies that this study was undertaken. CHAPTER II THE MODEL In this chapter a detailed discussion is made of the model and of each variable in the model. The model is a single equation model and the parameters are estimated by least squares regression analysis. W Yst = a + :31 Pi Xist + 11at s = l,2,...,l29 t = l,2,3,4,5 i = l,2,...,I where Yst is the average yield per acre in county 3 in year t, and Xist is the value of the ith independent variable for county 8 in year t. a is the overall constant term, Pi is the effect on Y of Xi increased by one unit and ust is the disturbance term for county 8 in year t. Necessary and sufficient conditions for obtaining best linear un- biased estimates of the 81's are: (l) The expected value of the disturbances be zero, (2) the disturbances be independent, (3) the disturbances have equal variance, (4) the independent variables in the model be independent of the disturbances, and (5) the matrix of independent variables be non- singular. It is assumed that the disturbances are distributed with mean zero. The non-singularity of the matrix of independent variables is verified by 23 24 the estimation procedure. The extent to which other conditions hold is discussed below. The second condition states that the disturbances are independent. In this study a criterion was established and all counties meeting this criterion were included in the sample. As a result, there are cases (about 300;/) where two counties included in the study have a common boundary. It is possible that observations on such adjacent counties may not be independent. To the author's knowledge, no tests have been devel- oped to determine if cross section observations in a combined time series- cross section analysis are independent. However, a naive procedure used by the author is discussed in the analysis chapter. Also, to the author's knowledge, no reports have been made showing what effects such dependence among the disturbances would have on the estimates of the coefficients.g/ These problems need to be investigated, but such investigations are beyond the scope of this study. The third condition states that the disturbances must have equal variances. It was recognized early in the study that the variance of the dependent variable (yield per acre) could be a function of the number of acres upon which it is based. That is, as the number of acres upon which the yield per acre is based increases, the variance would probably decrease. Awareness of this was the reason that no observations based on less than 1,000 acres were included in the sample. 1/ There are 8,256 possible distinct pairs of counties from the sample, of these about 300 pairs have a common boundary. To put it another way, of the 417,380 off-diagonal elements in the matrix of variances and co- variances of the disturbances, about 15,000 or four percent would have non-zero values, if disturbances for all adjacent counties were not inde- pent. g/ The special case of autocorrelated disturbances has been investigated but it is not known whether the consequences of other kinds of dependence of the disturbances would be the same. 1..” 25 The dependent variable is still based on acreages varying from 1,114 to 245,987 acres. However, it is believed that the data based on 1,000 or more acres is quite reliable and that the variance of the dis- turbances will not vary greatly for acreages greater than this. The fourth condition is that the independent variables in the model be independent of the disturbances. Marschark and Andrews have demon- strated that if firms maximize by differentiating current (actual) revenue with respect to inputs the input variables will be correlated with the disturbances of the production function (107). However, the author believes that farmers maximize by differentiating anticipated or expected revenue. Hoch has shown that if this is the case, then the input variables are not necessarily correlated with the disturbance terms of the production func- tion (71). Significance tests and confidence intervals for the estimated coefficients may be obtained by assuming that the disturbances are normally distributed. However, even without an explicit assumption of normality, the tests can be justified as being approximately correct by appealing to the Central Limit Theorem (89). The Dependent Variable The single dependent variable considered in this study is average pounds of grain sorghum obtained per acre of grain sorghum harvested. The Independent Variables Three sets of independent variables considered are: Man-controlled input variables (MCIV), dummy variables (for location and time), and weather variables. l0. 26 Man—Controlled Input Variables Acres of Grain Sorghum Harvested per Farm Harvesting Grain Sorghum This variable is included to determine the effect the size of the enterprise has on per acre yield. It is expected to "capture" indirectly the effects of specialization of machinery, changes in land quality used, and changes in management proficiency. As per farm acreage of grain sorghum increases, it is expected that better quality land will be used. This is because sorghum competes with wheat, cotton, or corn for land and these latter crops are grown on the best land. Government acreage controls on wheat, cotton, and corn may have "forced" an increase in acreage of sorghum grown (43, 84, 130). In any case, it is expected that the effects of using better land leads to increases in per acre yields. The effect of increased mechanization is measured explicitly by two other variables, tractor numbers and dollars spent on gas and oil. These two variables do not measure the effect of a shift to more special- ized equipment. An increase in acreage of sorghum is expected to lead to a shift to more specialized equipment. Such a shift is expected to lead to a very small increase in yields. A third factor related to this variable is management. Two opposing views exist concerning increasing acreage per farm of a particular crop and management of that crop. First, it is expected that management effort per acre is greater on small acreages than on large acreages. Thus, in- creasing acreages would lead to lower per acre yields. The second view is that given a particular size of farm, the specialization in production leads to more effective management of the remaining crops. It is believed that the effect on yields from such changes in management effect is small. .. .Va. 27 Number of Acres of Croplgpd Harvested Per Tractor or Number of Tractors Per Acre of Cropland Harvested These variables are indicators of the quantity of machines available for production operations. The only way that mechanization can affect yields is by timeliness and thoroughness of production operations (85). Thus, if quantity of machines increases relative to acres farmed, there should be better timing of production operations and an increase in yields. If, on the other hand, the quantity of machines decreases relative to acres farmed, there is likely to be poorer timing of production operations and a decrease in yields. It is believed that the change in quantity of machines available for production operations had very little effect on average yield. Dollars Spent on Gas and Oileer Acre of Cropland Harvested This variable is an indicator of the use and the change in size distribution of machines. That is, as large machines are substituted for small machines, the quantity of fuel used would increase even though the number of machines would not. In addition, the extent that machines available are used is reflected in fuel expense. The change in size and use of machines can affect yields only through timeliness and thoroughness of production operations. It is be- lieved that the effect of changes in the size and use of machines on average yield is very small. Acres of Grain Sorghum Irrigated as Percentage of Total Acres of Grain4§orghum Harvested Moisture is the most important factor limiting yields in almost all of the grain sorghum producing regions. The cost of varying this factor is high but the yield response to additional moisture is also high. 11,. .. u! ‘0 ~- “- -. - ~v ~ \. t. ‘1 ..~ “- 28 Increasing the proportion of the crop that is grown under irrigation is expected to greatly increase yields. Per Acre Value of Land The real value of land (value of land deflated by the consumer price index) was included as an independent variable because intuitively it seemed a good proxy variable for the interaction effect of technology with land. The changing per acre value of land is a priori related to the changing potential productivity of the land. The potential productivity is changing because (1) the quality, quantity, and mix of other factors of production available change over time, (2) accessibility of other factors of production varies cross-sectionally, and (3) there are basic differ- ences in soil structure and composition. It is the interaction effects of other factors of production with land that need to be measured. Value of land is used as a proxy variable for this. The fact that value of land may be a reasonable proxy for the inter- action effects is illustrated below. Suppose the production relation is: Q = pl 21 + $2 22 + 33 Z3 + 34 211 2:2 ZEB where Q = output, Z1 = acres of land, Z2 = amount of input 2, and Z3 = amount of input 3. The last term represents the "interaction." Yield per acre is: _ 1171 Y2 ‘Y3 Q/z1 - Bl + 32 x2 + 33 x3 + 54 Z1 Z2 23 Zi/Zl = amount of input i per acre. Y where Xi ‘.1 29 Now setting the value of the marginal product of land equal to its price, we have: -1 o Q/oz1 = 51 +‘Y1 BA 211 2:2 z;3 = (Plr)/PO where P1 = value of land P0 = price of product r interest rate on land (so that the "price" of land is rPl) -1 From this B4 Ell E:2 E;3 = (r/Yl)(Pl/Po) - fll/Yi and substituting this into the above expression for Y1 y = 51 (Yl'l)/Yl + 52 x2 + 33 X3 + (r/Yl)(Pl/PO). Note that: (1) The coefficients in the Y function are directly interpretable in terms of the parameters of the Q function. (2) Even though the Y function is simply linear in the X's and the price ratio, the Q function can display diminishing, constant, or increasing marginal products, and decreasing, constant, or increasing returns to scale (depending on the value of the y's). (3) The model is directly extendible to include any number of non-land inputs. (4) The price of land (rPl) is deflated by the price of the product (Po) rather than the consumer price index. The interest rate (r) has been relatively constant over timel/ and probably is relatively constant cross sectionally as well. Thus using the l/ The average for all lenders and for U. S. interest rate paid on mortgages was: 4.6, 4.4, 4.6, 4.6, and 4.9, respectively, in 1939, 1944, 1949: 1954, and 1959. 30 value of land (Pl) instead of the "price" of land (Plr) probably does not create any major biases. The coefficient estimated is approximately (r/Yl) instead of (l/Y1)° Using the wrong deflater probably does bias the resulting coeffi- cient. This can be shown by comparing the C.P.I. index used with a similar index based on the product price. The product price index is simply the price of the crop in each year divided by the price in 1949. The values of the index are 56, 94, 100, 111, and 88, respectively, for the years 1939, 1944, 1949, 1954, and 1959. The corresponding consumer price indexes with 1947-49=1OO are 59, 75, 103, 115, and 125. The major difference is in the 1959 indices. Deflating the 1959 value of land by the C.P.I. reduced the magnitude of the variable appearing in the model. Deflating by the product price, on the other hand, would have substantially increased the magnitude. Since yields also increased substantially between 1954 and 1959, the value of land variable deflated by the product price would probably have been more highly correlated with yield than was the variable used. A weakness of this variable for statistical purposes is that it is probably not completely exogenous with respect to, or unaffected by, the dependent variable. Clearly, if yields increase, other things constant, the value of land should (under competition) increase. However, since other things are not equal it is more correct to reason that as net returns per acre increase value of land would increase. Net returns per acre is a function of many things besides yield. To the extent that it is determined by things other than yield it may be reasonably exogenous with respect to the dependent variable. Also the value of land is determined by the demand for land for many purposes besides its value in the production of grain t..' a .rvo 31 sorghum. Among these demands for land are the demands for the production of other crops such as wheat and cotton, the demand for conservation and recreation uses, the demand for highways and urban growth, and the demand for land for speculative and investment purposes. All of these factors affect the value of land. In this model, treating the price of land as exogenous is essentially just assuming that the supply of land pp sorghum growing is infinitely elastic over the relevant range (other things equal). Thus, an increase in the demand for land for sorghum (such as would presumably occur due to an exogenous in- crease in sorghum yields, other things equal) would not by itself bring about an increase in price. To the degree that this assumption is ppp correct, there probably is some "simultaneous equations" bias in the least squares regression estimates. Although it was not possible to obtain an estimate of the extent of such bias, it was possible to obtain an indication of the effect of including the value of land variable on conclusions reached concerning the model and the other variables in the model. Although a detailed discussion of this point is left to Chapter V, it may be mentioned here that includ- ing the value of land variable apparently did not seriously affect any of the major conclusions reached concerning the model and the other variables. It is expected that if the relative value of land increases, yieli will increase. It is also expected that the influence will be significant.l/ Man-hours of Labor Per Acre A priori, increasing the amount of labor would increase the timeli- ness and thoroughness of the production operation and thus increase yields. 1/ Unless otherwise stated, "significant" means the estimated coefficient is significantly different from zero at the 0.10 level. -u I" “I ‘. 32 Decreasing labor would be expected, a priori to decrease yield. However, it is expected that the changing amounts of labor did not significantly affect yields. It is believed that the change in yield as a result of an increase in timeliness and thoroughness of the production operation is very small. Acres Cultivated Summer Falipp It has been established that in dry areas of the country, if grain is planted in fields fallowed the previous year, yields are up to 50 per- cent greater than yields on similar fields cropped the previous year (24, 130, 174). It is assumed that if acres fallowed increased, the proportion of sorghum grown on fallowed land would also increase. It is expected that increased acreage of sorghum on fallow would increase yields. Rapio: Acres Fallowed to Acres of Cropland Harvested This variable was constructed to remove the confounding influence of county size included in the acres fallowed variable. Both variables are not included in the same question. Pounds of Commercigi Plant Nutrients Applied Per Acre of Grain Sorghum For most of the grain sorghum producing region, moisture is the limiting factor of production. It has been shown that when this is the case the application of fertilizer will have very little effect on per acre yields. However, when a crop is irrigated, the yield response to fertilizer is very great. Since fertilizer is generally used in large quantities only when the crop is irrigated, the expected effect of in- creased fertilizer use is a large increase in per acre yields. a. 'fi 1 N. 33 Dummy Variables Three sets of dummy (zero-one) variables are considered. One set is concerned with years and the others with location. lees This set contains five variables, one for each year included in the study. The variable representing 1939 Will be dropped to allow estima- tion of the parameters. Nineteen thirty-nine was selected because the coefficients for the remaining variables will indicate the amount per acre yields have changed since the "base" period of 1939, due to factors that have changed over time and were not otherwise considered in the analysis. Several such factors known a priori to have changed with time and to have increased per acre yields are: Improved cultural practices, introduction and increased use of chemical weed killers, and improvement in varieties. The last factor, improvement in varieties, is believed to have in- creased per acre yields greatly between 1954 and 1959. This increase is due to the advent of commercial production of hybrid grain sorghum seed and the extremely rapid adoption of this new technology. Because of the development and acceptance of hybrid seed between the years 1954 and 1959, the coefficient for 1959 is expected to be sub- stantially larger than that for any other year. Other than this, the increase in per acre yields due to factors related to time but not included in the study is expected to be small. Crop Reporting Districts Within each State, counties are grouped into crop reporting districts which in turn generally reflect the different type-of-farming areas. It is 34 believed that the resulting districts are relatively homogeneous with respect to climate, soil type, topography, and so forth. Counties in- cluded in the study were located in 28 crop reporting districts. The number of counties included in a district ranged from one to thirteen. A set of 28 dummy variables is used to represent these districts. It is believed that the use of this set of variables will lead to meaning- ful estimates of consistent differences in productivity between districts. These differences in productivity are assumed to be related to difference in physical factors of production associated with location. Some such factors are: Soil type, topography, elevation, and climate. Growing Seasons The counties included in this study are located in widely differ— ing climatic regions. The seven growing seasons established for purposes of collecting relevant weather data reflect these climatic differences. A set of seven dummy variables is used to represent the different climatic regions. When this set is included in the analysis, it is expected that the coefficients obtained will give meaningful estimates of the consistent differences in yields due to climate. Weather Variables It has been recognized that weather is one of the primary factors influencing per acre yields. It is highly desirable that some technique be devised that can measure the effects of weather. In this study, three techniques will be developed and compared. While it is impossible to include all relevant weather variables in.an analysis of this kind, it is believed that the major influences of ‘weather can be measured by the principal weather factors, precipitation 35 and temperature. The effect of the distribution and interaction of these two factors over the relevant growing season will be taken into account. It was determined that the "relevant growing season" was a 23-week period beginning two weeks before the average planting date for grain sorghum. Because of the wide geographical spread of the counties included in the study, seven different average planting dates (growing seasons)l/ were used. weekly Estimates Technique The distribution aspect of precipitation is taken into account by constructing 24 precipitation variables. The first variable is preseason precipitation and is the total precipitation occurring in the 203- (in 1944 the 204) day period preceding the first day of the relevant growing season. Each of the next 23 precipitation variables represent one of the 23 weeks in the growing season. The value of each variable is the total precipitation in inches that occurred during a particular week. Each coefficient obtained from the analysis will indicate how much final per acre yields respond to a one-inch change in precipitation in that particu- lar week. Twenty-three variables for temperature were used, one for each week in the growing season. The value of each variable is the sum of the daily maximum temperatures that have occurred during a particular week. The coefficient obtained from the analysis will indicate how the per acre yield will respond to a one-degree change in the total maximum temperature in a particular week. i/ Growing seasons used are discussed in detail in Appendix A. L. 36 Twenty-three interaction variables were calculated, one for each of the weeks in the growing season. The value of a particular interaction variable is the total precipitation during that week times the total maximum temperature during the same week. The coefficients obtained will indicate how final yield per acre will change with a one-unit change in the interaction variable. It is not expected that all 70 coefficients will be significant, but it will be of interest to determine which of them are. Another rele- vant question is whether each weather factor with distribution taken into account is significant. To answer this, the 23 coefficients representing the weekly variables for each weather factor will be tested to determine if together they are significant. Weather PolynomigipTechnique One of the principal advantages of this technique over the pre- vious method is that it uses fewer degrees of freedom. In many studies, particularly those using only time series data, the number of observations may not be large enough to allow the previous method. For these cases the weather polynomial is suggested as an appropriate method to determine the influence of weather on yields. It is of interest to determine how well the polynomial method compares to the method of estimating coefficients for each week for each weather factor. The preseason precipitation variable is used in this technique in the same way that it was used in the previous technique. The 23 weekly totals of precipitation for the growing season are transformed in the following manner. 37 What is desired is a model that relates yield per acre to the amount and distribution of precipitation. Such a model can be written: ‘AJ H Yt = 50 + 2 f(h)rth + ut (1) h=l whereEY; is yield adjusted for the effects of nonweather variables; h designates the particular seven-day weather observation period, h = 1,...,H; t designates the year, t = 1,...,T; rth is precipitation in period h in year t; ut is a distrubance term; f(h) is assumed to be a polynomial in h (time), say, f(h) = a0 + alh + a2h2 + ...+ aphP (2) The value of f(h), h = 1,...,H gives the effect on final per acre yield of a one-inch increase in precipitation in period h (if rth is measured in inches). This may be rewritten as: ’11 = Bo + f(l)rtl + f(2)rt2 + + f(H)rtH + 11b (3) and substituting in the values for f(h) from 2 Yt = Bo + aort1 + alrtl + a2rtl + ... + aprtl 2 p + aort2 + al2rt2 + a22 rt2 + ... + ap2 rt2 + 0.. 2 p + aortH + alHrtH + a2H rtH + ... + apH rtH + ut (4) Rearranging and collecting terms leads to: N Yt - Bo + ao(rt1 + rt2 + rt3 + ... + rtH) + al(rtl + 2rt2 + 3rt3 + 0.. + HrtH) 2 2 2 f a (rt1 + 2 rt2 + 3 rt3 + ... + H rtH) 3'- 0.. ° P P P + ap(rt1 + 2 rt2 + 3 rt3 + ... + H rth) 1 (5) ut 38 Rewriting the terms in parentheses and redefining them as follows: H Xto = 2 rth = total season precipitation h=l H X .= 2 hr t1 h=1 th . H p .ti : 11‘; h 1‘th (6) and obtain: Yt = [30 + aoxto + alxtl + + apti + ut t = 1,...,T The values of the variables Xij are first computed from the weekly weather data (rth) in accordance with definitions (6). These calculated variables are inserted in the regression equation (generally along with other explanatory variables; here the effect of those other variables has been removed to obtain’Y) and the coefficients aj (i.e., the coefficients of the polynomial f(h)) estimated by least squares. In this study h = l,2,...,233 t = l,2,3,4,5; and p = O,l,2,...,7. In other words, all possible polynomials up to degree seven will be con- sidered and compared to determine which degree of polynomial is "best." It is apparent that the above derivation can be recalculated with mth = total maximum temperatures or ith 2 interaction = rthmth in place of rth and similar results will be obtained. In this study three weather polynomials (precipitation, temperature, and interaction) will be included in the analysis with the technology and dummy variables. The "appropriate" degree of polynomial for each factor will be determined by trial regressions. a. n 39 Seasonal Total Technique This technique is included to establish a benchmark with which to compare the other two techniques. Unless the other two techniques do much better, it may be wise for researchers to continue to use this easier technique. Preseason precipitation is handled in this technique in the same manner as in the other techniques. The weekly data is combined, however, to form a single variable representing the season total for each weather factor. It should be noted that the above variables are identical with the variables representing the zero degree term in each of the weather polynomials. It is expected that the other two techniques will do significantly better than this one. However, it is expected that the coefficients for these four variables will be significant. Conclusion Many regressions will be considered but it is expected that the "best" results will be obtained with a regression containing all of the technology variables, the weekly weather variables, and the dummy variables representing time and crop reporting districts. "Best" in the sense that it has a high R2 gpd that coefficients are meaningful. ,. l '!_l CHAPTER III PROCEDURE This chapter is divided into three major sections. The first section contains a discussion of the procedure used to determine (1) the unit of observation; (2) the selections of counties (observations); and (3) the data. The second section contains a discussion of the procedure used to select equations to be estimated. The third section contains a discussion of how the results are presented in the analysis chapters. The Unit of Observation,,the Observations and the Data The Unit of Observation It is desirable to have the geographical unit small to obtain as much homogeneity with respect to weather, topography, soil, climate, and production techniques as possible. It is necessary that the unit be one for which there are detailed and reasonably complete data. Such a unit, and the one used in this study, is a county. Use of the county as the basic unit of observation necessarily limits this study to agricultural census years as they are the only years for which there are reasonably complete and detailed data for counties. A combination of time series and cross section data are used in this study. The basic unit of observation being a county in a census year. The census years of 1959, 1954, 1949, 1944, and 1939 are included. Grain sorghum was a separate entry in agricultural censuses in 1929 and 40' a L. 41 1934, but many of the other variables in the model were not, and so these years were not included. The Observations It was not feasible to include all counties that had produced grain sorghum in these years. Since counties that had a "large" acreage contributed most to average yield, and one of the main objectives is to explain the change in average yield, it was decided that all counties that had a "large" acreage should be included. It was decided that the same counties would be included for all years.l/ _Thus, it was important to select counties that had produced enough grain sorghum in each of the years included in the study to provide meaningful data. All counties which had 30,000 or more harvested acres of grain sorghum in 1959 and 1,000 or more in 1954, 1949, 1944, and 1939 were included. A total of 129 counties from 6 States met this criterion. The acres and production of grain sorghum in these 129 counties and for the United States for years included in this study are given in Table 3-1. These counties contained over 50 percent of U. S. acres and produced over 50 percent of U. S. production of grain sorghum for all years considered. The average yield for these counties does not differ greatly from the U. S. average as shown in Table 3-2. The difference in 1959 suggests that the effect of hybrid sorghum on yields was greater outside the counties included in the analysis. This may also explain why the proportion grown in the 129 counties was less in 1959 than in 1954. i/ This symmetry is not a necessary condition for combined time series- cross section analysis. o.ep 000.com.mme.wm emm.omw.eom.mflm m.eo ooo.amm.ea «Hm.owm.o muuuuunomoa e.ee ooo.eme.eem.ma oom.mwm.mmm.o m o.me ooo.aom.aa ooe.mom.m muuuuuuemoe o.me ooo.oee.mmm.e ome.eefl.mam.m m o.He ooo.mmm.e wam.ome.e muuuuuuoeoa o.mo ooo.mam.amo.o Hme.moe.ame.o m o.eo ooo.amo.o eeo.oom.m wuuu--useoa m.mm ooo.mme.mmo.m ssm.oom.oem.a m H.mm ooo.mom.e sem.mmm.m muuuunuomofi mmmmmmm mmmmmm mUQSOm m psoonom mouse mummd m HMWMpmMMpWWOWOW mopmpm m moflesdoo mma WHMWMpmMMwaOWOW mopmpm m mofipqSoo mad m oma ommpsoonomm popfisb H mama mwmpsmonomm popes: m M meow noepodeHm H ommono¢ M modem sfi popsaosfi mofipnsoo mma a“ one mopmpm popes: one we adnmnom madam mo soapoSpopm pom popmo>nmn moaoeln.alm edema .. .1... a... . .... ...n. I ...: 43 Table 3-2.-—Average yield of grain sorghum 3 Yield for 129 counties 3 U. S. f . Year : in studya : yieldly, : Difference 3 Pounds Bushels Bushels Bushels l959---: 1,931.7 34.5 37.6 -3.1 l954---: 1,147.1 20.5 20.1 .4 1949---—: 1,291.7 23.1 22.5 .6 1944-“-: 1,119.6 20.0 1907 03 1939---: 636.1 11.4 11.2 .2 g/ Census of Agriculture. p/ Agricultural Statistics, U. S. Department of Agriculture. The Data A detailed discussion of data sources, data transformations, and procedures for estimating missing data is presented in Appendix A. The data used in this study on man-controlled inputs and yields are presented in Appendix A. Man-controiipd:lnputs and Yipidp The data on output and man-controlled inputs listed below were obtained entirely from the U. 8. Agricultural Censuses of 1959, 1954, 1949, 1944, and 1939: (1) Pounds of grain sorghum harvested (2) Acres of grain sorghum harvested (3) Acres of cropland harvested (4) Number of tractors (5) Number of farms harvesting grain sorghum (6) Acres of grain sorghum irrigated (except 1944; see Appendix A for discussion of procedure used to estimate 1944 values) .’~ Ni 44 (7) Dollars spend on gas and oil (current dollars) except 1944 (see Appendix A) (8) Value of land and buildings per acre (current dollars) (9) Acres cultivated summer fallow, except 1939 and 1944 (see Appendix A) The data on dollars spent on gas and oil were deflated by the Index of Average Prices Paid by Farmers for Motor Suppliesl/ to obtain dollars spent on gas and oil in constant dollars. The values for "value of land and buildings" were adjusted to con- tain only the values of land. Estimates of the proportion of value of land and buildings that was land for States were obtained from U. S. Depart- ment of Agriculture worksheets.g/' The value of land and buildings per acre by counties was adjusted by the appropriate State value to give value of land per acre. The resulting value of land was then deflated by the Consumer Price Index.2/ The data on man-hours of labor used per acre of grain sorghum were obtained from USDA.5/ Data were available only for farm production regions. The value for the region was used for each county in the region. Data on pounds of plant nutrients applied per acre of grain sorghum were not available on a county basis. values used were derived from.more i/ Obtained from USDA Statistical Bulletin No. 319, 1962. Values used are presented in Table A-6, Appendix A. g/ Obtained from William H. Scofield, Agricultural Economist, Farm Pro- duction Economics Division, Economic Research Service, USDA. The values used are presented in Table A-10, Appendix A. 2/ Obtained from Business Statistics, 1961 Biennial Edition of the U. s. Department of Labor. Values used from.this series are presented in Table A910, Appendix A. 4/ Obtained from personal correspondence with Reuben W. Hecht, Agricul- tural Economist, Farm Production Economics Division, Economic Research Service, USDA. Data used are presented in Table A-ll, Appendix A. l-o 45 aggregate data. Data by State parts of U. S. agricultural subregions were used for the 1959 estimates.l/ The 1954 estimates were derived from data for States.g/ The 1949 values were estimated from data for 1950 by farm production regions.2/ The value of this variable was estimated to equal zero for all counties in 1939 and 1944. Dpppy Variables The value for these variables is always either zero or one. A variable is set up to represent a particular class; if an observation belongs to the class, it is assigned a value of one, otherwise a zero. A set of four dummy variables to represent years was included. They represent the years 1959, 1954, 1949, and 1944. Counties from 28 crop reporting districts were included in the study.é/ A set of 27 dummy variables was used to represent all but one of these districts. A set of six dummy variables was used to represent six of the seven growing seasons (average planting dates).2/ A growing season is a 23-week period beginning two weeks before the average planting date. The average planting date is primarily a function of location. Thus, this set of variables and the set for crop reporting districts is expected to estimate i/ Data used for 1959 are presented in Table A-14, Appendix A. 2/ Data used for 1954 are presented in Table A-15, Appendix A. 2/ Data used for 1949 are presented in Table A-l6, Appendix A. 4/ A detailed list of the counties and the crop reporting districts in which they are located is presented in Tables A-17 and A—18, Appendix A. 5/ A detailed discussion of the growing seasons (average planting dates) and how they were determined is presented in Appendix A. The growing seasons used are listed in Table A-19 and the growing season appropriate for each county is listed in Table Ael7, Appendix A. J“ J. 46 the effects of location on yields. Both sets cannot be included in the same equation because they are linearly dependent; i.e., would create a singular matrix. weather variables All the weather data were obtained from Climatological Data, by States, by months, U. S. Weather Bureau, U. S. Department of Commerce. Preseason Precipitation Preseason precipitation was the total precipitation in inches that had occurred from the end of the growing season the previous year to the beginning of the growing season in the current year. The data were those reported by the Weather Bureau for the weather stations selected.l/ The weather stations were selected according to the following criteria: (1) If there were four or more weather stations in a county reporting precipitation, then three were selected; (2) if there has at least one but less than four, all were selected; and (3) if there were none, up to three nearby stations were selected. Seasonal Precipitation The average precipitation in inches was obtained for each week of the growing season. The selection of weather stations was the same as for preseason precipitation. Seasonal Temperature The total (sum of seven days) maximum temperature in degrees Fahren- heit was obtained for each week in the growing season. The stations used were selected according to the following criteria: i/ A list of the weather stations used is presented in Table A-20, Appendix A. The procedure used to estimate missing data is discussed in Appendix A. 47 (1) If there was at least one reporting maximum temperatures in a county, one was selected; and (2) if there were no weather stations in the county reporting maximum.temperatures, then the nearest weather station that did report was selected. ngpge in Data The dependent variable, yield per acre, was obtained in the computer prior to estimating the equations. Acres harvested and production in pounds were the raw data supplied to the computer. Because the dependent variable was generated in the machine, it was not until a list of residuals, per acre yields, and estimated per acre yields were examined that two errors in the raw data for production were discovered. For observation 482, yield per acre was calculated as 93.7 instead of the correct 941.3 and for observation 519 a value of 5275.5 was calculated instead of the correct 549.77. The data were corrected and equations 2 and 244 were re-estimated as equations 293 and 283, respectively. As expected, the change in R2 was large, from .77 (equation 244) to .85 (equation 283). The changes in the value of the coefficients were not large and it was decided that conclusions based on the equations using the incorrect data (equations 1-284) would be reasonably valid. Twenty-seven equations were estimated using the corrected data. Most of the conclusions in the analysis chapters will be based on these 27 equations. Procedure Used to Select Equations (Submodels) to Estimate The procedure used depended upon the particular objective being con- sidered. The two major objectives are: (1) Estimate the effect of changes 48 in inputs on yield of grain sorghum, and (2) estimate the effect of alternative model specifications. With respect to the second objective there are two minor objectives: (1) Estimate the effect of dropping variables or sets of variables from the model, and (2) estimate the effect of substituting variables or sets of variables with variables in the model. These are different because in the first case the question asked is, "Should this variable (set of variables) be included in the model?" In the second case, it is, "Which of the alternative variables (set of variables) should be included in the model?" Factors Affecting Changes in Yields An equation to meet this major objective was specified a priori. It was specified to include all the man-controlled inputs (MCI), years (Y), crop reporting districts (0), preseason precipitation (P), and weekly (during the growing season) precipitation (Ri), temperature (Ti), and inter- action (Ii) variables. The equation (equation 285, referred to as the "complete" equation) used to meet this objective differed from the one specified a priori in three respects. First, the man-hours of labor per acre variable (L) was dropped from the equation. Second, for ease of interpretation, the acres per tractor (A/T) variable was transformed to tractors per acre (T/A). Finally, to avoid the confounding influence of size of county, the acres fallowed variable (F0) was transformed to the ratio of acres fallowed to acres of cropland harvested (FO/A). Consequences of Leaving Variables Out of the Equation The principal reason for constructing submodels is to determine the effects (relative to the "complete" equation) of specifying alternative 49 models. The question being answered is, "How do the R2's and coefficients for variables in the submodels compare with those in the "complete" equa- tion and/or other submodels?" Although the information obtained from these submodels is not used in this study to determine the "complete" equation (except to drop the man-hours of labor variable), it is believed that it will be of value to others constructing models. Man-controlled Inputs Many production functions have been constructed that contain only man-controlled inputs as independent variables. It is of interest to compare several equations of this type with the "complete" equation. This comparison should provide some idea of the effect on our ability to explain yield of excluding the weather, years, and location variables. With this in mind, 21 submodels containing only man-controlled input (MCI) variables were estimated. All such submodels (equations) estimated are listed in Table 3-3 with a list of the variables each includes. The following "shorthand" will be used to facilitate presentation of the lists of submodels. Y = years 0 = crop reporting districts G = growing seasons (average planting dates) V = value of land L = man-hours of labor FO = acres fallowed FO/A = ratio acres fallowed to acres of cropland harvested FT = pounds of plant nutrients A/T = acres per tractor T/A A/F Tij Iij M(L) MT 50 tractors per acre acres of grain sorghum per farm percent irrigated dollars spent on gas and oil preseason precipitation weekly precipitation for each of the 23 weeks in the growing season; i.e., i = 1,...,23 weekly temperature, i = 1,...,23 weekly interaction, i = 1,...,23 all terms of the precipitation polynomial from ith through jth degree; i,j = O,l,...,7. A single superscript indicates the single variable. all terms of temperature polynomial from ith through jth degree; i,j = O,l,...,7. A single superscript indicates the single variable. all terms of interaction polynomial from ith through jth degree; i,j = O,l,...,7. A single superscript indicates the single variable. total precipitation during growing season total of maximum temperature during growing season total interaction during growing season represents the following set of man-controlled inputs: V, L, F0, FT, A/T, A/F, z, and 3 represents the set above except the labor variable is not included represents the following set of man-controlled inputs: V, FO/A, FT, T/A, A/F, %, and $. (All models containing this set are estimated using the corrected data.) 51 The submodels listed in Table 3-3 provide information about how the presence or absence of a particular MCIV affects the coefficients of the other variables, when only MCIV's are considered. It is also of interest to know the effect of dropping MCIV when other kinds (years, weather, and/or location) of variables are present. type were estimated in "sets.' Submodels of this Each set had a group of variables other than man-controlled input variables which was not changed and the MCIV were added one at a time. constant are presented in Table 3-4. The sets with the list of variables held Table 3-3.-List of submodels containing only man-controlled inputs as independent variablesé/ Variable A/T <: t“ F0 FT S 2° NMN NNNNN NNN NNN NNM N NNN NNNN NNN NM NNNNN >4 >< NNNNN NNN NM NNNN NNN NNNNN NNNNNN NNNN X X X X NNN NNNN X X p/ An X in a column means that the variable listed at the top of the column is included in the submodel listed in left hand column. tions estimated using the not corrected data. All equa- ...-I u.- 52 Table 3-4.-- List of sets of equations omitting some man-controlled input variablesa Other variables included Equations in set and equation numbers P, 307, T07, 107 103-109, 132-138 P, 307, T07, c, Y 12-18 C, Y 51-58 C 88-95 P, Ri’ Ti, Ii E 252259 P, R, R1, Ti ; 270-279 g/ All estimated using the not corrected data. To determine the effect on the coefficients of the MCIV of adding or dropping sets of other variables, it is necessary to compare equa- tions containing the same set of MCIV. Three major sets of MCIV were M, MT, and M(L). Table 3-5 contains a list of equations containing these sets. Table 3-5.-— List of equations containing a complete set of MCIVE/ Set Of MCIV f E uation numbers included I q M : 1, 4, 18, 26, 34, 86, 87, 88, 97, 109, 171, , : 179, 187, 205, 222, 223, 252, 276 M(L) : 2, 6, 9, 16, 61, 98, 100, 131, 170, 244, : 245, 248, 249, 263, 264, 277, 278 MT See Table 3-6 g/ All equations containing the sets M or M(L) were estimated using the not corrected data. To determine which sets of other variables have been omitted in these equations, it is necessary to look at the results presented in the Appendix. 53 Three additional equations estimated using the corrected data and containing the set M(L) were 283, 284, and 293. The "nearly complete" equations: This set was singled out for special attention because most of the discussion in the analysis chapters will refer to these equations. These 21 equations are listed in Table 3-6. Table 3-6.-The "nearly complete" equations Form of weather variables Set of vagi7bles . a . . . omitte : Weekly : 3:3:3? : Polynomialg/ 3 Equation numbers None : 285 302 294 MCIV : 286 307 295 C.R.D. : 288 308 296 Years : 287 309 297 Precipitation ---: 289 305 298 Temperature : 291 304 300 Interaction-- ------ : 292 303 301 Temperature and : precipitation--—--—: 290 306 299 p/ Relative to the "complete" equation (285), the "complete season total" equation (302), and the "complete polynomial" equation (294). p/’A polynomial of seventh degree. They are called the "nearly complete" equations because they omitted only one set (except for the two sets, precipitation and temperature) of variables. This entire set will be referred to as the "nearly complete" equations throughout the analysis chapter. Three other equations are given "titles" to make presentation of the results more understandable. Equation 285, which contained all sets of variables and the weekly weather variables is referred to as the "complete" equation. Since equations 302 and 294 differ only by the form of the weather variables, they will be referred to, :l‘ 54 respectively, as the "complete season total" equation and the "complete polynomial" equation. Year Variables The constants estimated for a year (say 1959) gives the consistent difference in yield (cross sectionally) between the year in question (1959) and the year omitted (1939), after the effect of all other variables in the model have been taken into account. It is of interest to see how these change as variables or sets of variables are dropped from the model. The effect of dropping individual MCIV or subsets of MCIV can be obtained by comparing the equations listed in Table 3-4 that also contain the set of year variables. Some equations that can be used to determine the effect of dropping sets of variables are listed in Table 3-6. Others estimated using the not corrected data are: 1, 2, 7, ll, 16, 18, 26, 33, 34, 35, 36, 43, 51, 59, 97, 99, 100, 101, 102, 244, 245, 246, 247: 261, 264, 269, 276, and 278. Crop Reporting Districts The constants estimated for a particular crop reporting district gives the consistent difference in yield (over time) between the district in question and the district omitted, after the effects of all other variables in the model (equation) have been taken into account. It is of interest to see how these constants change as variables or sets of variables are dropped. Some of the equations listed in Table 3—4 can be used to determine the effect of dropping a MCIV or a subset of MCIV. Equations, in addition to those listed in Table 3-6, that can be used to determine the effect of dropping entire sets of variables are: l, 2, 4, 6-11, l6, 18, 28, 33-38, 43, 51, 59, 60, and 244—251. All of these 55 equations contain the set of crop reporting district variables. To deter- mine which other complete sets were omitted, it is necessary to look at the results presented in the Appendix. Grogipg Seasons The constants estimated for growing seasons (average planting dates) have a meaning similar to that of the constants for crop reporting districts. Some of the equations listed in Table 3-4 can be used to determine the effect of dropping a MCIV or a subset of MCIV on the coefficients for growing seasons. Equations that can be used for this purpose with respect to dropping entire sets of variables are: 61, 85, 87, 88, and 96. Preseason Precipitation This variable is included in almost all equations estimated. How the coefficients change as a MCIV or subset of MCIV are dropped from the model can be determined by comparing equations listed in Table 3-4. The effect of dropping entire sets of variables can be determined by compar- ing equations listed in Tables 3-6, 3-7, and equations: 1, 11, 34, 35, 86, 87, 97, 130, 141, 161, 168, 203, 204, 221, 222, 261, 262, 266, and 267. Ppiynomiai_weather Variables Polynomials of degrees zero through seven are considered for each of the weather factors--precipitation, temperature, and interaction. This was done because there was no a priori way todetermine what the "correct" degree of polynomial should be. Equations estimated, including 'the various degrees of polynomials, are listed in Table 3-7. w‘ 56 Table 3-7.-- List of submodels containin weather polynomials of varying degreesa . : Variable Equation : number 3 M f C E Y E P f 307 f T07 f 107 213—220--: X Y 212-205--: X X Y 36—43 ----- : X X X Y 33—26---: X X X X Y 178-171---: X Y X 202—195---: X X Y 194-187---: X X X Y 44-50 , 11-: X X X X Y 25-18---: X X X X X Y 186-179--: X Y l68—16l--: X X X Y 3/ An X in a column means that the variable or set of variables listed at the top of the column are included in the set of equations listed in the left hand column. A Y in a column means for the weather factor listed at the top of the column, polynomials of degree zero through seven are included respectively in the eight equations listed in the left hand column. Since it is not necessarily true that the "best" degree of poly- nomial for one weather factor is also the "best" for another, equations were estimated where the degree of polynomial for the different weather factors differed. Three "sets" of 24 equations were estimated. In each set, the sequence of adding weather polynomials was the same. The weather polynomial variables were added singularly in the following sequence: R0, To, I°, R1, T1, 11, R2,...R7, T7, I7. In the first set estimated, the only other variable included was preseason precipitation. The equations estimated, following the sequence listed above, were: 213, 204, 141-160, 162, and 161. In the second set (equations 85—63), growing seasons and (M) were included. The third set (equations 212, 203, and 130-109) included (M) and preseason precipitation. 57 Additional equations estimated containing seventh degree poly- nomials are: l, 2, 4, 6-10, 16, 61, 79, 86, 87, 97-101, 131, and 170. A11 equations containing only the season total variable (zero degree polynomials) are included in the lists above. wppkiy weather vagiabigp Each of the weather factors (precipitation, temperature, and precipitation multiplied by temperature) was represented by a set of 23 weekly variables. Only complete sets were considered. Some equations containing these sets are presented in Table 3-6 and others are: 244-252, 261-267, 269, and 276—282. Location Variables Two sets of dummy variables for location (crop reporting districts and average planting date) were considered. Both sets could not be in- cluded in any one equation because they form a linearly dependent set, i.e., cause the matrix to be singular. The crop reporting districts represent different kinds of farming situations. The growing seasons represent different climatic situations. The question asked was, "Which set will do the "best" job of explaining cross sectional differences in yields?" Equations estimated to answer this question were: 61 and 6 (with M(L), P, R07, T07, and 107 included); 87 and 18 (with M, Y, P, R°7, and TO7 included); and 96 and 60 (with no other variables). Procedure Used to Present Results Because of the multiple objective and because of the large amount of information obtained from the 309 equations estimated, it was necessary to be selective in presenting and discussing the results.l/y The order of i/ All coefficients estimated for all 309 equations with an indicated level of significance are presented in the Appendix. 58 presenting the results and procedure used in selecting results to present are explained below. The results are presented in Chapters IV, V, VI, and VII. Chapter IV The objective of relating the change in the level of the inputs to yield is discussed. The "complete" equation (285) is discussed in detail. In the last section of this chapter, the R? for all the "nearly complete" equations (see Table 3-6) are presented and discussed. Chapter V This chapter has three sections. Models (equations) composed entirely of man-controlled input variables are discussed in the first section. The simple correlation coefficients among the MCIV are dis- cussed in the second section. The third and final section contains a discussion of the effects of dropping a set of variables from the equation on the coefficients of the man-controlled input variables. It is not feasible to present in the text (all results are presented in the Appendix) or discuss the consequences of all the combinations of variables considered. Presenta- tion in the text and discussion are limited to (l) the set of "nearly complete" equations, and (2) equations containing unusual or interesting results. Unusual in the sense of being greatly different from a priori expectations. Interesting in the sense of containing information that would be of value to other researchers when they construct models. Chapter VI This chapter is devoted to a discussion of the location and year variables. The effect of substituting the growing season variables for “the crop reporting district variables is examined. 59 Also considered is the effect on the coefficients for the location and year variables of dropping variables or sets of variables from the equation. All combinations included in the equations are not discussed in the text. Equations included in the text were selected on the basis of their containing unusual or interesting results. The effects on the coefficients for years when the man-hours of labor are dropped from the equation are given special attention. It is primarily on the basis of these results that the decision to drop the man-hours of labor variable from the "nearly complete" equations was made. Chapter VII The weather variables are discussed in this chapter. In the first section, the coefficients for the preseason, weekly, polynomial, and season total variables and how they are affected by model specification are discussed. In the second section the estimated effects of weather in each week of the growing season as obtained from the three forms of weather variables are compared. It was not possible to determine the "correct" or "best" degree of polynomial for the weather factors a priori. The third section con- tains a discussion of why the seventh degree polynomials were selected to be included in the "nearly complete" equations. Conclusion A large number of equations were estimated. Although each equation provides some additional information about the effects of specifying alter- native models, it was not feasible to present and discuss all of these in the body of the thesis. All results are presented in the Appendix. 60 This chapter was intended to provide the reader with an overall view of the study. Use of the tables and lists of equations presented in this chapter, with the results presented in the Appendix, should permit the reader to find equations of interest. T- CHAPTER IV THE "COMPLETE" EQUATION One of the major objectives of this study was to estimate how changes in inputs affected changes in yields of grain sorghum. How well this objective has been met by the "complete" equation will be the sub- ject of this chapter. The "complete" equation (equation 285) was chosen for detailed discussion because it is most comparable to the equation stated a priori as being of principal interest. This equation differs from the one stated a priori in three ways. First, the man-hours of labor per acre variable was dropped from the equation. The acres per tractor variable was trans- formed to tractors per acre and the acres fallowed variable was transformed to ratio of acres fallowed to acres of cropland harvested. A significantly higher R2 could have been obtained if the labor variable was included. The reasons for dropping it are presented in Chapter V. The two transformations did not materially affect the R2 but did make it easier to interpret the results. The "complete" equation contained 69 weekly weather variables, a preseason precipitation variable, variables for crop reporting districts, dummy variables for years, and seven man-controlled input variables. The coefficients obtained and an indication of their level of significance are presented in Appendix B, part 37, equation 285. 61 'u \ 62 The Coefficients Man—Controlled Input Variables Seven man-controlled input variables were included. The coefficient for each variable will be discussed briefly. Percent Irrigated The coefficient for this variable was 1,761 and was significantly different from zero at the one-percent level.;/ This coefficient indicated that irrigating an acre of sorghum increased the yields by 1,761 pounds per acre. In recent years grain sorghum has sold for about $1.80 per hundredweight. If this value is assumed, then irrigating one acre of grain sorghum increased gross income per acre by $31.70. The cost of irrigating varied greatly over the area covered by the analysis. Since the range was from well below to well above the marginal return figure shown above, no precise statement about net marginal return can be made. Acres Per Farm The coefficient obtained for this variable was .0332. This indi- cates that there were positive returns to size of enterprise. That is, if the acreage of grain sorghum per farm was increased one acre, yield per acre increased .03 pounds. This coefficient was not significantly different from zero. Tractors Per Acre The coefficient for this variable (209.9) indicates that as mechani- zation (as measured by tractor numbers) increased, yields per acre increased. This effect was not significant. 1/ Unless otherwise stated, significant means the estimated coefficient is significantly different from zero at the 0.10 level. Also this coeffi- cient underestimates the effect of irrigation as part of the effect is included in the coefficient of the interaction (value of land) variable. 63 Dollars Spent on Gas and Oil The coefficient obtained for this variable (-15.92) indicates that as mechanization (as measured by machinery operating expense) increased yield decreased. This effect was not significant. Value of Land The coefficient obtained for this variable was 2.858 and was significant at the .01 level. This indicates that if the value of land increased one dollar, yield increased 2.858 pounds. Of course, the value of land cannot directly affect yield. It was assumed here that the value of land variable was a proxy variable for the interaction effect of man- controlled inputs with land. Under this assumption, the coefficient can be interpreted as follows. If the value of land increased one dollar, the interaction effects of man-controlled inputs were such as to increase yields 2.858 pounds per acre. RatigfiAcres Falloweg to Acres of Cropland Harvested The coefficient for this variable (-129.15) was not significantly different from zero. The sign was contrary to what priori knowledge suggests. The reason the sign was negative is that where fallowing was practiced, moisture and yields were low. This variable clearly did not measure the influence of fallowing on yields. Fertilizer The coefficient for pounds of plant nutrient applied per acre was 11.29 and was significant at the one-percent level. The coefficient indicates that the addition of one pound of plant nutrients to an acre of A 1 64 grain sorghum increased the yield 11.29 pounds.l/ The marginal value of one pound of plant nutrient (assuming $1.80/cwt. for grain sorghum) was 20.32 cents. This compares to a cost per pound (in 1965) of $0.115 for N, $0.23 for P, and $0.07 for K. law—s Four dummy (0,1) variables were included to represent the years, 1944, 1949, 1954, and 1959. The coefficients obtained were assumed to primarily measure the effect on yields of changes in man-controlled inputs not included explicitly in the analysis. The coefficient for a particular year measured the net effect of such changes between the year omitted (1939) and the year in question. The coefficients obtained were 367.6, 419.5, 346.4, and 528.2, respectively, for the years 1944, 1949, 1954, and 1959. All were signifi- cant at the one-percent level. The 368-pound increase in yield between 1939 and 1944 was larger than expected. It is possible that this was due to the change to shorter combine varieties, the increased use of combines, and changes in other cultural practices. It is also possible (and likely) that some of the effects of "good" weather in 1944 were included. The average increase in yields of 52 pounds between 1944 and 1949 is consistent with the hypothesis of a gradual increase in yields due to improved varieties and improved cultural practices. The 73-pound decrease in yields between 1949 and 1954 was unexpected. It is possible that poorer varieties and poorer cultural practices were used in 1954. However, it is more likely that some of the effects of "bad" weather in 1954 were included. 1/ Of course this coefficient is an underestimate of the effects of fertilizer, as part of the effect is included in the coefficient for inter- action (value of land) variable. 65 The l82-pound increase in yield between 1954 and 1959 was smaller than expected. This is particularly true if the effects of "good" weather in 1959 were included. It was hypothesized that the yield- increasing effect of hybrid grain sorghum (which took place between 1954 and 1959) would be about 400 pounds. It is possible and likely that some of the yield-increasing effects of hybrid sorghum.are captured by other variables. It is also possible that the hypothesized effect of hybrids of about 400 pounds (as indicated by experiment station results) was not realized on the farms. Although the coefficients for years were meaningful (could be rationalized) they suggest that further refinement is necessary to com- pletely separate the effects of weather from those of "technology." Crop Reporting Districts Coefficients were obtained for 27 or the 28 crop reporting districts (C.R.D.). Eleven of these coefficients were significantly different from zero. However, since no hypotheses were being tested concerning the individual coefficients and the C.R.D. omitted was essentially arbitrary, the number of significant coefficients has little meaning. The individual coefficients are examined in more detail later in this chapter when the difference in yields between crop reporting dis- tricts is explained. It was hypothesized that including this set of variables would allow a significantly greater amount of the variation in yields to be explained. An equation (equation 288), differing from the "complete" equation only by the omission of the set of variables for crop report- ing districts, was estimated so that the significance of the set could be determined. -\. P.J ‘§ 66 The R2 for the complete equation (equation 285) was .8792 and for equation 288 was .8451. These were significantly different (F = 5.60 with 27 and 536 degrees of freedom), indicating that this set of variables was significant at the one-percent level.l/ Preseason Precipitation The coefficient obtained (20.34) was significantly different from zero at the one-percent level. It indicates that one additional inch of preseason precipitation increased yields by 20 pounds. Season Precipitation Coefficients were obtained for the 23 weekly precipitation variables. Statements of significance would have little meaning since no a priori hypotheses concerning the individual coefficients were made.g/ The significance of the set was important. An equation (289) which differed from the "complete" equation only by the omission of the set of season precipitation variables was estimated so that the significance of the set could be determined. An R2 of .8639 was obtained. This was significantly different from .8792 (from the "complete" equation) at the one-percent level. The F value was 2.94 with 23 and 536 degrees of freedom. l/ The following is from a mimeo "Procedure for Testing the Significance of A Subset of Regression Coefficients" by R. L. Gustafson, Michigan State Univ., Oct. 27, 1960. These formulas were derived from results presented in Anderson and Barcroft, Statistical Theory in Researgh, page 72. For con- 'venience, let the variable to be tested by represented by Xp+l to Xq. Let ‘the remaining variables in the model be represented by Xl,...,Xp. Obtain R? from the regression on Xl,...,Xp. Obtain R2 from the regression on LX ,...,XP, Xp+1,...,Xq. Then under the null hypothesis (i.e., Bp+l::Bp+2 := . . . = Bq = 0 and the assumption that the disturbances are normally distributed: 2 2 Fq—p, N-q-l : 59-:-EE ‘ E393; 1 - R3 q'P g/ The individual coefficients and an indication of their level of sig- Irificance are presented in Appendix B. olw L 67 Season Temperature Coefficients were estimated for the 23 weekly temperature variables. Individual coefficients are of little interest.l/ The significance of the set was determined by testing whether there was a significant increase in R2 when this set is added to an otherwise "complete" equation. The change in R2 (from equation 291 to equation 285) is .0225. This difference leads to an F value of 4.33 with 23 and 536 degrees of freedom which was sig- nificant at the one-percent level. Season Interaction The set of 23 interaction variables caused the R2 to increase from .8631 in equation 292 to .8792 in the "complete" equation. This increase was significant (F = 3.1 with 23 and 536 degrees of freedom) at the one- percent level. All four sets of weather variables were significant at the one- percent level. Explainingfthe Change in Yield One of the major objectives of the study was to estimate how changes in the level of inputs and the shifts in the location of production have affected yields. The change in weighted average yield is due to three components as shown below. we have: J Yit = Ci + 3E1 Xitj bj where Yit and Xitj are yield and levels of the J independent variables l/ The individual coefficients and an indication of their level of significance are presented in Appendix B. 68 respectively, in district i in year t; bj are the estimated regression coefficients, and Ci a constant estimated for district 1. For simplicity, we omit residual terms so "yield" really means estimated "expected" yield. It should be remembered that the four dummy variables for years are in- cluded in the Xj's as are all the man-controlled input variables and the weather variables. Weighted average yield in year t then is: _ n Yt = .§ Pit Yit l—l where Pit = Ait , i=1,...,n is the proportion of total 2 A k=l kt acreage in district i in year t. So: _ n J n Y : Z P. Co + E Z P. X. o be t i=1 1t 1 j=1 i=1 lt ltj j The change in average yield between two years, t-l and t, is: n t ‘ Yt—l = iii Ci (Pit ‘ Pi,t-l) AXt,t-l J n + jEi iii (Pit Xitj ‘ Pi,t—l Xi,t-l,j) bj n Let K = 15.1 Ci (Pit - Pi,t-l) * Let Xitj = Xi,t—1,;j + Xi,t-l,j J n - _ * AYt,t_l — K + jgi iii (Pit [Xi’t_1,j + Xi’t_1,j] - Pi,t-l Xi,t-l,j) bj 69 J n = K + jgl 1:1 (Pit ‘ Pi,t-l) Xi,t-l,j b5 J n * + 3,51 131 Pit Xi,t-l,i bi J n : + ' " ° ° . . K jg; iii (Pit Pi,t-l) X1,t-l,a b3 J n + iii iii Pit (Xitj ‘ Xi,t-l,j) bi Then, because weather, years and technology (man-controlled inputs) are represented by the Xj's, n K = iii Ci (Pit - Pi,t-l) is the effect on average yield of shifts in the location of production, independent of the effects of time, weather and technology. J n L : jél 1:1 (Pit' Pi,t-l) Xi,t-1,j bj is the effect on average yield of shifts in the location of production due to the fact that tech- nology and weather are not the same in all districts, based on their levels in year t-l. M = jgl 123 Pit (Xitj - Xi,t-l,j) bj is the effect on average yield of changes in the level of technology and weather between years, with the location of production as it was in year t. If Xi,t_1,j = Xitj - xgtj is substituted in place of xitj = xi,t_1,j + x;,t_l,j, K remains the same, but LS§YL* = g g (P- - P- ) X- - b- j=l i=1 it l,t-1 ltj j and 70 say J n M = M* 2 3'31 151 Pi,t-l (Xitj - Xi,t-l,j) bi Thus there are two estimates of the components of the effects of weather and technology on yield. Of course, L +-M = L* +-M*, so the total effect of weather and technology is the same regardless of how the components are estimated. Under certain conditions, some of the above values become equal and/or zero. If Pit = Pi,t-l for all i, i.e., if there is no change in the dis- tribution of acres, then K = L = L* = 0 and M =‘M* If Xitj = Xi,t-l,j for all i and j, i.e., if there is no change in the level of the independent variables (of course this is not possible if there are dummy variables for time included in the Xj's), L = L* and M = M* = 0 If Xitj = thj for all i, j and k, i.e., the level of each of the independent variables is the same (and necessarily the average, Kfj) in all districts, then: L = L* = 0 and M can be written J n M = Z 21 P- jzl 1:1 It (th - Xt-l’j) bj i=1 J _ n J — _ : .§ (th - Xt-l,j) bj Z Pit 2 jz (th - Xt-l,j) bj : M* n n Since 151 Pit = 25.121 Pi,t-l = 1 Clearly, if itj = it-l,j for all j, i.e., if the average level of each technology, weather and year variable (of course this is impossible for the year variables) is the same in the two years, then: 71 M = M* = O and the only non-zero term is K. In the thesis a constant (Ci) was not obtained for every district; rather an overall constant term was obtained. However, adding or sub- tracting (as is the case with an overall constant term) any constant to all the Ci would not change the results, since n 1:1 (Pit ‘ Pi,t-l) = 0 Constants (Ci) were obtained only for districts in this study because computer capacity was too limited to allow estimating the 129 county constants in the computer and hand computation of them would have been too tedious. However, there is no reason why county constants could not have been estimated. The procedure outlined above for deriving the components of change in average weighted yields was not developed until after the analysis was completed. It was decided that it was not worthwhile going back and computing L, L*, M and M*. Rather, the change in the unweighted yields was explained using the unweighted average level of each of the independent variables. That is, J it - Et-l : K + j§l (itj - it-l,j) bj where the V and i3 refer to unweighted averages and K is the same as shown above. Explainigg the Change in Yield Over Time One of the major objectives of the study was to estimate how changes in the level of inputs over time have affected changes in yields over time. ' The changes in yields over time that were to be explained are presented in Table 4-1. The changes in the level of inputs (except for seasonal weather) are also presented. 72 Table 4-l.--Changes in yields and levels of factors between yearsé/ Change between-- Factor 3 Unit . _ . . ; 1939-44 3 1944-49 ; 1949-54 ; 1954-59 ; 1939-59 Average yield---:Pounds 515.0000 90.3000 -223.0000 763.4000 1145.7000 Percent irri- : gated :Percent -.O240 .0290 .0230 .0210 .0490 Acres /farm ----- :Acres 47.7000 -10.4000 48.3000 7.9000 93.5000 Tractors/acre--:Tractors -.0001 .0010 .0014 .0005 .0028 Fuel expense-—-:Dollars -.0200 .5800 .2600 .1000 .9200 value of land---:Dollars Percent fallowed:Percent 6.0700 14.4200 10.5100 14.5900 45.5900 -.l234 .0608 .0451 .0162 -.0013 Fertilizer------;Pounds 0 1.3500 3.9100 4.3900 9.6500 Preseason pre- cipitation ----- :Inches 5.2300 -.1100 -3.6100 .6600 2.1700 g/ All values are unweighted; i.e., the distribution of acres between counties was p23 taken into account. The yield was given equal weight, regardless of acres harvested. Given the change in the level of inputs (Table 4-1) the effect on yield was determined by multiplying the change by the coefficient from equation 285.l/ This was done and the results presented in Table 4-2. The effects of changes in seasonal weather were not obtained directly, i.e., the change in each of the 69 variables between years was get obtained. The effect was determined as the residual amount explained. Because of the constant terms obtained for years, the average yield for a year exactly equaled the average predicted yield for that year: i.e., YA = H”) A. 1/ This procedure does not take into account the difference in the level of inputs between crop reporting districts, i.e., uses unweighted averages. 73 Table 4—2.--Effect of changes in level of factors on changes in yieldsfl/ :Coefificient: Effect of change between-- : rom : Factor - e uation - - - - - ; q 285 ; 1939-44 3 1944-49 ; 1949-54 ; 1954-59 ; 1939-59 Man-controlled ; Pounds per acre inputs: : Percent irri- : gated :2/1,76l.0000 -42.3 51.1 40.5 36.9 86.2 Acres/farm----—: .0332 1.5 -.3 1.6 .3 3.1 Tractors/acre---: 209.9000 0 .2 .3 .l .6 Fuel expense---: -l5.9200 .3 -9.2 -4.1 -l.6 -l4.6 Value of land---: 2/2.8580 17.3 41.2 30.0 41.7 130.2 Percent fallowed; -12.9100 1.5 -.7 -.6 -.2 O Fertilizer----—: E/ll.2900 0 15.2 44.1 49.6 108.9 TOtal M0001.-- ---- -2107 9705 111.8 126.8 31404 Years2/' —--- 367.6 51.9 -73.1 181.8 528.2 'Weather: Preseason pre- cipitation--—-- 2/20.3400 106.3 -2.2 -73.4 13.4 44.1 Season weatheré/ —--- 62.8 -56.9 -l88.3 441.8 259.0 Total weather- ——-- 169.1 -59.1 -262.1 454.8 303.1 Total = average . yield differencea/ —-—- 515.0 90.3 -223.0 763.4 1145.7 g/ All values are unweighted; i.e., the distribution of acres between counties was not taken into account. 2/ Significantly different from zero at the .01 level. 2/ Constants for years obtained in equation 285. g/ Instead of obtaining the 69 season weather values for each year and multiplying by the appropriate coefficient, this effect was obtained by sub- tracting the effects of all other factors from the total effect (see text). 2/ Average yield difference (Yj - Yk) equals éj - 7k) average predicted yield difference. 4 . e‘fl' .-.. v t.) (I) “'4... I 1“— . "-5-. . ‘ e 4. . "‘ II C . ‘- .. .._ ...w . . ... . \ u.» ‘1‘ ‘ . 'h -‘l '. it ‘. .5 ‘ .\ - a 74 _ _. ._ ._ 69 _ _ 80: (YA - YB) = f bi (XiA ' XiB) + 3:1 b1 (XJA ’ XJB) Where A and B are different years, the i's refer to independent variables other than seasonal weather, the j's refer to the weekly weather variable, ‘Y is average unweighted yield and ii or K3 unweighted average level of factor. When differences are taken, the overall constant drops out. Thus the effect of the changes in weekly weather is: ‘69 _. _ _ _ _ _ jii bj (X34 - XjB) = (YA - YB) - f bi (XiA - XiB) The Change in Yields, 1939-1944 The decrease in percent of acres irrigated between 1939-1944 completely dominated the influence of the explicit man-controlled inputs. The change in the level of the explicit M.C.I. 1939-1944 caused unweighted average yield to decrease 22 pounds. However, the effect of implicit M.C.I. (years) caused a substantial (368—pound) increase in yields. The weather in 1944 was better than in 1939, and enough better to have in- creased unweighted average yields 170 pounds. The explicit M.C.I. explained a -4.2 percent of the yield increase 1939-1944. Implicit M.C.I. explained 71.4 percent of the increase. Better weather explained 32.8 percent of the increase. Most of the effect of better weather (62.9 percent) was due to more preseason precipitation. The Change in Yields, 1944-1949 The increased use of irrigation, fertilizer and their interaction effect with land (value of land) caused yields to increase 107.5 pounds per acre. The effect of all explicit M.C.I. was to increase yields 97 pounds. The change in the implicit M.C.I. also caused yields to increase. weather in 1949 was worse than in 1944 and caused a decrease in yields of 60 pounds. 75 The change in M.C.I. explained 108 percent of the increase in yields. Implicit M.C.I. changes explained 57.5 percent of the increase. Poorer weather explained a decrease of 65.5 percent. Most of this decrease (96 percent) was due to poor weather during the growing season. With average weather the average yield would have increased 149.4 pounds in- stead of the 90.3 pounds actually achieved. The Change in Yields, 1949-1954 Unweighted average yield decreased 223 pounds between 1949 and 1954. The effect of changes in the explicit M.C.I. was to increase yields 112 pounds. The effect of implicit M.C.I. was to decrease yields 73 pounds. It is unlikely that the changes in the implicit M.C.I. would actually decrease yields. Rather, it is expected that some of the effects of "bad" weather were included in the years' coefficient. Poorer weather in 1954 caused yields to decrease 262 pounds. With average weather, and implicit M.C.I. at the same level in 1954 as in 1947, the 1954 average yield would have been 122 pounds greater than in 1947. Changes in M.C.I. "explain" a negative 50.1 percent of the decrease in yields. Changes in implicit M.C.I. explain 32.8 percent of the decrease. Poorer weather explained 117.3 percent of the decrease in yields. Of this, 72 percent was caused by poorer weather during the growing season. The Change in Yields, 1954-1959 The change in the level of explicit M.C.I. from 1954 to 1959 caused 'unweighted average yield to increase 127 pounds (l6fSpercent of the total increase). The change in the level of the implicit M.C.I. caused an in- crease of 182 pounds (23.8 percent). Better weather in 1959 than in 1954 caused yields to increase 455 pounds (59.6 percent of the total). 76 The Change in Yields, 1939 to 1959 The unweighted average yield increased 1,145.7 pounds between 1939 and 1959. Of this increase, 27.4 percent was explained by changes in the levels of the explicit man-controlled inputs, 46.1 percent by changes in the level of implicit man-controlled inputs;/, and 26.5 percent by changes in weather. Of the increase due to changes in explicit man-controlled inputs, almost all is due to changes in two inputs, fertilizer and irrigation and their interaction with land (value of land). Changes in the weather during the growing season accounted for 85.4 percent of the total weather effects. The relative importance of the implicit M.C.I. was unexpected. Although it was hypothesized that the effect would be large, it was not expected to be 60 percent mggg important than the explicit M.C.I. This is somewhat disturbing because it suggests that some of the most important factors affecting yields have not been explicitly identified or quantified. Explainipg Cross-Sectional Differences in Yields In addition to trying to explain changes in yields over time, it was also important to try and explain yield differences between crop reporting districts. The differences were determined relative to some base. In this case crop reporting district 19 was used as the base. The difference in yields between C.R.D. l9 and another district was explained by differences in levels of M.C.I., weather, and location (a constant which is really an un- explained residual that was consistent over time). 1/ Of course, the effect of other unquantified factors is also included, but it is believed that the unquantified man-controlled factors are by far the most important. 77 The average (over time and over counties within a crop reporting district) deviations in level of man-controlled inputs of crop reporting districts relative to C.R.D. 19 are presented in Table 4-3. The effects of differences in the level of M.C.I. are presented in Table 4-4. The effect of differences in weather and location are also presented in Table 4-4. Coefficients from the "complete" equation (equation 285) were used to calculate effects of M1C.I. and location. The effects of weather were derived in the same manner as were weather effects between years. Considering Table 4-3, it is apparent that the level of two of the most important M.C.I. (percent irrigated and pounds of plant nutrient per acre) were lower in all districts than in district 19. The level of the value of land (the only other important input) was lower in some districts and higher in others. The consequences of these lower levels oij.C.I. are shown in the second column of Table 4-4. The lower level of these M.C.I. in all districts relative to district 19 would explain substantially lower yields in these districts. Yield differences are not always great (see column one, Table 4—4) because the effects of location (soil, climate, topography) and weather gave yield advantage to some of these districts. Perhaps it will be clearer if one district as an example is dis- cussed. District 1 had an average yield of 74 pounds per acre greater than district 19. What explains this difference? The difference in the level of the M.C.I. would suggest that average yield in district 1 should have been 445.2 pounds less than in district 19. However, this effect was nwre than offset by a location which gave district 1 a 383.9-pound per acre yield advantage, and better weather which gave a 135.3-pound yield advantage. 78 Table 4-3.-Deviation in average (over time) level of man-controlled inputs for crop reporting districts relative to level in district 19 FactorsE/ 0.12.0.3 , , . . . . 3 5% 3 Ft 3 v ; A/F ; FO/A ; 8 ; T/A 3 Percent Pounds Dollars Number Number Dollars Number ---= -.264 -7.14 +34.73 —l32.99 -.O9l4 -.23 +.0029l 2---: -.271 -6.89 +10.37 -l24.03 -.O778 -.3O +.00212 3----: -.265 -6.72 -l4.79 -l26.l9 +.0516 -.27 +.00237 4---: -.274 -8.31 -25.18 -83.87 +.4307 -.48 -.00009 5----: -.278 -8.39 -12.73 -l27.25 -.Ol87 -.40 +.OO222 6---: -.279 -6.53 +5.47 -14l.25 -.O96l -.22 +.00346 7----: -.240 -8.31 -23.81 -54.86 +.4711 -.28 -.OOO26 8---: -.255 -8.21 +14.82 -l22.83 +.O30l -.21 +.00277 9----: -.279 -6.00 -lO.32 -l37.39 -.1022 -.13 +.00238 lO---: -.222 -8.31 -20.15 +44.47 +.40l8 -.50 -.OOO57 11---: -.270 -8.19 +21.60 -ll7.88 -.Ol58 -.42 +.00176 12--—-: -.279 -6.96 -2.64 -124.51 -.O709 +.lO +.OO428 l3---: -.271 -8.63 —30.59 —19.57 +.l352 —.7O -.OOO75 l4---: -.279 -7.92 +9.18 -l32.48 -.O88O -.48 +.OO308 l5---: —.278 -7.92 -6.67 -l34.37 -.O83O -.22 +.OO48O l6---: -.276 -8.96 -42.27 -59.62 +.4606 -.34 -.00023 l7---: -.l30 —8.98 -39.40 -38.36 +.4258 -.O8 +.OOO73 l8--—: -.208 —7.56 -31.99 -6.63 +.O489 -.O7 +.00148 19---: O O O O O O O 20--: -.219 -6.93 +6.26 -20.92 -.0584 -.30 +.OOO68 21--: -.276 -8.18 -26.06 -76.58 -.0820 -.43 -.OOl79 22---: -.277 -8.18 —lO.62 -94.50 -.O752 -.33 +.00280 23-—--: -.267 -7.38 -34.87 +81.73 -.O68l +.13 +.00312 24--—-: -.278 -7.03 +7.25 -133.67 —.O767 -.l7 +.00544 25---: -.273 -7.24 +3.55 -122.04 -.0632 +.56 +.OO768 26--: -.277 -7.38 +34.17 -15.48 -.O784 +.36 +.0032l 27--: -.276 -7.38 -20.27 -73.28 -.O86l +.l7 +.OO415 28--: -.l20 -7.63 +82.4l -61.52 -.O773 +1.68 +.OO704 g/ % means percent of grain sorghum acreage irrigated; Ft means pounds of fertilizer applied per acre; V means value of land per acre; A/F means acres of sorghum per farm; FO/A means ratio acres fallowed to acres of cropland harvested; $ means dollars spent on gas and oil per acre; and T/A means number of tractors per acre. 79 Table 4-4.--Yields of crop reporting districts relative to crop reporting district 19 and factors explaining the differences c R’D ; Yield ; Factors 3 difference 3 M.C.I. 3 Location Weather 3 Pounds per acre l----: 74 -445-2 383.9 135.3 2----: -126 -523.3 239.5 157.8 3"“""': “421 '5000 3 "lol 8004 4-----: -680 -505.0 -47.3 -127.7 5-----: -585 -545.0 -28.0 -12.0 6-----; -301 -548.6 182.8 64.8 8-----: -559 -499.9 -l31.l 72.0 9----: -336 -530.2 72.9 121.3 lO-----: -624 -422.8 -88.7 —112.5 11 ..... ; ’685 “50209 “23109 4908 12-—--- -576 -566.3 -ll8.4 108.7 l3----- -891 -478.3 -242.1 -l70.6 l4----- -690 -549.5 -20.3 -l20.2 15-----' -849 -558.8 -345.3 55.1 16- ----- -l,062 -468.9 -451.2 -l41.9 ' l7—--—-° -935 -223.1 -389.6 -322.3 18 ....... '1,008 -35906 -10806 -53908 19----: O O O O 20----° -673 -44l.0 -286.6 54.6 21-----§ -959 -498.2 -262.6 -198.2 22-—---. -885 -546.1 —227.2 -111.7 23--—--—: -728 -451.7 -7.8 -268.5 24"”"‘: -460 “-547 o 8 -13 o l 100 o 9 26-—--—-; 98 -478.0 115.7 460.3 27----: -442 -514.6 -l97.5 270.1 28--—---: -252 -88.2 -4l4.4 250.6 80 It is not possible to make any general statement about the relative importance of M3C.I., location, or weather in explaining yield differences between districts. The relative importance depends to a large extend upon which districts are being compared. Any two districts can be compared simply by taking the difference between the numbers presented in Table 4—4. For example, to compare district 18 with district 28: Average yield in district 18 was 756 pounds lower than in district 28. The difference in the level of M.C.I. would have explained yields being 271.4 pounds lower in district 18. However, locational factors caused yields in district 18 to be 305.8 pounds higher. Poorer weather in district 18 would have explained yields being 790.4 pounds lower. The net effect was to have yield 756 pounds lower in district 18. Effect of Shift in Acres on Average Yields The estimate of the effects of shifting the location of production is not entirely consistent with estimates of the effects of man-controlled inputs and weather. The estimate would be consistent if the effect of the man-controlled inputs and weather had been estimated taking into account the distribution of acres among the districts. The effect of shifts in location of production (the K terms discussed earlier in this chapter) on average yield was estimated by using the constants for crop reporting districts from the "complete" equation (equation 285). Since the constants were location effects, independent of weather and technology, the estimate of the effect of changes in the location of production using these constants is also independent of the effects of weather and technology. The effect of shifting acreages between any two years was computed by taking the change in the pr0portion acreage in a district is of the total mmltiplied by the constant for that district and summed over all districts. uy. 81 28 That is, KAB = :3 cfim - £113) where KAB is the change in yield due to shift AA AB in location of production between years A and B; C- is the constant from 1 equation 285 for crop reporting district i, AiA and AiB are the acreages in crop reporting district i in years A and B respectively, and AA and AB are the total acreages in years A and B, respectively. The effects for selected years are presented in Table 4—5. Table 4-5.--Effect of shifts in location of production between selected years Years Effect Pounds per acre 1939-1944, : 8.1 1944-1949 : 17.1 1949-1954 : 18.3 1954-1959 : 6.8 1939—1959 : 50.3 The effect between any two consecutive periods was small. The effect over the entire period was less than one bushel. This was less than 2.8 percent of the 1959 average yield. The effects have been positive over time, indicating that production has been shifting slowly toward higher yielding areas. lpdependence of_Residu§1§ If the residuals from an equation are to be examined, it seemed reasonable to use the residuals from the "complete" equation. That is why this section is included in this chapter. The residuals examined are from equation 285. ..W 82 In a combined time series and cross section analysis there is the possibility of dependence of disturbances in two dimensions-cross sectional and over time. The observations on the same county are separated by five years. Thus, the assumption that disturbances for a single county over time are independent seems warranted. This conclusion can be extended to the set of all counties. In addition, any consistent variation among all counties would be removed by the constants (coefficients for the dummy year variables) for years. Thus, it is concluded that there was no serious problem with auto-correlated disturbances in the time dimension. Cross sectionally, the counties included in the study were not selected at random and the disturbances, particularly in adjacent counties, might not have been independent. Of course, nothing can be done in terms of the disturbances, but the residuals are examined. If the disturbances in adjacent counties were not independent then one might expect to observe some relationship among their residuals. It is hypothesized that the residuals of two counties adjacent east to west (have a common north-south boundary) are correlated. In the test that follows, counties A and B are considered adjacent west to east if a person could move from county A straight east and immediately enter county B. The residual from county A (on the west) was considered a value of X and the residual from county B (on the east) a value of Y. There are 110 such pairs of values. The simple correlation coefficient of X with Y was obtained for each of the five years. The correlation coefficients obtained were: For 1959, r2 = .005; for 1954, r2 = .071; for 1949, r2 = .023; for 1944, r2 = .026; and for 1939, r2 = .040. It seems reasonable to conclude that there is very little relationship between residuals in adjacent (east to west) counties. 83 The above procedure is arbitrary. We could, instead of or in addi- tion to the above, have considered north-south adjacent counties. Other, more elaborate criteria (such as the common boundary must have a specified minimum length, etc.) could have been chosen in selecting pairs. With a complex situation such as we have here, it may be as appropriate simply to "look over" a map of the residuals to see if any pattern is apparent. This was done and no pattern was observed that would call into question the assumption that the disturbances are independent. Comparison of Alternative Models After the "complete" equation (equation 285) had been estimated, it was of interest to see how its explanatory power would be affected by omitting certain sets of variables and/or when different sets of variables were substituted. The sets alternately dropped were: (1) Man-controlled input variables, (2) crop reporting district variables, (3) year variables, (4) season precipitation variables, (5) season temperature variables, (6) season interaction variables, and (7) season temperature and season precipitation variables. The results of these seven omissions onR2 are presented in the first column of Table 4-6. Dropping the M.C.I.V. resulted in a decrease of .1454 in.R? which was significant at the one—percent level. As measured by R? deletes, this was by far the most important set (as expected). The second most important (as determined byR2 deletes) was the set of precipitation and temperature variables. This decrease of .0447 was significant at the one-percent level. Dropping any one of these seven sets of variables caused R2 to decrease significantly. 84 Table 4-6.-R2 for equations using corrected dataé/ Set of variables Season weather represented by-- .8432** (289) .7864 (305) .7985** (298) Precipitation----—- .8349** (291) .7843 (304) .8161 (300) Temperature omitted f . . , ‘ Weekly weather ° Season totals ' Polynomial Of : : : seventh degree None Q .8545 (285) .7861 (802) .8213 (294) M.C.I. ; .7094** (286) .6090** (307) .6445** (295) C.R.D. Q .8228** (288) .7462** (308) .7855** (296) Years Q .8420* (287) .7361** (309) .8022** (297) Interaction-—------' .8423** (292) .7861 (303) .8005** (301) Precipitation and temperature .8101** (290) .7840 (306) .7954** (299) g/ The equation with no sets of variables omitted contain 7 M.C.I.V., set of variables for crop reporting districts, set of variables for years, and weather variables for precipitation, temperature, and interaction. Number in parentheses is the equation number. * Indicates that the change in R2 when this set is omitted as compared to when no set is omitted is significantly different from zero at the 5- percent level. Test used was discussed earlier in this chapter. ** Same as for * except at l-percent level. It was also of interest to see how representing the season weather with different variables would affect the ability of the model to explain yield variation. The effects of substituting polynomials of seventh degree (24 variables) or season totals (3 variables) for the 69 weekly weather variables are presented in columns 3 and 2, respectively, of Table 4-6. The effect of substituting polynomials for the weekly variables was to reduce R? by .0335. This was a very small loss to gain 45 degrees of freedom. yThe .0687 reduction in R2 when the season totals were substituted was also quite small. This result suggests that no major error was made . ‘ 85 when simple season total weather variables were used instead of very detailed weekly or polynomial weather variables. Of course, this assumes that the remainder of the model is well specified. It is interesting to note that the manner in which weather was represented affects how R? changes when a set of variables were deleted. For the case of M.C.I. variables, when weekly weather variables were in- cluded the R2 decreased .1454. When polynomial weather variables or season total variables were included R2 decreased .1768 and .1771, respectively, when.M.C.I. variables were dropped. This suggests that the model was more sensitive to specification errors (of this type) when less detailed or less complete weather variables were included. The same pattern held for the other sets of variables. The order of importance of sets of variables (as judged by‘R2 deletes) was also affected by the manner in which weather was included. When weekly weather variables were used the set of precipitation and temperature variables was the second most important. However, when weather was represented by either of the other forms, the set of crop reporting districts was the second most important. Perhaps this was because when less complete weather variables were used the effects of consistent differ- ences in weather between districts tended to be "captured" by the district variables. It is also of interest to note that when season totals were used to represent weather, dropping the precipitation term increased R2 (see Table 4—6). In this case the value of an additional degree of freedom (even though 602 were already available) was greater than the value of the increased sums of squares explained. The reason for this may have been that the seasonal interaction variable was almost a perfect substitute for the seasonal precipitation variable (simple correlation .992). 86 Conclusion The "complete" equation did a reasonable job of explaining variation in yields. The coefficients, for the most part, were meaningful and consistent with expectations. The amount of variation explained by the model was significantly affected by dropping sets of variables, and greatly affected by substituting in new variables. The maximum effect of dropping variables (dropping the M.C.I. variables) was to reduce the variation explained by 17 percent. The maximum effect of substituting variables (substituting season totals for weekly weather variables) gag dropping the M.C.I. variables was 29 percent. The next three chapters will consider the question, Does dropping or substituting variables affect the coefficient (and thus the interpreta- tion) of variables remaining in the model? CHAPTER V MAN-CONTROLLED INPUTS This chapter has three sections. Models composed entirely of man- controlled input variables (M.C.I.V.) are discussed in the first section. The simple correlation coefficients among the M.C.I.V. are discussed in the second section. The third and largest section contains a discussion of the effects of dropping sets of variables from a model on the coefficients of the M.C.I.V. Man-Controlled Input Models Many models containing only'M.C.I.V. as independent variables have been estimated by agricultural economists. It is of interest to see how such models compare to the model containing weather variables, time variables, M.C.I.V., and location variables as independent variables. All the models discussed below were estimated using the not corrected data. However, the relative R2 should still be meaningful. The model containing the weekly weather, time, location, and M4C.I. variables was equation 244. It had an R2 of .779. The value (.779) will be compared with the R2 of models containing only M.C.I.V. The Rz's for selected M.C.I.V. models are presented in Table 5-1. The symbols used are listed in Chapter III. The highest R2 obtained (.454) was, as expected, with the model (equation 223) containing all the M.C.I.V. This, however, does not compare favorably with the .779 from equation 244. Mis-specification by not in- cluding weather, location, and time variables had a rather severe effect on -2 R . 87 88 Table 5-1.--R2's for models containing only :37-controlled input variables as independent variable a Equation : Man-controlled input 3 Man-controlled input 3 -2 number I variables included : variables excluded : R 230----: VL L,FO,FT,A/T,%,$,A/F .278 229------: VL,A/F L,FO,FT,A/T,%,$ .299 228----: VL,A/F,FT L,FO,A/T,%,$ .401 227-----: VL,A/F,FT,% L,FO,A/T,$ .440 226------: VL,A/F,FT,%,A/T L,FO,$ .440 225-----: VL,A/F,FT,%,A/T,$ L,FO .441 224--—--: VL,A/F,FT,%,A/T,$,L F0 .453 223——-—--: VL,A/F,FT,%,A/T,$,L,FO .454 237-----: F0 VL,L,FT,A/T,%,$,A/F .004 236-—----: FO,L VL,FT, T,%,$,A/F .163 235-----: FO,L,$ VL,FT,A/T,%,A/F .189 234----: FO,L,$,A/T VL,FT,%,A/F .189 233-----: FO,L,$,A/T,% VL,FT, F .356 232------: FO,L,$,A/T,%,FT VL,A/F .392 231--—--: FO,L,$,A/T,%,FT,A/F VL .396 238--—-—-—: A/F VL,L,FT,A/T,%,$,FO .015 239 ----- : A/F,FT VL,L,A/T,%,$,FO .258 240-—----: A/F,FT,% VL,L,A/T,$,FO .342 241-----: A/F,FT,%,A/T VL,L,$,FO .346 242 ------ : A/F,FT,%,A/T,$ VL,L,FO .346 243------ A/F,FT,%,A/T,$,L VL,FO .384 g/ Estimated using the not corrected data. Considering the models in Table 5-1, there are only three variables that greatly affected R2 when they were dropped. They are percent irrigated, fertilizer, and value of land. Not by chance, these were the same M.C.I.V. that had significant coefficients in the "complete" equation (equation 285). Simple Correlations The effects of dropping a variable or set of variables from an equa— tion on.R2 and on the coefficients of variables remaining in the model are influenced by the degree of intercorrelation among the variables. Because of this and because the intercorrelations among the M.C.I.V. are of interest in and of themselves, they are discussed here. '1 89 The simple correlation between the M.C.I.V. are presented in Table 5-2 0 Percent Irrigated The correlation of percent irrigated with yield was very high. This is not surprising as moisture is a principal factor limiting yields in the Great Plains. The positive correlation with acres per farm was not expected.l/ It was expected that large acreages of sorghum were found on farms with extensive operations. Irrigation is a form of intensive farming. There was very little correlation between number of tractors (T/A or A/T) and extent of irrigation. As larger irrigation operations require more fuel, it is not sur- prising that there was a positive correlation between the two. However, the extent (.353) of the correlation is surprising. The high correlation of value of land with percent irrigated is consistent with expectations. The negative correlation with labor is contrary to expectation. Irrigation certainly requires more labor per acre than non-irrigation. This result was probably due to the poor labor data used. This point is discussed in more detail later in this chapter. There is very little relationship between fallow and irrigation, contrary to expectations. This may be due to counteracting forces. Irriga- tion and fallowing are probable positively correlated to the extent that where moisture is limiting both irrigation and fallowing tend to be high and where moisture is not limiting they tend to be low. On the other 1/ "Expected" in all cases refers to the author's expectations. 90 =w= Hmvpmfi $39 .oocaaano poz a .eao._m a we Hosea 0H. as can .mmo. m a as Hosea mo. as abqa. 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Hao.- mnnuuuuuuoa ooo.H acme.- 6804.- * swam. amem.- aoNH.- ammm.- muuuuuuunnq ooo.H come. doom. amom.- Nmo.- comm. aamm. mssuuuuuuga ooo.H aamm. cooN.- ammo. ammm. seam. m a ooo.H * amea.- moooo.- amma. muuuunaua\e ooo.H aaam. 466.- amma.- muuuuuuue\< ooo.a anew. aoma. muuuuu--a\< ooo.H came. muuuvuuuuum ooo.H mutua\bfiow» as H axon H on M a H g> H a H «\e M e\¢ M a\< H a M<\6Howmw 1:. 323.85 11 633834 4/ \Mmoflnwfihw> adds“ poaaoppcoolcda can mo mGOfipoHoppoo oaeeamun.mnm canoe 91 hand, they are probably negatively correlated to the extent that irrigation is a substitute for fallowing. As hypothesized in Chapter II, high levels of fertilizer application were highly and positively correlated with high percent irrigated. This is probably because the practices are complements. Acres Per Farm Acres of grain sorghum harvested per farm harvesting grain sorghum was positively correlated with yield, contrary to expectations. It was expected that large sorghum operations were in areas where extensive dry- land operations were found. Yields in such areas tended to be low so a, negative correlation was expected. However, as also indicated by the correlation of acres per farm with irrigation, large sorghum operations tended to be found in the intensively farmed areas. The correlation with tractor numbers is as expected. That is, where the size of sorghum operations was large, the number of cropland acres to tractor (A/T) tended to be large or conversely the number of trac- tors per cropland acre (T/A) tended to be small. However, this indicates extensive type of farming where sorghum operations are large, which is contrary to conclusion reached in paragraph above. Fuel expenses per acre tended to be high when size of sorghum opera- tions were large. This again suggests that more intensive operations were associated with large sorghum operations. Value of land was not associated with size of sorghum Operations. This suggests that the value of land was not determined to any large extent ‘by the profitability of the sorghum enterprise. Size of sorghum enterprise was negatively correlated with man-hours of’labor. This is consistent with the hypothesis that large sorghum 92 enterprises are extensively farmed. However, because of the poor labor data, too much reliance should not be placed on this result. Sorghum enterprises tended to be large where number or proportion of acres fallowed was high. Since fallowing is practiced more in drier areas, it seems to follow that large sorghum enterprises would tend to be located in these drier areas. Given the correlation (.207) of irrigation with size of enterprise, it seems to follow that higher level of irrigation was also found in the drier areas. Rates of fertilizer application were positively correlated with size of enterprise. This is consistent with more irrigation being done on large sorghum enterprises and fertilizer being positively correlated with irrigation. Acres Per Tractor and Tractors Per Acre Both of these variables are discussed here because they both deal with the relationship of number of tractors to acres of cropland harvested. Yield was positively correlated with tractors per acre. Those areas or years that were more mechanized (as measured by tractor per acre) tended to have higher yields. The negative correlation of yield with acres per tractor leads to the same conclusion. As would be expected, as the number of tractors per acre increased, fuel expenses per acre increased. Or, the inverse relationship, when acres per tractor increased, fuel expenses per acre went down. Where value of land was high, the number of tractors per acre was also high. This indicates that land was more intensively farmed where land values were high. There was a high positive (.318) correlation between hours of labor per acre and number of acres per tractor. This is as expected because trac- tors are a substitute for labor. As the number of acres per tractor decreased (increased tractor numbers) the need for labor per acre of sorghum was less. 93 The positive correlation (.224) between acres fallowed and acres per tractor and the negative correlation (-.167) of percent fallowed with tractor per acre were consistent with each other. They were also consistent with the idea that if the number of acres fallowed was high the need for tractors would be less. Where the number of tractors per acre was high (more intensive farming) the level of fertilizer applied tended also to be high (also more intensive farming). However, the correlation (.066) was not very high. Dollars Spent on Gas and Oil Dollars spent on gas and oil (fuel expense) was positively correlated with yield. That is, use of machines was positively correlated with yields. Fuel expense was highly and positively (.436) correlated with value of land. This is consistent with a hypothesis that higher valued land is farmed more intensively. The negative correlation of fuel expense with labor is consistent with the fact that they are substitutes. There was very little relationship between acres fallowed and fuel expense. Fertilizer level and fuel expense were highly and positively (.424) correlated as expected. Both practices are a priori positively related to more intensive farming. Value of Land Yield was very highly correlated (.529) with value of land. This is as expected. Higher valued land would have to support higher yields if the higher value was justified (ignoring for the moment the effect of non- agricultural demand for land). Where (cross sectionally) or when (time series) the value of land was high, the amount of labor used per acre of sorghum was low. A priori, land which best accommodates the substitution of machines for labor would be the most valuable. \ 94 As expected, land which was fallowed was less valuable than land not fallowed. The relationship was clearer when the ratio of fallow land to cropland was considered. In Chapter II, it was hypothesized that value of land was in part determined by the interaction effect of fertilizer with land. If this hypothesis is true, then it follows that land values ought to be highly and positively correlated with fertilizer use. The simple correlation coefficient (.334) indicates that this was the case. Labor The variable man-hours of labor per acre of grain sorghum was nega- tively correlated with yield. Those areas or years where the man labor requirements have decreased the most tended to have the highest yields. Labor was negatively correlated with acres fallowed. In those areas or years where fallowed acres were high, labor requirement tended to be low. Fertilizer was negatively correlated with labor. Those areas and years which have done the most in terms of substituting machines for labor have also done the most in terms of increasing the amount of plant nutrients added. Acres Fallowed and Ratio: Acres Fallowed to Acres of Cropland Harvested These two variables are discussed together because they are con- cerned with the same factor, i.e., fallow. Fallow was negatively correlated with yields. Those areas that practice fallowing tended to have lower yields. Fallowing was negatively correlated with precipitation (-.34 for preseason precipitation and acres fallowed, and -.37 for preseason precipitation and 95 percent fallow). Thus, the negative correlations of fallow with yield can be interpreted as: Where precipitation was normally low, yields tended to be low. There was no significant correlation between fertilizer and fallow. Fertilizer Fertilizer was highly and positively (.509) correlated with yield as expected. Effect on Coefficients Mis-specification affects not only R2, but also the regression coefficients. But how do the M.C.I. variable coefficients (and thus their interpretation) change as the model is changed? Acres of Grain Sorghum Harvested Per Farm Harvesting Grain Sorghum The variable acres of grain sorghum per farm harvesting grain sorghum (acres per farm) was included to estimate the effects of size of the grain sorghum enterprise on yields. Quality of land, specialized equip- ment, and management effort are three factors expected a priori (l) to be related to changes in size of enterprise, and (2) to affect yields. It was decided that this variable should be included in the "complete" equation because on the basis of production theory, size of enterprise does affect yield and because its presence influences (and presumably improves) the coefficients of the other variables in the model. This variable was included in 192 equations. Its coefficient tested significantly different from zero at the ten—percent level or less in 13.9 percent of the equations (see Table 5-3). For the "complete" or nearly complete equations estimated using corrected data, the coefficient was 96 Table 5-3.-—Basic data concerning technology variables included in the analysis : Number of : Percent of coefficients significant at-—9/ Variableé/ ‘ times ‘ includedb/ :a _<_ .01: .01 < (15.05 .05 < c 5 .10 :a> .10 z i 180 100.0 0.0 0.0 0.0 A/F------: 192 5.2 2.0 6.7 86.1 A/T-----—-; 139 10.0 20.1 27.3 42.6 T/A—-----: 21 0.0 0.0 0.0 100.0 3 ; 159 3.1 8.1 22.6 66.2 VL---------: 189 100.0 0.0 0.0 0.0 L 2 139 78.4 16.5 0.0 5.1 Fa-----—---; 152 22.3 24.3 9.8 43.6 Fa/A--—--: 22 4.5 13.6 0.0 81.9 FT--—-—---: 188 100.0 0.0 0.0 0.0 g/ Symbols listed in Chapter III. p/ The number of equations in which this variable is included. All equa- tions, all coefficients, and the level of significance of all coefficients are presented in Appendix B. 2/ Since the equations estimated were not selected at random, no particu- lar significance can be given to the proportions reported. significant only when the year variables were omitted from the equation (see Table 5-4). The level of Significance, the Sign and the magnitude of the coefficient, was affected by model specification as shown in Table 5-4. The significance of the coefficient when year variables were omitted suggests a strong positive correlation with time even though the correlation with individual year variables was low (.223, .179, -.086, -.O30 for 1944, 1949, 1954, and 1959, respectively). Although it can be argued that when present 97 Table 5-4.-Coefficients§/ for acres of grain sorghum per farm variableh/ Season weather variables included Variables : omitted : Weekly : Season total : Polynomial of : : : seventh degree None : .333 é.l6l -.054 C.R.D. : "oOB? .250 .072 Years : .547a .593a .477a Precipitation--—--: .168 -.153 -.l87 Temperature-----: -.O27 -.O79 -.010 Interaction ----- : .171 -.l2l -.l63 Temperature and : precipitation---—: -.O61 -.ll9 -.l4l g/ The coefficient indicates how many pounds per acre yield will change with an increase of one acre of grain sorghum per farm. p/ Estimated using corrected data. Corresponding equation numbers are presented in Tables 3-6 and 4-6. the year variables "captured" the Significant effect of size of enterprise, it is more likely that when the year variables were omitted the acres per farm variable "captured" significant effects of other factors correlated with time. The amount of weather detail included also affected the coefficient. The coefficient became negative when season totals or polynomial variables were substituted for the weekly weather variables. Positive coefficients Significant at the one-percent level were obtained (see Table 5-4 and equations 95, 137, 138, 229, and 238 Appendix B). Negative coefficients Significant at the one-percent level were also obtained (see equations 12, 58, and 243 Appendix B). This indicates that any conclusion could have been reached concerning the effects of changes in 98 the Size of the enterprise, depending upon the model Specified. There was a tendency for the coefficient to be negative and significant when year variables were included or when only M.C.I.V. were included. The coefficient tended to be positive and significant when year variables were not included, when only location variables were included (equation 95), when only the value of land variable was included (equation 299), or when no other variables were included (equation 238). The significance of the coefficients in the nearly complete models (Table 5-4) suggest (1) there were no Significant effects of changes in size of enterprise, or (2) there were significant effects but they were not "captured" by this variable. In the latter case, this could be due to (a) this variable was not the apprOpriate variable for determining this effect, (b) the variable was measured with error, or (c) other variables masked (captured) this effect. It is believed that the first two (a and b) are not true. Examination and comparison of equations containing this variable (see Appendix B) and of the Simple correlation coefficients (see Table 5-2)l/ reveals no information to suggest that the third reason (c above) is true. Thus, it is concluded that changing the size of the enterprise does not significantly affect per acre yields of grain sorghum. Dollars Spent on Gas and Oil The variable dollars spent on gas and oil was included in 159 equa- tions. Its coefficient was Significant in one-third of the equations (see Table 5-3). For the "complete" or nearly complete equations estimated using the corrected data, the coefficient was not significant except when tempera- ture, interaction, and precipitation or crop reporting districts were omitted (see Table 5-5). l/ A table giving all the simple correlation coefficients can be obtained from the author. 99 Table 5—5.-Coefficients§/for dollars Spent on gas and oilQ/ Season weather variables included Variables : omitted : Weekly : Season total 3 Polynomial None E -15.9 —22.1 -18.7 C.R.D.--------; -23.76 -15.9 —25.8b Years : -4.9 -l5.0 -3.3 Precipitation---: -12.4 -22.0 -20.8 Temperature----: —27.2b -23.9C —20.3 Interaction-----: -l2.0 -2l.7 -20.l Temperature and : precipitation--: -22.7c -25.2c -24.Ob g/ The coefficient indicates how many pounds per acre yield will change for an increase of one dollar spent on gas and oil. p/ Estimated using corrected data. presented in Tables 3-6 and 4-6. Corresponding equation numbers are Positive coefficients significant at the one-percent level were obtained in two equations containing only a few M.C.I.V. and no other variables (equations 234 and 235). The level of significance, Sign, and magnitude of the coefficient were affected by model Specification. The results suggest a relationship between the fuel expense variable and weather and location variables. however, is only .26. The highest simple correlation, It is concluded that changes in mechanization (as measured by fuel expense) did not significantly affect per acre yields. This variable is retained in the "complete" model because production theory suggests that increased mechanization does affect per acre yield and because its presence influences (and presumably improves) the coefficients of the other variables in the equation. 100 Acres of Cropland Harvested Per Tractor The acres of cropland harvested per tractor (acres per tractor) variable was included in 139 equations. Its coefficient was Significant in 57.4 percent of the equations. This variable was included in only three equations estimated using the corrected data. The incorrect data affected both the magnitude and level of significance of the coefficients. In a model containing 0, Y, M(L), P, Ri’ Ti’ and Ii’ coefficients of .241 and -.001 were obtained for incorrect and correct data respectively. The incorrect data caused the coefficients to be larger and caused one to be Significant. Although the coefficients estimated using incorrect data are wrong, it is reasonable to assume that the relative magnitudes do measure the effects of omitting sets of variables. The Sign, magnitude, and level of Significance were affected by model specification. Significant negative coefficients were obtained in models containing only a few M.C.I.V. and no other variables (equations 241 and 242). Positive significant coefficients were obtained in many models (equations 1, 2, 34, 85, 205, and 222, for example). Significant coefficients were never obtained when weekly weather variables were included. For this variable, the form of the weather variables substantially affected the coefficients. It can be concluded that mechaniza- tion as measured by the acres per tractor variable did not significantly affect per acre yields. The inverse relationship of this variable to mechanization makes it difficult to interpret. Because of this, the variable "number of tractors per acre of cropland harvested" was included in most of the models estimated using the corrected data instead of "acres of cropland harvested per tractor." . o E‘ n..¥ '0”. - In . ‘Hw. . “z... . Vs I ‘- s I n“ “A, “U: 'u I 7 u \ V \ 5 - Nv‘g \ S u‘. . ‘ ~ ‘ “. ”a? “'1 x V \‘m . 5 ‘ \‘j . 101 Number of Tractors Per Acre of Cropland Harvested The variable number of tractors per acre of cropland harvested (tractors per acre) was included in 21 equations. The coefficient was never significant. The coefficient had its smallest value (210) in the "complete" equa- tion and its largest value (3,786) in the model using the least weather detail (season totals) and omitting the years variables (see Table 5—6). The Sign and magnitude of the coefficients were affected by model Specifica- tion. It is concluded that mechanization as measured by this variable had no Significant effect on per acre yield. Table 5—6,--Coefficients§/for tractors per acreH/ Season weather variables included Variables : omitted : Weekly : Season total : Polynomial of : : : seventh degree lhnc : 210 902 -661 CJLD.-—-—--—---: -1,573 1,813 -911 Years 3 1,947 3,786 1,333 Precipitation---: 852 861 l , 642 Temperature-—---: -l92 260 -863 Interaction ----- : 731 690 1 , 333 Temperature and : Precipitation--: 2, 251 386 l , 471 E é/The coefficient indicates how many pounds per acre yield will increase fbran.increase of one tractor per acre. E/Ebtimated using corrected data. Corresponding equation numbers are Presented in Tables 3-6 and 4-6. 102 Man-Hours of Labor The variable man-hours of labor per acre of grain sorghum was con- sidered in 139 equations. The coefficient was significant at a = .10 or less in 94.9 percent of the cases (see Table 5-3). On the surface, it would seem that this would be a very good variable to include in an equation. However, when the results of some of the equations were considered, it became apparent that a problem of multi- collinearity existed between the set of variables representing years and the labor variable. Some examples of how the coefficient for labor was affected by the inclusion of years in the equation are given in Table 5-7. It is interesting Table 5-7.--Effects of multicollinearity of yp7rs with labor on the estimated labor coefficienta Estimated coefficienth/ Egngépn 3 variables 3 for labor 109 ----- -: M, P, Ri,Ti,Ii -29.7a 97 ------- : M,P,Ri,Ti,Ii,Y —90.48 86-—------; 0,M,P,Ri,Ti -35.2a 87-----—---: G,M,P,Ri,Ti,Y -93.4a 5 2 C,M(A/T),P,Ri,Ti,Ii -23.3a 3 : 0,M(A/T),P,Ri,Ti,Ii,Y -4l.0 26-—-—--—---: G,M,Ri,Ti,IO'6 -50.8b 25-- ----- -: G,M,Ri,Ti,IO'§Y -52.0b g/ The equations are presented in pairs. The first equation does not in- clude the dummy variables representing years. The second equation is exactly the same except it does include years. See Appendix B for complete equations. Estimated using the not corrected data. p/ The coefficient indicates how many pounds per acre yield will change for an increase of one hour of labor per acre. 103 to note that the effect of years on the labor coefficient is markedly different depending on which of the other variables are included in the equation. When growing seasons, all technology variables, a seventh degree polynomial in precipitation and temperature, and a sixth degree polynomial in interaction are included in an equation (see equations 26 and 25 in Table 5-7) the addition of years had very little effect on the labor coefficients. This is surprising in that equations 86 and 97 were not much different but there the addition of years to the equation greatly affected the coefficient for labor. In general, the addition of years to an equation which already in- cluded the labor variable greatly affected (increase in absolute value) the coefficient for labor. The coefficients for the dummy variables representing years were also greatly affected by the inclusion of the labor variable in the equa— tion (see Table 5-8). In all cases, the inclusion of the labor variable greatly affects the magnitude, Sign, and level of Significance of the coefficients for years. The coefficients for years in those equations that include the labor variable were inconsistent in Sign, magnitude, and level of Significance. However, the coefficients for years in those equations that did not include the labor variable were consistent in Sign and (in general) the level of significance and differ only in magnitude. The labor variable is highly intercorrelated with the set of variables representing years. The simple correlations of the labor variable with the ilulividual variables representing years were large in magnitude and mono- txnricaflly decreasing in trend, i.e., .62, -.l9, -.41, and -.56 for 1944, 1949, 1954, and 1959, respectively. 104 Table 5-8.--Effect of multicollinearity of years with labor on the estimated coefficients for yearsé/ Equation : Variables : CoefficientsE/ number 1959 1954 1949 1944 97-----: Te,P,Ri,Ti,Ii,Y -l47 -232 -167 326a 98-—-—-—--: Te(L),P,Ri,Ti,Ii,Y 433a 297a 230a 246a 15------: C,P,Ri,Ti,A/F,FT,VL,L,Y 168 -l98 47 364a 14-----: C,P,Ri,Ti,A/F,FT,VL,Y 597a 178a 330a 323a l8------: c,P,Ri,Ti,Te,Y 226 —144 85 372a 16-----: C,P,Ri,Ti,Te(L),Y 647a 227a 367a 332a 54-------: C,A/F,%,FT,VL,L,Y 566a -25 359a 561a 55----: C,A/F,%,FT,VL,Y 828a 199a 528a 543a 271-----: P,Ri,Ti,Y,L,VL -76.5 —289 -176 328a 270 ----- : P,Ri,Ti,Y,VL 553a 263a 239b 308a 276----: Te,P,Ri,Ti,Y -276 -435b -299b 247a 278------: Te(L),P,Ri,Ti,Y 455a 210a 188c 230a g/ The equations are presented in pairs. The first equation contains the labor variable, the second equation is identical except the labor variable has been omitted. See Appendix B for complete equations. Estimated using not corrected data. p/ The coefficient indicates the pounds per acre yield differs from the 1939 yield. All the data on labor used in the study are presented in Table 5-9.l/ The problem of multicollinearity of years with labor was probably due 111 part to the aggregative nature of the labor variable.. It is believed that if"bhe value of the labor variable were determined for each county, the problem of multicollinearity with years would be reduced. y For more detail see Appendix A, Table A-ll. # ..h C v ‘ 105 Table 5-9.--Tota1 man work units per acre of grain sorghum Farm production region Year : Nebraska, Kansas : Oklahoma, Texas :Colorado, New Mexico l939---: 12.2 11.6 12.0 l944---: 13.4 11.6 12.2 l949---: 6.7 7.0 9.3 l954---: 5.0 5.7 7.9 l959---: 3.4 5.2 6.7 Because of the poor measure of labor used and because there was great interest in the coefficients for years, it was decided that the labor variable should be omitted from the "complete" equation. Omitting the labor variable does not mean that the influence of labor on yields is ignored. The effects of changes in labor on yields was (partially at least) included in the coefficients for years. 1 15.5.8 \10 NNN In spite of the problem of high intercorrelation with yearsl/, the coefficient for labor was almost always significant. This is contrary to expectations and to the hypothesis stated in Chapter II, i.e., that the coefficient for this variable would be non-Significant. The negative coefficient is also unexpected. The Significance and the Sign can be explained in terms of the multicollinearity with years. That is, the "labor" variable explained a significant part of the effects of technology that were associated with years. Since the technology effects being "captured" cannot be related to any Specific technology, it is better to let the effects be included in the coefficients for years. r ‘1/ One of the effects of multicollinearity is to cause the standard errors of the coefficients to increase. Thus, if there was a problem of high inter- correlation among variables, it would be expected that the estimated coefficients would not be significant. 1.1\ 106 Percent Acres of Grain Sorghum Irrigated It is well known that the irrigation of crops (particularly in the semi-arid parts of the country) will greatly increase yields. The variable percent irrigated did a good job of quantifying the effects of irrigation on yields of grain sorghum. This variable was included in 180 equations. The coefficients were Significant at less than .01 level in all equations. The coefficients from the nearly complete equations estimated using the corrected data are presented in Table 5-10. The magnitude of the coefficient was affected by the model specification, even though Sign and level of significance were not. Within the context of "nearly complete" (omitting only one set of variables) equations, the maximum increase or decrease in the size of the coefficient was about 100 pounds per acre or about one-seventeenth of the magnitude in the "complete" equation (see Table 5-10). Regardless of the form of the weather variables in the "nearly complete" equations (see Table 5-10), omitting the crop reporting district variables caused the coefficient for percent irrigated to increase. This is probably because there was great cross-sectional Variation in percent irrigated. The effect of omitting the year variables depends upon the form (detail) of the weather variables. When detailed weather variables (weekly and.polynomial) were included, omitting years had no effect. However, when season totals were used to represent weather, omitting years caused the (noefficient to decrease from 1810 to 1682. This may be due to the irriga- ‘biorlvariable (positively correlated with time) having to "explain" some of"the effects of bad weather in 1954 when weather is not included in detail and the year variables are removed. 107 Table 5-lO.-CoefficientS§/for percent irrigatedh/ Season weather variables included variables : omitted : weekly : Season total : Polynomial of : : : seventh degree None : 1,761a 1,810a 1,845a C.R.D. : 1,853a 1,866a 1,932a Years : 1,762a 1,682a 1,845a Precipitation--: 1,757a 1,812a 1,796a Temperature----: 1,753a 1,821a 1,867a Interaction--—-: 1,757a 1,819a 1,798a Temperature and : precipitation-: 1,678a 1,808a 1,788a g/ The coefficient indicates how many pounds yield per acre will change for a lOO-percent increase in amount irrigated per acre. p/ Estimated using corrected data. The corresponding equation numbers are given in Tables 3-6 or 4-6. There was very little effect of omitting any Single set of weather variables when weather is represented by either weekly or season total variables. However, the coefficient was affected when polynomial weather variables are included. Why this happened is not clear. For models containing only M.C.I.V., the absence of the fertilizer or value of land variables greatly affect the coefficient for the percent irrigated variable. For example, in equation 233 (see Table 5-1 for 'variables included) a coefficient of 2,057 was obtained. When the ferti- lLizer variable was added, the coefficient dropped to 1,619. In equation 1&L4 (see Table 5-1 for variables included) a coefficient of 1,663 was obtained. When the value of land variable was added (equation 245), the 108 coefficient decreased to 1,263. The effect of dropping these variables was reduced when a large number of other variables were included in the equation. For example, the 400-pound decrease caused by adding the value of land variable when only M.C.I.V. were included is reduced to 200 when the equation also contained location, years, and weather variables (equations 13 and 14). Model specification did affect the coefficient for the percent irrigated variable, even though it was always positive and highly Sig- nificant. Value of Land It was hypothesized in Chapter II that this variable reflected the interaction effects of land with the other M.C.I.V. It was further hypothesized that the coefficient would be positive and significant. These latter hypotheses were found to be true based on the results of this analysis. It is not possible on the basis of this study alone to determine the truth of the first hypothesis; but the results did not disprove it. The problem of this variable not being completely exogenous with respect to the dependent variable was also discussed. Six sets of equations were estimated where the only difference 'between the two equations in the set was the presence or absence of the vmiLue of land variable. By comparing the equations within each set it is grossible to obtain some idea of how the presence of this variable affected tune results. We can answer the question, W0uld the results or conclusions concerning the overall model or the other variables in the model have been greatly different if this variable were not included. 109 The most accurate estimate of the effect of dropping the value of land variable would be obtained from equations containing all the other variables included in the "complete" equation. The nearest we can come to this is the set of equations 13 and 14 which contain the percent irrigated, acres per farm, and fertilizer variables, the year variables, the crop reporting district variables, pre- season precipitation, the seventh degree precipitation polynomial and the seventh degree temperature polynomial. They were estimated using the not corrected data. Dropping the V of L variable from this equation caused the R? to drop from .758 to .752. The irrigation coefficient changed from 1,708a to 1,963a. The fertilizer coefficient changed very little from 14.28. to 14.43. The acres per farm coefficient changed from -.291 to -.408C and became significant at the 0.10 level. All the other coefficients exhibited little change (all the coefficients are listed in Appendix B, Table B, Part 2). Except for the acres per farm coefficient, all the other conclu- sions based on equation 14 (with the V of L variable) would have been essentially the same as if based on equation 13. The other five pairs of equations differing only by the presence or absence of the V of L variable are: 55 and 56, 231 and 223, 137 and 138, 92 and 93, and 269 and 270. Since essentially the same conclusion is reached concerning these sets, they will not be discussed here. The coefficients for all of them are presented in Appendix B. Of course, as 'the equation becomes more incomplete, dropping the V of L variable has a greater effect upon the R2 and on the coefficients of other variables. This variable was included in 189 equations and was significantly different from zero at the .01 level in all equations. The Sign was aJJways positive, but the magnitude varied. 110 The coefficients from the "nearly complete" equations estimated using the corrected data are presented in Table 5-11. Table 5-11.-—Coefficients§/ for value of land variablep/ Season weather variables included Variables : omitted : Weekly : Season total : Polynomial of : : : seventh degree None : 2.9a 2.1a 2.9a C.R.D. : 2.7a 2.2a 2.53 Years : 3.7a 3.9a 3.7a Precipitation--: 2.6a 2.18 2.3a Temperature ----- : 3.3a 2.38 3.0a Interaction----: 2.6a 2.2a 2.3a Temperature and : - precipitation--: 2.8a 2.48 2.58. g/ The coefficient indicates how many pounds per acre yield will change for a one-dollar increase in the value of land per acre. p/ Estimated using corrected data. Corresponding equations listed in Tables 3-6 or 4-6. The coefficient in equations containing weekly weather variables was not much different than the coefficient in the corresponding equa- tions containing the seventh degree polynomial variables. However, the coefficient was consistently smaller (except when year variables were omitted) in equations containing the season total variables. Only two sets of variables (years and temperature) greatly affected the coefficient. When the years variables were omitted the coefficient increased greatly. This was probably due to the value of land variable having "picked up" the effects of other variables also correlated with 111 time. The coefficient also increased when the set of temperature variables was omitted (except for the season total variable). The reason for this is not clear. In models containing only M.C.I.V., the coefficient was affected by three variables--man—hours of labor, fertilizer, and percent irrigated. Acres Fallowed This variable was included in 152 equations and was Significantly different from zero in 56.4 percent of them. The coefficient was obtained in only two equations estimated using the corrected data (equations 283 and 293). A coefficient of -.00013 was obtained in equation 283, which contained the weekly weather variable. Equation 293 contained the polynomial weather variable and a coefficient of .00011 was obtained. The form of the weather variables included affected the coefficient's magnitude and Sign, but not its non-Significance. The coefficient was not Significant in any equation containing crop reporting district variables. It tended to be Significant and negative in those equations not containing district variables. This indicates that the variation in yield explained by differences in acres fallowed can also be explained as consistent differences between crop reporting districts. The results also indicate the effects of fallowing were confounded with the effects of temperature as the coefficient tended to be significant only when temperature variables were included. Crop reporting district variables were not dropped from the "complete" equation even though they "coverqu' the effect of fallowing. When the coefficients obtained for districts are discussed, it will have to be remem- bered that one of the effects they include is the effect of fallowing. ~-: ~ 1‘ .n..- «zw .fi‘u a. N .W‘ v.. Q . ‘s W... -. ... Lu. :‘ h x ..n as A“. My.» n. ..n. .c. .7 .\..: :6. ..b\ .... .. s e: . 112 In future studies the effects of fallowing may be determined inde- pendent of the location effects if better input data are used. For example, if the proportion of grain sorghum actually planted on land fallowed the previous year were used, the effects of fallowing might be determined. The variable acres fallowed is influenced by the size of the county. In an effort to remove this undesirable effect a new variable, "ratio: acres fallowed to acres of cropland harvested" was developed and used in most of the equations estimated using corrected data. Ratio: Acres Fallowed to Acres of Cropland Harvested , The ratio of acres fallowed to acres of cropland harvested (percent fallowed) variable was included in 22 equations, all estimated using the corrected data. Its coefficient was Significant in 18.1 percent of the equations. The coefficients obtained in the "nearly complete" models are presented in Table 5-12. The coefficients were consistent in Sign (negative) but varied in magnitude and level of significance. The effect of fallowing on yield was hypothesized to be positive and significant. Experiment station studies have in fact demonstrated that yield of sorghum on land previously fallowed is up to one-third greater than for sorghum grown on land cropped the previous year. The negative coefficient can be explained. Percent acres fallowed is negatively correlated with the preseason precipitation variable and ‘with 19 of the 23 weekly precipitation variables. That is, acres fallowed 'were high where precipitation was low and very little fallowing was prac- 'ticed where precipitation was high. Because of the intercorrelation of 'these variables, both cross sectionally and over time, the effects of 113 Table 5-12.--Coefficients§/ for ratio:d;?cres fallowed to acres of crop- land harveste b Season weather variables included Variables : omitted : Weekly : Season total : Polynomial of : : : seventh degree None : -129 -126 -99 C.R.D. : --213a -100 —142b Years : -102 -64 -l35 Precipitation---: -171b -l27 -118 Temperature----: -95 -108 -84 Interaction-----: -174b —130 -117 Temperature and : precipitation--: -92 -98 -101 g/ The coefficient indicates how many pounds per acre yield will change for an increase of 100 percent in acres fallowed per acre of cropland. p/ Estimated using corrected data. Corresponding equation numbers are presented in Tables 3-6 and 4-6. precipitation and the effects of fallowing were confounded. This is un- fortunate because one of the objectives was to separate the effects of the M.C.I.V. from the effects of weather. The relationship of fallow to precipitation was also revealed by the coefficients obtained in the nearly complete equations (see Table 5-12). When the precipitation variables or the interaction variables which were highly correlated with the precipitation variables were omitted, the coefficient for fallow became significant. However, this only happened when the weekly weather variables were included in the model. The only other set of variables that greatly affected the coefficient for the percent fallow variable was the crop reporting districts. When the set of district variables was omitted, the coefficient for percent fallow 114 became significant. This was because there was a strong consistent cross- sectional pattern for percent fallow and when there were no location variables to pick up the location effects, they were "captured" by the percent fallow variable. Again the form of the weather variables affected the magnitude of this effect. When very detailed weather variables (weekly) were included, dropping the district variables caused the coeffi- cient to increase (in absolute value) 84 pounds per acre and it became significant at the .01 level. When polynomial variables were included, dropping the district variables caused the coefficient to increase 43 pounds per acre and it became Significant at the .05 level. When only season totals were included, the coefficient decreases 26 pounds and the coefficient was not significant. The reason why this happened as less detailed weather variables are included is not clear. However, it is clear that the conclusion reached concerning the level of significance and magnitude of the coefficient is greatly affected by model Specification. Fertilizer The variable pounds of plant nutrients applied per acre of grain sorghum (fertilizer) was included in 188 equations. Its coefficient was significantly different from zero at the .01 level in all equations. The coefficients from the "nearly complete" models estimated using the corrected data are presented in Table 5-13. The Sign and level of significance were consistent, but the magnitude varied greatly. Dropping the district variables from the equations caused the coeffificient to increase in Size. This indicates that in districts where :fertdlizer levels were high, there were other factors associated with the location that caused or allowed higher yields. 115 Table 5-13.--CoefficientS§/for the fertilizer variableh/ Season weather variables included variables ; omitted : weekly : Season total : Polynomial of : : : seventh deggee None : 11.2a 14.0a 12.4a C.R.D.-m-m-: 13.18. 16.03. 1505a Years : 13.4a 22.8a 15.0a Precipitation-—-: 11.7a 14.0a 14.7a Temperature----: 13.0a 14.0a 13.2a Interaction--—-: 11.4a 13.9a 14.5a Temperature and : precipitation--: 13.1a 14.3a 15.1a g/ The coefficient indicates how many pounds per acre yield will change for an increase of one pound of plant nutrient per acre of grain sorghum. p/ Estimated using corrected data. The corresponding equation numbers are given in Tables 3-6 and 4-6. Less detailed weather, whether by dropping sets of weather variables or by substituting in less detailed variables, caused the coefficient to increase. This indicates that where fertilizer levels were high, weather ‘was conducive to higher yields. Omitting the year variables also caused the coefficient to increase. Tlris suggests that in years when fertilizer levels were high, yields were lrigh. The combined effects of less detailed weather variables (season txytals) and omitting the year variables caused the coefficient to increase 1x31more than twice its Size in the "complete" equation, i.e., from 11.2 tn) 22.8. Even within the context of nearly complete equations, model specifica- 'ticu1 greatly affected the coefficient for fertilizer. 116 Within the context of models containing only M.C.I. variables, the percent irrigated and value of land variables greatly affected the coefficient for fertilizer (compare coefficient for fertilizer in equa- tions 227 with 228 and 228 with 239). The largest coefficient (47.1) was obtained in a model (equation 239) containing acres per farm and fertilizer as the only independent variables. Conclusion In this chapter man-controlled input variables and their coefficients were examined. It was determined that the RQ'S for models that contained only M.C.I. variables did not compare favorable with the RZ'S from.models that contained M3C.I., weather, location, and time variables. The simple correlation between the M.C.I. variables was discussed. The highest Simple correlation between any two M;C.I. variables was less than .5. High intercorrelations among the M.C.I. variables was not a problem. The effect of miS-Specification (determined by specifying alterna- tive models) on the regression coefficients was also examined. If the regression coefficient was Significant in the "complete" model, it was also significant in all submodels. However, if the coefficient was not Significant in the "complete" model, it may have been and often was significant in some submodels. In all cases, the magnitude of the cxaefficient was affected by changes in model specification. CHAPTER VI LOCATION AND YEARS In this chapter, the coefficients estimated for the dummy variables included in the analysis to represent location and time are discussed. Principal concern is with the effects of alternative model specifications on their coefficients. In the first section of this chapter the sets of location variables (growing seasons and crop reporting districts) are discussed. The year variables are discussed in the second and final section. Location Variables Two sets of variables (crop reporting districts and growing seasons) were considered to take into account consistent differences in yield between locations. Such consistent differences between locations are expected primarily to measure the effect on yield of the difference in the physical inputS-—soil, topography, climate, and cultural practices between the locations. The effect of consistent differences in levels of the M.C.I. and weather between locations may also have been captured ‘by these variables. The decision of which set of location variables to include in the 'Nzomplete" equation was not made a priori. Rather, it was made after the Iwesults of some equations had been obtained. The decision was to include ill the final equation that set which did the "best" job (in terms of R2) of? explaining yield variation. 117 118 There were three pairs of equations which differed only by the set of location variables included. The R2 for these equations are presented in Table 6—1. Table 6-l.-—P:2 for pairs of equations differing only by the presence of either growing seasons or crop reporting districtsa : -2 Other variables : R in equation 3 Growing seasons 3 Grggsiigzizlng M(L), P, R07, T07, 107"“: .720 (61) .740 (6) M, Y, P, 307, T07 --------- : .718 (87) .740 (18) None : .029 (96) .182 (60) g/ Estimated using the not corrected data. Numbers in parentheses are the equation numbers. The set of dummy variables for crop reporting districts consistently did a better job of explaining yields. Although the difference inR2 was not great for the "more complete" models, the following discussion of location variables is limited to the set representing crop reporting districts. The coefficients obtained in 21 equations estimated using the corrected data are presented in Tables 6-2-A, 6-2-B, and 6-2-C. The equa- tions were grouped for the tables according to the form of the weather ‘variables included in the equation. Table 6-2-A contains equations con- 'taining weekly weather variables; Table 6-2-B those containing season ‘total weather variables; and Table 6-2-C those containing seventh degree Imolynomial weather variables. Within each table, the equations are those cflibained by omitting selected sets of variables. 119 poserGOOI wdofll NHMMI DOHHI GHQMI wmmml ddmfil wbwml uIIIIIIIION III III III III III III III "IIIIIIIIOH cabal mma- Ho- owed- Hm- snob. moan "uuuuuuuuma omeI wNQQI mmal owns: ommml swoon comm: “IIIIIIIIbH QHONI wwOQI QHONI QOFfil wMMQI dNHOqHI MHMQI ullllllllQH ammo: emMMI poem: QHMMI comm: epmwl wmQMI “IIIIIIIImH nmbml as: naoml mm: wan wSOmI om: “Illlllllqa wfibml GOQNI wwwml wwNNI Qmwfll dwNwl dNfiNI “IIIIIIIIMH poem: Neal napml span DHNNI swam: mHHI "Illlllluma wHOQI wHwNI womMI wmoml mmOMI wmmol wNMNI ullllllllHH MHHI so: pedal as: mad: when: owl “IIIIIIIIOH moan mm: mm: bl v QmSMI ms ”Illllllllm wwmml mefll whom! 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NON .mHOm 0.3” 00mm «IIIIIIIIMN wH«mI N«I ngHI b«I oomm mo««I NH ”IIIIIIIINN doom: nmmmI wmoml nmmmI «mI wvmoI opmHI mIIIIIIIIHN ones nod meadow u “Mupwnomemm SOHpownoflnH ondpenmmaoe SOprpHdHoonm mnwow .H.o.z ocoz 30anme .p pH Hoonm.. . . . . . HmchnOan ooseaso nonncnssa no new H aone cossnasooun.oumIb canon or u- q.- u— .1. 14“ I.. fly h; H a: ... 2. v ‘1.- hno . s. . . h... ...s I. . r 1. CL A... 1‘ A 2. u. a Z» ‘\ . ‘ ‘ nhu \. .... .. .. ~\h ... 125 It would be difficult and nearly meaningless to discuss and compare the nearly 600 coefficients presented in these three tables. Thus, the discussion is restricted to general comments. The levels of significance, although presented in Tables 6-2, were of no particular interest in this study. This is because the level of significance refers to the significance of the difference relative to district 19. District 19 was arbitrarily omitted and no hypothesis was suggested concerning the significance of these locational differences. The only hypothesis concerned the signifi- cance of the set, and this was discussed in Chapter IV. Regardless of how the "complete" equation was modified, i.e., by form of weather variables or dropping sets of variables, the magnitude of the coefficients was affected and in many cases the Sign and level of significance were also affected. The largest change in the coefficients due to modifying the "complete" equation was when the M.C.I.V. were omitted. This was true regardless of the form of the weather variables. This suggests that the set of M.C.I.V. is highly correlated with the set of location variables. Dropping any of the four sets of weather variables did not greatly affect the coefficients of the location variables. Omitting the year variables did affect the coefficients. This is interesting because the year variables are independent of the location variables. This is evidenced by the fact that when year variables were added to an equation containing only crop reporting district variables, the estimated value of the cross sectional constants (coefficients for location variables) were not affected (see equations 59 and 60). Also, the simple correlation between any crop reporting district variable and any year variable was zero. 126 Although the presence of the year variables did not affect the estimated cross section constants when only the two sets were in the equa- tion, they did affect them if there were other variables in the equation (note results presented in Tables 6-2). This would be expected if the other variables in the equation were not independent of either the district or year variables. The magnitude of this indirect or second order effect was surpris- ing. It is second order in the sense that the presence of the year variables affects the coefficients of the weather and/or man-controlled input variables and they in turn affect the coefficients for the district variables. The magnitude of these indirect effects indicates that the simple correlation coefficients tell little concerning interrelationships or interdependencies of the independent variables in a multiple regression model. Year variables Time was taken into account in this analysis by a set of four dummy variables: One variable for each year except 1939 which was omitted to avoid a Singular matrix. The coefficient obtained for these variables measured the consistent difference in yields over all counties between the year in question and 1939. Time, of course, cannot directly affect yields. These variables 'were used to estimate the effect on yields of physical factors affecting :yields, changing with time and not otherwise included in the analysis. {the change in varieties was the major factor so related, but changes in Imanagement ability, changes in cultural practices, insecticides, herbi- cides, and labor used were also related to time. 127 The multicollinearity problem of the year variables with the man— hours of labor per acre variable was discussed in Chapter V. The relation— ship of the year variables with the location variables was discussed above. These discussions will not be repeated here. The coefficients for the year variables obtained in some equations estimated using the corrected data are presented in Table 6-3. The coeffi- cients presented in Table 6-3 are all positive and significantly different from zero at the .01 level. Altering the model by omitting a single set of variables or substituting different weather variables did not affect the Sign or level of significance of the coefficients. When the M.C.I.V. were omitted the coefficients for the year variables increased. The coefficients for all years were not affected the same. The coefficient for 1959 increased much more (as expected) than did other coefficients. The form of weather variables did not greatly affect the change in the 1959 coefficient. However, it did affect the coefficients for the earlier years. When detailed weather variables were included, the coefficients for 1944, 1949, and 1954 increased much more than when season total weather variables were included. Omitting the crop reporting district variables caused the coeffi- cient to decrease in size. This means that when the effects of cross sectional variation were taken into account (district variables included) Inore effects consistent over time were measured. This in spite of the independence of the year and district variables as discussed above. The (affect of omitting the district variables was not greatly affected by the zilternate sets of weather variables. 128 Table 6-3.--Coefficient§/for year variableSE/ Weather variables included Variables : Year omitted : : Wbekly Season total : Polynomial of : : : seventh degpee f 3 Pounds per acre None 2 1959 E 528a 676a 537a : 1954 : 346a 231a 313a : 1949 : 419a 271a 428a : 1944 : 368a 301a 376a M.C.I.V. 2 1959 2 968a 982a 994a : 1954 : 609a 409a 656a : 1949 : 532: 34%: 624: : 1944 : 377 30 395 Crop reporting : districts--—------: 1959 : 458a 510a 415a : 1954 : 340a 144a 280a : 1949 : 259: 107: 243: : 1944 : 259 179 255 Precipitation ------ 2 1959 ; 555a 672a 617a : 1954 : 225a 228a 210a : 1949 : 345: 270: 330: : 1944 : 342 300 314 Temperature--------: 1959 ; 528a 700a 541a : 1954 : 233a 215a 217a : 1949 : 370: 331: 43:: : 1944 : 341 337 35 Interaction -------- : 1959 : 553a 657a 609a : 1954 : 223a 216a 211a : 1949 : 364a 264a 341a : 1944 : 342a 297a 316a Precipitation and ; temperature-—-----: 1959 : 623a 743a 682a : 1954 : 208a 233a 180a : 1949 : 348a 367a 395a : 1944 : 300a 361a 331a jg/ Estimated using corrected data. listed in Tables 3-6 and 4-6. Corresponding equation numbers are .p/ The coefficient indicates the pounds per acre difference in yield relative to 1939. 129 Omitting any set of weather variables did not affect the coefficient for years greatly. Although, in almost all cases less detailed weather variables (either omitting a set of substituting a less detailed set) caused the 1959 coefficient to increase and the other coefficients to decrease. x . CHAPTER VII WEATHER VARIABLES This chapter is composed of three major sections. AIn the first section the coefficients for the preseason, weekly, polynomial, and season total variables and how they are affected by model Specification are dis- cussed. In the second section the estimated effects of weather in each week of the growing season as obtained from the three sets of weather variables are compared. It was not possible to determine the "correct" or "best" degree of polynomial for the weather factors a priori. The third section contains a discussion of why the seventh degree polynomial was selected for all three weather factors. Weather Coefficients in Alternative Models Preseason Precipitation The preseason precipitation variable was included because it was believed a priori to be an important factor affecting yields. The estimated coefficient obtained in the "nearly complete" equations are presented in Table 7-1. In all cases the coefficient is positive and significantly differ- ent from zero at the .01 level. The form of the weather variables greatly zrffects the coefficient. When less detailed weather variables are included, true coefficient is smaller. Omitting the M.C.I.V. when the weekly weather inuriables were included caused the coefficient to decrease a little (20.3 'to 2K3.1). It decreased more (19.1 to 15.2) if the form of the weather 130 131 Table 7-l.--Coefficients for preseason precip’tation variables for "nearly complete" equation a Set of variables Weather variables included omitted : Weekly : Season total : Polynomial of : : : seventh degree 3 Pounds per acre per inch None : 20.3a 11.6a 19.1a M.C.I.V.-----: 20.1a 6.9a 15.2a C.R.D. : 22.7a 21.4a 21.3a Years : 27.2a 13.3a 25.9a Precipitation---: 17.3a 11.7a 15.1a Temperature-----: 24.6a 14.1a 18.9a Interaction---—-: 17.8a 12.1a 16.0a Temperature and : ‘ precipitation--: 22.1a 14.1a 15.6a g/ Corresponding equation numbers are presented in Tables 3-6 and 4-6. Estimated using the corrected data. variables included was the polynomial. It decreased much more (11.6 to 6.9) if only the season total variables were included. The coefficient was increased when crop reporting district variables were omitted. The magnitude of the increase was affected by the form of the weather variables present. It increased a little when weekly weather variables were in- cluded, more when the polynomial variables were included, and almost doubled when season totals were included. It is interesting to note that 'when district variables were omitted the coefficient for preseason pre- cipfitation was nearly the same regardless of the form of the weather variables . 132 The coefficient for preseason precipitation was increased when the year variables were omitted. The magnitude of the increase was greater, the greater the weather detail included. Omitting the season precipitation and interaction variables had about the same effect. It caused the coefficient to decrease when weekly or polynomial variables were included but had almost no effect when season totals were included. When the temperature variables were omitted the coefficient in- creased for equations containing the weekly and season total variable, but remained nearly the same in the equations containing the polynomial variables. If the season precipitation variables are omitted in addition to omitting the temperature variables, the effect is nearly the net effect of omitting each set separately. Weekly Weather Variables When the weekly weather variables were included, coefficients were obtained for each week of the growing season for each weather factor. Coefficients for precipitation from the "nearly complete" equations are presented in Table 7—2; coefficients for temperature in Table 7-3; and coefficients for interaction in Table 7-4. When discussing the effects of alternative model specification on the estimated effects of increasing precipitation (or temperature), it is necessary to consider more than the coefficient for the precipitation (or temperature) variable. This is because of the presence of an interaction 'variable which also measures the effect of increasing precipitation (or tenmerature). To obtain a true picture of the effects of increasing pre- cipitation, it is necessary to consider the joint (or net) effect of the jprecipitation (or temperature) variable and the interaction variable. 133 Table 7-2.--Coefficients for weekly precipitation variables from the "nearly complete" equations Set of variables omittedé/ Week of growing : : : . None . M.C.I. C.R.D. . season : (285) 2 (286) (288) : Y(287) . T(29l) 1(292) 3 Pounds per acre per inch 1—-----: -486b -686b .6348 -397° -5soa 0.6 2-.-—---f 390° 207 472b 250 410b 14.4 3----- : 301° -073 5298L 335b 577a 63.9a 4-———-f -138 147 -148 ~311° -327b 9.5 5---..---: 192 006 205 274° -272b —21.3 6---———-: —057 —35 -048 -004 -101 -30.6b 7— ----- --505b -829° -550b -359 —408b 22.5 8—--..-: 212 666b 217 177 367b -13.0 9-----: -327 -523° - -495b -052 270 19.6 10—----: 292 305 551 578° 123 22.7 11-----: 254 130 203 153 212 50.78L 12 ------ f -238 -54.1 -237 -398 076 72.851 l3---—-: -150 076 226 —073 251 9.8 l4------: --796°L -489 -829a -910° ~410° 40.18L 15-----= -693&1 -339 -260 -8318 -479c 29.8 16----; -812‘=1 -9103 -496b -717° -827a 73.4a 17..-—-—f ..815a -291 —642b -818a -7809 -33.6 l8——----= -193 092 174 -233 018 22.4 l9-—---, -080 -163 083 -176 -263 -19.1 20..-...4 105 001 -039 135 2179 77.99 21 ————— 2 -027 -307 -090 198 204 -20.5 22-—---: -223 -379 -072 -l81 099 16.6 23-..._-. 026 166 -258 123 -185 -73.0b _a/ Number in parentheses is equation number. M.C.I. = man-controlled inputs; C.R.D. = crop reporting district; Y = years; T Estimated using the corrected data. I = interaction. temperature; and 134 Table 7-3.--Coefficients for weekly temperature variables from the "nearly complete" equations : . . g/ week of : Set of variables omitted growing : : : : : : season : (1)205?) : M(ZC8'6I)‘ : C(gg)‘ : 1(287) : P( 289) : I ( 292) 3 Pounds per acre per degree l---—--: -0.10 -l.28 0.34 -0.32 0.18 0.13 2----—: 1.46b 2.988 0.73 0.72 0.19 0.18 3------: -1.35b -2.21a -0.64 -1.20b -2.27a -2.128 4-----; 0.72 0.71 0.38 —0.17 0.93 0.83 5-----= 1.19b 2.468 1.30b 1.12b 1.12b 1.12b 6...-...-: -O.88 -2.24b -l.64b 0.06 0.13 0.09 7---- : 1.17° 0.44 0.83 1.268 1.62b - 1.57b 8—-----= -l.798 -0.37 -2.418 ~1.868 -1.978 -l.868 9—-—-; -2.338 -2.02° -2.438 -3.598 -2.198 -1.93b 10--—-—-= -l.34° 0.15 —0.17 -0.93 —1.55b -1.56b 11 = —0.22 —0.44 -O.28 -0.44° -O.48b -0.45° 12----; 2.60b -0.11 0.54 3.508 2.428 2.618 13.-.—--: -3.088 —2.86b —2.408 -2.288 -3.508 -3.408 14--—-—: -O.81 —l.38 -0.73 -O.85 0.30 0.25 15-----= -0.85 -0.06 -0.25 -0.93 0.08 0.21 16----= 2.53a 0.64 2.51a 3.51a 3.008 3.008 17--—-—; 0.30 -1.62 0.40 —0.60 0.83 0.77 18-----f -O.82 0.43 -0.12 -0.89 -0.60 -0.48 19----: 1.87a 3.22a 1.998 1.618 1.788 1.668 20—----: 0.25 -0.75 -0.99b -0.07 0.42 0.53 21—-_--—-: -0.79 -2.17a —0.44 -0.58 —0.45 -0.50 22-_---: -1.53b -2.48b -1.34° -1.04 -o.90 -l.Ol 23—-_---f -0.02 1.808 —o.92° —0.49 -0.53 -0.54 gy/ Number in parentheses is equation number. Estimated using the corrected data. 135 Table 7-4.-Coefficients for weekly interaction variablesé/ week of: Set of variables omitted owin : : : : : : g:easongf 102%?) 3 7.20801). . (15%? f Y(287) f P(289) f T(29l) f 12230? 3 Pounds per acre per unit 1-----§ 0.89b 1.30b 1.13a 0.72° .013 1.008 .033 2-.———-f -0.67° -0.29 -0.77° -0.44. .019 -0.71b .038 3—--—--; -0.44 0.30 -0.818 -O.48 .114a -0.928 .1228 4—---.f 0.24 -0.21 0.26 0.53° .016 0.52b -.037 5...---; -0.36 -0.06 -0.36 -0.49° -.040 -0.41° -.047° 6..-__-§ 0.07 -0.08 0.08 —0.04 -.0470 0.15 —.030 7------: 0.85b 1.358 0.958 0.63° .040 0.74b .0818 8----: -0.37 -1.148 -O.38 -0.31 -.025 -0.60b .004 9-—-—-: 0.54 0.98° 0.82b 0.09 .031 -0.40 .0728 10----= -0.42 -0.46 —O.83 -0.84 .037 -0.16 .039 ll----: -0.33 -0.20 —0.24 —0.17 .0788 -0.28 .075b 12—--: 0.46 0.91 0.43 0.73 .1098 -0.01 .1238 l3---—-: 0.26 -0.14 —0.31 0.14 .019 -0.37 .042b 14-—--: 1.358 0.87 1.418 1.548 .070a 0.73b .058a 15---= 1.20a 0.53 0.53 1.46a .057b 0.84b .027 16--—-: 1.518 1.618 0.998 1.368 .1278 1.498 .012 17---: 1.278 0.40 1.04b 1.308 -.045 1.198 -.042 18--—-: 0.38 -0.10 -0.25 0.45 .040 0.04 .073b l9—_--; 0.07 0.22 -o.21 0.27 -.030 0.41 —.022 20----: -0.08 0.09 0.21 -0.10 .1508 -0.24 .1838 21-_-; 0.01 0.46 0.11 -O.38 -.031 -0.40 —.014 22—_—_-: 0.39 0.64 0.13 0.31 .027 -0.17 .020 23-—_-: -0.22 -0.48 0.28 -0.36 -.132b 0.18 -.087° gy/ Estimated using corrected data. sented in first column of Tables 3-6 or 4-6. Corresponding equation numbers pre- 136 The following procedure was used to estimate the net effect of one inch of precipitation: Net effect = (precipitation coefficient) + (inter— action coefficient) X (average temperature).;/ If the precipitation variables were omitted, the first part of the equation would be zero and the effect would be measured Simply by taking the interaction coefficient multiplied by average temperature. If the interaction variables were omitted, the effect is simply the coefficient for precipitation. Similar statements hold for the net temperature effects. The average total maximum temperature and the average precipitation for each week in the growing season, needed to estimate the net effects, are presented in Table 7-5. The net effects for precipitation for some equations are presented in Table 7-6. The great change in the magnitude of the precipitation coefficients between the equation (292) not containing the interaction variable and equations containing the interaction variables apparent in Table 7-2 are not present in Table 7-6. Table 7-6 also contains estimates of the effect of precipitation for some equations not containing the precipitation variables. Although the estimated net effects of precipita- tion differ between the alternative equations, there is great consistency in direction (Sign) and relative magnitude. In all cases, omitting a set of variables caused some net effects to change greatly and others to change very little. Also, the net effect for some weeks was greatly affected when some sets of variables were omitted, but affected very little when other sets were omitted. Except for this, very little of a general nature can 'be said concerning the effect of omitting sets of variables on the estimated ‘net effect of precipitation. ‘1/'N0te that this is Simply the partial deviation of yield with respect to precipitation evaluated at the mean of temperature. 137 Table 7-5.--Average temperature and precipitation for each week in growing season . Week in 2 Average total : Average growing season 2 maximum temperature I precipitation ; Degrees FO Inches 1 = 549 .787 2 : 577 .797 3 : 577 .815 4 : 600 . 838 5 : 623 .706 6 : 633 .695 7 : 645 .560 8 : 638 .867 9 : 649 .672 10 : 644 .625 11 : 630 .510 12 = 648 . 539 13 : 648 .808 14 : 639 .696 15 = 641 . 579 16 : 642 .474 17 : 618 .355 18 : 617 .344 9 : 593 .448 20 : 566 .724 22 : 546 .316 23 : 534 .173 The same general statement applies to the net effects of tempera- ‘ture. They are presented in Table 7-7. The major exception is that when 'the temperature and precipitation variables were omitted (equation 290), tkma Sign and magnitude of the coefficients were greatly affected. This is dtua to the interaction variable having to explain with a single coefficient true effects of both temperature and precipitation. The precipitation efflfiects dominate the determination of the Sign and magnitude of the inter- action coefficients (the signs are the same as for the net precipitation effectsL Table 7-6.-Net effects of one additional inch of precipitation-by weeks for several alternative equationsé/ lon number Set of variables omitted and equat Week in (290)i Ti & P 11(292) Ti(291) Pi(289) 1(287) C.R.D. (288) M.C.I. (286) None (285) Pounds per acre 18.1 21.9 70.4 -22.2 -29.3 \OQO‘QM o o c or; 0 MC) Elsa oz OMNOO HOxOan \‘I‘Tc—I Howpm O O O O O b-Hquoxsi Ho N L‘O‘OOM r-I'ITwFI-i \OL\\OOC‘\ o o o o o Ml\t—Iw0\ r-IN\O l—|l Ljijr-iOVI [\O\OI—|I—I \OfiflOm NML‘ON \T m 1-------_ 2--—--: 3---—-: 4--—--: 5----: 138 ON\OL\I—l on... O\NN\OU\ 1“ \TN -30.6 22.5 -13.0 19.6 22.7 wad’o o o o o o mmoo [\I—JHN m«00«o O\L\O\(3L\ m 'QN \OOO«NI.n 0.... 020130be) \ONmr—I \OOOMOOO o o o o 0 Ln r-Imoo H 9----: 10--—--: 6----: 7----: 8----: ijOOI—Im ONOOO‘ “\L‘ \TN \OmNm« 5 69. 11. 56. 9. m I—|\O(“\L\Lr\ 88888 C\OL\r-10\ O O O O O mm1\«\1 \‘I‘L‘HL‘IC—J‘ -Continued 139 .Moo: SpH now onSpsnomSmp SSSHSSS hHMooz prop mo omwno>w mH Mm USS Moo: SHH SH oprHnw> SOHpownopSH nom pSoHOHmmooo mH HHQ «Moos SHH SH oHannw> SoprHHQHoonm nom pSmHonmooo mH QHD axon: 6S9 op mnemon H ences MAHHV N AHHQV + AQHQV nopSSS SOdeswo 6S8 poppHSo moprHnwn mo pom u poommm poz .noeSSS SOprsvo mH momoSpSmnwm SH nonedz .opmp pmpoonnoo mSHmS popwermo mSOHpmsvo HH< \m n.0«l o.mbI o.mmI m.o>I N.®©I m.w0HI m.omI m.HmI ”IIIIIIImN 0.0H 0.0H «.0 b.«H b.HHI o.HI 0.0NI H.0HI "IIIIIIINN b.bI m.omI «.oHI H.5HI «.HHI «.mmI m.mmI m.HmI "IIIIIIIHN p.mOH o.bs N.Hw m.«w «.wb m.mb m.mm b.0m uIIIIIIIom o.mHI H.0HI m.mHI m.bHI m.mHI m.H«I m.NmI m.wmI "IIIIIIImH o.m« «.mm s.m« e.«m be«« w.mH m.om m.H« ”IIIIIIIwH 0.0NI o.mmI o.««I m.bNI ©.«HI v.0 w.m«I H.omI "IIIIIIIbH v.5 «.mb 0.0NH m.Hw H.0mH ©.mmH p.mmH «.bmH "IIIIIIIoH mnos non mpSSom m mass. m A a m a m 4 m m sac m sac m 388 m . . Nomv H . AHva a . Hommv.m . Abwmvw . . . . . . . . . . Somwom m ow .H. u u u u u a m o u H o z u oSoz u MSHsonm " SH #603 HmSSHpSOOII.oI> mHnna 140 poSSHpSOOI noQSSS Soprsvm USw poppHSo moprHnw> mo pom 0H.o n0.o 00.0 HH.0 00.0- 00.0 00.0 00.0- u----0H 00.0 00.0 H0.o 00.0 00.0 00.0 00.0- 00.0 u----«H 00.0 00.0- 00.0- 0«.0- 0H.0- 00.0- 00.0- 00.0- u----0H 00.0 no.0 no.0- 0«.0 00.0 00.0 00.0 00.0 ”----0H «0.0 m«.OI «H.0I ««.OI mm.0I o«.0I «0.0I mm.0I "IIIIIIIHH 00.0 00.n- 00.0- 00.H- 00.n- 00.0- «H.0- 00.n- ”-------0H 00.0 mm.HI 00.0I 0H.0I mm.mI 00.HI pm.HI 00.HI “IIIIIIIIm no.0 00.0- 00.0- 00.n- 00.0- 00.0- 00.n- 00.0- ”----0 00.0 00.0 00.0 «0.H no.0 00.0 o0.n 00.H “--------0 00.0- 00.0 on.0 on.o 00.0 00.n- 00.0- 00.0- .----0 00.0- 0H.H 00.0 00.0 00.0 00.H 00.0 «0.0 .----0 00.0- 00.0 «0.0 00.0 00.0 00.0 00.0 00.0 .----« 0H.o 0H.0- 00.- 00.0- 00.n- 00.n- 00.n- H0.H- “--------0 00.0 00.0 00.- 00.0 00.0 00.0 00.0 00.0 “----0 00.0 00.0 00.0 0H.o 00.0 00.H 00.0- 00.0 .----n onow nmm moSSom « A000v H n H n H H H H A000V H A0000 H A0000 M an 0 n m 0000 H m 3000 n m 0000 0 M 000000 N 0.0.0 m .H.o.z m ocoz m 0%me » mo Moo: \MmSOHpmswo oanmSnopHd Honobom nom 0x003 hnIIonSpsnoSSop mo omnmop HwSOHHprw oSo mo mpoommm poZII.bIb oHnwa .xmm3 SHH SH SOHpmpHmHoonS omwnm>0 0H Hm oSw .Mmoz SHH SH mHann0> SOHpownmpSH nom pSoHOHHHmoo mH HHQ aMoo: SpH SH oprHnwn mnSpwnomSmp now HSoHOHMHooo mH HHQ axon: map 0H H mnon: MAHmv N AHHQV + AHHQV 141 nmnSSS SOHposvo 0S0 omppHSo mmeanwn mo pom n poommo pmz .nmeaSS SOHpmsvo 0H mmnmepSmnmm SH nmnesz .0900 empomnnoo mSHmS oopwsHpmm mSOHpmsom HH< \m N0.0| ¢m60l No.0 mm.Ol mm00l bw.Ol NboH ©0.0l “IIIIIIIMN Hooo HO.HI mO.OI mm.O| QwoOl Omofll omoml HfioHl «IIIIIIINN H000! Om.Ol ON.OI b<.0| Db60l OM.OI «wed! ©b.OI “IIIIIIIHN MHoo mmoo DH.OI Mmoo QH.OI #w00l m©.Ol ®Hoo ullllIIION HO.OI wooH mHoO bbofl MboH O®oH mmom omofl «IIIIIII®H MOoO mfioOl HOoO ©m.Ol #bocl HN.O| Ofioo ©00Ol ullllllle H060! bboo Nfioo Hwoo «HoOl bboO wfioHl mboo ullllllle HOoO Doom Hboo coom mH-Q wmom OQoH mN-M ”IIIIIIIQH mnom non mpSSom u 808 . a . n u a . . awe . $000 . $08 . n . 0000 H " 300V 0 . A0000 0 " A000; “ . . . u . . . u n 0808 Hm 0 H u u u u u m m o u H o z. ” oSoz ” mSHzonm “ no #003 aaaeneaoo-.0-0 annan 142 Polynomial Weather Variables Only polynomials of seventh degree will be considered in this section. Thus, eight coefficients were estimated for each weather factor. Because the coefficients were for transofrmed variables, they have no direct interpretation. The following discussion would have been more meaningful if the polynomial had been evaluated to obtain the weekly estimates. The weekly estimates then could have been used to obtain the net effects as was done in the previous section. However, this was not done because the rounding error, particularly for weeks late in the season, was so large that it made meaningless the resulting estimates. This was because enough Significant digits were not obtained for the coefficients for the higher order polynomial variables.l/ Even though coefficients containing eight places after the decimal point were obtained, in many cases there were only two or three Significant digits (for example, see Appendix B, Part 22). It was discovered, after the bulk of the analysis was done, that this was not sufficient to give meaningful weekly estimates. The discussion will necessarily be limited to the coefficients obtained for the polynomial variables. Polynomial Precipitation Variables Six equations were estimated using the corrected data and containing the polynomial precipitation variables. The coefficients obtained are presented in Table 7-8. In all equations the coefficients are consistent in Sign. Except for the equation omitting the interaction variables, the coefficients are i/ The coefficients for the "complete" polynomial were re-estimated to obtain additional significant digits. The results are discussed on pages 151 and 153. 143 Table 7-8.--Coefficients for polynomial precipitation variablesé/ Set of f Polynomial precipitation variables variables I , , , , , , , omitted. 3 R0 3 R1 3 R2 3 R3 3 R4 3 R5 3 R6 3 R7 -39748 5455a -23268 4518 -45.58 2.478 -.0688 .000758 2 O :3 (D A N \O .(.\ V I .0 M.C.I.V.(295.:-4672a 6022a -256581 50481 —51.78 2.8681 -.081°L .000928 C.R.D.(296); -4733a 6250a —26.4a 500a -49.9a 2.69a ‘-.074a .00081a r(297)---; -3559a 48118 -20178 3878 -38.78 2.088 -.0578 .000618 T(300)----: -3762a 48858 -20658 4028 -40.78 2.228 -.0618 .000688 I(301)---; -1178 2278 -1O68 20.78 -2.018 .103b -.0026b .00003 2/ Estimated using corrected data. consistent in level of significance (all being Significant at the .01 level) and quite consistent in magnitude. When the interaction variables were omitted, the magnitude and level of Significance of the coefficients were affected. This was probably due to the high intercorrelation between precipitation and interaction variables (never less than .97). Adding the interaction variables to the equation containing the precipitation variables (obtaining the "complete" polynomial equation, 294) caused the coefficients for the precipitation variables to become larger (in absolute value) and to pppppg Sigpificant at a pigppp level. This result is interesting because one of the "problems" generally associated with multicollinearity is that it is pppg difficult to Show the Significance of the coefficients of variables that are highly correlated. With respect to the Significance of the set of precipitation variables in the "complete polynomial" equation, it seems intuitively clear that if all the coefficients in a set are Significant, the set would be Significant.;/ i/ That is, the hypothesis: B0 = Bl = 02 = $3 = [34 = 85 = B6 = B7 = 0 would be rejected. 144 This was checked using the test discussed in Chapter Iv, and an F value of 10.42 with 8 and 581 degrees of freedom was obtained. This set was Significantly different from zero at .01 level. Polypomial Temperature Vgiiables Coefficients were obtained in six equations estimated using the corrected data. The coefficients obtained are presented in Table 7-9. In the "complete polynomial";/equation, all coefficients are Sig- nificantly different from zero at the .05 or less level. Omitting the M.C.I.V. caused all the coefficients to increase (in absolute value) and all became significant at .01 level. Omitting the crop reporting district variables resulted in coeffi- cients only a little larger (in absolute value) than the coefficients obtained in the "complete polynomial" equation. The level of Significance of the two highest order terms was reduced. Only one of the coefficients was significant in the equation omitting the year variables. This means that if the effects of factors correlated with years were not taken into account, it was not possible to pick up (measure) the significant effects of temperature. If the polynomial for either precipitation or interaction was omitted, the coefficients for the temperature polynomials were not significant. In the "complete polynomial," the set of variables explained a significant amount of the variation in yield. Using the test discussed earlier, an F value of 3.16 was obtained. With 8 and 581 degrees of free- dom, this is Significant at the .05 level. i/ Contains polynomial weather variables and with no set of variables omitted. l .0900 popomnnoo mSHmS pmpmermm \m H00000000.I N000000. H000.I m00. bm0.I 0H. mm.l Ob. m AHomVH H0000000.I H0000. m000.l 000. 000.I m«. H0.HI H0.H m Amomvm m0000000. m0000.I HH00. «N0.I ohm. m©.HI owH.« Hm.ml m Abwmvw enoooooo. pnnooo.- 00000. 0000.- a000. 000.0- 000.0 80H.0 - m---A0000.o.m.0 0m000000. 00N000.I 00000. mNmH.I 00H«.H 0mm.0I 0«0.mH 00H.0HI WIIIIAmmNV.>.H.0.S 80000000. 0280.- 00000. 8000.- 0000. 000.0- a0H.0 o00.17 m----A000vmsoz 00 H on M 00 M 00 H 00 H 00 M an H on H ooeeneo . moprHnwn moHannmn mnSpwanSop HwHSoSHHom m Ho pom \mmondponmmeop HdHSoShHoa now mpSoHOHmHmOOII.mIb oHnwe 146 Ppiynomigi_lnteraction Varigbles Coefficients were obtained for the interaction polynomials in seven equations estimated using the corrected data. The coefficients obtained are presented in Table 7-10. The coefficients were significantly different from zero at the .01 level in all equations containing polynomials for all three weather fac- tors. When the precipitation polynomial variables were omitted, either alone or with the temperature polynomial variables, some coefficients were not significant and the magnitude of the coefficients was greatly reduced (in absolute value). Omitting only the temperature polynomials did not greatly affect the magnitude or level of significance of the coefficients. The set of interaction polynomial variables in the "complete poly- nomial" equation was Significant at the .01 level. An F value of 9.61 was obtained with 8 and 581 degrees of freedom. Temperature and Precipitation PolynomiaiTVariablep A test was made to determine if the set of variables composed of the temperature and precipitation variables explained a Significant amount of variation. Omitting these sets of variables from the "complete poly- nomial" equation significantly reduced variance explained. An F value of 6.43 with 16 and 581 degrees of freedom was obtained. Any F value greater than 2.75 is significant at the .01 level. Season Total Weather Variables Season Total Precipitation Variables This variable is the sum total of precipitation that fell during ‘the growing season and corresponds to the zero degree term in the 147 .0000 copomnnoo mSHmS UmpwaHpmm \m 000000000. 00000000.- 00000. 0000.- 0000. omn.- n00. 0H.- m----A00000 0 0 0H000000.- 0noooo. 0000.- 0000. 0000.- 000.0 000.0- 000.0 m A00000 000000000. 0000000.- 00000. n0oo.- 0000. n0n.- 000. 0n.- m A00000 0noooooo.- 000000. 0000.- 0000. 0000.- 000.0 000.0- 000.0 m A000vn 0H000000.- 00Hooo. 0000.- 0000. 0000.- 0n0.0 000.00- 0nn.0 m---A000v.o.0.0 00000000.- 00Hooo. 0000.- 0000. 0000.- 000.0 000.0H- 000.0 m--A000V.>.H.o.z 0Hoooooo.- 0Hnooo. 0000. 0000. 0000.- 0H0.0 000.0- 000.0 m----A000000oz a a a s n an n an ass. . mme0Hn0> moHQ0Hn0> SOHpo0nopSH H0HSoShHom N no pom \MmmHn0Hn0> SOHpo0nmpSH HOHSoShHom nom mpSmHOHmmmooll.0HIb 0H909 148 precipitation polynomial. The coefficient gives the average effect on yield of one additional inch of precipitation regardless of when it falls during the growing season. A coefficient was estimated in six equations using the corrected data. The coefficients obtained are presented in the first column of Table 7-11. The coefficient is only significant when the interaction variable or the year variables are omitted. The interaction variable and the precipitation variable have a simple correlation of .992, which explains why its presence caused the precipitation coefficient to become insig- nificant. However, the net effect of one inch of precipitation based on the "complete season total" equation (equation 302) and assuming average temperature, was 25.43 pounds per acre or nearly the same as the 24.82 obtained when the interaction term was omitted (see Table 7-ll). In fact, omitting any of the sets of weather variables had very little effect on the net effect of one inch of precipitation. Omitting the M.C.I.V. or the crop reporting district variables caused the net effect to decrease about 20 percent. The season total precipitation variables explained a large amount of the variation in yield between years unless that variation was explained by year variables. Season Total;Temperature Variables This variable is the sum over all the days in the growing season of the daily maximum temperature. Its coefficient indicates how much yield would increase as a result of a one-degree increase in the season total maximum temperature. The coefficient is significant in all equa- tions presented in Table 7-11. The sign is always negative, indicating that on the average, higher temperatures decrease yields. 149 .010 00 0:0 000900 n0 00000H 090 0903830 00090300 0000000009900 .0Hn0flh0> 00002000000 039 Mo 0005 009 0900 H09 00n50m \m .000000 H00. mo coap0pfim00mnm 0Hx003 0m0h0>0 no 0000m \m .0009000 000 mo 0Hzp0n0gamp H0pop 0Hx003 0w0H0>0 so 00m0m \m .0900 009009900 00005 0090afipmm \m IIGH 0000n000 00Hn0090> 9000003 H0pop 000000 90m mpn0fi0fimM000 00HQ0HH0> no 900 0000. 00.0w 000000. WIIA000V0MMHWWMMMWMMMMH 0000.- 00.00 00000.- 000.00 mun-nwfl000v 0000000000H 0000.- 00.00 00000.- . 00.00 m ..... A0000 00000000000 0000.1 mm.0m 00000. 00000.: WIIIA000V moap0pfimfio0hm 0000.- 00.00 000000.- 00000.- 000.00 m A0000 00000 00mm.| 00.00 0HN00. 00N0N.I 00.0: WIIIIIIIIIIAoomv .m.m.0 0H0H.I 00.0H 000000. 00N0N.I 00.00: WIIIIIIIIA0000 .>.H.0.z 0000.: 00.00 wHH00. 30000.: 00.0: m Amomv 0002 \m0900 n09 m0ndom m \M0Hdp0y0ma0a H \mc0000p00000hm 000p00h0psH M 0Adp0h0ma0a M 00090009000nm N 0090080 9005:000 0 mo mp00mmm p02 \Mm0an0fih0> 0000003 H0000 000000 pom 000000000000II.HHI0 0HQ0B 150 The net effect is also negative except when both the precipitation and temperature variables were omitted. This is probably due to the fact that a single coefficient (for the interaction variable) had to measure the effect of a one-unit increase in precipitation and temperature. The effect of precipitation clearly dominated. The net effect of temperature relative to the "complete season total" equation (equation 302) was not greatly affected by omitting any single weather variable. However, omitting the M.C.I.V., the district variables or the years variables caused the net effect to increase (in absolute value) greatly. Thus, the importance of temperature differentials in "explaining" yield differences was greater when the effects of location, years, or M.C.I.V. were not taken into account. Weekly Estimates from Alternative Sets of Weather Variables It was possible, regardless of the form of the weather variables, to obtain estimates of the effect of changes in a weather factor in any week on yields. When the weekly variables were included, their coefficients were the estimated effects. When season total variables were included, their estimated effect was their coefficient and was constant for the entire growing season. When polynomial weather variables were included, it was necessary to evaluate the resulting polynomial for each week of 'the growing season. It is of interest to compare these alternative estimates. The Inesults discussed here are limited to three equations (equations 292, 303, axui 301) estimated using the corrected data and differing only by the form of?‘the weather variables. Equations were selected which did not contain tkma interaction terms to simplify the comparisons. All equations selected 151 contained the seven M.C.I.V., the crop reporting district variables, the time variables, and the preseason precipitation variable. The effects from the weekly and season total variables were the coefficients and are presented in Table 7-12. The weekly estimates from the polynomial variables are also presented in Table 7-12. It was discovered when trying to evaluate the temperature poly- nomial for the weekly effects that the eight digits after the decimal obtained for each polynomial variable 3232 not sufficiently accurate.l/ The basic temperature data was transformed (inputed as thousands of degrees instead of degrees) and the equation re-estimated. The resulting temperature and precipitation coefficients are presented in Table 7-13. To obtain the weekly precipitation estimates, the first column vector was multiplied successively with the rows in the matrix presented in Appendix C. In effect, it is taking the number of the week raised to the degree of the variable and then multiplied by the corresponding coefficient. For example, to obtain the weekly precipitation estimate for the third week we have: (-ll7.04644798) (30) + (227.15416386) (31) + (-105.o9337465) (32) + ... + (0.0000261?) (37) = 63.9 (see Table 7-12). The same procedure is used for the temperature estimates. To facilitate the comparisons, the coefficients from the three techniques (and presented in Table 7-12) are presented in Figure 7-1 and 7-2. Let's consider precipitation first. The season total estimate is :not a function of time (within the growing season). There is no direct relationship between the 23 within-season estimates from either the weekly (Ir polynomial techniques and the season total technique. It is worth Iuyting, however, the average effect of one additional inch of precipitation ;/'N0ne of the coefficients presented in Appendix B for the higher order Ixilynomial terms are accurate enough to derive meaningful weekly estimates. n‘ v 5.. .1»..- --\ u. ... 152 .wpwp copomhhoo moans Umpwsflpmm \m oHSpwhomamp ma omwmnosw mmnmmplmqo w you mmwmnosfl vaHM soapwpfimfiompm 2H omwohoca sosfllmco w pom omwohosfi camflw common mnfizohm mo #83 HH.OI 5m.Ol «n.0I w. Hwfiaoshaom cosmop zpsobom no haxmms mmfi deHosH mamwos Scam ohdpdpomsmp no sofipmpfimfiomnm s“ omwonosa was: w mo mpoommo mo mopdfiflpmo haxmmznu.maub magma 153 Table 7-13.--Coefficients for precipitation and temperature polynomial variables, equation 301 Degree : Coefficients variible : Precipitation polynomial : Temperature polynomial 0------: -ll7.04644798 0.78974512655 lst- ----- : 227.15416386 —0.54797708289 2nd-------: -105.90337465 0.19485726587 3rd ------ : 20.71714078 -0.03674830948 4th-----: -2.01418461 0.00335175379 5th-----—: 0.10260923 -0.000142l7109 6th-—-----: -0.0026147l 0.00000236237 7th-----: 0.0000261? -0.00000000509 is 24.8 pounds per acre for the season total estimate, 21.16 pounds per acre (simple average of the 23 polynomial estimates) for the polynomial estimates, and 14.59 pounds per acre (simple average of the 23 weekly estimates) for the weekly estimates. The estimates from the weekly and polynomial techniques make a similar pattern (see Figure 7-1). They show that additional precipitation (above average) during the time the plants are seedlings and during the harvesting season decreased yields. Additional precipitation during the planting seasonl/and during most of the growing season increased yields. The maximum effect was obtained by additional precipitation about the time 'the plant was in bloom. Although the estimates "tend" to make the same pattern, there are sxxne major discrepancies. As expected, the estimates from the weekly lj'Of course, this may decrease acres planted. 154 Figure 7-l.-—Estimated effects of precipitation——for a one—inch increase in weekly total-~alternative techniques Pounds per acre 80- 70~ A h 60‘ 50’ 40- 30L 1 ! ! + ! 20‘ I I I \ i I I I i ‘ I \ I I I 3 --.P -——- ."' Polynomial Estimates ————— weekly Estimates —'—'—'—‘- Season Total Estimate 5 5 5 1 i w‘ I ‘r f 3 v I I I 0 I PI! 6 7 8 9 10 ll 12 13 14 15 16 17 18 19 20 21 22 23 Week of Growing Season H P N ’ b.) ;\ . U‘ . 155 Figure 7-2.-—Estimated effects of temperature—-for a one—degree increase in weekly total-~alternative techniques Pounds per acre 3.5L 3.0” A 2.5- ,1 2.0- l.5~ A (\ (I 1.0” /" ‘1 ‘ 0.5+- —0.5L -1.0~ Polynomial Estimate Weekly Estimate _3 5- _____________ Season Total Estimate Illiillllgnlntgg 1 I 1 l I l l 1 , e - 1 2 3 4 5 6 7 8 9 D3 11 12 13 14 15 16 17 18 19 20 21.22 23 week of Growing Season 156 techniques were more extreme and changed (difference in estimates for consecutive weeks) more rapidly than those from the polynomial technique. The same general conclusions can be made about the temperature estimates (see Figure 7-2). However, the pattern made by the weekly estimates is only vaguely related to the pattern made by the polynomial estimates. The patterns indicate that above average temperatures during the planting season and during grain development caused yields to increase. Above average temperature during plant growth from seedling until after blooming and during the harvesting season decreased yields. These results agree with our a priori knowledge concerning the effects of temperature on yields. The reasons why the weekly estimates deviate from the polynomial estimates and why the deviations are so much greater for the temperature estimates than the precipitation estimates need further consideration. The principal reason for the discrepancies was intercorrelation among the weather variables. Consider the precipitation variables. Mggt of the simple correla- tions of a precipitation variable in one week and the precipitation variable in either the preceding or following week were very low, less than 0.01. However, the correlation between the 12th and 13th week (where a major shift in the size of the coefficient occurred) is .209. Similarly, for the 15th and 16th weeks the simple correlation was .206; 16th and 17th weeks, .186; 19th and 20th weeks,.501. These higher intercorrelations correspond to the major discrepancies between the weekly and polynomial estimates (see Figure 7-1). There is a very positive relationship between the size of the simple correlation coefficient and the extent of the 157 discrepancies between the polynomial and weekly estimate. It is surpris- ing that such low intercorrelations would have this effect. When it is realized that the simple correlations of a temperature variable with the temperature variables in the preceding or following period were high (as high as .857), it is not difficult to understand the discrepancies between the temperature polynomial estimates and the weekly estimates. The intercorrelations among the weekly precipitation variables or the weekly temperature variables do not affect the coefficients estimated for the polynomials. One of the principal advantages of the polynomial technique is that it does take into account the preceding and following weather values in "estimating"l/the effect for a particular week. Conclusions There are several advantages of using a polynomial to estimate the effects of weather during the growing season, as compared to the weekly variable technique. (a) It uses fewer degrees of freedom, in this case using only 24, while the weekly variable techniques used 69. (b) It takes into account the preceding and following weather values in "estimating" the effects for a particular week. (0) The resulting estimates are more meaningful and are consistent with a priori knowledge that the effects of a weather factor should change slowly from one week to the next. There are also disadvantages of the polynomial technique (advantages in.using the weekly variable technique). ;/'The technique does not estimate the effect for a particular week Iiirectly, rather the weekly effect must be derived from the estimated coefficients for the polynomial. 158 (a) The weekly variable technique results in a higher E2. If the weather coefficients are of no particular interest, it may be more valuable to obtain a higher E2. (b) There are additional data transformations required for the polynomial technique. Of course, these can be done within the computer as was done in this study. (c) More likely to accurately measure particularly critical week (no smoothing) if degrees of freedom are large. The advantages of the season total technique are: (1) It has simple data requirements, and (2) it uses very few degrees of freedom. The disadvantages are: (1) It produces a lower E2, and (2) its coefficient ignores the timeliness or distribution of the weather factors. The Polynomial Model; No information existed on which to base an a priori hypothesis concerning the "best" degree of polynomial to include. Many trial equa- tions were estimated in an attempt to resolve this. How E2 changes as the degree of the polynomial was varied is discussed. The discussion is necessarily limited to equations estimated using the incorrect data. Precipitation Polynomials Five sets of precipitation polynomials were estimated. A set con- tains a polynomial for each degree, zero through seven. Each set was estimated with different "other" variables included. The resulting 'fiz's are presented in Table 7-14. In all cases, the seventh degree precipita— tion polynomial did "better" than polynomials of lower degrees. Here "better" simply refers to magnitude of E2. The fact that E2 increases as degree terms are added indicates that the gain in increased variation ex— plained is greater than the loss due to decreasing degrees of freedom. 159 Table 7-l4.--E2 for models containing recipitation polynomials of varying (212) (211) (210) (209) (208) (207) (206) (205) .555 .558 .559 .561 .561 .562 .569 .571 C,Y,P------ (36) (37) (38) (39) (40) (41) (42) (43) degree a Other 3 Degree of precipitation polynomialsy/ variables I . . . . . . . included ; 0 g 1 ; 2 ; 3 I 4 I 5 I 6 : 7 : R2 P E .112 .134 .133 .134 .139 .149 .161 .172 = (213) (214) (215) (216) (217) (218) (219) (220) M,P--------; .613 .622 .626 .626 .627 .628 .632 .638 M,C,Y,P--"-: 0725 0725 0727 0728 0729 .730 0737 .738 : (33) (32) (31) (30) (29) (28) (27) (26) M,T07 —————— Q .660 .662 .668 .670 .671 .670 .678 .684 (178) (177) (176) (175) (174) (173) (172) (171) 2/ Symbols used are listed in Chapter III. Estimated using the incorrect data 0 b/ Number in parentheses is equation number. It is interesting to observe the pattern presented by the coefficients in equations 213-220 (Table 7-15). With the only other variable in the model being the preseason precipitation variable, consecutive terms of the precipitation polynomial were added. In the first two equations, the high- est degree term (zero degree and first degree respectively) were significantly different from zero at .01 level. In the next two equations, no coefficients were significant. Beginning with the fifth and the sixth equations, some of the higher order terms became significant. In the seventh and eighth equations all coefficients are significant. All were significant at a < .01 in the last equation. The above investigation considered the coefficients for the precipi— tation polynomial without taking into account the effect of temperature and .IF)I\'. CII-IIII: uIilI.-I‘£i-‘IIII rquIFII.$. I. n-,IP\IIIF,’I§I§I.;IID‘I\ FII ...-Ahai-IILO.-III.I --..<-.fi Qi.‘ _nI-'-9-0.-~iI.‘-\ ‘I‘\\h AvI‘~.~\\.‘s 160 8600.: same. emHI send awaoau aenmm somsmu.m nIOH.6Ioe.a m oma mao.u man. m.mu w.sm m.ns wow. u mIOH.mIoa.m " 6mH amsH.I em.m send- seep oases: m «IOH.quoa.m m mma sHH.I w.a was- new " muoH.mIoe.d “ ems 0 I- u n a u on e defies so “ NIOH muse a ” sea 0 .I u a a u am am seam . HIOH duos a . sea 0 Q fl 0 ssma H OH 09 m “ sea amoooo. smoo.u new. sem.eu use smmmn menu seem: H a M omm smooo.u sumo. semm.n as.HH sm.sou sssa emmfl m m m saw amoo.u sass. ssH.mI 6H.ma mm- N.mo n a ” mam sHHo.I cos. emo.6n m.sm m.H¢I ” a u dam wmo.u mm.H cm.en o.Hm H a m ram mmo. mm.m mo.mn u m u mam www.m H.0HI m a N «am an.mm . a ” mam hm u om u mm u «m « mm u mm » Hm « om u moansfins> nonpo u Gmmmmmmm mGOHpmsvo Umpomfiom scum mpquOHHMmoo .Hdfisoshaom soapmpfimfiomnmll.malb manna 161 interaction variables. Every third equation beginning with equation 141 and extending through equation 162 (see Table 7—15) gives the coefficient for the precipitation polynomial when polynomials of the same degree for temperature and interaction are present. Considering the levels of sig- nificance of the coefficients, the pattern is much different than when temperature and interactions effects were not considered. None of the coefficients in the polynomial of third or fifth degree are significant. The coefficients for the last two terms of the seventh degree polynomial are not significant. 0n the other hand, all the coefficients in the fourth and sixth degree polynomials are significant and all but one at the a < .01 level. The fact that this pattern exists raises some interesting problems. For example, if we drew our conclusions about the effect of introducing polynomials of equal degree in temperature and interaction on the coefficients of the precipitation variables, based on the fifth degree polynomial, we would conclude that the effect was to make the coefficients insignificant. If based on the sixth degree polynomial, the effect was to make the coefficients more significant. If based on the seventh degree, the effect was to make some coefficients less significant and some insignificant. In all cases the magnitude of coefficients was greatly affected. Because the E2 in all equations containing the seventh degree poly- nomial was higher than in any comparable equation containing a polynomial of lower degree, it was decided to include a polynomial for precipitation of seventh degree in the "nearly complete" equation. Temperature Polynomials Five sets of temperature polynomials were estimated. The resulting 115-2's are presented in Table 7-16. 162 Table 7-16.--E2 for models containingst7mperature polynomials of varying degree a Other 3 Degree of temperature polynomialsy/ variables I . . . . . . . included 3 0 ; 1 ; 2 3 3 j 4 j 5 3 6 g 7 i 82 P,RO7-----: .244 .257 .255 .266 .266 .295 .316 .315 : (202) (201) (200) (199) (198) (197) (196) (195) M, P,RO7---§ .669 .674 .674 .687 .688 .691 .691 .691 : (194) (193) (192) (191) (190) (189) (188) (187) C,Y,P,RO7—-: .573 .574 .575 .575 .575 .575 .575 .583 : (44) (45) (46) (47) (48) (49) (50) (ll) M,0,Y,P,RO7§ .738 .738 .738 .740 .740 .740 .739 .740 : (25) (24) (23) (22) (21) (20) (l9) (18) M —————————— 2 .613 .618 .618 .620 .632 .635 .635 .635 : (186) (185) (184) (183) (182) (181) (180) (179) 2/ Symbols used are listed in Chapter III. b/ Number in parentheses is equation number. The results for the temperature polynomials were quite different from the results of the precipitation polynomial. There is no consistent pattern. The E? for equations containing the seventh degree temperature polynomial in one case is higher, in one case lower, and in three cases the same as the H? for equations with lower degree polynomials. The pattern made by the coefficients in equations 195-202 is inter- esting (see Table 7-17). When only the zero degree term or the zero and first degree term are included, all coefficients are significant. However, where the second degree term is added, only the coefficient of the zero degree term is significant. The addition of the third degree term.makes all coefficients significant again. The fourth degree term causes the coefficient for the two highest degree terms to become insignificant. F .$I.I.I 163 0000000. 00000.- 000. 000.- 000. 0.0- 0.0 0.0- H 00.0.A008B.».0 . 00 80000000. 80000.- 8000. 8800.- 80.0 80.0- 80.00 80.0- m 00.0.0.0 m 00 80000000. 80000.- 8000. 8000.- 8000. 80.0- 80.0 00.0- “ 00.00.m.10080.0.0 " 0 000000. 80000.- 0000. 0100.- 800.0 800.0- 80.00 80.00-m 0H.00.m m 000 800000.- 8000. 8000.- 8080. 80.0- 80.00 80.0H-m 0-0H.0-0m.m m 000 0' 0 0| cl. 0 o A a o 0000 000 000 000 00 H e-0H 0-00 m H 000 8000.- 8000. 8000.- 0.0 m 0-0H.0-00.m m 000 0 0|. ol. 0 a a o 000 000 0000 H 0-0H 0-00 m H 800 o 0' o a a o 8000.- m 00.00.0 m 000 0000000. 00000.- 000. 000.- 0000. 800.0- 80.00 80.0H-m 00.0 m 000 800000.- 8000. 8000.- 8000. 800.0- 80.00 800.0- “ 00.0 n 000 800000. 8000.- 8000. 8000.- 880.0 800.0- m 00.0 m 800 00000. 000.- e000. 8000.- 800.0 a 00.0 n 000 8000.- 8000. 8000.- 800.0 m 00.0 m 000 0000.- 000. 8000.- n 00.0 n 000 8000. 800.- m 00.0 m H00 8080.- n 00.0 n 000 m m m m m m m m . 8888 be . we . ma . we . ma . me . Ha . OB . moanwfinw> 90:90 "soapQSUm 0:00pmsvo vopooaom sonm npsofiowmmmoo .waaoshaom endpwymmamall.balb mamas 164 Adding the fifth degree term makes them all significant again. The sixth degree term affects the magnitude, but not the levels of significance. The coefficients for the four highest degree terms are significant when the seventh degree term is added. One can only wonder if an eighth degree term were added. The above results are from a submodel which has other than the temperature variable only the precipitation polynomial (of seventh degree) and preseason precipitation variables. When the remaining variables are added, the coefficients of all terms in the temperature polynomial are significant (see equation 2, Table 7-17). The second set of equations presented in Table 7-17 shows how the coefficients for the temperature polynomials are affected by adding the next highest degree term to each of the weather polynomials. It is interesting to note that none of the coefficients of the fourth degree temperature polynomial are significant. All are significant in the fifth and sixth degree polynomial and all but the highest order term in the seventh. It is interesting to note that when the M.C.I.V. and the interaction polynomial variables were omitted (equation 11, Table 7-17), all the temperature polynomial coefficients were significant at the .01 level. When.the M.C.I.V. were added all the coefficients were not significant. .Adding, in turn, the interaction polynomial variable (equation 2) caused a11.the coefficients to again be significant. In any case, all coefficients were significant in equation 2 and on 'bhis basis it was decided to include the seventh degree polynomial for temperature in the "nearly complete" equation. 165 Interaction Polynomials Only one set of interaction polynomials was estimated. The Ez's are presented in Table 7-18, and the coefficients in Table 7-19. -2 Table 7-18.-R for equations contaigiyg interaction polynomials of varying degree a Other Degree of interaction polynomialS/ variables . . . . . . . included 10 j 11 : 12 : I3 z I4 1 I5 1 16 3 I7 R .315 .315 .317 .318 .319 .321 .339 .339 P,Ri,Ti---- (168) (167) (166) (165) (164) (163) (162) (161) O. O. O. O. O. O. O. C. O. O. I N g/ Symbols used are listed in Chapter III. b/ Number in parentheses is equation number. The E2 increased or stayed the same as additional terms were added. The pattern of coefficients when only other weather variables were included is interesting. With two exceptions, all the coefficients in the fifth or lower degree polynomials were not significant. Most of the coefficients in the sixth and seventh degree polynomials were significant. However, if instead of seventh degree precipitation and temperature polynomials, polynomials of the same degree for all three weather factors were included (equations 141-161, Table 7-19), more interaction coefficients ‘were significant. In the zero and first degree polynomials, all the coefficients were significant. Two of the three coefficients in the second (degree polynomial were significant. In the third and fifth degree poly- Inmnials, none of the coefficients were significant. All the coefficients iml‘the fourth and sixth equations were significant. Seven of the eight coefficients in the seventh degree polynomial were significant. 166 80000000.- 80000. 8000.- 8000. 8000.- 80.0 80.0- 80.0 00.00.m.00v80.0.0 0 80000000.- 80000. 8000.- 8000. 8000.- 80.0 80.00- 80.0 m 00.0m.m.0 M 00 80000000.- 80000. 0000.- 0000. c000.- 80.0 80.0- 80.0 n 00.0m.m.0 u 0 80000000.- 80000. 8000.- 8000. 8000.- 80.0 80.0- 80.0 m 00.00.m.00080 m 000 wHOOOO. dHOO-l GHNO. deN-I d5oH d5ofil wmofi u ©lOBq®IOM0m u OmH 00000. 000.- 000. 000.- 000.- 000. m 0-00.0-0m.m m 000 wMOOOo mmflool meNo wNofll QOoH m filOBafilomum m MmH o 0.! 0 cl 0 A A o 000 000 00 000 8 0-00 0-00 m n 000 o o o o A A o wwOO I OOMH NwH I H NIOB Nlom m H FQH 0' o o A A o 8000 8000 0 0-00 0uom.m u 000 0000.- m 00 00 m m 000 00000000.- 800000. 80000.- 8000. 8000.- 800.0 80.0- 00.0 m 00.00.m m 000 0000. 000.- 8000. 0000.- 00.0 00.0- 00.0 m 00.00.0 m 000 00000.- 000.- 000. 000.- 000. 000.- m 00.00.m m 000 000. 000.- 8000. 000.- 0.0 ” 00.00.m u 000 000.- 000. 000.- 000. m 00.00.m m 000 8000.- 000. 000.- ” 00.00.m n 000 000.- 000. m 00.00.m m 000 000. m 00.00.0 m 000 m m m m m m m m 8 8. 8 m 8888 5H m 0H m mH m ..wH m MH m NH m HH m OH m Ho, ..0 .0930 H QOfiPgUm chHpmswo wmpomaom Bonn 0920000mmooo “Hafisosmaom sOHpomnmpsHII.mHI0 00909 167 When non-weather variables were included in the equation, all coefficients in the seventh degree polynomial were significant (see equa- tions 2, 8, 99, and 131, Table 7-19). On this basis, it was decided to include a seventh degree polynomial for the interaction variables in the "nearly complete" equation. Conclusion Many combinations of weather polynomials were considered. It was concluded that a polynomial of the seventh degree in each of the weather factors (temperature, precipitation, and interaction) would be included in the "nearly complete" equations. Polynomials of degree higher than seven were not considered. The coefficients for polynomial variables changed as the power of the polynomial was increased. However, the net effect of such changes may have been very small. ‘; ha 1:0 ‘.\.- CHAPTER VIII SUMMARY AND CONCLUSIONS There were two principal objectives in this study. The first was to estimate how changes in inputs have affected yield, and the second was to determine the effect of specifying alternative models. A single equation model was developed. The parameters were estimated by least squares regression analysis. The dependent variable was yield of grain sorghum per acre. There were 645 observations—-observations on 129 counties in each of the agricultural census years, 1939-1959. Three kinds of independent variable were included--man-controlled input variables, dummy (0, 1) variables, and weather variables. The seven man-controlled input variables were: (1) Percent of grain sorghum acreage irrigated, (2) dollars Spent on gas and oil per acre of cropland harvested, (3) pounds of fertilizer nutrients applied per acre of grain sorghum, (4) ratio of acres fallowed to acres of cropland harvested, (5) average acres of grain sorghum per farm harvesting grain sorghum, (6) number of tractors per acre of cropland harvested, and (7) per acre 'value of land (to measure the interaction effects of land with technology). Two sets of dummy (0, 1) variables were included--27 variables to stpresent the crop reporting districts and four variables to represent years. 168 169 Four sets of weather variables were included: (1) Preseason precipitation, (2) season precipitation, (3) season temperature, and (4) season interaction (temperature times precipitation). Three forms of the seaggn weather variables were considered in detail: (a) A weather variable for each week of the 23-week growing season for each weather factor, (b) a polynomial of seventh degree (8 variables) for each weather factor, and (c) a season total variable for each weather factor. Also considered but in less detail were: (1) Man-hours of labor per acre of grain sorghum, (2) a set of seven dummy variables to represent different climatic regions (based on average planting date), and (3) season weather polynomials of degrees two through six. Results Estimates of the effect on unweighted average yield of changes in the level of inputs were obtained from the "complete" equation. This equation as specified a priori contained the seven man-controlled input variables, the 27 dummy variables for crop reporting districts, the 4 dummy variables for years, the preseason precipitation variable, the 23 season precipitation variables (one for each week), the 23 season tempera- ture variables, and the 23 season interaction variables. On the basis of this equation it was estimated that of the 1,146- poundl/ per acre increase in yield between 1939 and 1959, 27.4 percent was explained by changes in the level of the explicit man-controlled inputs, 46.1 percent by changes in the level of implicit man-controlled inputs, amd 26.5 percent by changes in weather. 1/ This is unweighted average yield. The other effects were also esti- mated using unweighted averages of the explanatory variables for each year. 170 Of the increase due to changes in explicit man-controlled inputs, almost all is due to changes in two inputs, fertilizer and irrigation and their interaction with land (value of land). Changes in weather during the growing season accounted for 85.4 percent of the total weather effects. It is important to note that the implicit (unquantified) man-controlled inputs (as measured by the dummy year variables) were 60 percent more important in explaining yield changes than the explicit man-controlled inputs. Shifts in the location of production, 1939 to 1959, caused average yield to increase 50 pounds. The second objective can be broken down into these four sub—objectives: (1) What is the effect on R? of omitting sets of variables from the "complete" model, (2) what is the effect onR2 of representing factors in alternative ways, (3) what is the effect on the coefficients of variables in the model when sets of variables are omitted, and (4) what is the effect on the coefficients of representing some factors in alternative ways. A set of 24 equations was used for the most part to meet these objectives. This set of "nearly complete" equations includes the "complete" equation described above. Seven equations were obtained from the "complete" equation by omitting respectively the following sets of variables: (1) Man- controlled input variables, (2) crop reporting district (dummy) variables, (3) year (dummy) variables, (4) weekly season precipitation variables, (5) weekly season temperature variables, (6) weekly season interaction variables, and (7) weekly season precipitation and temperature variables. The "complete" equation was then modified by substituting the poly- nomial weather variables for the weekly weather variable. The seven sets of variables listed above were omitted in turn from this modified equation. This gave eight additional equations. Season total weather variables were 171 then substituted for the weekly weather variables. The seven sets of variables were again deleted in turn. With respect to the first sub-objective, omitting any set caused the E? to decrease significantly. The largest effect was obtained by omitting the man-controlled input variables which caused R2 to decrease from .8548 to .7094 (17 percent). The effect on R2 of substituting a polynomial of seventh degree in each weather factor for the weekly variables was to reduce R? from .8548 to .8213 (4 percent). Substituting in the season total variables caused R2 to decrease from .8548 to .7861 (29 percent). The largest effect of omitting a set of variables and substituting in a set of weather variables was obtained by omitting the man-controlled input variables and substituting in the season total 2 of .609. variables and resulted in an R With respect to the third and fourth sub-objective, the results were too diversified and extensive to allow a simple summary. As expected, the magnitude of the coefficients was affected in almost all cases. In many cases the sign and level of significance were also affected. For some variables, it would have been possible, under different model specification to have the coefficient test (1) not significantly different from zero, (2) significantly less than zero, and (3) significantly greater than zero. The effect of omitting a set of variables on the coefficients of’variables remaining in the model was reduced as the number of other 'variables remaining in the model was increased. Other Conclusions Although not the principal objectives, there are several aspects of“the study that warrent attention. 172 These results demonstrate that a crop yield model containing several explicit man-controlled input (technology) variables and several explicit weather variables can do a good job (R2 of .85) of explaining yield variation. It also demonstrated that it is feasible to use real world (aggregate) data by counties. It is also demonstrated that combined time series and cross section data break up the multicollinearity problem often faced when using only time series data. By using combined time series and cross section data it is also possible to include dummy (0, 1) variables for years and loca- tions. The advantage of using a set of dummy variables for years instead of the usual trend variable is that no particular functional form for time is forced. The advantage of using dummy variables for locations is that the coefficients "pick up" consistent difference between locations. These differences can then be used to determine the effect of shifts in the location of production, net of the effects of weather and man-controlled inputs. Comparing the estimated effects of each weather factor in each week of the growing season from the weather polynomial technique with the weekly weather variable technique revealed these advantages and dis- advantages. The polynomial technique: (1) Uses fewer degrees of freedom, (2) results in a lower R2, (3) requires more data manipulation, and (4) results in more meaningful estimates. Model Flexibility Although this model was set up for a particular crop, with particu- lar objectives, and for particular data, the basic ideas of including several weather and technology variables and using time series, cross section data admit a wide variety of models. Some of the characteristics that can vary are discussed below. 173 Functional Form The principle of fitting a physical production function using several explicit technology and weather variables does not specify any particular function form. A linear, Cobb-Douglas, quadratic, or other (form could be used. Time Series-Cross Section Data Although there are advantages in using combined time series and cross section data, this method may be used with either time series or ' cross section data. If the combined data are used it is not necessary that the same number of units be used in each time period. Aggregation of Data The basic unit of analysis in this study was the county. Data from firms, plots, States, etc., could be used instead. The counties were selected to represent the major area producing grain sorghum. Units could be selected to represent the Nation or selected to represent a very small (local) area. Type of Crop Considered The crop considered is one of the major feed grains. Fruits, vegetables, forage, etc., could also be analyzed by a model similar to this. Number and Kind of Weather Variables The relevant growing season was assumed to be 23 weeks for grain sorghum. Clearly the model can be adapted to any length of growing season. The model could be fitted for time periods other than one week in length. 174 Precipitation, temperature, and temperature multiplied by precipi- tation (interaction) were considered. Other weather variables (including a different form for the interaction term) could be employed. The weekly weather variables were considered in a linear form and in a polynomial (of degrees zero through seven) form in this study; other forms could be used. (l) (2) (3) (5) (6) (7) (10) (ll) (12) Vol. 41, No. 10, pp. 1515—16, 1913. BIBLIOGRAPHY Airy, John M., Totum, L. A., and Sorenson, J. W., Jr. "Producing Seed of Hybrid Corn and Grain Sorghum," Seeds, The Yearbook of A iculture, U. S. Department of Agriculture, 1961. Anderson, Elna, and Martin, J. "Werld Production and Consumption of Millet and Sorghum," Economic Botany, 3:265-88, July 1949. Alsberg, Carl T., and Griffing, E. P. "Forecasting Wheat Yields from the Weather," Wheat Studies of the Food Research Institute, Stanford University, Vol. 5, No. 1, 1928. Atkeson, F. W., and Fountaine, F. C. Storage and Utilization of Grain Sorghums in Dairy Cattle Feeding. Kansas Agricultural Experiment Station Circular 356, 1957. Auer, Ludwig. 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(183) "The Third National Fertilizer Practice Survey," The Fertilizer Review, Jan.-Feb.éMarch, 1946:7-10. (184) Epited States Census of Agriculture. 1939, 1944, 1949, 1954, and , 1959, U. S. Department of Commerce. r4— I” (II APPENDIX A THE DATA: PROBLEMS, SOURCES AND METHODS OF ESTIMATION The Dependent Variable Yield per Acre. This value was obtained by taking the ratio of total pounds of grain sorghum produced in a county by the total acres of sorghum harvested for grain or seed in that county. The data for the latter two variables were obtained from the U. 8. Census of Agriculture. The specific source is shown in Table A-1. Table A-l.--Sources of data on acres of grain sorghum harvested and pro- ‘duction of grain sorghum Year 5 Volume 3 Parts . County table number 1959---: I 20,21,36,37,41,42 ll 1954--° I 12,13,25,26,29,30 9 l949---: I 12,13,25,26,29,30 5 1944--: I 12,13,25,26,29,30 2/1 The Independent variables Acres of Grain Sorghum Harvested per Farm Harvesting Grain Sorghum. These values were obtained by taking the ratio of total number of acres of sorghum harvested for grain or seed in a county to the total number of farms harvesting sorghums for grain or seed in that county. The data for the latter two variables are found in the sources indicated in Table A-1. Acres of.Sorghum Harvested for Grain opiSeei. These values were used as published. The specific sources are listed in Table A-1. 188 189 Acres of Cropland Harvestei per Tractor. These values were obtained by taking the ratio of total acres of cropland harvested in a county to the total number of tractors in that county. The data for these variables were obtained from the U. S. Census of Agriculture. The specific source is given in Table A-2. Table A—2.--Sources of data on cropland harvested and tractor numbers County table number Year : Volume : Parts : Harvested : Tractor : : : croplang : numbers 1959--: I 20,21,36,37,4l,42 1 6 1954---' I 12,13,25,26,29,30 l 5 1949--: I 12,13,25,26,29,30 l 3 1944-_- I 12,13, 25, 26, 29, 30 1/1 1/2 Proportion o§_§orghum Acreage Irrigated. These values for the years 1959, 1955, 1949, and 1939 were obtained by taking the ratio of acres of grain sorghum irrigated to the total acres of grain sorghum harvested. The data for total acres of grain sorghum harvested were obtained from the sources listed in Table A-1. The data for acres irrigated were obtained from the sources listed in Table A—3. Table A—3.--Sources of data on acres of grain sorghum harvested that had been irrigated Year 3 Volume 3 Parts fCounty table number 1954“--: I 12,13,25,26,29,30 9-A 1949---: I 12,13,25,26,29,30 5-A l939---: I 2,5,6 15 190 No census data on irrigation by counties were available for 1944. However, State totals of irrigated land in farms were available. They were obtained from the 1949 U. S. Census of Agriculture, Vol. 1, Parts 12,13,25,26,29, and 30, State Table l, and are reported in Table A-4. Table A-4.--Data on acres irrigated by States, and values used to estimate acres of grain sorghum irrigated in 1944 5 1... farms 32:09:? 3.5.2.128: State ; ‘ , , ; 195% ; 195% ; 0 = B/A = 1949 g 1944 : 1939 : acreage , acreage : Nebraska---: 904,492 631,762 473,775 430,717 157,987 .36680 Kansas---: 145,334 96,248 82,872 62,462 13,376 .02142 0klahoma-—-: 34,857 2,237 4,437 30,420 -2,200 -.07232 Colorado--:2,902,ll8 2,698,519 2,467,548 434,570 230,971 .53149 New Mexico-: 663,195 534,640 436,402 226,793 98,238 .39789 Texas-----:3,l67,536 1,320,216 894,638 2,272,898 425,578 .18724 From the State total it was possible to determine (for the State) what proportion of the total change from 1939 to 1949 had occurred by 1944. The assumption was made that the change in acres of grain sorghum irrigated in each county had changed in the same proportion as the change in the total acres irrigated in the State. The estimated acres of grain sorghum irrigated in 1944 for a county was set equal to the 1939 acres of grain sorghum irrigated in that county plus C (selected from Table A—4) times the total change in acres of grain sorghum irrigated in that county between 1939 and 1949. Dollars Spent on Gas and Oiigper Acre. These values were obtained as the ratio of total dollars spent on gas and oil (in constant dollars) to acres of cropland harvested. The sources of data for cropland harvested is given in Table A—2. The data on dollars spent on gas and oil for the years 1959, 1954, 1949, 191 and 1939 were obtained from the U. S. Census of Agriculture. Specific sources are given in Table A-5. Table A-5.--Sources of data on dollars spent on gas and oil Year 5 Volume . Parts . County table number 1959---: I 20,21,36,37,41,42 7 l954---: I 12,13,25,26,29,30 6 As all census figures were in current dollars, it was necessary to deflate them. The values were deflated by the Index of Average Prices Paid by Farmers for Motor Supplies. These data were obtained from USDA Statistical Bulletin No. 319, 1962, and are listed in Table A-6. Table A-6.--Index of average prices paid by farmers for motor supplies for years used in study Index of average prices paid by farmers Year for motor supplies, 1910—14 = 100 1939 : 102 1944 : 115 1949 : 146 1954 : 162 1959 : 173 There are no data by counties or States for 1944. However, there are some U. S. values and they are presented in Table A-7. They are ob- ‘tained from the Farm Income Situation, July 1964, Table 53-H, page 53. Of the total change in dollars (deflated) spent on gas and oil txatween 1939 and 1949, 27.37 percent had occurred by 1944. By assuming truat the change in dollars (deflated) spent on gas and oil in each county ckuxnged in proportion to the change at the national level, it was possible .. Q0 00 - .2 Cu 0 A. 0 ad h x. [...-d VIN IMIH NIII‘ n.“ a» a~\ ‘- 1,1 . a ..a A: . up“ — r!“ h-w - u A.- ~...- 0 WI uI-\ «x; x ....O. .... ...: . r .. I. .. .0 e 192 to estimate values for each county for 1944. The 1944 estimated value for a county was set equal to the 1939 dollars (deflated) spent on gas and oil in that county plus .2737 times the change in dollars (deflated) spent on gas and oil between 1939 and 1949 in that county. Table A-7.--Data for U. S. on dollars spent on gas and oil, 1939, 1944, and 1949 Dollars spent on gas and oil Year 3 Current 3 Constanté/ 1939 : 323,000,000 316,700,000 1944 : 509,000,000 442,600,000 1949 : 1,134,000,000 776,700,000 g/ Deflated by the index given in Table A-6. Value of Land per Acre. The values for this variable were derived from the statistic, value of land and buildings per acre, which is re- ported in the U. S. Census of Agriculture. Specific sources are given in Table A—8. (Table A-8.--Sources of data for value of land and buildings per acre Year 3 Volume 3 Parts 3 County table number l959—--: I 20,21,36,37,4l,42 l l954--° I 12,13,25,26,29,30 1 194 ---: I 12,13,25,26,29,30 1 1944--: I 12,13,25,26,29,30 1/1 The value of land and buildings per acre is reported in current ciollars. To make the data more meaningful they were deflated by the Chansumer Price Index. These data were obtained from Business Statistics, 15961 Biennial Edition of the U. S. Department of Labor, Bureau of Labor Statistics, and reported in Table A—9. 193 Table A-9.--Consumer price index for years used in study Year f Consumer price index (1947-49 = 100) 1939 : 59.4 1944 : 75.2 1949 : 101.8 1954 : 114.8 1959 : 124.6 Estimates of the proportion of value of land and buildings that was buildings were obtained from photostats of USDA worksheets on farm real estate, selected statistics. These photostats were made available by William.H. Scofield, Agricultural Economist, Farm Production Economics Division, Economic Research Service, USDA. The data for the States and years included in the study are presented in Table A-lO. Table A-lO.--Percent buildings are of land and buildings, for years and States used in study Percent buildings are of land and buildingsa/ State : . , , , ; 1939 ; 1944 ; 1949 ; 1954 ; 1959 Oklahoma-—-: 17.1 (82.9) 17.2 (82.8) 13.5 (86.5) 14.1 (85.9) 8.0 (92.0) Texas---—-: 16.3 (83.7) 17.5 (82.5) 16.9 (83.1) 14.3 (85.7) 11.9 (88.1) Nebraska---: 22.4 (77.6) 17.8 (82.2) 16.4 (83.6) 17.5 (82.5) 12.5 (87.5) Kansas--—-: 18.4 (81.6) 15.3 (84.7) 15.7 (84.3) 13.3 (86.7) 14.6 (85.4) Colorado---: 21.8 (78.2) 20.1 (79.9) 19.5 (80.5) 18.3 (81.7) 15.6 (84.4) New Mexico-: 15.1 (84.9) 13.1 (86.9) 12.5 (87.5) 10.9 (89.1) 8.0 (92.0) g/ Numbers in parentheses are the percent land is of land and buildings. The values of land and buildings per acre for counties were adjusted by the appropriate State value to give value of land per acre. Man:hoursgper Acre. The values used for this variable are the sum of preharvest and harvest man work units used per acre of grain sorghum. The data were obtained from personal correspondence with Reuben w. Hecht, 194 Agricultural Economist, Farm Production Economics Division, Economic Research Service, USDA. The data were available only for selected farm production regions.l/ The data used in the study are presented in Table A—ll. Table A—ll.--Data on total man work units per acre of grain sorghum, farm production regions and years used in study Farm production regions Year 3 Nerthern Plains 3 Southern Plains 3 Mountain I Nebraska, Kansas 2 Oklahoma, Texas : Colorado, New Mexico 3 Man work units l939-—-: 12.2 11.6 12.0 1949-“_g 607 7.0 903 1954“--: 5.0 507 709 l959---: 3.4 5.2 6.7 Acres Cultivated Summer Fallow. The values used for this variable are as published in the Census of Agriculture for the years 1959, 1954, and 1949. For the years 1944 and 1939, the value used is derived from the variable, acres idle and fallow, appearing in the Census of Agricul- ture for those years. The specific sources are shown in Table A-12. Acres cultivated summer fallow was not obtained as a separate entity in the agricultural census of 1944 and 1939. However, an estimate of the total number of acres in summer fallow for 10 Great Plains States 'was available for these years. It was published by Sherman E. Johnson 111 USDA Bureau of Agricultural Economics Bulletin F.M. 58, revised June 1948 . gp/.A map delineating the farm production regions and showing the States ttuarein is shown on the inside front cover of USDA Statistical Bulletin No. 233, September 1959 revision. 195 Table A-12.-—Sources of data on acres fallow and acres idle and fallow Table number Year :Volume: Parts : Acres summer : Acres fallow r : : : fallow : and idle l959--—-: I 20,21,36,37,41,42 County Table l l954-—-: I 12,13,25,26,29,30 County Table l l944--: I 12,13,25,26,29,30 County Table 1 19 a --: I 11,12,13,25,26,28,29,30 State Table l a/ State totals only. All six of the States included in this study were included in the 10 Great Plains States used for these estimates. The total number of acres in idle and fallow for these same 10 States was obtained from the 1944 Census of Agriculture. The specific sources are shown in Table A-l2 for the year 1944, footnote _/. Using these data, the proportions that fallow acres were of total acres fallow and idle was obtained for each year. The results are given in Table A-lB. Table A—13.--Data on acres fallow and acres fallow and idle for 10 Great Plains States, 1939 and 1944 Item 3 1939 f 1944 Acres fallow : 17,400,000 10,800,000 Acres fallow and idle ------ : 29,237,205 16,602,644 Percent land fallow is of : land idle and fallow---—-- 59.51 65.05 The county estimates for these years were obtained by multiplying the acres fallow and idle in the county by the appropriate percent from Table A—13. It should be noted that the above data are not what is really desired. The most appropriate data would be proportion of total grain 196 sorghum acreage planted on fallowed land. Such data are not available. Given that total acres summer fallowed have to be used it would be desirable to have this data for the year previous to the one being studied as these are the acres that will influence yields in the current year. The data used will lead to meaningful results only if the follow- ing two assumptions hold: That the proportion of acres fallowed in the current year to acres fallowed in the previous year is nearly the same for all counties included in the study, and that the proportion of total acres fallowed used for grain sorghum is the same for all counties in- cluded in the study. Pounds of Fertilizer Nutrientsgper Acre of GraingSorghum. The values used for each county are as published for the aggregates which contain the county. For 1959 the aggregate is the State part of U. S. agricultural subregions. The data for this were obtained from USDA Statistical Bulletin 348, "Commercial Fertilizer Used on Crops and Pasture in the United States, 1959 Estimates." These estimates are based on 1959 Census of Agriculture data. The data used in this study from this source are presented in Table A-l4. Data for 1954 were available only at the State level. The data are obtained from USDA Statistical Bulletin 216, "Fertilizer Used on Crops and.Pasture in the United States, 1954 Estimates." The data from this source used in this study are presented in Table A—15. Data for 1949 was not available. However, estimates for 1950 were arvailable in USDA Agricultural Handbook No. 68, "Fertilizer Use and Crop 'Yields in the U. S., 1950 Estimates." The 1950 estimates are used as good approximations of 1949 values. They are believed to be close estimates 19 7 Table A—l4.-Ferti1izer data for 1959, for economic subregions and State parts used in study Economic: : Acres I Tons of fertilizer applied : Average sub- : State I harvested 2 to grain sorghum : pounds of region 2 part : of grain I , , , I nutrients I I sorghum. I N 2 P205 2 K20 I Total I per acre 68 ----- :Kansas 425,117 898 1,429 340 2,667 12.55 69----:Kansas 193,699 670 879 130 1,679 17.34 70 ----- :Kansas 203,700 1,362 1,008 114 2,484 24.39 :Nebraska 266,091 940 260 19 1,219 9.16 76-----:Kansas 418,419 757 281 0 1,038 4.96 :Nebraska 1,143,593 6,679 859 77 7,615 13.32 77------:Kansas 876,414 1,655 1,336 0 2,991 6.83 80------:Texas 787,588 1,294 1,147 229 2,670 6.78 81 ------ :Texas 1,246,995 1,698 1,291 129 3,121 5.01 82------:Texas 177,200 335 0 0 335 3.78 83----:0klahoma 319,167 180 617 44 841 5.27 :Texas 1,031,770 269 228 23 520 1.01 84----:Texas 2,135,831 6,893 893 0 7,786 7.29 85--—--:Kansas 2,678,212 6,770 1,012 0 7,782 5.81 :New Mexico 233,795 1,035 3 0 1,038 8.88 :Texas 1,684,691 33,720 1,605 0 35,325 41.94 :Oklahoma 438,777 278 88 10 376 1.71 :Colorado 576,921 606 98 0 704 2.44 Table A-15.--Fertilizer data for 1954 for States used in study 3 f Fertilizer in tons :Pounds of State ; hAcreSt d; . . . :nutrients ; arves e 1 N ; P205 ; K20 ; Total ; per acre jNebraska----: 540,000 841 97 15 953 _ 3.53 Kansas-——--: 3,567,000 2,381 3,004 138 5,523 3.10 0k1ahoma—--: 614,000 577 866 247 1,690 5.50 Colorado----: 396,000 --— —- —-- —-- a/ New Mexico--: 281 , 000 27 8 945 5 528 3 . 76 fTexas-—-----: 5,782,000 15,472 7,167 1,725 24,364 8.43 2/ Used Kansas average. Imecause (1) the 1950 values are quite low, and (2) the values are for a ,faJnn production region and changes in averages for a whole region would be very small. The data used for this study are presented in Table A—16. 198 Table A-l6.--Fertilizer data for 1949 by farm production regions :Acreage off Pounds of plant nutrients sorghum . ,per acre harvested N P205 ' K20 States included Farm production region Total Southern States--—--- Oklahoma} 8,185,000 0.5 0.6 0.3 1.4 Texas Nebraska} 3,190,000 .5 .6 .2 1.3 Kansas North Central States-- New Mexico and Colorado belong to the Mountain States. Estimates for the Mountain States were dominated by sorghum production in California and Arizona where most of the crop is irrigated and heavily fertilized. Thus these estimates were rejected as being unreasonably high for New Mexico and Colorado. The values used for New Mexico were set equal to those of Texas, whose production practices are similar and the values used for Colorado were set equal to those for Kansas, whose production practices are similar. The total pounds of nutrients applied per acre of grain sorghum for all counties for 1944.and 1939 was estimated to be zero. Not a single source of data about fertilizer applied to sorghum prior to the 1950 esti- mates was found. The value zero was used because it was believed to be a very close approximation to the true value. This belief is substantiated for 1939 by the fact that of the 129 counties included in the study, 82 used no fertilizer on any crops in 1939. It is also substantiated by the fact that for dryland farming the recommended fertilizer practice was to use no fertilizer. The assumption that no fertilizer was applied to grain sorghum in 1939 was extended to 1944 because: (1) Nitrogen which was the principal 199 nutrient applied in latter years was scarce during the war years; (2) the low values for 1950 indicate that fertilizer use, which has increased over time,must have been very near zero for 1944; and (3) the fact that no fertilizer data were obtained for this crop while they were obtained for other crOps;/ indicates that fertilizing sorghum was not an important practice. l/ "The Third National Fertilizer Practice Survey," The Fertilizer Review, National Fertilizer Association, Inc., Jan., Feb., and March 1946: 7-100 ) U N1— iv. —.~,. 1. e. . ~ s ...u \..... o r 200 wasnflmeOI 6mm Hos NeN mcH «H an N m nausea muuuuuuuuuescHo on cos HeN NHH mH no N NH menace WunuuuuuuacHeam mNm mom oeN HHH NH mm N m nausea muuuuuuuuccpacm eNm mam ch 62H HH on H m cancnccz Wuuuuuuuacrncez cNm can ch amH 6H 66 H N cancaccz muuuuuuuuacacce mNm cam ecN mmH a on H N cancaccz muuuunuunHHcasz «Nm mam ch emH m on H H cancaccz muuuuuaeenceccH NNm cam ch cmH e 68 H H cancaccz WuuunuuccpHHacm NNm mom ch mMH c on H N cancaccz m ewes HNm Nam ch cmH m 68 H m caucaccz Wuuuunnuuaccasm ONm Hem NcN mmH c on H m cancaccz muuuuuncHHHacaa on com HcN NmH m 68 H N cancacez muuuunucacaHHHa mHm own ocN HmH N on H N cadencez m acHo eHm mmm emN omH H on H m cancaccz m mecca as m is m as m as “inflame? 98.88... m e 4%....“wa 8.8 888 \manSSG soapw>homno H owsosoom M mcazohw H .WOHU H H 50x gonads mpnsoo was anonsfia noapmbaomno a.3560 gowo op wopwaoh spec Spas human map Ga popsaocfi mofipndoo mo pmfigll.bal< manwe ~.._.~:~s,\-\..~7 .o\\.~.¥—. ...\_\~...\_ 201 ecacHeceo- cam ch cmN emH wN on N m nausea ”------IIHHcsce mam HHH mwN cmH 8N an N HH menace muuuuuuuuaceacm Nam MHc cmN mmH 6N an N HH armada Wuuuuuuuuaeaacm Ham NHc mmN cmH mN mm N CH menace muuuuuuaeeHHanm 6cm HHH NmN mmH 8N mm N e nausea mauuunnuachcau mmm OHc HmN NmH MN mm N OH nausea muuuuuuuuuuacac NNm doc omN HmH NN mm N 6H nausea wnuuunuunueccae emm mos meN OmH HN mm N a menace muuuuuuuuacacao 6mm soc weN 62H 6N mm N a menace m 0866 mmm 66c eeN meH NH mm N 6H menace muuuuuunuuucaea emu mac ceN eeH NH an N OH menace wuuuuuuuuacccHa mmm soc meN 60H eH mm N m nuance muuuuunuuunHHHm Nmm mos ceN mcH 6H mm N HH nuance muuuuuusmenesam Hmm Noe meN 82H mH we N NH tandem muuuunuulacHseo momma .30..” 0..me .Nmma ”Hangmnmmtmwgoow \mdoamohoBm o, omwom mwgnpmflc m oasocoomn m wcHBOHc m mwmmwmn m opwpm m hpqsoo \onQSBn coapw>Hmmpo eceaHeaeo--.eH-a eHcce 202 UossfipcOOI wmm 0N9 oom ASH Nu mm N HH mmqum mulllllllooc3sm bmm qu omN 05H Hm mm N m mwmqu WIIIIIIIonhonmo 0mm qu NON 00H 0% mo N m mwmsmx WIIIIIIIIIommmo mmm oNq me woa mm mm N b mwmssm m mmoz Qmm me pmN boa mm on N o nausea WIIIIIIIImnmsmz mmm «Nd mmN 00H hm mm N CH momsmm millsltllnophoz Nmm MNH «ON m®H om mm N m mwmcsm WILIIIIHHonosz Hmm NNQ mmN «ca mm 05 N o mmmqwm WIIIIIIHHwanmz 0mm HNq NmN moH QM bu N m mamawm WIIIIIIIIQONHNE ®homno eeecHeceo.-.eH-a eHcee 3.1.5.: . ...lu'o , ~ \.\.....~ \\.\l—\ .-..\.\.V. 203 pondenoon me MQH «Hm me om mm N CH mamqmm WIIIIIIIaopqum Hbm qu MHm qu mm mm N HH mwmcwm WIIIIIIUhommwpm Ohm qu NHm me «m 05 N m mwmnwm willllllllszsw mom OQH HHm NmH mm mm N Q mmmnmm WIIIIIIIqwaHonm mom mmq OHM HmH Nm mm N w mmquM WIIIIIIquHHosm bow wmq mom owH Hm mm N OH mwmnwm WIIIIIIIIpHm3om com bmq mom 05H Om vb N HH mwmcwm WIII|IMOHzompom mom omq bom wbH 0% mm N b mmmcwm WIIIIIIIIIppoom «on mmq mom 55H wq mm N w mwmqwm m nmdm mom Homno ecauaececuu.eHua chce 204 woanpcOOI owm bmq wNm 00H Ob mm H 0H OUNHOHOU WIIIIGomaoo pHm mmm omq bNm on mo mm H 0H ovaOHoo WIIIIIIIIINBOHM «mm mmd ©Nm bmH mm mm H SH opwHOHoo m momm mwm qu mNm ©®H >0 mm N «H maondeo WIIIIIIImpHSmm3 Nwm mmq «Nm mmH we mm N mH Naongxo WIIIIIIIIImmxoB Hmm va mNm de mo mm N mH msonsto WIIIIIcOHHnwsHo own qu NNm mmH <0 mm N mH maoanMo WIIIIIIIIIoppmo mum om¢ HNm NoH mo mm N mH maosdeo WIIIIIIIIHobwom mum mud ONm HoH No mm N b mmmnmm WIIIIIIImpHJOHz bum wQQ mHm owH Ho on N m mmmmwx WIIIIqopmcHnmmB ohm qu mHm owH 00 mm N b msmssm mlullllllnomope mam oqq me me mm mm N v mmmcmm willullllmmsone «hm mqq on bmH mm be N HH mwmssx WIIIInIIIHosasm Mbm qfifi mHm omH hm mm N CH mmmqwm WIIIIIIImco>opm ommH 30H «6qu lwmmvH “HmnaHmHmmMpgooM \chmennHa n ommmm ms 9.36.ch m oHsocoomfl m mcHsoHc m smwmmwmn m opspm m hassoo \mnmnssn qOpr>Hompo noqupaooun.>Hn< oHnsa 205 UossHpsOOI OON HNH NHN NHN «N «N H ON ncxce muuuuuuuuacnoao NNN Oec HHN NHN NN NN H HN waxes mureacszcHHHeO NNN NNN OHN HHN NN «N H ON naxce munuuuuucaaaecO NNN NNH NNN OHN HN mN N NH maxee muuuuuuuueaenaO NNN NNN Nmm NON ON mN N NH maxme muunuuunuconaNO NNN NNN Nmm NON Ne mN N NH mcxca muuuuuuucchHaN NNN mNN NNN NON Ne HN m NN nnxce munuuuuuuuncxcm NNN «NH mmm NON NN ON c «N waxes m HHmm NNN NNN «mm mON Ne HN m NN aexce m com HNN NNN NNN NON Ne «N H ON ntxoa muuuuuuunacHHNN ONN HNN NNN NON Ne mN N NH eethz sez muuuuueHcacnecN NNN ONc HNN NON Ne NN N NH ecHacz_3ez muuuutunuuaaasO NNN ch ONN HON Ne NN H NH ccaaOHcO m NasH NNN ch NNN OON He mN H NH ceaacHeO muuuuuuuunacscm memmH 30H @me .vmmOH ”Honusnmmegoo” \mOHOHmoHnHHm o, omwom N. 84.99me u oHaoqoom ” mcHBOHw " wsznomoh « opwpm u handoo « u u mono « u \mamnasn GOHpN>nomno eceaasan--.eH-H cheH \ O- .Nsln 206 code.“ .50 on NHN mNN NNN NNN NN ON N «N naxce m HHHN NHN NNN NNN NNN NN NN N NN ncxce muuuuuuucNHNNHm NHN NNN NNN NNN NN MN m NN maxce mnuuuuuuHHaxaNN HHN NNN NNN NNN mN NN N NH ancH WuuuuuuuacHeacN OHN HNN NNN NNN NN NN H HN mnxce m HHNN NON ONN HNN NNN NN NN N NH maxme m cHnm NON NNN ONN HNN NN ON N NN nmaca wnuuuucaaHNNNaO NON NNN NNN ONN HN NN H HN mmxce muuuuuuuuucaaao NON NNN NNN NHN ON NN H ON nnxce muuuunuuunmcHNN NON NNN NNN NHN NN NN N NH meme muunIIIIIINNOHm NON NNN NNN NHN NN ON N NN mmxee wuuuuuuuuuaHHHm NON NNN NNN NHN NN NN N NH maxme WII--NNHaN NNNO NON NNN .NNN NHN NN NN H ON atha muuuuuuuucensz HON NNN NNN NHN NN NN N NH nwxme WuuuuauuuaaHHNO ommH 30H 01‘9” «moH “HmQaHWQmmqugooH \HOdOHmoHQSm . Q ommom \NPOHHPNHU m OHEocoom m mcHsoHu m quWHMWoH m opwpm m hpnsoo \mnonasq GOHpNPHomno NcscHrchuu.NHu< canH \-s‘-N‘I\-\~n\Vul-III. .\\.\l—\\ .5‘.‘~n.~\. 207 voanPQOOI wNO OOQ Ohm HHN NHH mm N OH waxes m onooz NNN woq mom OdN HHH om q «N waxes WIIIIIIIIIENHHE ONO NOH mom OMN OHH Hm m mN mwxoe WIIIIIIIINanoz mNO OOQ pom wMN OOH Hm H ON mmxoa WIIIIIIIIGHpHmz «No mow 00m NMN wOH ow q QN mmxoa willlllaoqaoaoz MNO wow mOm OMN NOH «m H ON mwxoa m mama NNO mow qom mMN OOH Hm H ON mwxoa WIIIIIIIMoonQdH HNO Nov mom QMN NOH Hm O NN waxes WIIIIIIMNO o>HH ONO qu Nom MMN «OH «w H ON mwxoe m mafia OHO OOH Hem NMN mOH mm m NN waxes m Noam wHO mmq oom HMN NOH mm m NN mmxoe m ammoh NHO wmv Omm OMN HOH Hm O bN mwxoa WIIIIINHHoz EH5 OHO qu wmm ONN OOH «w H ON waxes willlllllvhwzom mHO owq bmm wNN mm «m H ON mmxoe WIIIIIIIhmeoom OMOH 30H NGOH «NOH “Honfismammwmpgoom \HIOHHOHmoHoBm a, omwmm \WWOHHOPWMWNHHN mpmpm .3560 ofiaonoom . manonc .QOHO \mnmpss: qOHpN>HmmpO OmschqOOII.NHu< NHQNB 208 OoanPGOOI HNN NHN NNN NNN NNH ON N NN amxca munuuuuuumHNNaN ONN HHN NNN NNN NNH HN N NN memN muuuuuccnaO aeN NNN OHN HNN NNN NNH NN H ON mecN munuuuuunuNaatN NNN NON ONN HNN NNH NN N NN antN muununnuuchNNN NNN NON NNN ONN HNH NN N NH nmxte WuuuuuuuacNNHsm NNN NON NNN NNN ONH NN N NN ancN muuuuuunnuNnaaN NNN NON NNN NNN NHH HN N NN meoN wuuchHNme sNN NNN NON NNN NNN NHH NN N NN antN wuuuuuuunHmacsN NNN NON NNN NNN NHH HN N NN anxce muuuuunucHNaNmN NNN NON NNN NNN NHH NN N NH aNxtN muuuuuuuHHNNdNN HNN NON NNN NNN NHH NN N NH mecN muuuuuunuhmaawm ONN HON NNN NNN NHH HN N NN NNNNN mnnuuuunumcetsz NNN OON HNN NNN NHH NN N NN mecN muuuuuuunucNch NNNH m NNNH m NNNH m NNNH m NNNHN m m m ”0288 n u . . . .nmpsss handoom wmmmmmmmwm m mwhmwwww m mWHpnomon H opwpm H Npqsoo . . . . mono . . \mnopssn QOHpN>pomnO NtscHNan--.NH-N NHNNN .5 hfiu c\\.\ 209 .mnonesu hundoo omHN ohm OmOH Mom whenssa GoprbpomQO \m .NNN .62 cHemHHsm HNeHNmHNNNN NONO =.nmpNaHpnm NNNH .NNHNHN NepHdO tap cH onspmmm Odd mmopo so Ummm HoNHHHpAmm HmHohosso0= mo N owmm so quom mp ado mGOHonQSN oHaoqoom mGHmeqHHmu mm: \m .NHuN NHNNN NNN \m .NHnN NHNNN NNN \N NNN NHN NNN NNN NNH NN H ON mmme muuuunuuaaNNcH NNN NHN NNN NNN NNH ON N NN mNacN mnuucemaNHHHHz NNN NHN NNN NNN NNH NN N NN mecN muuuuuuNaNHHHg NNN NHN NNN NNN NNH NN H HN magma muuuuuuachcgz OMOH M NNOH M ONOH H NmmH Mnons%mwwwqdoow GOHmoHQSm H omwom H \MpOHHPmHMU M H ”.NWHacaceN memMM:ONO m NcmwmmwNa m tpmpm m NpcacO \mnonsfiz QOHNN>HomnO NmssHHaoouu.NHn¢ NHQNB 210 The Dummy;yariables. Three sets of dummy variables were considered in this study. The data on these variables is presented by counties in Table A-17. The last part of Table A—l7 contains the county numbers and observation numbers which are keyed to other tables to be presented later The year variables need no explaining. Table A-18.—-Key to crop reporting districts used in study Crop reporting district p reporting district, number used in study 3 State 3 State number or name 1 : Nebraska East 2 : Nebraska South East 3 : Nebraska South 4 : Kansas North west 5 : Kansas North Central 6 : Kansas North East 7 : Kansas west Central 8 : Kansas Central 9 : Kansas East Central 10 : Kansas South west 11 : Kansas South Central 12 : Kansas South East 13 : Oklahoma No. 1 l4 : Oklahoma No. 4 l5 : Oklahoma No. 7 l6 : Colorado District No. 6 l7 : Colorado District No. 9 18 : New Mexico North East 9 : Texas l-N 20 : Texas l-S 21 : Texas 2-N 22 : Texas 2-S 23 : Texas 7 24 : Texas 4 25 : Texas 8-N 26 : Texas 8-8 27 : Texas lO-N 28 : Texas lO-S Since the counties included in the study came from different Skates and different climatic regions, it was decided that a single ggrowing season representing all counties was not appropriate. Data as 211 to the average planting date are not available on a county basis. Data as to the average planting date are available for crop reporting districts in Texas. There were seven different growing seasons considered in the study but because two seasons were affected by leap year (1944) two extra growing seasons were included. The growing seasons considered are shown in Table A-19 and are keyed to the numbers given in Table A—17. Table A-l9.-Key for growing seasons Growing season number 3 Begins f Ends 1 : May 11 October 18 2 : May 20 October 27 3 : April 19 September 26 4 : March 25 September 1 5 : March 11 August 18 6 a/ : February 27 August 6 7 b/ : February 17 July 27 a/ For 1944 February 28 to August 6. b/ For 1944 February 18 to July 27. Temperature. A weather station reporting daily maximum temperatures was selected from each county, if there was at least one in the county. If there were no weather stations in the county reporting maximum tempera- tures, then the nearest weather station that did report temperatures was used. A list of the weather stations used to obtain the weather data for each county is presented in Table A—20. If maximum temperature was not reported for (1) one day or for two consecutive days, the missing values were estimated by simple interpola- tion, and (2) three or more consecutive days, the missing values were 212 estimated by using the actual reported maximum temperatures of the nearest reporting station. Precipitation. If there were four or more weather stations in a county reporting precipitation, then three were selected for use in this study. If there were at least one, but less than four reporting weather stations in a county, then all were used. If there were no reporting weather stations in a county, up to three nearby stations were used. A list of the weather stations used is presented in Table A-20. If there were any days for which precipitation was not reported, the missing value(s) was (were) approximated by using precipitation occurring on that day (those days) at the nearest weather station. 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M MOOOOOOM . popowm .mpmpm.» k32.900 . . .9869 chprpm nonpwms uHmNPNmB. . . . . meNNNwCOQII .ONI.‘ QHDQB 230 Omsch:OO| coach m NONNHO mmwm mmmqmn< mmmm mmmzwh< copnwm mopaHm m NOONOOO OOOOOO NOONOOO OOOOOO NOONNOO OOOOOO ONHNOOON ONNN>OON a OONOO ONH ONONOOON OON mpoquz mnmchz mumchB m NONONNNON OONONNHON NONONNHON OONOOHNON OONONHNON N OONONHHON OONONHNON OONONHHON OOONNON OOOHNOO a OONOB NHH OHOOOON ononmwooz ononmoooz m 0demmm camsmmm OHmsmmm ononmvooz ononmcooz m onopmwooz .Q.3 HHm3pmd¢ HHmzpms< HHmzpms¢ HHmzpms¢ m .H.3 HHm3¢m5< .Q.3 HHmzpmd< HpmHHNO mumpoo mOHHHOrmmm mHHH>mmm a mwxma NHH 0Hmdmmm hthwnab Nownmnab Oannmsm m qohcwo somcmo cohqwo achcwo qohswo m OHHHpma< OHHHhmad H whompmm 3mfl>sHmHm 3mH>aHNHm e mmxma OHH Hwaqwm H upomuom .m mmnob ppwm m puma muoHNm H Opomnmm muoflhm “:0Hpm m monmHsz mogmmsz H unomnmm 3mH>anHm 3mH>anHm a mmxms mHH nmanwm azopmnom nsopmnom nzopmnom m .N.N NoOONN OOOONN OOOONN OOOONN NOOONN N Hpmfihzo msmnoo HpmHHflO msmhoo fipmHNfiO msahoo Hpmflnno mSQNoO HpmHNflO manhoo m HpmHHSO msmnoo HpmHnno magnoo Npmfinno magnoo Hpmego mumpoo HpmefiO msmpoo a mmxme QHH mmomdz comma comma comma m , moomom moomom moomom moomom moomom m moomom moomom moomom mcmHHn« mcmHHn¢ a mmxme mHH :NHoz Omwa u «NOH " NOON u NOON " ONON “ \m " “OOOOOOO "hopQfiwumpmpmuhpndo . hpQSOO chHpmpm poprms .Nm3pwm3. . O” OOOONOOOO--.ON-N ONOON 231 OmdquQOOI OHHNNOONOHON OHHOONONONON OHHNOOONOHON N wowsocwz Newsccwz wownoawz m anms< qfipms¢ qums< anmd< cfipm5< m qfipms< aHpms< :Hpms< aHpNSN chmd¢ B mmxma mH>NNB NONHON OOOON NONHON OOOON N Hw>opmHH£O Hm>opmflhno Hw>opmfiuzo m OHmmq< cam OHmmq< cam OHmmq< 2mm hmeflHme OHmmq< 3mm m onwq< mam onmq¢ cam onmq< mam manHn< OHmmq¢ saw 9 mwxma nmmNO Boa chHmazonm OHmwmsaonm OHmHmczopm OHmHmnsopm meflmnaonm m OGNHHm>mQ OcmHHm>oH OQNHHm>mA mHocHamm mHonHamm a mwxma hhpme pawns pamhe m nzwq nzmq m mquHON oqufip< mquHQ< mquHQ< manHp¢ m mcmHHn¢ manHn¢ mamHHn< muoHHQN mamHHn< B mmxme NOHNNB OOHSB dHHS. m ONNOB OOHOB OOOON NOOON ONHOO N NHHSB NHHSB 3mfi>quHm zmfibanHm 3mH>qflme a mwNme HNNNHBO pmchqm pmvmmm pmvhnm Hochcm hmwmqm m nmwhqm hocmnm pmuhqm ocmHHn< mamHHp¢ a mwxme ONH hhnsm ONOH m «NON m OOOH m NOON m ONON m \m m MOOOOOOM « .HoONomm umvamuhpgoo. .3880 OOONOOON OOOOOON .NOOOOON. . ” OOOONpNoonu.omuN NHQNB 232 .HHNMGHNN n m NmNSPwNmQSmP n a .mhopowm hmnpwmz \m OHmHmcson OHmHmnzonm m mafime mGHme mnHme mHocflamm mHquamm m mafime mqame mHonHamm mHonfiamm mHosfiamm a mwxme ONH adxwow NNNN NOOOONN NNNN NONOONN NNNN NOOOONN N HHmGwh HHmcwh Hquwh m NOthe NlowB NOthB NOHONB NlomB m NOHmwB NOHONB NOHhmB NOHhmB NOHANH a mwxme ONH sowewHHHHz OHmHmanz phom m ONNNOOOOaNON ONNNOOOOaNON ONNN>OOOaNON ONNNOOOOONON ONNNOOOOaNON N mHHN>anamwm mHHH>Onoazwm mHHN>Oqoshwm mammopmz mmwmopmz a mmxma ONH howHHflz Mochadnm Mochawnm m mfipmonoz mfipmmnoz m Hawfiz flawflz Mochawnm Moonawgm Noohfimnw m stHz mHnQamz mangamz covsmeHO cocconHO a mmxma ONH NmHmmnz OOON ” OOON u OOON u OOON " OOON " \m u “OOOOOOO .OOOQNN.OOOON.N . NOOOOO . . .pSS? macapwpm Nmapmmz .Nmapwmz. . . OOOONNOOQII.omuN OHQOO 233 The Basic Data Tables containing the following data by year and by county have been prepared and are available from the author. Data included are: Acres of grain sorghum harvested for grain or seed; production of grain sorghum in hundredweight; yield per acre; number of farms harvesting grain sorghum; number of tractors; acres of grain sorghum irrigated; acres of cropland harvested; value of land per acre; acres fallowed; total expense for gas, oil and lubricants; acres of grain sorghum harvested per farm harvesting sorghum.for grain or seed; acres of cropland harvested per tractor; percent of grain sorghum acreage irrigated; and dollars spent on gas, oil and lubricants per acre. APPENDIX B The results of the regression analysis are presented in the following table. The equations are presented in numerical order. The variables in the equation are also presented in numerical order. However, if some variables are not considered in any equations presented on a particular page, they were dropped from the list. The smallest degree of freedom available for the test of sig- nificance (t test) of the estimated coefficient was 536. Thus, the values listed for infinite degrees of freedom in the table of percentiles of the t distribution, as presented in Table A—5 (page 384) of "Introduc- tion to Statistical Analysis," 2d ed., by Dixon and Massey, were used. In order to conserve space, the following symbols (appearing as superscripts to the coefficients in the table) were used to indicate the levels of significance of the estimated coefficients. In all cases the hypothesis tested is: HO : bi = 0 HA : bi # O and the level at which the null hypothesis was rejected is indicated by: aztb32.57O=Og .01 b = 1.960 5 tb _<_ 2.575 = .01 < a g .05 O = 1.645 5 tb g 1.959 = .05 < a 5 .10 nflwre a is the probability of rejecting the null hypothesis when it is true. 234 235 Also to conserve space, the variables are identified by numbers rather than by names. The numbers are keyed to the list of variables as follows. It should be noted that the coefficients for variables 84-110 have a different meaning in some equations than in others. The reason for this is that crop reporting district 9 is omitted in some equations and crop reporting district 19 is omitted in others. When district 9 was omitted, the space following variable number 92 (see list below) would be blank and the space following number lOl-A would be occupied. If district 19 was omitted, the space following variable number 92 would be occupied and that following lOl—A would be blank. The coefficients for crop reporting districts from equations with district 9 omitted can be com- pared to those with district 19 omitted by subtracting a constant equal to the coefficient for district 19 from the overall constant term and from the coefficients for all other districts (including district 9 which has a value of zero) in the equation in which district 9 had been omitted. This was not done here because statements of significance cannot be made after the coefficients are transformed. Variable No. Variable O Constant term 1 Pounds of grain sorghum per acre of grain sorghum harvested 2 Percent of grain sorghum acreage irrigated 3 Average acres of grain sorghum harvested per farm harvesting grain sorghum 4 Number of acres of cropland harvested per tractor 4—A Number of tractors per acre of cropland harvested 5 Dollars spent on gas and oil per acre of cropland harvested 236 variable No. Variable 6 Per acre value of land 7 Man-hours of labor per acre of grain sorghum 8 Acres cultivated summer fallow 8-A Ratio acres cultivated summer fallow to acres of cropland harvested 9 Pounds of fertilizer nutrients applied per acre of grain sorghum 10 1959 11 1954 12 1949 13 1944 14 Preseason precipitation 15 Precipitation, lst period week 16 Precipitation, 2d period week 17 Precipitation, 3d period week 18 Precipitation, 4th period week 19 Precipitation, 5th period week 20 Precipitation, 6th period week 21 Precipitation, 7th period week 22 Precipitation, 8th period week 23 Precipitation, 9th period week 24 Precipitation, 10th period week 25 Precipitation, llth period week 26 Precipitation, 12th period week 27 Precipitation,13th period week 28 Precipitation, 14th period week 29 Precipitation, 15th period week Variable No. 30 31 32 33 34 35 36 37 38 39 4O 41 42 43 45 46 47 48 49 50 51 52 53 54 Precipitation 16th period week Precipitation, Precipitation, Precipitation, Precipitation, Precipitation, Precipitation, Precipitation, Total Total Total Total Total Total Total Total Total Total Total Total Total Total Total Total Total 237 Variable 22d 23d temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, temperature, period week period week 17th period week 18th period week 19th period week 20th period week 2lst period week lst period week 2d period week 3d period week 4th period week 5th period week 6th period week 7th period week 8th period week 9th period week 10th period 11th period 12th period 13th period 14th period 15th period 16th period 17th period week week week week week week week week 238 Variable No. Variable 55 Total temperature, 18th period week 56 Total temperature, 19th period week 57 Total temperature, 20th period week 58 Total temperature, 2lst period week 59 Total temperature, 22d period week 60 Total temperature, 23d period week 61 Interaction, lst period week 62 Interaction, 2d period week 63 Interaction, 3d period week 64 Interaction, 4th period week 65 Interaction, 5th period week 66 Interaction, 6th period week 67 Interaction, 7th period week 68 Interaction, 8th period week 69 Interaction, 9th period week 70 Interaction, 10th period week 71 Interaction, 11th period week 72 Interaction, 12th period week 73 Interaction, 13th period week 74 Interaction, 14th period week 75 Interaction, 15th period week 76 Interaction, 16th period week 77 Interaction, 17th period week 78 Interaction, 18th period week 79 Interaction, 19th period week 80 Interaction, 20th period week 239 variable No. Variable 81 Interaction, let period week 82 Interaction, 22d period week 83 Interaction, 23d period week 84 Crop reporting district 1 85 Crop reporting district 2 86 Crop reporting district 3 87 Crop reporting district 4 88 Crop reporting district 5 89 Crop reporting district 6 9O Crop reporting district 7 91 Crop reporting district 8 92 CrOp reporting district 9 93 Crop reporting district 10 94 Crop reporting district 11 95 Crop reporting district 12 96 Crop reporting district 13 97 Crop reporting district 14 98 Crop reporting district 15 99 Crop reporting district 16 100 Crop reporting district 17 101 Crop reporting district 18 101-A Crop reporting district 19 102 Crop reporting district 20 103 Crop reporting district 21 104 Crop reporting district 22 Variable No. 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 240 Variable Crop reporting district 23 Crop reporting Crop reporting Crop reporting Crop reporting Crop reporting Growing Growing Growing Growing Growing Growing season 1 season 3 season 4 season 5 season 6 season 7 district 24 district 25 district 26 district 27 district 28 0th degree term, precipitation polynomial 1st degree term, precipitation polynomial 2d degree term, precipitation polynomial 3d degree term, precipitation polynomial 4th degree term, 5th degree term, 6th degree term, 7th degree term, 0th degree term, 1st degree term, precipitation polynomial precipitation polynomial precipitation polynomial precipitation polynomial temperature polynomial temperature polynomial 2d degree term, temperature polynomial 3d degree term, temperature polynomial 4th degree term, temperature polynomial 5th degree term, temperature polynomial M112. 131 132 133 134 135 136 137 138 139 140 241 Variable 6th degree term, temperature polynomial 7th degree term, temperature polynomial 0th degree term, interaction polynomial lst degree term, interaction polynomial 2d degree term, interaction polynomial 3d degree term, interaction polynomial 4th degree term, interaction polynomial 5th degree term, interaction polynomial 6th degree term, interaction polynomial 7th degree term, interaction polynomial Table B, Part 1 242 Variable; Equation number number i 1 2 3 : 4 5 6 7 : 8 §§----: .783 .782 .781 .768 .767 .764 .626 .531 R ----: .759 .758 .758 .745 .743 .740 .590 .489 D.F.---: 580 581 581 584 585 585 588 592 0-—--—.-: 29358 2234b 3101a 38338 4005a 3555a 5085a 7778a 2------: 1855a 1862a 18149 18723 1832a 1843a 3-----: -.368 -.340 -.216 .019 .151 .276 4----: .3410 .357b .3420 .176 5-----: -20.1 -20.7 -21.8 —11.4 -13.3 -.54 6-----: 3.0a 3.1a 3.1a 3.3a 3.3a 3.9a 7------ -39.6 - .0 -27.89 -23.3a 8-----° .0002 .0001 .0003 .0002 .0002 .0004 9---_—-» 13.3a 13.3a 13.3a 13.5a 13.4a 15.7a 10-—--- 297 5619 252 981a 11—---: 91.9 327a 47.6 631a 12-----: 242 432a 222 609a 13-----: 400a 383a 403 38981 14-----: .2118 18.8a .204a .291a .282a 26.9a 13.1b 17.6a 84------: 479a 502a 487a 440b 449a 4309 18.1 -258 85-----: 559a 579a 577a 519a 537a 509a 82.9 —2620 86----: 128 145 138 154 166 152a .426a -696a 87------: —67 -58.8 -49.8 -82.4 -64.4 -73.1 -763a -9763 88------: 4.5 18.0 13.1 -42.5 —34.8 —11.2 -668a -886a 89-----: 13 169 145 27.9 33.8 80.4 -468a -599a 90-----: -32.6 -34.2 -15.4 -51.2 -32.8 -56.4 -647a -852a 91 ----- : -81.7 -80.5 -78.1 -105 -100 -112 —555a -7279 92--—--: 119 57.1 -447b -6279 93-----: —90.1 -98.3 -74.1 -104 -83.0 -132 .6409 -773a 94---: -234c -2319 —225C -278b -267b -287a -674a -864a 95-----: —104 -87.3 -101 -176 -169 -168 -623a —772a 96----: -180 -184° -153 -134 -103 -127 -8013 -863% 97----: -497 -45.4 -45.1 17.0 24.8 41.3 -552a -457 98-----: -295 -290° -282 -245 -228 -220 -763a -713a 99---—-: -204 -2490 -184 —125 -109 —197 —9659 -1219a 100--—-: —123 -164 -109 —12.0 -.894 -84.9 .6229 -8049 101-----: 20.2 -50.3 33.1 52.7 56.1 11.8 —64oa —6269 102-——-: 23.3 -49.5 47.5 31.8 58.6 41.0 -395a -3648 103----: -35.4 -99.1 -29.3 96.1 105 24.0 -615a -691a 104---: -90.2 42.1 -83.8 49.7 57.7 250c -4189 -306 104-A--: 25.6 26.6 248 258 -Continued "f ."'.' I”? I 524,1 I [—1 .. N- ,--. F4 243 Table B, Part l.--Continued Variables Equation number number i l 2 3 4 5 6 7 8 105---§ 336 3620 336 536b 546b 524b -123 -156 106----: 230 273 231 369° 384° 336° 48.9 —100 107-----: 78. 4 135 97. 6 217 249 204 -184 —283 108-----: 568b 632a 559b 614a 621a 561a 322 162 lO9----: 248 292 242 365 371 332 -83.3 -265 110---; -134 -89.6 -138 -27.2 -12.3 -99.7 32.0 17.4 117----: -4l77a —4l25a -4089a -41sia -40699 -3790a -47029 —3989a 118----: 5531a 5594a 54498 5427a 5371a 5071a 60768 4598a 119----: -2356a -24129 -23279 -23068 —2293a -2161a -2623a -18188 120--—--: 46.1a 476a 456a 452a 451a 424a 523a 342a 121----; -47.1a -48. 92 -46.5a -46.2a -46.1a —43.4a -54.5a -34.1b 122-----: 2.6a 2. 7a 2.63 2.5a 2.5a 2.4a 3.1a 1. 8b 123--—--: N072a — .075a - .071a N070a — .070a -.066a -.087a - .050b 124----: .001a .001a .0008a .001a .001a .001a .001a .0005° 125 ----- : -6.6a —5. 9b —6. 2b —5. 3b -4. 9b —4.2° —11.4a —11.8a 126----; 10.3a 9.4a 9. 92 8.22 7. 7b 6.22 16.9a 13.4a 127 ----- : -4.6a -4.2a -4.48 -3. 6a —3. 3: -2.6 -7.7a -5. 2a 128-----: .924a .832a .887a .691a .6402 .4650 1.6a .943a 129———--: -.096a .086a N093a - .069b -.063b -.043 -.1689 —.0922 130—---: .0058 .005 005 .004 .003 .002 .010a .005 131---:- 0082a - .0002a -.000 a -.000 b -.0001° -.0001 —.0003a N0001b 132-----. 2x10-9 2x10- 9 2x10- 8 1x10 b 1x10-69 1x10-6 3x10-6a 2x10-6a 133-----: 7. 2a 7. 0a 7.0a 7.0a 6.98 6.4a 8.0a 6.7a 134——---: -9.3a 94a -9. 2a -9.0a -8.98 -8.4a -1o.oa -7.5a 135 ----- : 3.9a 4.0a 3. 9a 3.83 3.7a 3.5a 4.3a 2.9a 136----—; -.7618 .785 -.753a -.729a -.727a -.686a -.846a N538b 137----: .077a .080a .076a .0742 .074a .070 8 .088a .0532 138--—-—: -.004a —.004a —.0042 - .0042 -.004a —.004a — .0052 -.003 139-----: .000 a .000 .000a .0002a .000 a .000 a '0006a .000%° -1x10' -1x10‘ 2 -1x10'a-1x10 a -1x10‘ 2 -1x10'-2x10 a -1x10' ° l40----: 244 Table B, Part 2 Variable: Equation number number : 9 : 10 ‘ 11 ‘ 12 ‘ 13 ‘ 14 ‘ 15 ‘ 16 a: ------ : .736 .502 .614 .623 .752 .758 .761 .760 R ----: .713 .466 .583 .592 .731 .737 .739 .738 D.F.---: 593 600 596 595 593 592 591 589 0------: 2990a 70958 2603b 29478 1821b 1240 2175b 1211 2- ----- : 1759a 1963a 1708a 1741a 1815a 3----- .204 -1.0a -.408° —.291 -.247 -.422° 4---"... .128 .354c 5------: -5.8 -21.5 6------: 3.78 2.5a 2.58 2.5a 7----—: -62.8a 8----—: .0003 .0001 9----: 18.58 14.48 14.28 14.2a 15.2a 10 ----- : 1014a 1131a 698a 597a 168 647a 11-----: 515a 605a 255a 178a -198 227a 12--—--: 544a 584a 340a 330a 46.7 367a 13------: 360a 433 329a 323a 364a 332a 14----: 21.22 12.5° .124b .105° .178a .159a .187a 15.7a 84------ 293 -339° 574a 536b 597a 524a 505a 386a 85__.___. 352a -337b 606a 580a 592a 572a 564a 425a 86----- 96.3 -6922 163 132 186 218 208 77.0 87----- -91.1 -943 -89.0 -77.7 60.8 84.2 92.9 -67.6 88--—-- -128 —9322 -88.3 -104 31.5 36.0 37.3 -98.4 89 ------- -169 -761a 54.4 42.4 112 82.7 70.6 -43.6 90 —————— : -41 —804a 17.0 75.2 112 131 137 -20.3 91---—-: -169° -743a 22.2 13.9 60.3 -12.9 -5.6 -146 92------: —85.9 -66Ba -39.2 93-----: -102 -717a 18.2 178 97.7 87.9 94.5 -69.3 94-----—: ~334a —8629 -97.7 -89.9 -61.4 -153 -147 -2949 95—-----: -224 —745a -38.4 -27.1 -58.6 -85.5 -86.6 -172 96------: 123 -848a -171 -81.8 -24.1 -14.3 -8.4 -188° 97----—-: -22.6 -475b 15.1 17.2 93.6 10.9 15.7 -129 98—-----: -189° -626a -147 -134 -127 -173 -l68 -310C 99------: -230 -1200a -374° -342° -206 -136 -50.4 —296b 100----: —119 -796a -58.1 -11.1 -155 -64.5 12.1 -221 101----: —52.7 —694a -28.2 62.3 14.3 44.7 144 -102 102--——-: 26.3 -392a 663a 7868 188 192 191 -88.5 103 ----- : -25.3 -738a 142 236 127 57 66 -188° 104—-—-: 225 -306 —147 -116 —60.2 -50 -39 -38.1 104-A---: 68.3 86.7 130 107 82.5 -Continued 245 Table B, Part 2.--Continued Variable; Equation number number ‘ 9 f 10 11 12 13 14 ‘ 15 16 105——--: 569: —33.7 42g 443 450° 466° 426° 328 106----: 393 13.1 518 520° 434b 383C 335 245 107----: 301 -177 336 354 304 273 203 138 108---: 655a 253 879a 989a 859a 761a 676a 646a 109-----: 428° -157 485 5330 394 418° 364 298 110 ————— 2 123 208 631b 671° 353 147 66.9 44.0 117 ----- : -232a —289a -204b —199b -1440 -157b —145b -167b 118----: 399a 491a 393a 377a 278a 295a 283a 298a 119---: -1838 -2209 -187a -178a -131a —1409 -1369 —1409 120 ----- : 36.48 42.78 38.1a 36.18 26.18 28.281 27.5a 28.0a 121----; --3.7a -4.2a -4.0a —3.7a -2.68 -2.8a -2.8a -2.8a 122———--: .179a .224a .221a .206a .137a .151a .149a .151a 123----: -.005a -.0062 -.006a -.006a —.00 b -.0042 —.0042 -.004% 124--—-: .00018 .0001 .0001a .0001a .00004 .00004 00004 .00004 125—-—--: —1.3 -9.73 -7.6a -7.58 -1.2 -1.4 -2.2 -1.5 126----; 1.7 10.8a 11.4a 11.3a 2.4 2.6 4.0 2.8 127 ----- : -.48 -402 -5.2a -5.28 —1.2 —1.2 —1.9 -1.3 128---: .034 .670 1.1a 1.1a .237 .230 .405 .254 129----—: .002 -.O61° -.117a -.1168 —.025 -.024 -.044 -.027 130-—--: -.0004 .003 .007a .007a .001 .001 .003 .002 131-----; .00002 -.0001 Noooga -.0002a -.0000 -.0000 -.0001 -.0000 1210-6 2x10- 9 2x10 1x10- 5x10- 1x10-6 1x10- 132—----:-2x10-7 Table B, Part 3 246 Variable; Equation number Number ; 17 18 19 , 20 21 I 22 23 24 55 ----- : .762 .763 .762 .762 .762 .761 .759 .758 R ----: .739 .740 .739 .740 .740 .740 .738 .738 D.F.----: 589 588 589 590 591 592 593 594 0—-----: 20888 1920a 1534° 1531a 16148 l637° 1380a 1397° 2—----: l780° 1817° 1842° 1841° 1844° 1845° 1824° 1829° 3------: -.271 -.408 -.400 -.400 -.408 .389 -.366 -.375 4-----: .318° .299 .298 .296 .301c .311c .317c 5------: -22.4 -20.7 _20.9 _20.9 -21.0 _20.7 -20.4 -20.1 6- ----- : 2.6a 2.6° 2.6° 2.6° 2.6° 2.6° 2.7° 2.7° 7------: -63.4a -61.4° -47.8b -47.6° -48.5° —5l.9° -52.3 ° -49.0b 8----- .0002 .0001 .00007 .00007 .00008 .0001 .00004 .00004 9-----—- 15.2a 15.18 15.39 15.39 15.58 15.28 l4.7° l4.7° 10 —————— 180 226 287 288 291 262 269 277 11-----— -190 -144 -95.2 -93.6 —102 —110 -127 -123 12----- 60.7 85 98.9 99.6 88.2 85.6 77.5 86.0 13---- 374a 372a 341a 340° 341a 330a 314a 303a 14 ------- 18.8b 19.4a 17.8a 17.8° 17.6a 17.8a 18.1a 18.3a 84 ------- 504° 497° 520° 521° 520° 525° 570° 554° 85 ------ 561° 546° 558° 559° 559° 564° 624° 608° 86---- 203 194 212 212 212 216 272C 257° 87---- 80.8 67.6 71.8 73.0 72.8 61.3 58.8 42.3 88 ------- 38.4 31.8 40.1 40.7 41.5 35.6 28.5 15.1 89 ------ 75.0 68.7 81.3 81.4 84.0 76.7 86.6 77.4 90-—--——- 132 118 123 124 125 117 115 103 91-—---- -9.3 -11.8 -10.9 -10.5 -8.5 -12.9 -32.3 -43.8 92 ------- 97.8 74.2 71.7 72.1 74.4 64.9 57.0 49.6 93 ------- -156 -163 -l67 —166 -166 -172 —190 -201 94----- -52.5 -53.7 -47.4 -47.1 -49.8 -54.5 -50.8 -57.6 95 ------ _24.6 -48.6 -5l.2 -51.0 -50.6 -58.9 -6l.8 -67.2 96.-.-.- 2. 5 '106 .719 0400 0313 '5 .2 -200 8 -1904 97 ------ -176 -185 —179 -180 -181 -183 —184 —18O 98----- -68.5 -88.4 -81.0 -80.1 -80.5 -76.8 _24.7 -40.2 99...... —8.9 -21.3 —14.0 -13.5 -12.8 -4.9 28.4 18.3 100 ————— 149 138 122 121 122 125 145 142 101 ————— 181 160 150 149 148 145 151 151 102 ----- 61.8 56.4 51.8 50.9 49.9 64.4 68.0 80.8 103--- -47.1 -54.8 -53.3 -54.1 -52.2 -37.5 -33.4 -28.9 104-..-- 62.1 59.1 66.0 64.0 61.6 59.2 91.1 101 -Continued 247 Table B, Part 3.--Continued Variable: Equation number number 3 17 f 18 f 19 f 20 f 21 f 22 f 23 f 24 105----§ 413° 412° 421° 419° 414° 409° 432° 446° 106--—--: 306 303 302 299 302 301 339 355 107---: 194 180 215 212 212 217 245 267 108---: 664° 674° 708° 705° 704a 713° 732a 756a 109----: 351 357 365 362 359 372 413° 434° 110 ----- E 75.5 81.0 111 108 108 112 98.9 124 117----: -142° -152b -137° -137° -131° -131° -132° -135° 118----: 278° 284° 258° 259° 252° 253° 247° 247° 119----: -134° -135° -121° -121° -119° -120° -116° -116° 120---: 27.1° 27.2° 23.8° 23.9° 23.5° 23.9° 23.3° 23.2° 121----; -2.7° -2.8a _2.3° -2.3° -2.3° -2.3° -2.3a -2.3° 122----: .147° .148° .120b .120b .119b .122b .123b .123b 123-----: -.004° —.004° -.003° -.003° -.003° -.003° -.003° -.003b 124-----:.00004b 00004b .00003° .00003° .00003° .00003° 00003° 00003° 125-----: _2.2 _2.4 .690 .613 .901 1.2° .263 .106 126--—--; 4.0 4.2 -.119 -.025 -.286 -.5002 -.O48 -.014 127-----: -1.9 -2.0° -.025 -.058 .010 .048 .001 128----: .406 .429° .003 .008 .001 -.001° 129---: -.044 -.047° -.00002 -.0004 -.00005 130----: .003 .003° -.000007 .000006 131----; -.0001 -.0001C .0000002 248 Table B, Part 4 Variable; Equation number number ; 25 3 26 3 27 28 29 30 31 f 32 52-—-—- .758 .758 .756 .749 .748 .746 .745 -743 R ------ .738 .738 .737 .730 .729 .728 .727 .725 D;F.--- 595 596 597 598 599 600 601 602 0------ 1390 608° 635° 666° 692° 647° 600° 440 2--————- 1837° 1836° 1858° 1860° 1858° 1866° 1837° l88l° 3 -------- -.370 -.357 -.388 -.342 -.358 -.360 -.416 .350 4—---- .319° .330° .334° .366b .409b .456b .4768. .484° 5 ——————— _20.9 -21.8 —24.0 -23.0 -23.8 —22.8 -23.4 _24.1 6—-----: 2.7° 2.8° 2.7° 2.8° 2.8° 2.7° 2.7° 2.7° 7------: -52.0b -50.8b -56.2° -56.4° -59.0° -55.4° -52.8b -42.6° 8- ----- : .0001 .0002 .0001 .0002 .0002 .0001 .0001 .0002 9 ------ : l4.9° 15.19 15.03 14.9a 15.2a 14.9a 15.1a 14.7° 10-----—: 258 288 279 282 287 306° 341° 382° 11------: —152 —146 -170 —167 —183 -142 -94.9 -61.5 12—-----: 74.2 115 105 96.3 68.4 73.2 100 109 13 NNNNN 309° 326° 361° 351° 351° 340° 357° 338° 14 ----- l8.1° 19.1° l9.6° 17.2° 16.5° 16.7° 16.5° l7.7° 84------ 5525L 590a 595a 543a 526’3L 505a 512a 539°L 85------: 599° 633° 630° 573° 550° 529° 540° 561° 86------: 247° 275° 278° 244° 231 219 228 251° 87-----—: 58.7 79.8 107 85.9 97.7 93.8 104 - 121 88 —————— : 35.2 56.3 69.3 51.9 48.2 38.2 66.5 77.4 89-----: 97.2 116 128 46.2 22.9 14.8 45.3 68.1 90--——--: 117 134 164 152 163 157 176 192 91------: -30.9 -29.0 -6.8 -32.7 -24.6 -28.2 -8.6 10.3 92--—--: 60.1 62.2 96.5 61.2 78.6 77.8 96.0 109 93 ------ : -192 -195 —175 _204 —l88 —192 -170 —158 94 ----- : -54.1 -61.5 -39.8 —80.0 -52.2 -57.5 -34.0 -33.2 95 ------ '6504 “6703 '3909 “8304 ’6404 “6904 -6398 ”4106 96 ------- ~19.5 -49.5 -23.5 -79.5 -66.9 -72.2 -61.1 -67.4 97—---——: -184 _223 -202 _242 -231 -241 _219 -236 98-----: -65.0 -37.2 -1.7 -43.3 -43.1 -38.6 ~50.4 -51.4 99—-----: -16.6 .889 31.5 2.9 6.9 6.1 -2.3 -30 100—----: 156 169 196 175 164 148 148 172 101 ----- : 157 162 183 143 144 131 141 166 102---—: 61.8 55.8 63.0 14.2 20.9 15.1 24.3 34.3 103---—-: -59.4 —64.6 -65.4 —102 —101 —105 -98.3 -95.4 104—-——-: 5.0 -21.1 -13.8 -67.9 -68.8 -87.5 -94.8 -62.0 -Continued 249 Table B, Part 4.--Continued Variable: Equation number °°m°°r ‘ 25 26 27 28 ‘ 29 30 31 32 105 ----- § 308 308 352° 265 251 250 227 267 106----: 202 181 210 103 100 113 75.0 97.7 107----: 109 100 113 50.9 57.9 46.1 39.3 80.6 108----: 603° 610° 631° 590° 597° 595° 609° 637° 109-—--: 276° 279° 308° 250 255 263 265 297° 110----; -22. -38.6 7.5 -65.8 -53. 8 -57.9 -28. 2 17.6 117----: —l45 ~145° -39. 1029 56. 6° 27.4 -. .672 21.5° 118---: 253° 254° 104 -70.1° -27. 9° -6.7 .254 119----: -118° -119° -49.1° 16.1° 5. 0°c 1.1 -.248° 120—---: 23.7° 24.0° 9.0° -1. 5° - 293° -.038° 121--—--; _2. 4° _2. 4° -.758° .061° .005° 122---: .126 .l29° .030a -.0009° 123----: .003°b— .003: -.0004° 124—---:- .00004 .00004 125----: -.051 Table B, Part 5 Variable; Equation number number : 33 34 35 36 : 37 38 39 40 g:---- .743 .728 .569 .578 .582 .583 .585 .586 R ————— .725 .710 .547 .555 .558 .559 .561 .561 D.F.--- 603 604 612 611 610 609 608 607 0-----: 458 889° 531° 221 178 208 225 222 2—---- 1800° 1776° 3 -------- .353 -.463° 4- ------ .475° .455b 5------ -23 .9 -26. 9 6-------: 2.7"1 3 . 0a 7-------: -44.5° -46.8° 8-------: .0003 .0002 9.. ------ 14.6° 14.4° 10----- 375° 489° 1125° 1029° 1072° 1094° 1085° 1095° 11-—--- -70.1 -73.1 368° 364° 387° 406° 379° 368° 12.-.---: 98.9 24.5b 557° 438° 435° 446° 452° 464° 13-—----: 341° 494° 446° 352° 366° 374° 355° 363° 14------: 17.8° 15.6° 9.3 10.4° 10.0° 9.0 9.1 8.7 84 ...... : 539° 480° 633° 666° 660° 640° 634° 653° 85------: 563° 547° 627° 632° 638° 621° 608° 625° 86----: 249° 140 151 236 228 210 198 208 87-----: 117 -49.6 -91.9 47.2 33.0 8.2 -5.8 -.630 88------: 75.4 5.0 -19.7 33.6 21.2 11.7 -23.2 -14.6 89-----: 65.1 18 248 14 121 105 69.2 77 90---—-: 190 37.9 35.7 169 163 138 116 121 91--——--: 10.4 -78.5 9.1 72.5 71.9 54.7 32.9 ' 37.2 92---—-—: 108 -52.8 -27.1 121 111 85.1 69.5 68.5 93 ------ : -158 _232° -131 -80.5 -80.4 -92.5 -117 -113 94---—--: ~31.4 -21 . 6 -44. 8 -50. 6 -33. 5 -33 .4 -59. 6 ~55 . 5 95------: -43.3 -234 -297 —133 -141 -170 -176 —173 96 ------ : -69.2 -162 —122 -58.8 -73.9 -69.1 -81.3 -76.0 97---—--: -234 —307 -304 -250 _235 -223 -249 -241 98-----: -50 -259 -452 -269 -263 -290 -283 -291 99----: -2.6 -201 -155 30.9 31.0 1.9 6.1 3.6 100-----: 178 19.5 -73.3 71.2 94.1 55.2 52.6 58.7 101----: 169 9.2 580° 735° 731° 704° 696° 703° 102----: 38.7 -144 —61.9 86.8 103 88.1 83.3 84.6 103-----: —95.1 -222 -325C -223 -225 -230 -239 -238 104----: —648 -222 -288 -168 -192 -218 -208 -192 -Continued 251 Table B, Part 5.--Continued Variable: Equation number number ‘ 33 34 ‘ 35 f 36 37 38 39 f 40 105-——--; 270 47.9 -49.3 127 150 118 147 147 107----: 85.1 -50.7 -16.0 90.2 123 91.9 102 112 108 ----- : 645° 483° 578° 707° 758° 733° 727° 730° 109---: 303° 120 65.6 216 272 247 246 240 110-----§ 25.1 -199 266 437 482° 446° 421° 426° 117----: 24.4° 18.8° 39.8° 23.2 56.5b 81.2b ll8----—: -1.8° 2.3 -12.6 -30.7° 119 ----- : -.180 1.4 4.7 120 ----- : -.045° -.262 121----; .005 252 Table B, Part 6 Variable: Equation number num°°r ; 41 ° 42 ; 43 ; 44 45 46 47 48 g ----: .587 .595 .598 .600 .601 .603 .604 .604 R .__-—: 0562 0569 0571 0573 0574 0575 0575 0575 D.F.---: 606 605 604 603 602 601 600 599 0——--- 221 180 227 2328° 2369° 2201° 2361° 2357° 10 ------ 1077° 1061° 1037° 961° 951° 995° 990° 987° 11 ------ 372° 359° 357° 348° 362° 412° 421° 432° 12---- 480° 480° 467° 363° 355° 377° 381° 393° 13 ----- 363° 369° 333° 279° 270° 296° 306° 297° 14- ----- 9.4 12.3° 11.8° 9.3 10.3° 9.6 9.2 9.1 84 ------ 668° 717° 714° 599 604° 650 618 622° 85----- 644° 700° 705° 605° 621° 667° 627° 631° 86----- 219 253 248 169 186 228 193 196 87 ------- -9.5 .169 -25.2 .101 .143 -98.1 -92.8 -98 88 ----- -11.6 4.7 -11.1 -71.7 -108 —70.9 -67.2 -71.6 89----- 97.4 180 165 111 75.6 102 94.2 88.7 90 ------ 111 112 81.8 22.5 -12.1 20.6 23.5 21.1 91-—-- 31 52.9 29.9 14.3 -10.4 21.8 31.6 28.9 92----- 53.1 71.7 40.3 14.0 -15.3 6.2 11.2 8.8 93 ----- : -127 —103 -124 —129 -147 -115 -106 -110 94-----: -79.3 -41.0 -61.3 -41.0 -46.1 -28.9 -32.1 -35.0 95-----: -l92 -160 -188 -198 -212 -196 -193 -195 96-----: -87.2 -30.9 -60.7 14.8 14.3 11.4 18.1 14.8 97------: -251 _211 -235 -132 -127 -138 -139 -139 98---—--: -285 -257 -280 -380° -345 -314 -341 -337 99 ------ : 8.1 22.8 3.5 -68.2 -l7.6 —1.6 ~14.1 -6.0 100-----: 72.3 88.1 67.2 22.9 5.0 -3.5 -16.4 -17.0 101----: 700° 730° 706° 678° 661° 657° 654° 653° 102----: 78.0 121 118 124 160 127 119 132 103----: _240 -202 -201 -186 -131 -142 -146 -134 104-----: -191 -134 -l43 -71 98.3 77.1 56.8 57.9 105---: 159 252 199 200 444 41% 400 400 lO6-—---: 157 266 232 285 547° 514 492° 497° 107 ----- : 104 165 153 174 447 399 383 392 108-----: 725° 757° 740° 706° 969° 9208 906° 920° 109----: 236 294 260 251 524° 47 457 472 110----: 419° 484° 439° 465b 721b 664 668b 677° 117---: 121° _25.3 -154° -156° -138 -132 -131 -132 118-----: -68.3° 112° 296° 294° 281° 285° 289° 291° -Continued Table B, Part 6.--Continued 253 Variable; Equation number number i 41 42 g 43 ‘ 44 45 46 f 47 48 119----: 14.7° -52.9° -138° -136° -132° -134° -137° -138° 120---—: -l.3° 9.5° 27.8° 27.1° 26.2° 26.7° 27.1° 27.5° 121----: .054 -.794° -2.8° -2.7° -2.6° -2.7° -2.7° -2.7° 122----: -.001 .031° .151° .145° .140° .141° .141° .143° 123-—--: 0004° -.004° -.004° -.004° -.004° -.004° -.004° 124---: .00004° .00004° .00004° .00004° .00004° 00004° 125---: - ° .134 .566 1.2 1.4° 126..-“: "o 018 -01-170 .4030 "o 560 127---: .004 .003 .061 128---: -.001 -.003 00004 l29---§ 254 Table B, Part 7 Variable: Equation number number ' 49 50 51 52 g 53 54 55 56 §§---- .606 .606 .724 .722 .720 .720 .719 .709 R ------ .575 .575 .707 .705 .703 .704 .703 .692 D.F.--- 598 597 605 606 607 608 609 610 0—---- 2085° 2085° 8512 948° 919° 918° 524° 613° 2—---- 1711°1660° 1605° 1605° 1578° l898° 3.. ..... -.508° —.302 -.265 -.265 —.307 -.414 b 4----- .433 5 ----- —30.4° -32.9° 6---- 3.2° 3. 3° 3. 2° 3.2° 3.0° 7 ------ -31.3 -33. 5 -33. 6 -33.5 8---- -.0002 — .0001 .0001 9-—--- 14. 0° 14. 1° 12.9° 12.9° 12.8° 12.8° 10----. 951° 951° 649° 586° 567° 566° 828° 961° 11- ----- 445° 445° 58.2 .041 _24.2 -25.3 199° 294° 12---- 423° 423° 403° 375° 359° 359° 528° 593° 13----- 301° 301° 562° 560° 560° 561° 543° 575° l4—----- 10.0 9. 9 84—--- 627° 628° 461° 468° 478° 477° 483° 620° 85- ..... 634° 634° 527° 544° 553° 552° 555° 620° 86---- 197 197 120 133 137 135 136 122 87---- -97.9 -97.7 -60.4 -34.9 -33.5 —43.1 -42.6 -79. 6 88---- —72.0 -71.9 -10.3 1.9 7.6 6.1 6.0 3. 0 89----- 80.1 80.2 189 196 202 202 203 261 90------: 1907 1909 1909 4501+ 4304 36.6 40.0 0565 91-----: 25.8 25. 8 -98. 3 -93.9 -89.2 -91.8 -88.0 -13.7 92----: 5.1 5.1 ~79. 50 -54.6 -58.5 -65.4 -57.0 -80.8 93----: -108 -108 -233° -222 -211 -213 -208 -107 94------: -21.7 -21. 7 18. 8 18.9 -31.6 —31.8 -3o.5 -2.1 95------: —194 —194 -262 _223 _215 _222 -217 -264 96-----: 20 19.9 -190 -186 -174 -175 -170 -107 97------: -132 -132 _292 -288 -286 -287 -284 -269 98-—---: -335 -335 -317° -284 -290° —302° -351° -436° 99------: -10.2 -10.1 -249 -228 -235 -246 -291° -413° 100---: —12.7 -12.9 -85.3 -69.8 -77.4 '80. 6 -125 -196 101--—-: 659° 659° -33.3 -1.5 11.7 9. 6 23.1 -14.1 102-—--: 142 142 -226 -215 -216 -216 _210 -160 103—-—--: -l40 -140 -287° -247° -270° -270° -272° -311b 104-----: 65.5 65.1 -296° -296° -293° -293° -294° -287° -Continued 255 Table B, Part 7.-—Continued Variable; Equation number number ‘ 49 ‘ 50 51 52 53 54 f 55 56 105----—: 413 413 -16.7 -20.0 -32.6 ~32.5 -34.5 -71.1 106-—--: 477c 4760 44.3 40.3 44.1 43.4 43.3 112 107----: 383 382 2.0 10.9 —11.2 -11.4 -11.3 36 108---: 921° 920° 599° 576° 561° 561° 568° 718° 109----: 471 470 199 187 173 173 172 151 110---; 679b 678b -145 -162 _211 -212 -198 42.2 117----: -158c -158° 118----: 319a 319a 119---: -147° -147° 120—----: 28.8° 28.8° 121----; -2.8: -2.8° 122—---: .147b .1472 123-—-—-: -.004 -.004 124 ----- : .0004c .00004° 125 ----- : o 278 o 264 126-----; o 474 o 492 127 ----- : -.211 -.217 128----: .026 .027 129----: -.001 -.001 130--—-: .00002 .00002 131-----; -3xlO-8 256 Table B, Part 8 Variable: Equation number number ‘ 57 58 59 60 61 62 63 64 33------- .698 .576 .568 .216 .736 .741 .728 .727 R -—--- .682 .553 .546 .182 .720 .724 .711 .711 D.F.-—-- 611 612 613 617 606 603 607 608 0—----— 621° 604° 618° 1148° 4087° 5691° 5404° 5176° 2------. 2074° 1910° 1828° 1821° 1840° 3-----: -.313 -.976° .300 -.4310 -.048 -.056 4- ----- .210 .350c .3520 .324 5------ -7.7 -20.1 -15.2 -14.8 6—- _____ 3.8° 3.3° 3.5: 3.5: 7..-.... -77.1° -27.5 ~24.5 8------- -.0010 -.001° -.001° -.001° 9- ------ 17.9° 15.1° 15.1° 15.5° 10-—-- 1072° 1236° 1146° -53.8 11---- 351° 465° 382° -130 12 ------- 606° 641° 605° -24 13- ----- 579° 561° 515° a 367° 14---— 22.1 84 ------ 610° 616° 627° 627b 85-—--—- 615° 610° 616° 616° 86--—-- 118 128 132 132 87---- -107 -88.3 -127 -127 88 ------ _20.1 —38.8 -32.4 —32.4 89---- 263 233 252 252 90". ----- -3609 "7908 103 1.3 91---- -39.0 -7.4 -6.1 -6.1 92"..." -117 102 “’7009 -7009 93----- -131 -124 —132 -132 94----- -7 -26 -22.7 -22.7 95-—---- -303C -230 -338C -338 96 ....... -123 -147 -137 -137 97---- -286 -308 -296 -296 98---- -476° -436b —509° -509° 99----— -479° -130 -213 _213 100-—-- _233 -23.3 —140 -140 101—.... 17.3 673 534° 534b 102-.... -187 -3.7 —114 -114 103--—- -3362 -325C -367b -367 104--- —311 -306c -332c —332 -Continued Table B, Part 8.--Continued 257 Equation number Equation: number i 57 58 59 f 60 f 61 62 63 64 105—---: —87.1 -58.4 -90.6 -90.6 106---: 107 85.8 93.0 93.0 107--—: 24.8 29.0 15.8 15.8 108---: 696° 752° 651° 651 109----: 135 156 110 110 110----: -2.4 329 301 301 111--—--: 187° 39.1 113° 128b 112---—-: 507° 162 317° 330° 113----: 766° 524° 659° 657° 114----: 590° 369c 522° 558° 115-—-: 846° 757° 865° 887° 116----: 309 196 306 342 117---: -4l96° -2599° -2433° -2358° 118--—--: 5564° 2949° 2809° 2667° 119----: —2340° -1047° -1031° -970° 120----: 452° 166° 170° 159° 121-----: —45.6° -13.1° -14° -13° 122----: 2.5° .512° .584° .535° 123——---: —.068° -.009° -.011° -.010° 124-----: .001° .00003 .0001° .00005° 125-----: -5.4° -7.1° -4.8c -2.0 126---“': 8 0 3a 10 0 8a 7 0 3b 3 0 3b 127----: -3.5° -4.8° -3.lb -1.3° 128----: .675° .942° .575° .201° 129 ----- : -.067° -.097° -.055° -.015° 130---: .004: .005° .003c .0005° 131----: -.0001b -.0002: -.0001-.00001° 132---—: .000001 .000002 .000001 133----: 7.1: 4.4: 4.0: 3.9: 134-"'“'"': '9 o 3 '40 9 ‘40 4 -40 2 135-----: 3.9° 1.7° 1.6° 1.5° 136-----; -.743° -.258° -.245° -.232° 137---: .075° .020° .019° .018° 138--—-: -.004° -.001° -.001° -.001° 139-----: .0001° .00001° .00001° .00001° _lxlO-oa 140..-"; Table B, Part 9 258 Variable; Equation number °°°°°r ; 65 66 67 68 69 70 71 72 3§----: .724 .712 .711 .710 .710 .708 .708 .703 R —----: .708 .696 .696 .694 .695 .694 .694 .689 D.F.---: 609 610 611 612 613 614 615 616 0..-..-—: 5311° 5454° 5460° 5416° 5460° 5558° 5478° 5313° 2---—-: 1864° 1873° 1881° 1888° 1886° 1888° 1883° 1894° 3-----: -.028 .095 .126 .130 .132 .136 .172 .22 4-—----: .3250 .345C .352C .364c .363b .346c .3570 .354 5---—---: -16.9 —18.5 —18.9 —18.2 -18.0 -19.1 -18.5 -19.3 6 ------- : 3.4° 3.2° 3.2° 3.3° 3.3° 3.2° 3.3° 3.2° 7-------: -23.2° _22.4° -23.1a -22.2a -21.9a _22.oa -21.7a -23.8° 8-----: -.001° -.001° —.001° -.001° -.001° —.001° -.001° -.001° 9 ------- 15.4° 16.7° 16.8° 16.9° 16.8° 17.8° 17.8° 17.8° 111 ------ 108° 146° 146° 151° 152° 161° 153° 140° 112-----: 328° 382° 401° 403° 407° 409° 391° 390° 113—----: 675° 751° 776° 741° 748° 771° 747° 837° 114 ----- : 556° 624° 660° 649° 654° 683° 680° 748° 115—---: 901° 1009° 1040° 1047° 1051° 1070° 1054° 1093° 116—---: 381c 556° 583° 567° 570° 591° 577° 605° 117----: -2549° -462 -432 -522 -415 -41% -382 428C 118----: 2793° 493 468 505 405 395 390b -104 119----: -966° —120 —113 -103 -76.6° -73.8° -78.5° 4.5 120—---—: 149° 12.2 11.3 7.5 4.6° 4.3° 5.1° -.004 121-—---: -11.4° -.661 -.6 -.213 -.082 -.06 -.105° -.0002 122-—--: .418° .019° .018c .002 -.0003 -.001 123---—-: -.006° -.0003° -.0002c b 125 ----- : -2.5§ —.217 .717 .527 .688 1.8b 1.8: 2.8° 126-----: 3.8 1.25 .132 .219 .087 -.852 -.898 -1.4° 127-----: —1.4° -.555 -.162 -.169 -.136 .102c .107° .184° l28-----: .217° .083 .024 .024 .020 -.005 -.005 -.009° 129-——-: -.015° -.006 -.001 -.001 -.001 -.0001 -.0001 -.0001° 130---: .001° .0002 .00002 .00002 .00002 131----:-.00001 —.000002 133----: 4.5° .771 .721 1.0 .815 .846 .729 -.697C 134..---: -4.8° -.727 —.692 -.916 —.742° -.750b -.693° .166 135-—-—-: 1.6° .150 .143 .187 .141° .143° .138° -.006 136-—--: -.250° -.011 -.011 -.014 -.009a -.009° -.009° -.0001 137----: .019° .0003 .0003 .0004 .0002° .0002° .0002° 138----: -.001°-.000004 —.000003 -.000004 00001° 139----—:. 259 Part 10 Equation number °°m°°r f 73 f 74 f 75 f 76 f 77 f 78 f 79 ‘ 80 2 R __ _____ 8° ...... D.F.---- 0____.._. 2 _____ 3--..... 4 ______ 5 “““““ 6------ 7 _______ 8- _____ 9------: lll----: ll2----: 113----: 114 ----- : 115-----: 116 ----- : 117---: ll8-----: ll9----: 120----: l2l---—: l25-----: 126---: l27--—--: 128-—--—: 133 ----- : 134-----: 135-----: 136----: .700 .687. 617 5376° 1880° .247 o 316 -l9.5 “1405 -.001: 19.3 108° 416° 866° 767° 1078° 608° 4060 -103 4.1 .067 -.002 1.9° -.816° .079° -.002° -.668 .173 -0007 3.3° b .700 O 687 618 5388° 1881° .247 .306 -19.8 3.3: -15.0 19.3 111° 431° 870° 780° 1085° 622° 3920 -106 515 -.045° 1.9° -.829° .080° -.002° -.618 .161 -0 007 -.00002 -.00002 .700 .688 619 5392° 1881° .247 .305 -l9.7 3.3° -15.0b -.001: 19.3 111° 430° 870° 781° 1085° 623° 403b -110° 6.0° -.059b 1.9° ‘0 833a .0812 -.002 .169° 0 691 o 679 620 5094a 1881° .207 .358c -1904 3.3° -18.8b -.001° 19.2° 179° 535° 980a 888° 1187° 688° 20 -5900 3.1° -.017 .110 -.025 -.001 -0 331 .1020 -.005 .691 o 679 621 5025° 1887° .199 .353c -19.3 3.3° —18.4b -.001° 19.2° 171° 526° 956° 878° 1185° 691° 23 '5904 2.8° .138 -.028 -.001 -.401 .113 -.005 .688 .677 622 490581 1880° .17 .419 -19.0 3.1° 24.0a .001° 19.2° 178° 515° 931° 856° 1171° 717° -89 OZ 15.6 -.152 .035 .160 “'0 0160 .686 .675 623 4868° 1830° .13 .397 -19.5 3.2° -27 o la -.001° 19.5° 162° 483° 881a 788° 1106° 644° ”59.4 12.6° --.249C .226 .184 -.045° -.045° .114 .2310 -.012 -.017° 260 Table B, Part 11 Variable; Equation number °°m°°r = 81 82 = 83 84 = 85 I 86 87 88 _2—---- .682 .673 .670 .670 .624 .719 .734 .490 R ----- .672 .664 .661 .661 .615 .705 .718 .479 D.F. ----- 625 626 627 628 629 613 609 630 0- ----- 4887° 5527° 5429° 5296° 170 4024° 4768° 1020° 2-—-—-- 1818° 1820° 1805° 1809° 1917° 1979° 1984° 1363° 3- ----- .15 .125 .188 .198 .135 .146 -.150 -.544° 4 ------- .393 .423 .347 .347c .562° .463b .408b .577b 5------ _22.9 -29.3 _28.7° -28.6° -36.7° -16.3 -18.4 -44.6b 6-—---- 3. 4° 8 4° 3. 3° 3. 3° 3. 4° 2.6° 2.4° 5. 3° 7------: -22. 4° -20. 6° _22. 7° -22. 1° -28. 1° -35.2° -93.4° -34. 2° 8-----: - .001b — .001 - .0005 -.001 .001 -.001° -.001-.001 9---—-- 20.5° 21.8° 21.6° 21.5° 22.2° 16.6° 16.6° 23.4° 10---- -116 11 ------ -337° 12—----- -191 13- ----- 302° 14.. ----- 20.2° 18.0° 111 ————— 168° 32.7 42.9 41.9 90.8° 223° 133° 14.6 112--—: 563° 138C 116 115 -78.g 474° 293b -228° 113----: 972° 288° 290° 288° 176 817° 643° 64.9 114-----: 895° 178b 210° 210° 207° 665° 467b 43.3 115--—--: 1207° 480° 526° 529° 699° 967° 785° 465° 116-----: 757° 96.5 141° 142 198 558° 3720 -182 117-----: 27.5 -6.1 19.3 30.0° 52.0° -250° -165b 118----: .078 1.7° 400° 277° 119 ----- : -179° -123° 120————-: 35. 5° 23.9° 121--——-: -3. 6° -2.4° 122-—---: .196° .123° 123 ----- : —.005° -.003 124 ----- : b .0001° .000030 125 ----- : .371 -.358° -.351° -.342° -1.7 -2.0 126-----: -.058a 3.7 4.4 127----: -1.8 -2.3g 128-----: .356 .521b 129—---: -.037 -.059b 130----: .002 .004 131 ----- : -.0001 -.0001bb 132---—-: .000001 .000001 -.002 .028 .018 133-----: Table B, Part 12 261 Equation number Variable: number i 89 90 91 ‘ 92 I 93 f 94 95 96 £2----; .486 .481 .480 .471 .383 .277 .053 .038 R ----: .475 .471 .471 .463 .374 .268 .042 .029 D.F.----: 631 632 633 634 635 636 637 638 0——----: 1070° 980° 939° 680° 992° 1013° 1094° 1184° 2------: 1335° 1256° l245° 1124° l545° 2262° 3------: -.261 -.223 -.256 -.072 -.4280 .192 .842° 5------: -50.l° 6—-----: 5.2° 5.0° 5.2° 6.1° 7------; -28.l° -24.1° -22.3° 8------: -.0004 -.0004 9-------: 23.2° 21.6° 2l.9° 25.6° 34.2° lll---"-: 807 4.2 1005 6.]. 310% 250 -3505 -4207 112 ----- : -251° _253 _237° -228° -226 -214 -333° -368° 113---—-; 40.7 34.5 45.0 40.7 123 190° 90.3 31.1 115----: 423° 393° 408 384 536° 542° 389° 399° _210 -296 -296 -355° 147 64.1 298 265 116----: 262 Table B, Part 13 Variable; Equation number number ‘ 97 98 99 100 101 f 102 f 103 104 R -—--: .725 .718 .461 .699 .439 .347 .547 .554 D.F.---: 607 608 615 616 623 640 617 616 0------: 6208: 43:8: 6761° 4g76: 5237° 644° 7244° 7903° 2-----: 1934 l 9 l 92 3-----: -.258 -.267 —.131 4..."---- o 3160 o 3500 O 3620 5-----: -29.50 -29.00 -25.1 6----..- 207: 2.8a 2.5a 6.8: 6.4: 7---——- -90.4 -46.5 ~48.6 8- ----- -.001c -.001° -.001c -.001° 9------ 16.l° l6.3° 18.1° 10----- ~147 434° 941° 458° 952° 1146° 11-——--- -232 297° 687° 153° 576° 382b 12—---—-- -167 230° 402° 137c 363° 605° 13----- 326 246° 272° 185° 275° 515° 14 ----- 20.1° 18.8° 14.6° 17.6° 13.9° 35.8 0.3 117 ----- -4563° -4774° -4777° _223° -268° -483l° -5ll6° 118---- 5740° 6200° 619l° 328° 442° 5937° 6175° 119 ------ -2344° -2593° -2687° -142° ~199° -2430° -2498° 120--- 441° 499° 532° 26.9° 39.6° 455° 466° 121 ----- -43.5° -50.l° —54.4° -2.6 —4.l° -44.1° -45.2° 122 —————— 2.3° 2.7° 3.0° .135 224° 2.3° 2.3° 123—--- -.063° -.075° -.083° -.004° -.006° -.60: -.062° 124 ------ .001° .001° .001° .00004c .0001° .001 .001° l25—--- -8.0° -6.9° -l2.5° -1.2 -7.8° -12.3° -l2.3° 126-—--- 11.8° 10.1° 19.0 ° 2.0 12.5° 17.7° 17.2° 127 ...... -5038. -4048 “8.5a -0957 -5068 -7058. -7028. 128-----; l.l° .870° 1.7° .186 1.1° 1.4° l.3° 129—---: -111° -.089° —.l72° -.019 -.117° -.139° -.131° 130----—: .006° .005° .010° .001 .007° .007° .007° 131----: -.0002° —.0001° -.0003° -.00003 -.0002° -.0002° —.0001° 132-----:.000002° .000002° .000003° .0000003 .000002° .000002° .000002° 133----§ 7.8° 8.1° 8.0° 8.0° 8.5° l34----: -9.7: —10.4: —10.1: .9.g: -10.0: 135 ----- : 309 403 403 30 400 136--—--: -.737° -.833° -.855° -.709° -.730° 137-----: .073° .084° .087° .068° .070° 138----; -.004° —.005° -.005° -.003: -.004° 139----: .000%° .0001° .0001° .0001 .0001° 140-—---:-lx10' ° -1x10‘°° -1x10‘°° -lx10‘°° ~1xlO'6b 263 Table B, Part 14 Variable; Equation number number ‘ 105 106 107 108 109 110 111 112 33 ------ .633 .634 .634 .726 .727 .724 .721 .719 R -—--- .616 .616 .615 .711 .712 .709 .707 .706 D.F.--- 615 614 613 612 611 612 613 614 0 ««««« 6796° 6640° 6608° 5143° 5200° 4985° 49o9° 4933° 2 ————— 1865° 1904° 1899° 1903° 1933° 3...-..-. .222 .191 .10 .096 .056 0035 0057 4-- ----- .074 .359 .333c .351b .302 .308c 5----- -27.2° -25.8° -24.0 -25.3 6 ------- 5.6: 5.6: 5.7: 2.8: 2.9: 2.8: 2.9: 2.8: 7-------: -17.1 -15.6 —16.6 —27.9 -29.7 -29.7 -24.2 -23.8 8--—----: -.001° -.001b -.001b --.001b -.001b -.001b -.001° -.001b 9------: 28.8° 28.2° 28.2° 15.4° l6.4° 16.6° 17.l° 17.0° 14—----: 6.6 7.3 7.4 21.2° 21.4° 21.8° 19.4° 20.7° 117---; -4174: -4142° -4148° -4122° -4184° —2464° —2385° -2555° 118----: 5186 5180° 5178° 5293° 5364° 2968° 2782° 2889° 119--—-: -2122° 2l27° -2124° —2176° -2204° -1106° -1026° -1022° 120--—-—: 400° 402° 401° 409° 414° 182° 167° 160° 121----: —39.2° -40.0° —39.5° -39.9° -40.4° -15.0° ~13.6° -l2.2° 122—---; 2.1° 2.1° 2.1° 2.1° 2.1° .614° .544° .451° 123---: -.055° -.056° -.056° -.056° -.056° -.011° -.009° -.006° 124---: .001° .001° .001° .001° .001° .0001° .00004b 125-—--: -7.9° -7.9° -8.0° -6.9° -7.l° -6.5° ’2°5§ -2.9° 126----: 11.6° 11.6° 11.8° 10.l° 10.3° 9.4° 3.6 4.0° 127---; -5.0° -5.0° -5.1° -4.4° -4.5° -4.0° -1.4° -1.5° 128----: .949° .965 .977° .847° .870° .758° .210° .222° 129---: .094° -.096° -.097° -.085° -.087° -.075% -.015° -.016° 130---: .005° .005° .005° .005° .005° .004b .0005° .001° 131---: -.0001 -.0001° -.0001° -.0001° -.0001° -.0001 —.00001° 00001° 132----;.000002b .000002° .000002b .000001° .000002°.000001b 133----: 6.8° 6.8° 6.8° 6.9° 7.0° 4.0° 3.9° 4.4° 134-—--: —8.3° -8.3° —8.3° -8.7° -8.8° —4.6° -4.4° -4.9° 135----: 3.3° 3.4° 3.4° 3.5° 3.6° .1.7° 1.6° 1.7° 136-—-—-: -.623° -.627° -.626° -.661° -.670° -.266° -.249° -.264° 137----; .060° .061° .061° .064° .065° .021° .019° .020° 138--——-: ’-OO3% -.003° —.003° -.003a —.003° -.001a -.001a -.001° 139--—-—: .0001 .0001° .0001° .0001° .0001° .00001° .00001° 00001° -1x10'°° -lx10‘°° —lx10'6° -1x10'°° 140——--: -lx10—6a 264 Table B, Part 15 Variable; Equation number °°m°°r ‘ 113 114 115 116 117 118 119 120 3% ------- .705 .704 .701 .701 .698 .698 .692 .689 R ------ o 691 o 691 o 688 o 688 o 686 o 687 o 680 o 678 D.F.--- 615 616 617 618 619 620 621 622 o------ 5324° 5304° 5491° 5511° 5782° 5744° 5454° 5292° 2 ----- 1921° 1931° 1923° 1923° 1913° 1911° 1934° 1938° 3 -------- 0204 0226 0245 o 24.6 .243 0259 o 286 029]. 4-----: .310 .319C .324C .323C .298 .307 .324C .306 5———--—— -25.6 _25.9 —24.8 _24.7 -26.0 -25.4 _27.0° -27.50 6- ----- 2.8: 2. 8°a 2. 9°b 89°b 2.9% 2. 9g 2.8: 2. 8° 7 ------- -20.8b _21. .6b -19. 9: -19.7b -19.4b -19. .3b -23.3 -16. 6b 8---—-—- - .001a — .001a — 001° —.001 —.001 -.001 .0005 - .0004 9 ------ 18. 3° 18. 3° 18. 4° 18.4° 19.8° 19. 8° 20.0° 2l.3° 14.. ----- 19. 7° 20.3° l8.4° 18.5° 17.3° 17.3 18.1 18.9° 117----§ -295 -275 -387 -319 -247 -338 527 495b 118---: 420 402 433 370b 367b 369b —163b —160b 119---: -120 —115 -95.6 -79° -76. 7° -79. 5° 10.5 10.1 120-----: 13.7 13.0 7.1 5. 4a 5. 0° 5. 4° -16.1 -.101 121 ----- : -.811C -.767C -.2 -.118b -.094°- .115° -.0004 -.002 122—----; .026° .024° .002 .0002 —.0003 123-—---: -.0003° -.0003° b . 125----: -.550 .171 -.070 .032 1.5b 1. 5° 2.6° 1. 7° 126—----: 1.3 .442 .523 .440 -.891 - 909° —1.5° - .967° 127-----: -.575 .267 -.265 -.245 .095c .097c .19° .099° 128---; .083 .036: .035 033° -.003 -.003 .009° -.003° 129---: -.005 -.002b -.002° - .0022 .00001 .00001 00001° 130—-—--: .00001 .00003 .00003 .00003 13l---:-.000002 b b 133--"-: 0338 0302 .680 o 560 .643 .602 '0921 -0877 135 ----- : .114 .108 .163 .134° .140° .138° -.016 -.018 136----: -.009 -.008 -.012 -.009° -.009° -.009° .0002 .0003 137----: .0002 .0002 .0003 .0002° .0002° .0002° .000002 -.000001 -.000002 138--—--:- 265 Table B, Part 16 Variable: Equation number °°m°°r f 121 ‘ 122 ‘ 123 f 124 ’ 125 126 127 128 B§----§ .689 .689 .676 .676 .673 .672 .670 .665 R —----: .679 .679 .666 .666 .664 .664 .662 .658 D.F.--: 623 624 625 626 627 628 629 630 0-----; 5317° 5278° 5235a 5241a 5217a 5135a 5700° 50423 2—-—--: 1937a 19348 1893b 1893° 1888° 1862° 1860° 1848° 3------: .290 .291 .231 .232 .211 .19 .226 .206 4-----: .296 .307 .345c .346c .401b .387 .386b .385b 5-----: _27.70 -28.7C -30.4c -30.5° _29.2° -29.8° -30.7b -34.6b 6------: 2.8: 2.8: 2.88 2.8a 2.8a 2.8a 2.9° 3.0° 7----: -17.1 -16.9 -20.6° -20.7° -25.4° -27.3° -24.6° -21.3° 8------: -.0004 -.0004 -.001° —.001° -.001 -.001 -.0004 -.0004 9-----: 21.3° 21.4° 21.4° 21.4° 21.1° 21.3° 21.2° 23.0° 14—----: 18.6° 18.3a 14.4° 14.4° 13.6° 13.8° 13.8° 14.3° 117----; 477: 352° 135 133 -162° -140° —197° —16.8 118----: -162 -107° -48 -47. 20.5° 18.5° 15.5° 2.3° 119-----: 11.5 5.9: 2.4c 2.5 -.313° -.283° 120-—--: -.211 -.052 .002 l26-----; -.982° -.940° .013 .014 .070 .020° .025° .012° 127----: .101° .096° .0004 .0004 -.002 128-----: -.0033 -.0033 133-----: -820 -.602 -.281 -.273 .240c .21 .3488 .030 134-—---: .266° .172a .098 .097c -.020° -.018 -.024° 135----; -.017 -.008° -.005° -.005b .003 136-----: Table B, Part 17 266 Variable; Equation number °°m°°r f 129 130 f 131 132 133 134 135 136 a: ----- .661 .654 .721 .726 .724 .720 .718 .630 R ----- .654 .648 .706 .711 .710 .706 .705 .613 D.F. ----- 631 632 612 612 613 614 615 616 0 ------ 4467° 4544° 4771° 5364° 5316° 4867° 4535° 5984° 2---- 1803° 1757° 1846° 1882° 1837° 1809° 18lO° 3 ------- .211 .311 .292 .237 .258 .350C .3280 .262 4------: .3290 .207 .133 5-------:-27.8c -23.4 _20.0 _29.5° 6 ------- ; 3.08° 3.1° 3.5° 2.9° 2.7° 3.3° 3.4° 6.1° 7 ------- -218 -23 o la -25 04% -23 o 1% b 8---- -.0005 -.001 -.001° -.001 -.001 -.001 9------ 22.8° 22.4° 19.1° 16.4° 15.3° 18.0° 18.6° 30.4° 14----—- 19.8° 20.2° 19.8° 20.6° 20.3° 19.5° 21.4° 8.7c 117-----§ -8.6 20.9 -3737° -4164° -4095° -3907° -3750° -3854° 118----: 2.2° 5123a 5377° 5301a 5103a 4967a 4909a 119-—--: -2110° 2217° -2187° -2106° _2066° -2032° 120----: 398° 417° 412° 397° 391° 386° 121----: -39.1° -40.9° -40.3° -39.0° -38.4° -38.1° 122—--; 2.1° 2.1° 2.1° 2.1° 2.0° 2.0° 123----: -.055° -.057° -.056° -.055° -.054° -.054° 124—--: .001° .001° .001° .001° .001° .001b 125----:-.286° -.291° -5.7b -6.7° -6.5° -86b -85b -7.2° 126-—--—: 7.9° 9.8° 9.5° 7.8° 8.1° 10.7° 127 ----- ; -3.3§ -4.3° -4.2° -3.3° -3.4° -4.6° 128----: .608 .825° .797° .599b .639b .871° 129----: -.058° -.082° -.079° -.057° -.O61° -.085° 130-—--: .003° .004° .004% .003c .003° .005: 131----: -.0001° -.0001° -.0001 -.0001° -.0001c -.0001 132-----; .0000090 .000001b .000001° .000001 .000001°.000001b 133 ----- : .014 .006 6.6° 7.0° 6.9° 6.5° 6.2° 6.3° 134---: -8.5° -8.9° -8.7° -8.4° -8.2° -7.8° 135--..”: 305a 3068 306a 304a 304a 3.28. 136---—-: -.649° -.677° -.667° -.648° -.635° -.602° 137---; .064° .066° .065° .063° .062° .059° 138-----: -.003° -.003° -.003° -.003° -.003° -.003° 139-—--: .0001° .0001° .0001° .0001a .0001a .0001b l40----: -1x10'6° -1x10’6° -1x10’6° -1x10-6° -1x10"6° .1x10-6b Table B, Part 18 267 Variable: Equation number number ; 137 138 139 140 141 142 143 144 3§----: .557 .380 .724 .708 .218 .233 .238 .256 R ------ .537 .354 .709 .693 .213 .227 .231 .248 D.F.--: 617 618 612 613 640 639 638 637 0-----: 6013° 6486° 4984° 5349° 5228° 5429° 6208° 7389° 2---.-—-: 1899° 1885° 3------- .873° 1.0° .056 .194 4.. ...... .351C .332c 5..--_--. -25.8° -25.2 6------ 8.0° 2.8° 2.8° 7 ------ : -29.7: -24.5: 8------ -.001 —.001 9------ b 16.6° 18.2° b 14-----. 3.4 16.1 21.8° 19.7° 10.1 8.60 1.8 .8 117 ----- -4489° —3261° -2464° -111 134° 73.6 62.2 -289° 118---- 5658° 4005° 2968° 36g 3.1° 3.1° 29.3° 119---- -2332° —1656° -1106° .156 120-____. 442° 313° 182° 29° 121----: -43.5° —30.5° -15.0° -2.9° 122-----: 2.3° 1.6C .614° .165° 123—---: -.061° -.040C —.011° -.005° 124----: .001° .0004 .0001° .0001° 125—----: -10.9° -13.5° -6.5° -2.4 -.309° -.321° -.518° -.893° 126---—-: 14.8° 17.3° 9.4° 4.2 .0120 .037° 127-----: -6.1° -6.9° -4.0° -2.0 128----: 1.1° 1.2° .758° .388 129 ----- : -.105° -.118° -.075° .388 130—----: .005° .006° .004b .002 131-----: -.0001° -.0002b -.0001b -.001 132-----:.000002° .000002b .000001b .000001 133-—--: 7.4° 5.3g 4.0° -.250 -.205b -.162 -.140 .479° 134—---: -9.1° -6.3 -4.6° -.019 -.047° 135----: 3.7° 2.6b 1.7° .002 136-----: -.699° -.479° -.266° .002 137----: .068° .046c .021° -.0002 l38----: -.004° -.002C -.001° .00001 139----: .0001° .0001 .00001° '°° -.000001 l40---:-lxlO 268 Table B, Part 19 Variable: Equation number number ‘ 145 146 147 148 , 149 150 151 152 RS .257 .258 .266 .266 .278 .278 .278 .278 D .F.---: 636 635 634 633 632 631 630 629 0- ----- 2 7790° 7973° 8056° 8123° 8362° 8370° 8363° 8371° 14‘--.--: OZ .6 2.7 2. l 4.9 5. O 5.4 5.3 117---: -325 -364° 97 76. 6 256 297 320 340 118----: 27.7° 30.6° -787c -78 60 -130° -l48 -146 -148 119—--: .152 .104 4.6° 5.0° 7.9° 9.8 7.9 7.9 120—---; -.016 -.O65° -.117 .035 .030 121-—---: -.003 -.003 125----: -.950° -1.3° 8796b 88369 1. 2 L 2 L 2 L 4 126---: .039° .1030 -.004 .001 — .865° 8879° 8858° 8997b 127----: -.002 .002 .002 .089° .090° .088° .112 128----; -.002° -.002° -.002° -.004 129---: .00003 133 ----- : .563° .622° -.182 -.124 -.390 -.462 -.538 - .573 134-----: -.050° —.053° .136c .126c .189° .220 .240 .242 135--—-: -.008° -.007 -.010° -.013 -.014 8014 .001 .0001 .0001 136----§ Table B, Part 20 269 Variable: Equation number number 3 153 154 155 = 156 157 158 .f 159 160 Egmu-3 .295 .297 0315 0317 0321 0339 0358 0362 D;F.--: 628 627 626 625 624 623 622 621 0-----; 8579a 8615a 77988 8014a 7757a 7761a 7135a 7039a 14 ------ : 6.1 6.10.3C 10.8c 12.9b 9.1 11.10 9. 7 117----: -1042b -1065 -937b -292 -161 -338 -2789a -25268 ll8----: 677a 669b 644a 46.8 13.1 173 2864a 2694a 119 ----- : -130a —121a —121a 37.8 16.6 -31.1 -1018a -1016a 120—--—-§ 8. 5a 7.2a 7. 8a -9.2 -2.6 4.0 164a 174a 121--8 - .178a -.112 8165b .613 -.025 —.442 -13a -l4.8a 122----: -.001.0002 -.013 .01 .025 .491a .619a 123---: -.0004 — 0005a -.007a -.011a 124----: .0001C l26----: -.215 -.172 3. 2a 2. 4: 2. 3 9.1a 11.7a 11.1a 128 ----- : .003 .003 094a .073a W075 .440a .579a .558a 129----: -.0001 -.0001 -.004a -.003a 8003a -.030a 80418 -.039a 130—---; .0001a .0001a .0001a .001a .001a .001a 131 ----- : —.00001a 00002a -.00002a 133----: 1.9b 2.0b 1.70 .557 .116 .437 4.8a 4.1b 134----: -1.2a -1.38 -1.1a -.077 .248 .015 -4.7a -4.0a 135----: .225a 229a .204a -.071 .248 .015 -4.7a -4.0a 136-----; —.015& -.015a -.013a .017 .021 .015 8265a 8243a 137 ----- : .0003a 0003a .0003a -.001 -.001 -.001 .021a .020a 138-----: .00002 .00002 .00002 8001a 80018 .00001a .00001a 139----: Table B, Part 21 2'70 Variable; Equation number number 3 161 162 163 164 165 166 167 168 g§-----: .364 .365 .345 .342 .340 .338 .335 .334 R ...-n: 0339 0339 032]. 0319 0318 0317 0315 0315 D.F.---: 620 619 621 622 623 624 625 626 0-—---: 7103a 72118 7687a 7483a 7312a 7328a 7211a 6847a 14—--—-- 11.60 11.4c 8.6 8.2 8.3 9.1 6. 7.8 117----: -2590a -3451a -80.4 -868° -403 -115 -483 -370a 118----: 2867a 4066: 80.3 787a 542a 418a 526a 514a 119-----: -10903 -1639 -70.8 -245a _221a _211a _227a -226a 120-.--§ 187a 3032 23°8 38.6a 40.98 40.78 43.1a 43.1a 121 ----- : -168 —28.8 -3.2 -3.5a -4.1a —4.0a -4.2a —4.2a 122—-—--: .68a 1.4c .205b .184b .214a .210a .219b .220b 123——---: -.013a -.036 -.006b -.005b -.006b -.006b -.006b -.006b 124-----:.00007b .0003 .0001b .00010 .0001b .0001b .0001b .0001b 125-----; —13.5a -13.8a -9.3a -11a -1L2a -10.3& —10.1a —10.5a l26-—--—: 16.5a 17.1a 11.4: 12.9: 13.5a 12.5a 11.5: 12.2: 127----: -6.29a -6.5a -4.2 -4.7b -5.0: -4.6: -4.1 -4.3 128----: 1.07% 1.1: .695C .751 .809 .743 .617c .665c 129—--—: -.094 -.101 -.059 -.063 -.068° -.062 -.O48 -.052 130---; .0052 .0052 .003 .003 .003 .003 .002 .002 131-----:-.0001 -.0001 -.0001 -.0001 -.0001 -.0001 -.00004 -.00005 132———-:.00000% .000001 .000001 .000001 .000001 .000001 .0000003 .0000004 133-----: 4.1 5.7 -.372 1.1 .191 -.354 .277 .083 134-----: -4.3a -6.4b .679 -.666 -.122 .128 -.015 135-----: 1.563 2.5b -.240 .1120 .020 -.006C 136-----: -.258: -.460: .031 -.006 -.001 137-—---: .021 .043 -.002 .001 138-----:-.0008: -.002 -.00003 139-----:.00002 .0001 140—.——---§-5x10‘7 Table B, Part 22 271 132 ----- :.0000007 -1x10'7 Variable; Equation number number 3 169 f 170 171 172 173 174 f 175 176 32----: .703 .696 .695 .690 .681 .681 .680 .678 R """""‘-""" : o 691 o 684 o 684 o 678 o 670 o 671 o 670 o 668 ILF.---—: 619 620 620 621 622 623 624 625 0...-..-3 4884a 45418 5844a 57228 58458 5861a 57398 5829a 2——--—— 18868 18098 17728 18188 18308 18318 18338 18218 3 ------- .171 .343C .112 .129 .194 .193 .206 .192 4 -------- .371b .149 .287 .263 .324C .3220 .3609 .369C 5----- -25.5 -18.0 -23.3 -24.3 -23.8 -23.8 -23.6 —22.3 6-------: 2.67a 3.3a 3.15a 3.11a 3.07a 3.06a 3.09a 3.12a 7 ------- : -30.38 -26.28 -22.18 -26.68 _26.68 -25.38 -27.48 8------: 800070 -.0005 -.0018 -.0018 -.0009b -.0009b -.0009b -.0018 9 ------ : 18.08 21.18 17.68 17.78 17.58 17.58 17.78 17.48 14—-----: 18.48 16.08 117----§ -297a -28Ba _29sa -115b 41.9 44.8 6.76 .276b ll8-----: 431a 413a 433a 172a _26.9 -29.BC -4.31 11.1a 119--—-: -1898 —1788 -1938 -70.58 4.97 5.76b 1.23 -.4248 120-——--: 37.0a 34.28 38.28 11.9a -.255 -.342° -.O48b 121--—-: -3.738 -3.48 —3.888 -.9398 .002 .0060 122---; .2018 .1768 .2128 .0358 .00007 123-----: -.0068 -.0058 -.0068 -.00058 124----: .000063 .00005a .00007a 125-----: -2083 -109 -1093 0430 -1003 “'0992 -1029 ”1.029 126 ----- : 4.520 2.5 3.01 -.390 1.58 1.55 1.65 1.41 127’"--"; “2.050 ”0891 “1033 0254 "o708 -070 -0729 -060 l28----: .392 .120 .233 -.104 .114 .114 .124 .103 129-----: -.039 -.007 -.021 .017 -.009 -.009 -.010 -.009 130 ----- : .002 .0001 .0010 -.001 .0003 .0003 .0005 .0004 131—---: -.00006 .000001 —.00002 .00004 -.000006 -.000006 -.00001 -.00001 .0000002 —6x10'7 3xlO-8 4x10"8 .0000001 .0000001 Table B, Part 23 272 Equation number Variable: number 3 177 178 179 180 181 182 183 f 184 53--m-: o 672 o 669 o 644 o 644 o 643 o 639 o 627 o 625 R “--.“: 0662 .660 .635 0635 0635 .632 0620 .618 D.F.--: 626 627 628 629 630 631 632 633 0 ...... Q 59798 60408 80518 78638 78968 79778 81298 80268 2—-—-—- 1804a 1767a 1553a 15588 1559a 1533a 1463a 14648 3 ....... .27 .324 .027 .027 .057 .074 .008 -.016 4 ....... 0373 .287 0247 o 233 0236 0218 .120 0135 5-----: -25.3 -21.1 -24.5 -24.0 -24.8 -25.8 -29.4° -32c 6 88888 E 3.268 3.268 3.688 3.718 3.708 3.72a 3.93a 3.898 7- ----- : -24.7: _26.28 -30.78 _28.58 -30.98 -34.28 _22.48 -23.4a 8-----: -.0008 -.OOO8b -.0028 -.0018 -.0028 -.0018 -.0018 -.0018 9------: 17.38 ‘ 16.2a 16.5a 16.8a 16.7a 18.4a 22.11a 22.5a 117----: 8.16 26.8a 118-----; 1.71b a b 125---: -2.08 -3.0 -2.46 -l.08 .039 1.73 ’22? -.481 126----: 2.81 3.76 3.26 1.34 .009 -1348 -.342 -.022 127----: -1.32 -1.61 -1.45 -.583 -.115 .2118 .034b .001 128----: .263 .302 .272 .092 .021 8012a 80009b 129----; -.027 -.030 -.026 ~.007 8001b .00028 130----: .002 .002 .001 .0002 .00002b 131----: —.00004 -.00004 -.00004 -.000003 132----:.OOOOOO5 .0000005 .0000004 Table B, Part 24 273 Variable; Equation number number ‘ 185 186 187 188 . 189 , 190 191 192 B§------ . o 624 o 618 o 703 o 703 o 702 o 698 o 697 o 684. R ------ .618 .613 .691 .691 .691 .688 .687 .674 D.F.---- 634 635 619 620 621 622 623 624 0——-——-- 8041a 79768 48848 4697a 46478 50068 48878 50148 2 —————— 14818 13958 18868 18918 19028 18838 18798 18428 3----..- . o 004 o 014 o 171 o 157 o 181 o 148 o 131 o 066 4 -------- .139 .049 .371b .341c .356c .3260 .306 .333c 5 -------- -3688 -22.0 -25.5 -24.8 -25.6 -28.2° -29.28 -31.3 6-—--——- 3.89a 4.12a 2.67a 2.73a 2.70a 2.73a 2.76a 2.848 7----: -21.58 -20.28 —30.38 -26.98 -28.48 -29.38 -24.28 -25.68 8--——--: -.0018 -.0028 -.0007° -.0007° -.0007b .0006C -.0006 -.0008b 9 ———————— 22.5a 21.98 188 18.28 18.38 20.38 21.68 21.88 14 ------ 18.48 17.28 18.08 16.38 16.98 13.68 117 ----- _297a _283a -285a -2548 —2678 —2718 118---- 4318 4078 4048 3708 3878 3528 119---- -1898 —176a —1758 -1668 -173a -1528 120---- 372 33.9a 33.78 338 34.58 3038 121—_——-: -373 —3.34a -3.34a -3.348 -3.51a -3.118 122—-—--§ .201a .177a .1778 .1808 .1918 .1718 123--—--: -.0068 -.0058 -.0058 -.0058 -.0058 -.0058 124--—-—: .000068 .000058 000058 000058 .000068 .000058 125----: -.6538 -.4748 -2.83 -.681 ..253 2.158 1.528 -.5938 126 ----- : .013a 4.520 1.47 .332 -1.29a 88928 .036 127-----§ —2.05° -.648 - 24g .163a .094a -.0008 128 ----- : .392 .096 .035 -.0078 -.0038 129----—: -.039 -.006 -.0028 000090 130----: .002 .0002 000038 131-----: -.00006 -.000002 132-----; .0000007 Table B, Part 25 274 Variable; Equation number number 3 193 ‘ 194 f 195 f 196 f 197 198 199 200 52""“; .684 .678 .334 .333 .311 .282 .282 .269 R -----: .674 .669 .315 .316 .295 .266 .266 .255 D.F.---: 625 626 627 628 629 630 631 632 0—-----: 5g2l: 4588: 6284a 6126a 5974a 7138a 7112a 7077a 2—----: 1 32 1770 3------: .059 .043 4—-——-: .325C .275 5-----: -31.60 -23.4 6 _____ : 2.85: 2.93a 7------: -26.5 -28.2 8-—----: .0008b 0009 9------: 21.8a 21.7a 14----: 13.6a 18.8a 8.4 7.8 10.90 5 5.2 3.0 117----; -2698 _2738 -3358 -324b -3428 -258b -261b _256b 118----: 3538 3408 5268 507a 504a 500a 413a 3708 119—---: -153: -l46: _231: -2212 —219: -194: -1962 ~1712 120—---: 30.5 29.4 44.1 41.7 39.8 40.2 35.3 121—----: -3.138 —3.058 -4.318 -4.018 -4.148 -4.13a -4.17a -3.708 122———-; .172a .1708 .2268 .207b .219a .227a .2298 .2068 123-----: 80058 -.0058 -.0068 —.005b -.0068 -.0068 -.0068 -.0068 124----:.00005: 000058 .00006: .00006: .00006: .00007: .00007: .00006: l25-----: -.492 -.282 —lO.6 -8.82 -3.53 1.34 1.24 -.773 126-----: .015a 12.5a 10.1a 3.37a 8919b 8846a .037 127 ----- ; -4.48§ -3.428 89768 .1020 .0898 -.0006 l28—----: .703 .483a .108a —.003 —.0028 129 ----- : -.057 8033a -.0058 .00002 130--—--: .002 .001: 00008a 131-----: -.00006 -.00001 132—---; .0000005 275 126 ----- : Table B, Part 26 Variable; Equation number number ‘ 201 202 203 204 205 206 207 208 B:------ .269 .256 .654 .211 .647 .641 .636 .636 R ------ .257 .244 .648 .207 .638 .632 .628 .627 D.F.--- 633 634 633 641 627 628 629 630 0 ------- 7091a 6183a 4500a 6699a 274b 207c 207c 206c 2--—-: 17588 1917a 19718 19808 19668 3-------: .315 .103 .144 .179 .189 4-----: .207 .554a .5688 .5948 .6168 5----: -23.3 -29.4C -32.10 -32. 0c -32.5C 6- ------ 3.09a 2. 91a 2.77a 2.80a 2.868 7 ....... _22.98 -32. 48 -3.l68 -32.38 -32.38 8------ -.0005 .0001 .0002 .0004 .0004 9-----: 22.38 22.68 22.98 22.78 23.18 14—-----: 2.9 12.98 20.28 8.70 30.08 33.18 30.68 30.08 117 ----- Q -254b _2428 24.48 13.68 -3298 -1448 _20.1 —46. 7c ll8----: 3718 331b 4178155a 8.32 32.2b 119 ----- : -1728 -152b -1738 -52. 2a 1.72 -4.610 120 ----- : 35.48 31. 7a 33. 8: 8. 24a -.387 .291C 121----: -3.71a -3.37 a 3. 46a 8644 .025 80060 122----; .206a .191: .191a .0248 -.0005 123---: - .006a — 805 80058 -.0004a 124 ------ ..00006b 80006b .000068 125--—-: -.7008 -.3778 -.2888 -.4148 .002a 276 Table B, Part 27 Variable: Equation number number f 209 210 211 'f 212 213 214 215 216 a; ------ 2 .633 .633. .629 .619 .115 .138 .139 .141 R ——---: .626 .626 .622 .613 .112 .134 .133 .134 D.F.---: 631 632 633 634 642 641 640 639 0--—-———: 198c 1960 95.2 92.1 5468 6068 6048 6068 2 ------ 1978a 19808 19848 1934a 3-------: .182 .180 .262 .374g 4—------: .594: .5988 .584: .451 5-------: -34.3 -34.38 -34.7 -29.9° 6 ------ 2 2.84a 2.84a 2.988 2.988 7------: -32.18 -32.38 -25.28 -28.28 8---—-: .0004 .0004 .0006 .0006 9 -------- 2305a 2305a 2305a 23.28. 14- ----- 29.68 29.68 31.18 31.98 16.68 14.88 15.18 15.38 117-----; -11.4 -17.2 11.1 40.28 33.68 -10.1 -5.05 21.0 118-----: 7.38 10.18 ,2.598 3.888 2.58 -9.50 119-—--: -.049 -.348 .058 1.35 —.009 -.038 120 ----- : Table B, Part 28 277 Variable: Equation number number 3 217 218 219 220 221 . 222 223 224 3%------: .147 .158 .172 .184 .051 .538 .461 .459 D.F . —--—: 636 637 636 635 643 635 636 637 0—-----; 629a 5948 5608 6188 8838 6528 9968 9468 2......-: 16998 12638 1233a 3n-m-: .- 356 "'0 384 4-----..: .6268 .465: .412C 5---—--: -51.68 -39.5 -39.80 6—--—-—-§ 3.658 5.518 5.778 7-----: -37.48 -30.98 -27.88 8-----: .0003 -.0006 9------: 25.78 23.58 23.98 14 ——————— 16.28 18.98 24.08 21.18 29.58 42.48 117----§ -41.3 62.2 135: -3908 -.188 118----: 34.3 -59 174 5378 119----: ~6.69° 18.1c -67.2b _2358 120-----: .490: _2.17b 11.48 47.08 121---: -.011 .112b 8939a -4.878 122----; -.0028 .0378 .2708 123----: 800058 -.0088 124—---: .000098 Table B, Part 29 Variable: Equation number num°°r ‘ 225 226 227 228 229 230 . 231 232 §§----- .446 .445 .444 .405 .301 .280 .402 .398 R ------ . 441 . 440 . 440 . 402 . 299 . 2'7 8 . 396 . 392 D. F . --- 638 639 640 641 642 643 637 638 o---——-_ 6708 6348 6758 6278 5378 6398 15118 14758 2 —————— 10808 10518 10578 16638 16198 3-—--- -.040 -.062 .027 .205 1.018 -.580b 4"" ------ o 142 o 20]. o 291 o 039 5 ....... -27.0 -8.85 -8.34 6 —————— 6.468 6.268 6.098 7.108 9.288 9.208 7..-..--. -51.98 -47.28 8 ........ -.0028 -.0028 9 ....... 27.68 26.68 26.18 338 23.58 22.08 278 Table B, Part 30 Variable; Equation number number ‘ 233 234 , 235 236 . 237 238 239 240 33.....- o 361 o 194 o 193 o 166 o 005 o 017 o 260 o 345 R --.-... 0356 .189 0189 0163 .004 .015 0258 0342 D.F.--- 639 640 641 642 643 643 642 641 0------: 16188 15308 1561a 18548 1207a 10888 10318 10188 2--—---- 20078 15008 3------ .898a -.218 -.386C 4------ .160 .253 5-----' 15.5 109a 1053 7-—-----: -64.88 -62.68 -6O.48 -74.68 8-----: -.0028 -.0028 —.0028 -.0028 80009c 9 ..... g 47.18 34.48 Table B, Part 31 Variable; EQuatlon number number : 241 242 243 B: ------ o 350 o 3 51 o 389 R ----—- .346 .346 .384 D.F.---- 640 639 638 0 nnnnn 1097a 10518 14398 2.. ------ 14838 14458 1633a 4----: 8508b 8435c .108 5-----: 24.4 -5.45 7.1-1-; -458 9 ------- 32.68 31.58 24.98 279 Table B, Part 32 Variable; Equation number number ‘ 244 245 246 247 248 249 250 251 B ...-u.- o 816 o 798 o 697 o 677 o 805 o 785 o 642 o 619 R ------ .779 .767 .640 .632 .768 .754 .578 .570 D.F.--- 536 559 543 566 540 563 547 570 0—---- 31618 822 57878 31298 35638 18258 7615b 57128 2.. ...... 1764a 17498 1759a 17308 3-----.. - .249 -.O88 .315 .445° 4- ------ .241 .210 .064 .068 5--——- -18. 9 —15.7 -7.2 -5.0 6-----; 3.18 2. 9a 3.9a 3.68 8-----—-: -.0001 - 8002 .0003 .0001 9------: 12.08 11.78 13.98 13.48 10—--—--: 5368 5688 9438 9258 11—-----§ 3558 2288 5838 4448 12_——---: 455 4058 5428 4828 13 ------ : 3788 3458 374a 3498 14------: 20. 68 18.28 18. 78 18.78 27.98 24.3 29. 5: 26.18 15-----: -514°1.5 -669° 20.9 -418 6.7 -7. 9° 17.9 l6-—-———§ 258 25.3 23.5 49.0b 163 24.3 -496 26.9 17------: 3820 58.78 21.9 83.78 408C 80.58 390 1098 18------: —159 11.2 140 24. 3 -339 13.3 -71.2 39.30 19 ----- : 193 -25.6 13.1 -22. 271 -17.3 183 -31 20-——---: -22.9 -4l.8 62.6 -46. 3 35.2 -47.8 -83.1 -95.68 21 ----- ; -512° 17.7 -820b -8.6 -348 40. 98 .499 14.8 22 ----- : 137 -12.6 559c -20.8 100 -5. 6b 561 -31.1 23-----: -360 20.9 -638C 45.3b -90.8 18. -318 28.5 24-----: 205 21.7 262 12.2 508 41. 7 .173 32.2 25 ------ : 201 41.70 83.7 -6.4 95.6 45. 4b -30.3 9.4 26-----; -380 64.58 -582 60.6b -543 73. 68 -833 75.5b 27 ------ : -88.6 14.4 110 -18.7 -49.0 23. 9 432 13.8 28-----: -771b 38.7b -464 41.38 -9098 36. 2c ~644 30.8 29---—-: -9428 22.6 -524 _29.0 —10678 39.10 -541 -3.6 30------: -10128 57.0 -10948 18.2 -9218 68.78 -9548 23.1 3l---—-§ -815b -ll.8 -261 -20. 7 -826b -8.0 _279 -l7.5 32----: -322 14.2 .533 8. 4 -335 12.2 -1.1 -3. 6 33 ------ -39.6 _22.4 —111 —34.6 -138 -10.4 -523 -4.4 34 ------ 108 79.68 -4.1 52. 5% 129 96.18 236 66. 6b 35 ------ -36.6 -23.4 -313 -69.6 196 -11.0 172 -30. 7 -Continued 280 Table B, Part 32.-—Continued variables Equation number number ‘ 244 245 246 247 . 248 249 250 251 36 ------ -613b -27.2. -761° -45.9 —600° -38.4 -443 -68c 37---""- “'32 o 5 “'61 o 5 109 '77 o 2 940 6 -25 o 2 35 -53 o 7 38--—--— .115 .336 -1.1 -.741 .018 .238 -3.48 -2.58 39------ 1. 3C .359 2.88 2.38 .629 -. g 1 4 1.5C 40 ------- — .894 —1.78 -1.88 -1.88 -.826 -1.5 -.927 -1.4° 41—----- .987 1.2 1.0 .262 .02 .560 -.263 -.432 42--—-- .947 .78 2.2b 2. 48 .821 .214 2.6a 2.18 43------ -.984 - .089 -2.3C -2. 0c .103 1.1 -1. 9 -1.1 45----- —2.38 -2. 38 -1.1 -L70 -2.48 -2. 68 -.703 -1.50 [56------ "2 o 2 "l o 5 "l o 9 - o 569 "'3 o 68 "2 0 9a "4 0 3a -30 3a 47...-..”0 -0997 ”1.1 0452 .003 “0456 -0645 0410 .681 48------ -0116 -033 -0357 '06500 -0376 -04510 -0384- -0556 49—--——- 2.00 2.1 -.647 .217 2.7b 3.08 .620 1.6 50—--—- _2.98 -3.38 -2.8 -2.8b _2.ob -2.98 -1.5 -2.5b 51------; -1.3 -.094 -1.7 —1.2 -1. 3 .006 -1.9 -1.3 52----"-: 0799 o 313 o 057 0972 -0987 0496 lo 2. 6a 53------: 1.90 2.2b .205 .745 2. 98 3.48 2.7 3. 38 54 ------ : .593 1.2 -1.4 -.775 .294 .216 -3.58 -2. 9b 55------: -.606 —.105 .597 .821-.590 .008 .484 .827 56-_———-§ 1. 9a 1.8b 3138 2. 98 1. 6b 1. 48 3.5a 3.58 57------: .006 .208 -1.0 - .938 .288 .231 -2.08 -2.28 58 ————— : -.227 -.004 -1.6C -1. 0 .082 -.377 -1.2 -1.50 59----: -1.68 -.974 —2.5 -1.6 -1.1 -.644 -1.8 —1.0 60-----': -0387 -0956 1.40 0954 '0088 "loéa 0396 -0409 61—-----; .941 -1.38 .768 1.3c 62 ------ : -.427 .040 —.264 .939 63"----- -0 597 o 110 "o 620 ""o 520 64 ------ o 277 - 200 o 576 -0173 65---“- .367 “.077 ‘0484 -0375 66-————-§ -.004 -.139 -.117 -.002 67----: .8520 1.3b .606 .828 68----—: -.251 -.967c —.191 -.975c 69-—----: .585 1.10 .142 .522 70 ...... : “.283 0399 '0 731 0319 71---—-—; -.254 -.139 —.097 .052 72 ----- : .663 .958 .945 1.4 -Continued 281 Table B, Part 32.-Continued variable; Equation number number f 244 245 246 247 248 249 250 251 3----: .16 -.199 .106 -.683 74“-—": 103 .826 105a lo]. 75—--—--: 1.68 .827 1. 8: .903 76-—--—-: 1.8a 1.9a 1. .7b 1.6a 77-----: 1.3b .378 1.3 .459 78--—--§ .577 .033 .605 .023 ’79 ------ : -. 014 . 116 . 190 . 893 80-—---: — .082 .092 -.O85 -.337 81------: .013 .450 - .382 -.368 82------: 1.001.2°.976c .641 83-—————§ —.117 -.381 —.305 -.229 84------: 539a 2.273 207 161 515a 363b 250 92.3 85------: 625a 1.61a 391b 220 575a 385a 275 10 86-----: 134 65.5 -175 -28l° 136 50.5 -230 -342 87-----: -60.6 -51.1 -66a .660a -64.6 -88.6 -839a -839a 88-—---: 30.9 -54.2 -5913 -639a 16.6 —100 -793a -840a 89- ————— 3030 27.4 -218 -392b 240 -92.1 -340 -546a 90- ----- -35.6 -32.3 -588a -597a -65.8 -79.2 -801a -779a 91 ------- -75.8 -125 4.72a -5483 -131 -169° -677a -684a 92 ----- 201 52.1 -235 -377° 122 -41.1 -450b -569a 93----- -103 -93.9 -557a -5662 .1500 -1516 -687a -648a 94 ------- -184e _255a —55oa -6542 -265a -326a -771a -813a 95 ------ -l8.6 -111 -4843 -559a-136 -206 -693a -7078 96 —————— -245a -255b --811a —833a -206° -222b -874a -858a 97 ------ -2.6 —52.6 -469b -537a -10.8 -4.7 -540b -519a 98 ----- Q -321 -335b -789a -8003 —362b -301c -921a -818a 99-----: -3803 -435a —9003 -1006a 4.01a -440a -1023a —112/.a 100——-—-: -3000 -389b -587a -702a _288° -363b -639a -766a 101—----: -78.8 -12 -660a .660a -35.7 -92. 6 -597a -6633 102-----: -1990 _265 -368a -475a -152 —174° -222 ~331b 103----; -172 -293b —559a -732a -133 -202c -568a -703a 104----: -130 _259 -560a -760a 6. 7 -62.2 -420° -5603 105---—: 13.5 -31.7 -346 —546° 55. 8 110 -459 -5o9 106 ----- : —10.2 -138 -62. 6 -391 -99.8 -109 -389 -5170 107-----: -276 -l87 -388 -460 -257 -79 -561° -489 lO8----; 127 140 —35.3 -111 -47.8 104 -375 -29 109 ----- : —188 -214 -392 -533° -258 -130 -6580 -651 —401 -354 -102 -191 -568b -362 -337 -224 110-----: Table B, Part 33 282 Variable; Equation number number ' 252 253 254 255 256 257 258 259 3§-----: .777 .776 .702 .702 .702 .676 .670 .650 R ---: .747 .746 .662 .663 .663 .634 .628 .607 ILF.---: 566 567 568 569 570 571 572 573 0------: 51283 5050a 6463a 6333a 6292a 7182a 64698 54438 2------: 1862a 1830a 3-------: -.222 -.l97 -.168 4------: .258 .269 -.011 -.058 5-------: -26.0° 6------: 2.8a 2.6a 5.7a 5.7a 5.78 6.2a 6.5a 7.8a 7-------: -49.4§ -47.7§ -33.2: -30.6% -31.1ba -56.4a -58.1a 8—-—---—: -.001 -.001 -.001 -.001 -.001 -.002 9--—-—-: 13.28 12.0a 22.1a 21.8a 21.8a 14—-----: 22.1a 22.1a 8.11 8.41 8.63 3.82 5.94b 4.42 15-----—: -630b -6l9b -664a -648b -651b —802b -740 ..581c 16 ------ : 357 35 361 348 348 288 526 405 17------: 428a 424 266 268 264 227 108 393 18 ------ : -292 -281 -153 -144 -138 43.2 -27.9 -67.7 19----: 223 223 124 132 131 34.7 33.9 168 20-—---: 208 190 390 366 364 342 338 147 21---—-: —672b -637b -731b -720b -725b -724b —688b —382 22 ----- : 161 134 274 27 284 282 310 261 23—--———: —511° —519° -744b -719 -712b -740b -726b -713b 24—----—: 839b 828° 860° 872° 879° 1023 1118b 770 25-----: 271 286 273 259 248 341 262 185 26---——-: —456 —410 —912 -882 -896 —912 -981 -800 27 ------ : 312 291 265 251 255 318 403 567 28-----: —694b -697b —624° .611° -597° -718° 648° -698° 29 ------ : -527 -529 -432 -426 -434 -329 -362 -304 30 ------ : -414 -40 -779a -772a -764a -765b -788a -794b 31 ------ : -712b -714 -510 -482 -473 -541 -507 -537 32---—-—: -273 -49.7 -281 -250 -242 -372 -347 -328 33--—---: 157 185.2 78 91.4 92.9 -97 8.20 -164 34—-----: -85.6 —88.1 —175 -162 -161 -236 -257b -99.8 35-----: —68.6 -43.7 —77 -76.6 -79.4 -216 -184 8.40 36 ------ : -237 _240 -413 -400 -395 -207 -185 184 37------: -171 -190 -318 -314 -311 -364 -314 -387 38------: .841 .871 .665 .663 .642 .301 .754 -.51 39 ------ : .894 .929 1.34C 1.28° 1.28 1.71b 2.30a 2.12b -Continued 283 Table B, Part 33.--Continued Variable; Equation number number 3 252 253 254 255 256 257 258 259 40----: -.702 -.712 -.916 -.930 -.92 —1.12 -1.26 -.493 41 ------ .013 .047 .138 .11 .112 1.04 1.0 .402 42----- .495 .560 1.588 1.57 1.56b 2.06b 1.90 2.67a 43------ -.713 -.886 -.397 -.444 -.451 -1.11 -.908 -1.47 44----- .253 .327 -.812 -.775 -.791 -1.60 -1.60 -2.39b 45-----: -2.87° -2.96° -2.42° -2.46a -2.438 -2.53b -2.55a -2.56a 46-----: -2.87a -2.80a -4.22a -4.09a -4.06a -3.82a -3.62a -4.18a 47---- .166 -.008 2.37b 2.37b 2.39b 2.14b 2.39b 2.46b 48-—--- -.332 -.337 -. 72° -. 71° -.572° -.783b -.805b -.886a 49.. ————— 1.14 1.33 1.33 -1.21 -1.19 -1.62 _2.12° -1.71 50—---: —1.740 -1.75° —1.44 -1.58 -1.61 -1.10 -.474 .217 51--—-: -1.33 -1.33 -1.78° -1.73° -1.74° -1.91° -2.16b -1.44 52—-—---: -1.05 —1.07 —.716 -.682 -.664 -.291 —.633 .562 53------: 1.58° 1.69 1.36 1.52 1.53 .862 .618 2.32° 54-----: 1.39C 1.440 .120 -.009 -.040 -.208 -.022 -1.35 55 ...... .181 .140 .348 .381 .381 1.06 1.10 1.07 56 ------ : 2.37a 2.438 3.04b 3.10a 3.11a 3.95a 3.99a 4.15a 57—-----: -1.52a -1.66° -1.79a -1.78a -1.76a -3.00a -2.91a -3.02a 58 ————— -.697 -.673 -.832 -.873 -.897 —1.530 -1.500 -.89 59--—- -1.22 -1.17 -1.73° -1.72° —1.73° -1.88° —1.74c -1.43° 60----: -1.24b -1.34b -.978 -.977 -.940 -.221 .02 -1 26b 61---: 1.12b 1.10b 1.16b 1.13b 1.13b 1.41b 1.33 1.01 62------: -.541 -.531 -.536 -.514 -.514 -.397 -.796 —.588 63----: -.658° -.647° -.356 -.355 -.348 -.26 —.046 -.549 64--——--: .498 .481 .243 .228 .218 -.087 .034 .102 65-----: -.381 -.376 -.217 -.228 -.228 -.070 -.054 -.270 66-----: -.345 -.31 -.682° -.643c -.641° -.599 -.580 -.332 67---"': 1.14a 1.08 1.20b 1.18b 1.19b 1.17b 1.10b .608 68 ----- : -.290 -.244 -.502 -.510 -.517 -.506 -.552 -.480 69-----: .839° .854° 1.21b 1.18b 1.17b 1.21b 1.19b 1.15b 70-----: -1.27 -1.250 -1.31c -1.33° -1.340 -1.59b -1.25b -l.20 71-—--: -.350 -.373 -.346 -.325 -.309 -.455 -.332 .204 72—-----: .776 .706 1.45 1.41 1.43 1.46 1.55 1.29 73-----: —.449 —.416 —.402 —. 82 -.387 -.495 -.623 -.866 74 ----- : 1.20b 1.21b 1.08° l.06° 1.04c 1.24b 1.130 1.21c —Continued 284 Table B, Part 33.--Continued Variable; Equation number number ‘ 252 , 253 254 255 , 256 257 258 259 75---“‘: 09660 09670 .7680 .760 0773 a 593 .638 o 557 76------: .8480 .829C 1.49a 1.48a 1.47a 1.47a 1.51a 1.52a 77------: 1.18b 1.19b .784 .738 .726 .818 .783 .862 78----: .062 .096 .494 .445 .432 .624 .611 .589 79--—-—: -.327 -.356 -.205 -.224 —.225 -.052 -.074 .245 80-----; .295 .291 .437b .411c .411° .488b .535b .253 81 ————— : .048 -.005 -.004 -.003 .003 .206 .170 -.147 82------: .346 .352 .620 .600 .594 .248 .226 -.390 .134 .166 .385 .381 .380 .481 .415 .486 83-----: 285 Table B, Part 34 Variabl; Equation number numberf 260 261 262 263 264 . 265 266 267 32......g o 774 o 588 o 514 0767 o 779 o 511 .482 o 365 R ---—...: 074-3 0 534 0455 0735 0747 0452 0442 0315 D.F.--—-: 565 570 574 567 563 575 597 597 o------: 3993° 6366° 6699° 4174a 4539° 7456° 6284° 724a 2 ——————— 1801a 1780a 1820a 3--—-—--: -.311 .192 -.403c 4—-—-—--: o 008 o 250 5—--—-—-: -29.1° -18.4 -24.7a 6—-----; 3.2° 3.6: 2.9° 8-----: -.001 -.001a 9------: 15.0a 16.7a 13.1a 10 ..... : 418° 862° 474° 11 ..... Q 309° 592° 372° 12-----: 224b 272c 257b 13----: 239a 15 261° 14----: 22.2: 16.2 14.1b 21.7a 20.9: 11.2C 19.0° 15-—---: -549 -588 -410 -458 —646 -477 246 16----: 466c 451 179 249 273 167 -274 17 ------ 557a -3.13 227 622a 645a 111 397 18----. -262 62.9 -107 -330 -156 -194 -193 19-----—: 218 -134 62.3 341C 203 88 16.4 21-.—--§ -517 -6710 -372 -443 -580b -309 713c 22------: 18 5720 464 131 174 501 879a 23 ----- : -601 -722° -680 -467 -619b -645 760° 24--—--—: 546 870 167 604 507 117 309 25 ----- : 244 147 6.79 206 289 -334 227 26-——..—§ -316 —918 -83% -243 -316 —95 467 27-----: 316 567 1138 34 242 1110 1310b 28 ------ : -761b -980b -921b -701 —815b -881° —575 29----: -45g 11 33 -477 -456 407 -5.00 30----: -521 —871 -862 —429° —490° —835b -848b 31-_--—-§ -594 -337 -480 -662° -594 -432 -275 32 ------ : 70.0 153 136 47.4 55.6 153 324 33 ----- : 134 —17 -5650 103 131 -561 -744b 34-—---: —50.5 -306 -7.06 51.1 -44.6 -.795 579b 35 ----- : -17.1 133 363C 88.6 -66.6 335 912a -Continued Table B, Part 34.--Continued 286 Variable: Equation number number ‘ 260 261 262 263 264 265 266 267 36—----: -184 -240 506 -31.6 _200 438 1142: 37-----: _220 .275 .412 -215 -270 -421 -865 38---—: .832 .438 -2.08b .147 .397 —1.83° -1.0 39-----: 1.19C 2.99a 3.30° .570 .767 2.93a 1.56 40----: -.198 -.884 -.156 -.253 .009 -.718 -1.38 41----- .062 1.1 .423 -.606 .193 .43 .631 42 ————— : .657 1.83 2.51b .905 .791 2.26 2.75a 43------: -1.33 -1.80 -l.88 -1.1 -1.60 -1.40 —1.3 44--—--: .180 —1.22 -3.11° .257 .335 -3.18° _2.54 45-----: -3.22° 3.21a -2.98a -3.0a —3.2a -3.27a -3.858. 46—----—: -1.97° -2.38° —4.21° -3.1° -1.9° 4.15° -3.63a 47---—-: .121 2.160 1.91 .224 —.084 1.6 1.880 48-----: -.293 -.802b -.804 - .346 -.282 —.840 —1.20° 49------: .22 -3.538 -2.24 1. 7C .628 .2.440 -1.49 50__-—--: -2.01 —.814 .835 -2. ob -2.7° 1.14 -.746 51-—----: -1. 01 -2.69b -2.70b -.717 .890 2.85b -2.422 52—----: -.902 -.584 1. 45 -.316 -.557 1. 3 2.13 53-----: 2. 24b .552 2. 77; 3.2° 2.4b 3.15b 3.50a 54------: L 23 -.592 -2. 46 .311 1.1 -2. 87b -1.62 55—----: .051 .262 — .242 .103 .004 -.227 -.138 56 ..... . 2. 33° 5.08° 6.43° 2. 4° 2.4° 6.25° 6.91° 57-----: —L 38b -2.99° -3.63° -1. 4b -1.5a -3.45° -3.42° 58-——---: -.025 —.857 .961-.250 -.160 -.844 -1.54C 59-----: -1.19 -1.47 -.735 -.964 -1.4 -515 -.37 60--——-—: -1.76° .148 —1.02 -2.3° -1.8° -1.09 -1.59 61—-——--: .992° 1.130 .782 .787 1.1b .916 .00 -.539 62-----: —.726 -.644 -.159 -.359 -.402 -.134 .120 .558 63-—----: -.880b .170 .266 - .997° -1.0° -.070 .159a -.544 64 ...... .450 -.065 .241.564 .274 .383 .056 .303 65-———-- -.361 .224 -.12 -.567° -.350 -.156 -.023 -.019 66-—---: -.077 -.528 -.290 -.119 -.07 -.372 -.1433 .008 67-17---: .880C 1.05C .576 .779C .984 .469 -.013 -l.06: 68 ------ : -32 -.952° .794 -.246 -.302 -.852 -.038 -1-37 69-----: .989 1.20° 1.10 .764 1. 0b 1.04 .01 -1.22° 70-—---: —.814 -1.36 .237 -.888 - .754 -.158 .021 --51 —Continued 287 Table B, Part 34.--Continued Variable; Equation number number ‘ 260 261 ‘ 262 263 . 264 ‘ 265 266 f 267 71-----; -.312 -.239 -.001 -.240 -.391 .065 .0001 -.437 72-----: .55 1.47 1.39 .462 .554 1.5 .082 -.572 73-----: -.44g -.886 -.176 -.501 -.335 -1.71 .04 -2.13b 74-----: 1.31 1.69b 1.61b 1.2b 1.4° 1.54b .099 1.05 75-----: .830 -.167 -.499 .891 .839 -.60 .038 .058 76------; 1.03b 1.64b 1.62b .878b .077b 1.58b .194° 1.56b 77 ------ : 1.010 .573 .844c 1.1° .994c .765 .026 .463 78-----: -.081 -.208 -.179 -.051 -.077 -.206 -.02g -.50g 79------: -.299 .328 1.09 -.216 -.302 1.100 .144 1.25 80—---: .24 .613b .023 .047 .222 .011 .026 -. 79a 81------; -.024 .109 -.733 -.209 .040 -.672 -.067 —1.71a 82 ------ : .298 .363 -.950 .015 .313 -.832 -.054 -2.10a .188 .323 .521 .173 .272 .554 -.184a 1.48b 83-----: Table B, Part 35 283 Variable: Equation number number 3 268 269 270 271 272 273 274 3 -----: .480 .556 .645 .651 .656 .686 .686 R ---- .439 .518 .613 .620 .625 .656 .656 D.F.--: 597 593 592 591 590 587 588 o-----: 596o° 4839° 4097° 5609° 5788° 5307° 5232° 3------ .056 .066 4----— -.106 6---- 6.4° 6.5° 6.2° 5.7° 5.7° 7------ —94.2a -66.4 -70. b -68. 7b 8---- -.001° -.0008 - .0008c 9.. ..... 23.0a 23.1a 10--—- 899a 553a -76.5 126 -124 —102 11---- 489° 263° -289 -102 —298 -277 12----- 309a 239b -176 -30.9 -86.3 —73.1 13 ------- 259° 308° 328° 323° 318° 319° 14----- 12.5C 15. 4b 3.51 4.52 1.62 6.42 6.68 15----- -5.73 - .115 -45.3° -39.1 -55.8 -48.7° -51.1b 16---- 717° 81.1° 63.8a 60.9° 46.9° 43.1b 43b 17------ 88° 91.1° 109° 91.8° 95.7° 78.2° 78.5° 18---- 30.8 18.0 -7.88 -5.13 -6.21 -2.56 -2.26 19--- -11.3 -6.94 -4.8 -13.2 -18.7 _22. -22.8 20 ------ —90.6° -32.0 —41.8° -43.1° -46.6b -53.2 -53.3b 21—----: -9.22 -23 -15.1 -15.4 -12.8 2.61 2.72 22----: _20.1 -1.2 -l7.8 -19.3 -19.9 -28 -27.7 23----: 4.95 23.8 17.1 19. 6 16.4 21. 4 21.6 24----: 13.8 -3.58 -5.02 -.767 -.508 18.4 18.3 25-----: .320 -20.6 29.4 27. 3 31.1 43. 4° 41.2° 6-—----: 50. 9 38.3 22. 4 22.1 25.6 23.3 22 27-----: 33. 6 -4.5 -.467 -7.18 -7.70 1.08 1. 36 28------: 56. 7b 58.1 46.1b 50.1b 41.5° 43.2b 42. 6b 29——--—-: 21. 6 .3 11. 3 4. 42 9.45 17.8 18. 9 30-----: 113° 105 111a 111a 104a 108a 109a 31——----: 11.7 17.6 -14.8 -27.4 -37.6 -32.6 -31 32-—--—: -12.g -7.09 -6.03 -12.3 -25.0 —8.4 -7.74 33------: 76.0 27.7 -19. 9 -28.7 -32. 0 -2 _24 5 34————-—: 11.6 13. 34. 6 26.2 25. 8 50. 5 51. 9b 35-----: -35.1 -73.5 -77. 6b -89.3° -96. 3° -74. 8b -75.1b -Continued Table B, Part 35.--Continued 289 Variable; Equation number number ‘ 268 269 270 271 . 272 '273 274 36-—--: _20. -15. 7.2 5 -13.0 -21. -32.2 -32.g 37---- -109 -95.9 -98.3 -91.3b -97.1 -89.8b -87.6 38 ..... —107 1972 1.01 1.22° .932 .985 .948 39.----- 1.63 1.410 .366 .553 .193 -.16% -.165 40—-.-— -1.23 —1.40° —1.51° —1.59 -1.57 -1.52 —1.48 41----- .596 1.1 1.11 1.01 1.05 .812 .810 42-——--— 2.79° 2.7% 2.74% 2.46° 2.56° 1.75b 1.76b 43----- -1.58 -2.61 -1.99 -1.65 -1.90b —1.06 -1.08 44 ..... _2.67b -.311 -.868 —1.07 -.724 -.298 -.306 45----- -3.61° -3.67° -3.15° -3.12° -2.99° -2.76° _2.72° 46———--- —3.63° -.798 -1.35 1.16° -1.44 -1.64 -1.64 4 -—-——- 1.83 1.3 1.63 1.93 1.77° 1.74° 1.76° 48----- -1.22° -1.31° -1.27° -1.44° -1.37b -.971° -.962° 49---..- -1.41 -3.14° _2.45 -274b -2.33b -1.68b -1.63 50----— -.593 -1.25 -1.74 -1.28 —1.90° -2.27 -2.29 51----- _2.38° -2.45b —1.33 -1.93b -1.59 -1.57° -1.60° 52-----— 2.25b .259 .704 .166 .528 .125 .170 53 ------- 3.55° 1.13 1.870 1.18 1.40 1.82 1.81 54--—-- -1.65 .137 ‘.161 .333 .141 .728 .716 55 —————— -.006 .279 1.11 1.490 1.34° .638 .606 56------: 6.93° 5.53° 4.17° 4.39° 4.31° 3.40: 3.39% - 57------: -3.47 —2.5° -2.30° _2.35° -2.50° -l.28 -1.25 58----—-: -1.56° -.972 -.741 -1.23° -1.19 -.937 -.947 59-----: -.301 -.932 —1.26 -1.52 -1.42 -1.34 -1.35 —1.69b .264 -.922 -.184 —.526 -1.05 -1.02 60—---—-: 290 Table B, Part 36 Variable: Equation number number ‘ 275 276 f 277 278 “f 279 280 281 f 282 .53 ------- .765 .765 .748 .758 .428 .237 .478 .422 R ------- .742 .742 .725 .734 .406 .209 .438 .401 D.F.--- 586 585 590 586 620 621 598 621 0 ------- 4774a 4782° 3513° 3276° 7544° 819° 6557° 8464° 2 -------- 1862° 1882° 1772§ 1823° 3"“""’"‘" "' o 009 "’ o 024 o 4.10 ""o 078 4----- .179 .169 .060 .242 5"- ----- -1601 -1009 ’1507 6-------: 2. 5a 2. 6a 3 . 5°1 2. 8° 7 ------- : -10. 7a —10.7° 8-----.- .0007°- 0007° -.001° -.001° 10.....- _282 -276 455a 11 ------- -441° -435b 210° 12----- -309b -299b 188° 3------: 243° 247° 230° 14-—----: 21° 21° 18.6° 18.7° 15------: ~20.8 -20.8 -33.4 -35.3 ~60.1° 1.1 16—--——-§ 41b 40.6b 39.1b 38.9b 52. 7b 74.5a 17------: 73.6° 72 86.7a 82.9° 82. 4° 81.7 18------: 9.81 9128 8.0 5.9 -47. 6° 29.3 19-----: -16.1 -l7.8 -4.9 -ll.8 19. 6 —10.2 20—.-—--: —21.8 _22.5 -35.9° -23.4 -62. 4b -84.5° 21-----; 30.8 32.1b 45.1b 32.2 104° -12.6 22 ----- : -13.1 -14.1 -16.5 -12.9 56. 2b -19.6 23----: 15.1 14.8 901 1104 290 3 301 24—-——-: 26.5 26 360 21.3 -2. 6 16.0 25—----: 38.9C 39c 56.3° 40.9° -40. 6 1.4 26---—-§ 47. 9b 47.3 53.9b 47C 64° 58.6° 27 ------ : 17.1 17. 2 32.00 23. 23.4 37.7 28----: 50. 6° 50. 6° 45.2b 42.8 23.5 54.6b 29 ------ : 43b 44.1b 61.6° 52. 5b 39.0 33.6 30—-—--: 89. 7° 90. 0° 95.8° 87. 8° 580 125° 31-—---; 11.1 10.5 14.7 19.7 8. 4 13.6 32-‘-‘"‘": “'8 o 72 "7 o 63 “l o l -5 o 4 490 O l""8 o l 33-----: -15.8 —16.3 6.7 -8. 7 20 81.3b 34------: 63.9° 66.8° 81° 75. 7° 150° 9.9 35 ----- 3 .4105 -3903 '1809 -310 2 -904 -3305 -Continued Table B, Part 36.--Continued 291 , ‘ Equation number Variable: number ‘ 275 276 277 278 279 f 280 281 282 36---—-: -12. 7 -13. 1 -16.6 -11.1 -6 5 -23. 6 37----: -84. 6b -83. 8b -75.7b -91.7b 41.4 -102b 38-—---: 1. 39b 1.37b .659 1.00 1.30 - .811 -1.0 39------: - .437 -.467 -.810 -.762 1. 6° 1.4° 1. 2° 40-----: -1.50b —1.51b-1.5b —1.4b -1. 6b -1.5° -L 8b 4l------: .579 .592 .361 .723 -.694 .734 -599 42-----: .437 .386 .452 .762 2.7° 2.5° 2. 4° 43------: -.778 -.700 .312 —1.2 -.106 —L 4 -.059 44 ----- : .837 .825 .461 1.1 -3.0° -2. 8° -3.3° 45----: -3.20° -3.16° -3.16° —3.1° -3.4° -3. 9° -3.6° 46-----: -.933 -.967 _2.5° -1. 3 _2.0° -3.5° -1.7 47------: -.173 -.092 -.25 - .368 1.9° 1. 4 1.4 48—-----: -.823° -.808° -.552 -.621b -1.3° -1. 2° -1.2° 49 ----- : .113 .028 1.6° .400 -1.6 -1.5 -1.6 50----—-: -2.25b -2.26b -2.8° _2.9° -.756 -.118 -.421 51—-----: -.769 -.790 .045 —.115 -2.0° -2.7b -2.1b 52 ----- : .492 -.462 .654 .257 1. 6° 2.1b 1.3 53—-———-: 1. 63° 1. 58 - 3.2° 2.3b 1. 8 3.9° 2 1° 54 ----- : 2. 02b 2. 01b 1. 2 L 7b -2. 6° -1. 9° -3. 0° 55-—----: .578 .589 .097 .138 .847 .096 .910 56-----: 2. 62° 2. 59° 2. 4° 2.4° 5. 5° 6.7° 5.1° 57-----: —1.19b -1.11b —1.2b —1.1b -4. 0% -3.4° -3. 9g 58-----: -.767 -.762 - .699 .266 -1. 9 -1.40 -L 6 59----: -.86 -.897 -.261 -.618 -.288 —.15 -.116 -1.46 -1.42° -2.7° _2.2° -.338 -1.7 —.333 60-----: 292 Table B, Part 37E/ Variable; Equation number number : 2839/ 284 285 286 287 §:--——--§ .8787 .8792 .8792 .7550 .8675 R —----: .8542 .8548 .8548 .7094 .8420 D.F.----: 536 536 536 543 540 0 ——————— § 2644.3° 2665.0° 2661.9° 5235.3° 2986.8a 2 ------ : 1741.7a 1758.9a 1761.2° 1761.9a 3-------: .01138 .04363 .03317 .54712a 4------: -.00103 —.02722 4-A----: 209.90 1946.8 5 —————— 3 -17.111 -15.866 -15.923 —4.8790 6------ 2.9891° 2.8569° 2.8578° 3.6994° 8 ——————— - 00013 8—A-——-— -130.67 -129.15 —102.33 9 ------- 11.333a 11.270° 11.289° 13.388° 10—--—-—§ 524.22a 526.53a 528.20a 967.56° 11 ------ : 347.44a 344.13° 346.37° 609.10a 12 ------ : 423.11a 417.90° 419.46° 531.55° 13------: 371.12° 368.14° 367.60° 376.95° 14------: 21.398° 20.257° 20.344° 20.090° 27.170° 15------; -487.86b -483.l9b -485.90b -686.19b -396.88° 16—-----: 413.660 392.230 389.840 206.56 250.2 17----: 280.450 302.446 301.150 -73.231 334.94 18 ----- : -150.71 -138.44 -137.52 146.99 -310.850 19------: 189.52 192.09 192.35 6.0363 274.470 20-—-"-'-: -68015% ’56093g -5606“ 340959 —400270 21-—----: -502.43 -503.60 -505.13b -828.53° -358.730 22 ------ : 220.25 210.58 212.06 665.58b 176.99 23------: -229.24 -328.55 -326.93 -588780 -51.567 24——----: 255.08 289.06 291.70 304.66 578.170 25------§ 248.57 255.03 253.96 130.15 152.78 26 ...... : “2620 52 “230.11 -237-67 -540052 -397062 27 ------ : -137.72 —154.38 -150.49 75.604 -72.961 28—-----: -809.86° -797.17§ -795.73° -488.82 —909.94° 29-----: -717.65a —687.99 .692.59° -339.08 -831.06° 30-—---; -834.00° -811.87° —811.94° —910.16° -716.65a 31——----: -827.04° —816.37° -815.21° -291.27 -818.31° 32 ------ : -205.89 -190.08 -193.37 91.919 -233.49 33------: -83.940 -79.714 -80.082 -163.38 -176.33 34----—-: 116.97 105.15 104.76 .72457 135.26 —Continued 293 Table B, Part 37§/.--Continued Variable; Equation number number : 2839/ 284 285 286 287 35 ————— -9.0958 _24.607 -26.633 -307.41 198.45 36----- -230.25 -223.30 _222.63 -378.96 -l8l.32 37----- 37.991 26.010 25.716 166.45 122.92 38 ------- -.06280 -.0927 -.1008 -l.2778 —.32137 39 ...... 1.5160b 1.4594 1.4558 2.9779° .72919 40--——--: -1.4091b -1.3549b -L3515b -2.214la -l.2049b 41-----: .67002 .7116% .71854 .70915 —.16640 42 ----- : 1.2386b 1.1927 1.1939b 2.4598° 1.1159b 43------: -.92115 -.86338 -.87645 --82415b .06132 44—----: 1.1126 1.17770 1.1686C .44349 1.2621C 45-——-: -1.7183b —1.8064° -1.7918° -.36733 -L8553a 46------: -2.3113° _2.3395° _2.3292° -2.01580 -3.59218 47----: -1.3935° -1.3375° -1.3379° .14920 -.92535 48-----: -.22706 —.21971 —.2182 -.44307 -.439970 49 ..... : 2.5652° 2.5992° 2.6008 -.11398 3.5013a 50------: -2.9286a -3.0812° -3.0774a -2.862'7b -2-2833° 51 ----- : -.86374 -.81095 -.81356 -1.3772 -.84688 52--—---: -.93882 -.85167 -.84896 -.05656 —.92720 53 ...... 2.4821° 2.5418a 2.5329a .63743 3.5065a 54 ------ .28534 .30000 .29712 -1.6170 -.60224 55_———-— -.73766 -.81380 -.81700 .42868 -.88692 56 ....... 1.8913a 1.8750° 1.8736a 3.2210° 1.6118° 57----- .22077 .24357 .24589 -.75344 -.07360 58----- -.7911g -.77707 -.78866 _2.1711° -.57849 59--—---: -1.5026 -l.5208b -1.5250b -2.4800b -1.0360 60 ------ .0125 -.0306g -.01931 1.8014° -.49283 61 ------ : .90120 .88785 .89201b 1.3043b .72237c 62————-: -.714600 -.67863° -.67476C -.29495 -.43722 63 ------ —.39866 -.44057 -.43802 .29577 -.47684 64 ------- .26241 .24014 .23864 -.21203 .527780 65------; -.35487 -.36180 -.36262 -.06039 -.48923° 66-----—: .08454 .06686 .06617 -.07939 -.04264 67-----: .84382b .84767b .85039b 1.3490a .632450 68------: -.38699 -.37024 -.37254 -1.1395a --31465 69------: .49335 .54164 .53906 .98085° .08578 -Continued 294 Table B, Part 37§/.-Continued Variable: Equation number number 3 2832/ 284 285 286 287 70-‘----: '0 35919 ‘041522 ‘041943 '046340 -' 84384 71------: -.31823 -.32813 -.32681 —.20404 -.16653 72 ...... = .49897 .44617 .45769 .90701 .73452 73 ------- .23850 .26352 .25754 -.14415 .14160 74----- 1.3796° 1.3577° 1.3553° .87128 1.54398 75-—_-- 1.2394° 1.1943° 1.2021° .53362 1.4626° 76-----: 1.54669 1.5081° 1.5085° 1.6050° 1.3566° 77 ----- : 1.28736 1.2668° 1.2655a .39882 1.3031° 78------: .40002 .37049 .37636 —.10284 .45007 79 ----- : .07337 .06582 .06665 .21509 .27172 80 ------ : -.10055 -.07953 -.07851 .08695 -.10258 81---—--: —.02122 .00153 .00535 .45521 -.37803 82------: .40692 .39397 .39267 .63670 .30629 83-----: -.23651 —.22005 -.21887 -.47660 -.35560 84------: 383.71° 383.14° 383.87° 99.119 364.83° 85 ______ 239.96b 238.72b 239.46° -6.9717 195.510 86 ------- —10.118 -1.9825 -1.1370 -331.58b 9.5765 87-----— -90.675 —46.758 -47.277 -712.33° -22.044 88------. -38.463 -28.831 —28.036 .664.18° —32.111 89------: 176.90 180.87 182.83 -340.77b 133.56 90-———-: —61.358 -6.6425 -7.2118 .619.47° ~23.739 91------: -153.790 -131.800 -131.08 -552.85° -177.03b 92—-—---: 60.776 71.943 72.877 -375.40b 3.8427 93----: -131.86C .88.553 -88.744 -577.38a -118.87 94------: -253.17° _232.33° -231.95° -624.90° -305.13° 95 ------ : -134.65 —119.29 —118.41 -591.64° -221.34b 96-----—: _253.25° -241.53° -242.09° -827.88a —182.69b 97 ------ : -37.654 _21.273 -20.295 -507.413 -l8.183 98-—--: -361.90° -345.26a -345.34° -836.29° -378-74° 99------: -488.818 -450.66° -451.17° -1012.2° -432.90° 100--_——: —416.01° -389.83° -389.56° -697.88° -332.97° 101—--—-: -110.31 -lO9.48 -108.61 -677.39° -51.108 102-—---: -277.74° -288.60° _286.60° -434.02° -235.47° 103—--—-: -260.09° -264.38° -262.65° -635.47° -214.87b -230.09° -228.98° -227.20° -650.40° -78.152 lO4----: -Continued 295 Table B, Part 372/.-Continued Variable: Equation number number 3 2832/ f 284 f 285 f 286 ‘ 287 105---: -5.3315 -10.054 -7.7613 -345.20 49.257 106----: -25.376 —14.801 -13.075 -72.637 -79.898 107----: -293.17 -280.74 -282.32 -410.12 -271.09 lO8-----: 100.40 114.73 115.69 ~43.848 -42.182 109 ..... , -204.38 -198.67 -197.45 -391.39 ~250.54 110----: -425.89b -415.86b —4l4.43b -121.96b -567.82a a/ All equations in parts 37, 38, 39, 40, and 41 were estimated using the corrected data. b/ Equation 283 is the same as equation 244 except the corrected data were used for equation 283. 296 Table B, Part 382/ Variable: Equation number number = 288 289 290 291 292 3 ----- .8451 .8639 .8284 .8567 .8631 R ---- .8228 .8432 .8101 .8349 .8423 D.F --- 563 559 582 559 559 0------ 3694.2a 1170.6° 239.60° 202.60° 802.65 2 ———————— 1853.4a 1757.3a 1677.6° 1753.2° 1756.5a 3 ----- : - 03687 .16777 -.06079 -.02708 .17053 4A-----: -1572.6 852.35 2251.4 -192.1g 731.22 5-------: -23.690° -12.436 .22.7029 -27.205 -12.019 6-------: 2.6610° 2.5539a 2.8174a 3.3457a 2.6132° 8-A-----: -212.66° —170.59b -91.744 -94.878 -l73.76b 9------: 13.1358 11.664a 13.090a 13.042° 11.412° 10------: 457.80a 554.63° 623.21° 527.52° 552.908 11-----: 340.23a 224.84a 208.44a 232.80° 222.85a 12 —————— : 259.38° 345.13° 347.78° 369.57° 364.35° 13——----: 259.27a 324.34° 299.81° 340.86° 341.96° 14------: 22.728a 17.265a 22.078a 24.623° 17.822a 15------: .633.57° —550.15° .63074 16------: 472.25b 410.12b 14.413 17--—---: 529.11° 576.87° 63.892° 18 ————— : -l48.23 -326.60: 9.4584 19—-—--: 205.41 -272.08 _21.30 20------: —47.732 -100.60 -30.644 21-—-—-: -550.18b -408.44b 22.472 22—-----: 216.84 366.84b ~13.022 23----: -495.43 270.11 19.583 24—-—-—-: 550.78 123.49 22.677 25------: 202.62 212.14 50.658° 26-—---: -236.96 75.781 72.837° 27----: 225.66 251.22 9.7896 28-----: -828.70° -409.85° 40.070° 29------: -260.3g -478.75C 29.820 30-----: -495.50b -827.04° 73.424° 31---—--: -642.06 -780.23a —33.587 32—---: 174.56 18.030 22.400 33-----: 82.732 -263.29 -19.149 34------: -38.894 216.81° 77.895a 35----: -9O . 199 203 . 55 -20 . 489 36-----: -’72. 091 99.411 16.647 -Continued 297 Table B, Part 389/ Variable: Equation number number ‘ 288 289 290 291 292 37-----; -258.20 -185.13 -73.040b 38-----: .34330 .18372 .12655 39------: .73338 .18769 .18959 40—-—---: -.63752 -2.2733a —2.1256° 41 ------ : .37935 .9349 .83307 42 ————— 3 1.3007b 1.1192b 1.1175b 43----—-: -1.6389b .1255 .08597 44--—-. .83058 1.6216 1.5717b 45-----: -2.4128° -1.9726° -1.8615° 46—-----: -2.4329° -2.1897° -1.9343b 47----- -.17224 -l.548lb -1.5572b 48-—--- -.28238 -.47515b -.44578° 49----- .54204 2.4200° 2.6075a 50——-—-—: -2.4000° —3.5043a -3.39843 51 ------- -.73159 .30213 .24654 52 ----- E -.25010 .07735 .21352 53 ------ : 2.5144° 3.0049° 2.9982a 54------: .40427 .83408 .76727 55------: -.11935 -.59752 -.47635 56-----: 1.9909° 1.7756° 1.6637° 57-----§ -.98687b .41507 .53164 58-----: -.43981 -.45393 -.49623 59-----: -1.3381C -.89633 -1.0105 60-----: -.91531° -.52802 -.54266 61-----: 1.1285° .01277 .03347 .99652° 62----- -.77176° .01919 .03807 -.714l4b 63------ -.81393a .11364° .12233° —.91775§ 64-----: .25836 .01580 -.03661 .52172 65-—-—--: —.36192 -.03956 -.04703C .41232C 66-—---: .07970 -.O4670° -.02981 .15326 67-----' .94566° .04000 .08077° .73825b 68 ------- -.37952 -.02496 .00385 -.60441b 69----- .82191b .03088 .07214° -.39827 70—-—-—- -.82571 .03726 .03933 -.15864 71 ------ -.24426 .07836° .07453b -.27999 72-----§ .43014 .10858° .12302° -.00161 73------: -.31277 .01949 .04231b -.36839 -Continued 298 Table B, Part 382/.--Continued Variable; Equation number number ‘ 288 289 290 291 292 74----; 1.4101° .06978: .05781° .732392 75-----: .52984 .05722 .02747 .84407 76-----: .98764% .12733° .01161 1.4886: 79-----; -.21213 -.03023 -.02163 .41136 80-—----: .21238 .14974a .18289a -.23735 81------: .10649 -.03118 -.01397 -.39937 82-----: .12839 .0273g .02036 -.17184 83-----: .27769 -.13217 -.08725° .18418 84-----: 354.50a 433.16° 459.20a 351.37° 85-----: 184.740 294.78° 322.20a 180.38° 86-----: -2.6348 112.97 132.82 -4.9980 87....--: -140822 -6509g .700853 '250606 88----: -81.827 -157.36 -68.392 -96.218 89----§ _25.231 -118.58 92.948 -34.799 90 ------ : 30.108 -31.227 -46.912 13.574 91 ------ : -142.01° -258.26° -207.00° -159.33b 92—----: -6.5329 -165.20 -59.39g —22.699 93"""'": -47 o 102 -113 o 65 -139 o 51 .640 197 94----—-§ -262.89° .401.33° -356.38a _280.53° 95-----: -164.40 -345.98° -26l.48b -l72.26 96-----: -228.78° -273.81a -285.95° -239.75° 97------: -37.5 -271.90b _261.29b -47.47 98-----: -331.01 -557.59° —570.30° -335.18a 99-----; -469.82° -228.37b -200.56b -468.15: 100-—---: -438.368 —184.50° —158.82 -442.45 101-----: -147.920 —16i.67° -60.91% -134.89 102----: -340.80: -163.61: -1l8.66b -331.00: 104 ----- Q -325.29° -281.40° -184.33b -326.38° 105----: -82.710 41.305 16.057 -66.22 lO6-----: -167.37 -80.842 -55.327 -158.36 107----: -268.94 -185.69 —194.85° -238.14 108----: 42.185 221.18b 168.78 104.84 109—---; -304.8% -58.891 -90.270 -250.72 110-..—-: -423.95 -242.28b -424.93a -384.54b a/ All equations in parts 37, 38, 39, 40, and 41 were estimated using corrected data. 299 Table B, Part 393/ Variable; Equation number number - 293 294 295 296 297 298 B§------= .8385 .8388 .6754 .7975 .8203 .8157 R -—-—-- .8210 .8213 .6445 .7855 .8022 .7985 D.F.---- 581 581 588 608 585 589 - 0..———-— 1662.8° 1727.8b 4417.0° 3903.0a 3049.5a 1736.3b 2 ——————— 1844.1 1844.5a 1931.8a 1844.5a 1796.0 3 -------- —.10584 -.05355 .07224 .47664a -.18655 4- ----- 09515 4—A--- -660.72 -910.69 1332.8 1642.4 5-----—_: -19.919b —18.754 -25.795b -3.3319 -20-760 6—----—-: 3.0063a 2.9041° 2.5418a 3.7075a 2.3076° 8------: .00011 b 8-A---—: -99.172 -141.55 -135.31 -117.64 9 ........ 12.654a 18394a 15.467° 14.973a 14.675° 10 ------- 540.45a 537.19a 993.57a 415.22° 617.20° 11----- 320.43: 312.88: 656.48: 280.04: 210.13: 12-—--. 433.94 427.63 624.15 243.00 330.00 13 ————— : 383.44° 376.12° 395.37° 255.43a 313.58° 14------: 20.462a 19.071° 15.167° 21.266° 25.896° 15.056° 84--—--: 413.86° 411.97° -76.446 343.97° 336.85° 85-----: 287.19° 287.33° -220.10° 217.70b 184.24b 86-----: 82.684 92.864 -492.39° 111.45 47.649 87-----: -90.120 -39.087 -799.64a -8.2390 -54-49§ 88-—---: -53.999 -52.106 -74o.32a -68.042 -155-13 89 ----- : 41.712 43.495 -595.21° -30.821 -133.70 90---—-—: -68.757 -18.799 -679.91a -2.3759 8.747% 91..--_-: _140.58° -130.16° -614.02° -l4l.83° -191.28 92 ------ : 9.8550 13.747 -554.52° -35.607 -l28.23 93--—--: -140.76b -98.727 -669.85° -93.546 -48.680 94-----: _294.02° -283.14° -738.84° -317.43§ -337.48: 95----: -175.37 -170.71 -704.838 -232.02 -245.31 96-----: -203.21b -183.59b -824.84a -104.74 478.40b 97------: -105.6% -98.89 -612.61° 6.7152 -169.71 98----—-: -339.52 -327.02b -8l7.l7° -239.75° -344.45b 99------: -280.75° -226.40b -1000.3° -132.27 -261.50b 100----: -204.700 -157.24 -658.01° -34.750 -191.90° 101----: -79.558 -83.892 -657.74° -17.357 -128.06 102--—-: -84.l67 -96.414 -419.81° -8.3036 -117.30° 103----: -147.120 -155.92C -657.47° -34.426 -222.62b -Continued Table B, Part 392/.-06nt1nued 300 Variable; Equation number 104 ----- : 10.705 12.237 ~444.73° 229.94c -46.772 105----: 332.75C 330.450 145.72 501.43% 301.57 106---: 242.48 262.31 14.310 343.17 208.88 107—-—--: 134.05 159.09 -198.91 218.54 113.50 108----: 598.44° 615.12° 302.96 569.65° 604.23° 109-—---: 283.46 293.90C -82.240 349.400 260.85 110-----: -106.79 -88.974 18.453 -8l.299 30.317 117—-——-: -4025.7° -3973.6° -4672.2° -4732.9° -3558.5a 118 ----- : 5486.2a 5453.7a 6022.1a 6249.9a 4810.6a 119----: —2337.l° _2325.9° -2564.6° -2613.7° _2016.9° 120—-—--: 453.61° 451.35° 503.54° 500.21° 387.348 121.--—-: -45.796° -45.542° -5l.660° -49.936° -38.742a 122-—--: 2.4840° 2.4686° 2.8587° 2.686l° 2.0782° 123-—---: -.06857° -.06810a -.08081° -.07373° -.05657a 124----: .00076° .00075a .00092° .00081° .00061° 125..--_: -4.4307b -4.3633b -1o.188° -5.1197b -2.8126 1.0078 126 ----- : 7.2729° 7.1732° 15.042° 7.7006° 4.18110 -1.0108 127---: -3.262i° -3.2034° -6.8287° -3.3900° -l.6334 .43290 128-—---: .65079° .63635° 1.4102° .65903° .27574 -.08967 129—----: -.06778° -.06605° —.15207° -.06684° —.02401 .00931 130--—--: .00383° .00372° .00886° .00369° .00113 -.00050 131----: -.0001 a -.00011° -.00026° -.00011b -.00003 .00001 132-—---: 1x10' a 1x10-68 3x10-6a 1x10-6b 3x10-7 -1x10-7 133-----: 6.9371a 6.8424a 7.9751a 8.1126° 6.0676° -.13468 134—---: —9.3001° -9.2447° -10.022° -10.605° -8.0410° .30187b 135-----: 3.9252° 3.9076° 4.2191° 4.4116° 3.3344a -.144.74b 136----: -.75753° -.75417° —.82287° —.84208° -.63574° .02852b 137---: .07621a .07584a .08406a .08393a .06326° -.00276b 138-----: -.004l2° -.00410° -.00464° -.00452° -.00338° .00014c l39----: .00011° .00011a .00013° .00012° .0000 ° -3x10‘6 140—---: -1x10-6a -1x10-6a -1x10-6a -1x10—6a -1x10- 9 3x10-8 a/ All equations in parts 37, 38, 39, 40, and 41 were estimated using corrected data. Table B, Part 402/ 301 Variable; Equation number number 3 299 300 301 302 303 304 E§-----= .8103 .8318 .8175 .8000 .7997 .7981 R ...... : .7954 .8161 .8005 .7861 .7861 .7843 D.F.---: 597 589 589 602 603 603 0 -------- 235.80a 210.32a 1363.5° 2224.5a 1808.4b 220.67a 2-------: 1787.7a 1867.2° 1797.5a 1810.2° 1818.5a 1821.0° 3-----: -.14101 -.00979 -.16347 -.l6131 -.12078 -.07878 4-A---—: 1470.8 -862.82 1332.7 901.77 690.34 260.22 5---—-——: _24.029b —20.288 -20.l37 -22.096 -21.678 -23.863° 6 ...... : 2.5202° 3.0195° 2.3425a 2.1376° 2.1746° 2.3440a 8-A-----: -100.72 -84.240 -117.49 -126.30 -129.60 -107.71 9————--—: 15.106° 13.204° 14.488° 14.024° 13.865a 14.026° 10-——---: 682.20a 541.278 608.688 675.88a 657.28a 700.48a 11 ————— : 180.03° 216.93° 211.01° 230.81° 216.45a 215.31° 12..-..-: 395.37° 432.17a 340.60° 271.20° 264.21a 330.62° 13-----: 330.53a 357.98° 315.55a 300.87a 296.66° 336.99° 14----: 15.563° 18.878° l5.992° 11.558° 12.093° 14.118a 84-----: 431.258 454.50a 332.38° 284.20° 279.93a 340.55a 85-----—: 273.12° 307.25° 181.310 131.23 120.59 171.69b 86----: 118.07 117.37 47.967 24.156 23.891 65.076 87------: —1.0378 —39.891 -56.284 -26.478 -21.40 12.244 88------: -107.37 -50.303 -159.55° -l48.53b -150.71 -120.59 89------: -73.504 62.228 -131.39 -176.04 -179.36 -155.29 90-—---: 48.577 -13.726 -1.0494 59.039 56.887 75.915 91——---: -193.30b -165.67b -194.l5b -164.97b -167.40b -171.99b 92------: -108.81 -13.031 -131.03 -115.38 -119.28 -ll7-89 93 ----- : -43.77 -115.00° -57.254 -3.7865 -9.5163 -16.427 94-—---: -356.71° -343.85° -342.74° -312.68° -319.31° -333.25° 95-----: -269.93b -229.29b -249.63b -215.36° _225.27b -246.21b 96-----: -190.30b _207.29b -181.86b -164.06° —166.46° -174.43b 97----.., _256.94° -174.38 -169.16 —193.0 -l97.8g -265.99° 98----: -440.53: -407.8l: -348.36b -334.93 -344.36 -427.93° 99------: -202.02 -212.10 -256.23b _275.96° -270.47° -2l8.62b 100 ..... : ~170.62 -180.73° -193.44c ~214.90b -216.36b -186.25° 101—----: -91.998 -30.04 -127.05 -114.61 -110.61 -89.384 102—-—--: -154.74a —114.46 -116.34C -141.08b -l46.64b -167-79° 103—---—: -268.85° -202.00% -222.63b -276.72° -281.83° -303.82° 104-----: -240.57° -171.09 -41.534 —214.10° —221.69° -287.28° 105-----: 93.507 78.08 311.85C 54.717 48.139 39.739 -Continued Table B, Part 40§/°--06ntinued 302 Variable; Equation number number 3 299 300 301 302 303 304 106----: -16.403 -28.855 222.98 -32.071 -44.665 -99.762 107---: -106.08 -53.186 136.67 -72.134 -77.064 .-98.689 lO8----: 422.68° 414.23° 618.10° 446.70° 444.84° 453.18° 109 ----- : 72.878 62.963 278.98 92.287 95.350 98.625 110——-.-: -215.57 -315.98° 38.427 -111.57 -1l7.22 -148.28 117----: -3761.9° -117.05° -5.7404 24.819° 35.683 118 ----- : 4884.6° 227.15° 119----: _2064.9° -105.90° 120----: 401.79° 20.717° 121---: -40.734a -2.0142° 122-—--: 2.2179: .102612 123----: -.O6l40 -.0026l 124-----: .00068a .00003 125----: .78975 -.13671b -.10719b 126----: "' 0 54798 127 ----- : .19486 l28---: -.03675 129-----: .00335 130----: -.000 131----: 2x10" 132-----: -15.:10‘8 133-----: -.1363§ 6.4641a .05118 -.01509 134-----: .29335b -8.2785° 135----: -.l4633 3.4706° 136---: .03028° -.67199a 137----: -.00309° .06796° 138----: .0001 b -.00370° 139-----: -4x10 .0001 a 140--—--: 55:10"8C -lxlO ° a/ All equations in parts 37, corrected data. 38, 39, 40, and 41 were estimated using Table B, Part 41E/ Variable: Equation number number 3 305 306 307 ‘ 308 309 53 ----- : .8000 .7974 .6302 .7521 .7517 R --—--—: .7864 .7840 .6090 .7462 .7361 D.F.--: 603 604 609 629 606 0—--—-—: 2142.9° 201.24b 4397.6° 3909.4° 2862.29 2-------: 1812.08 1808.8a 1865.6a 1682.1a 3---—-: -.15340 -.11850 .25040 .59276a 4-A---: 860.78 386.19 1812.7 3786.2 5-------: -22.019 -25.2130 -l5.905 -15.021 6 ------ 2.1441° 2.3629° 2.2180° 3.8770° 8-A---- -126.58 -98.283 -99.800 -63.686 9-------: 13.996° 14.273° 15.949a 22.750a 10 ------ : 672.12° 743.10° 982.90° 509.55a 11------: 228.12° 233.49a 409.37a 143.67° 12..---_: 269.57° 367.27° 340.92° 106.53b 13 ....... 299.73° 360.66° 308.32° 178.86° 14 ----- 11.673a 14.053a 6.9387 21.442a l3.306° 84 ------- 283.23a 370.24: —152.60 161.26 85 ------- 128.92 207.34 -344..66a 22.158 86------: 24.017 81.564 -535.10a 53.446 87---—--: _25.537 18.291 -728.68° 43.420 88 —————— : —l49.16b —104.55 -7l8.03° -113.0% 89—-----: -177.52 -134.34 -616.88a -287.34 90—-—-—-: 58.550 86.553 -577.16° 90.973 91 ------- —165.70b -168.l5b -627.16° -130.78 92----- -1l6.81 —106.09 -581.48a -135.42 93 ——————— -4.9491 -lO.783 -583.70° —2.5396 94------ -3l4.25° -326.92° -763.32° -305.19a 95--_-_. -217.81° -235.94° -703.42a -215-34° 96—-----: -l64.44° —174.88g —818.67° -39.204 97------: -l94.3g -281.63 -634.46a ~49.163 98----: -337.18 -442.54° -785.49° -169-50 99 ..... : -274.73° -208.21b -1069.61° -185.57° 100 ————— : -215.03b -174.43° -720.55° ~104.00 101—---: -113.81b -87.176 -672.14a -20-560 102----: -142.07 -169.01° -564.47° -76.974 103 ----- : -277.79§ -304.423 -833.503 -136-02 104 ————— : -215.67 -300.56 -730.99 -24.670 53.700 43.211 -525.60a 212.60 105 ----- : -Continued Table B, Part 4l§/.--Continued 304 Variable; Equation number number = 305 306 307 308 309 106 ----- 2 -34.779 -lOO.87 -453.06° 35.126 107-—--: -73.196 -99.13 -555.51° -22.200 lO8---: 446.34a 458.82° —32.438 379.35a 109---—-: 92.915 95.397 ~487.96° 208.36C 110—---: -112.52 -153.49 -2l6.80 -124.96 125---: -.13099° -.24238° -.26289° -.19337° 133----: .041708 .04457a .11794° .04215 -.10819b g/ All equations in parts 37, 38, 39, 40, and 41 were estimated using corrected data. wmsufipaool 305 8Hm.m68.~6 866.6mm.6 mmN.H8m H6m.wm 88H.~ 86H . MH H muuunuumH mom.Hmm.mm 668.8m8.m mmw.m6m 6m8.om mm8.H . 66H NH H muuuuuumH H8H.866.8H H6m.H88.H Hmo.H6H H66.6H Hmm.H HNH HH H muunuuuHH 666.606.6H ooo.ooo.H 666.66H ooo.oH coo.H 66H OH H muuuquOH 868.N68.6 H66.Hmm 866.88 H6m.6 8N8 H6 8 H Wuuuunuuo mmH.8oo.m 66H.m6m 668.mm 680.6 NH6 66 m H muuuuuuum m6m.mmm 866.8HH 866.6H H66.m m6m 86 8 H mnuuuuu.8 6m8.88m 666.66 688.8 68N.H 6Hm 6m 6 H muuuuuuu6 mmH.w8 mm6.mH mmH.m 8N6 mmH mm m H msuuuuuum 6mm.6H 686.6 6mo.H 6mm 66 6H 6 H muuuuunu6 8mH.m 8N8 m6m H6 8m 8 m H muuuuuuum 66H 66 mm 6H 6 6 N H Wuuuuuuum H H H H H H H H WuuuuuuuH 8 u 6 . m u 6 . m . m . 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