‘” “W“ “*FFFFF FF FFFFF I CONTROLLED FERTILIZER INPUT OUTPUT " " XPERIMENTS IN MICHIGAN ' Thus. fir thobogrgo aI mp; MICHIGAN STATEUNIVERSITY Wuloy Buflon Sundquisi" . . 1957 . This is to certify that the thesis entitled An Economic Analysis of Some Controlled Fertilizer Input-Output hbcperiments in Michigan presented by Wesley B. Sundquist has been accepted towards fulfillment of the requirements for PhD degree in gricultural Economics 1", v4 ,‘ '/ ~V'Ltrw‘m‘,"\ \J... VWW Major professor / Date 8/21/57 0-169 L [BRA R Y Michigan State University _._ _._< 44 _,_.___— AN ECONOMIC ANALYSIS OF SOME CONTROLLED FERTILIZER INPUT-OUTPUT EXPERIMENTS IN MICHIGAN By ‘wesley Burton Sundquist A THESIS Submitted to the School of.Advanced Graduate Studies of Michigan State University of.Agriculture and.Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of.Agricultural Economics 1957 1‘ A . ‘ r" I.) t (1 ‘1 ca -’;‘1-‘ "" I I", /“ I ’ y L'. [I (7/ {1‘ Li.- ACKNOWLEDGMENTS The author is indebted to several members of the departments of Agricultural Economics and Soil Science of Michigan State University who aided in organizing and conducting the research reported in this thesis. Dr. R. L. Cook, Head of the Department of Soil Science, Dr. J. F. Davis, L. N. Shepard and J. C. Shickluna of the Department of Soil Science and C. R. Hoglund of the Department of.Agricultural Economics aided in numerous ways in conducting the research reported here. Dr. L. L. Boger, Head of the Department of.Agricultural Economics at Michigan State University and C. W. Crickman and H. L. Stewart of the Farm Economics Research Division,.Agricultural Research Service, United States Department of.Agriculture, provided financial aid without which the author could not have conducted this study. Much of the computational work was performed by the girls in the statistical pool of the Department of.Agricultural Economics under the direction of Mrs. Iantha Perfect. Beverly Hamilton deserves a vote of thanks for typing an earlier draft of this thesis. The author's sincerest appreciation is due to Jack L. Knetsch, formerly of Michigan State University, now with the Tennessee Valley Authority with whom much of the experimental work was planned and con- ducted and to Dr. Lynn 8. Robertson Jr. of the Department of Soil Science who aided in planning the experimental work and who has super- vised the field work. ‘Without a tremendous amount of work on his part, the data utilized in this thesis would not have been produced. Above all, the author appreciates the encouragement and guidance of Dr. Glenn.L. Johnson, not only while writing this thesis but through- out the course of my graduate studies. Association with him has helped to make graduate work a really enjoyable and profitable experience. \I \I \I \I \I \I \< V \I \I \I \I \I \l \I .. —. — .- ~ ~ — ~ — - - .~ — — — l\ u'\ I\ I\ a\ I‘ I\ l\ I\ l\ l\ I\ I\ I\ l\ ii AN ECONOMIC ANALYSIS OF SOME CONTROLLED FERTILIZER INPUT-OUTPUT EXPERDIENTS IN MICHIGAN By wesley Burton Sundquist AN ABSTRACT Submitted to the School of.Advanced Graduate Studies of Michigan State University of.Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of.Agricultural Economics Year 1957 Approved wfipfl'Z/‘W ABSTRACT The three primary plant nutrients, nitrogen, phosphoric acid and potash are major farm resource inputs. In l9Sh, farmers in the United States paid over a billion dollars for various commercial forms of these plant nutrients. In order to allocate optimally their resources, farmers need information as to the productivity of expenditures made for various production inputs including the three primary plant nutrients. In the spring of l9Sh the Michigan.Agricultural Experiment Station, aided in part by resources contributed by other interested agencies, initiated a series of plant nutrient input-crop yield output experiments which has been expanded in each succeeding year. Experimental input- output information analyzed in this thesis included data for the following crops: (1) a rotation of oats, wheat, alfalfa and corn on a Kalamazoo sandy loam soil in Calhoun and Kalamazoo counties (2) a rotation of corn, field beans and wheat on a Simms loam soil in Gratiot county (3) corn produced in continuous culture on a'Wisner clay loam soil in Tuscola county and (h) potatoes grown on a Houghton muck soil at the Experiment Station muck farm near East Lansing. In total, over 1150 individual experimental plots were contained in these experiments in 1956. The primary objectives of this thesis are (l) to estimate plant nutrient input-crop yield output production surfaces and then (2) to provide an economic analysis of the physical input-output relationships iv derived. Continuous function analysis is utilized to estimate the input-output relationships of interest to researchers and farmers. Two general formulations of the production functions for plant nutrients are fitted for most crops. These are a polynomial of the type: I - a + blN + sz2 + b3P + b4P2 + b5K + b6K2 + b7NP + bBNK + bQPK where N, P and K represent pound per acre inputs of nitrogen, phosphoric acid and potash. The second production function formulation is an exponential of the Carter-Halter type: r - allolclNPb202PKb3c3K Both equations are fitted by least squares techniques, the latter being first converted to logarithms. Significant yield reSponse to applied nitrogen was found for corn, wheat, oats and field beans. Corn, wheat and field beans showed a signifi- cant yield response to applications of phosphoric acid. Only potatoes showed a significant reSponse to applied potash for crOps produced during the growing seasons for which experimental data were analyzed. DeSpite statistically significant response to applied plant nutrients for several crops, applications of plant nutrients were profitable for only two crops assuming current crop and fertilizer prices. Nitrogen applications were profitable for corn produced on a Kalamazoo sandy loam soil in 1955 and for field beans produced on a Simms loam soil in 1956. In computing high~profit plant nutrient inputs, however, no credit was made for residual fertility or benefits derived from seedings in the small grain crops. Mid and late summer drouths in 1955 and 1956 very probably reduced the crop yield benefits which might have been derived from applied plant nutrients particularly on the lighter soils. Further information on input-output relationships over time and with varying weather conditions is needed to establish a probability distribution of these relationships. The experimental results analyzed in this thesis are from a very limited number of soil types. These soils tend to be either very fertile or very unproductive. One might expect the largest yield reSponses to applied plant nutrients on soils with a high production potential but depleted in fertility; such soils are not included in the experiments analyzed here. However, as individual low-treatment plots in the experi- ments become depleted and if treatments are rerandomized, a wide range of combinations of residual fertility and applied nutrients should be observed. The adjusted coefficients of multiple correlation between applied plant nutrients and crOp yields ranged from .28 to .78 for the various production function formulations for the different crOps studied. Further analysis indicated substantial amounts of yield variance not associated with regression were due to experimental error and inability to control entirely unstudied variables. Limited analysis to relate residual fertility, as measured by soil tests, to the deviations of predicted from observed yields (Yi - Ii), was relatively unsuccessful. vi However, further extension of this type is needed in order to provide conclusive results. vii T.’ TIE OF CCI’T'” T CHAPTER - Page I TIIE IL: -TU-I.E PL.“ D 1313. .II\‘4IT[JDE OF FIRTH-T .L/"LR. Us?) FTC/73.4.13 o o o o o o o l H Current Research Problems in Fertilizer Use............ The Impor*"nce of Fertilizer as an.Agricultural Production Factor................................... Reasons for Increased Fertilizer Use................... Long-Run Agricultural Production Needs................. The Type of Infornation Needed by Farmers.............. l OCOO\I\) II I'ETEIODS OF AI':1:LLYSISO.....‘.................‘C............. 13 Methods of Collecting Data............................. 1h The Concept of Functional Relationships................ 16 Determining Economic Optima............................ 20 Alternative Types of Analysis.......................... 22 Continuous Function unal\sis........................... 23 Analysis of Variance................................... 2o Discrete Point Analysis................................ 30 III EXPERILELTLL WCZLK CCFDUCTED BY THE KICH_ CAN A"1IL ULTURLL MILLILILFP SrifxmiLC'I‘looo000000000000...ooooooooooooooooooo LL) DJ Characteristics of minerim,L a1 Dcsi;ns................ 32 The Cats, Hdi3at, Alfalfa and Corn Rotation............. 33 Continuous Corn........................................ 35 Fie d B ans, U7 eat and Corn iotatien................... 38 Potatoes............................................... 39 The lj;h Potato Expe riment............................. h2 Tm Lflirot. EM),imam.uu.n..uu.u..u..u..u.. MI Thefmallgengwma Pm) anun.u.u.n.n.u.u.n. h] IV HIRLYSI CF Tn} LLLZ'L...................................... hf; The Cats, wheat, iljalfa and Corn Rotation............. h? iralysis of the Late Data........................... L? Interpre to .tion cf the Statistical Results........ 53 High Profit Ccrcinations of Plart Int ients...... F"CF final sis of the wheat Data.......................... 5; ;aximum Yieles and High Profit Plant Lutrient applications.................................. 75 aralfi SLS of tie Spin Data........................... 75 viii ,I .73! .l» t. ‘ l‘_f"!'T—V~" (V L I .. w «ifiLb Or deth n3 - corhiunied CHAPTER Page 1&1f.alfja0 O O O O O 0 O O O O O O O O O O O O O I O O 0 O O O O I O O C O O O O O O O O O O O O O O 79 Analysis of theC ontinuous Corn Data.................,,. 79 fiaximum Yield and High Profit Combinations of Plant Putrients................................ BC Analysisc of t1e Bean Data Is on the Corn, Beans and Wheat Rotation....................................... cl -3x1nun Yields and Optimum Ir put sof Plant Nutrients...................................... 83 Analysis of the Potato Data............................. 90 Laximum Yiel;ls and Iiign Profit Plant Nutrient npplications................................... 92 V SlUi-CES C‘F UNEIPIJI I VIIRLLIV‘“ IN YIELDS III-ID BLIS OF 1 SI iLOEfiIfiICILJ:::‘OOOOOOOOOOOOOOOOOOOOOO0.00.0000... 9L1. Sources of Unexplained Variance in Yields............... 05 Experimental Error................................... 95 Uncontrolled and Unmeasured Variables................ 97 Affects of Within Treatment Variance on Stauistical Estimates......................................... 93 Factors Related to the Independent Variables Used in Regression Analysis.................................. 100 Incidence of beads, Lodging and Plant Disease........ 100 Relationships Between Residual Fertility and Crop Yields............................................ 102 Affects of Residual Fertility on Wheat Yie lds..... lOU Affects of Res’dual Fertility on Bean Yields...... 10? Conclusions............................................. 110 VI EYJILLUA'LTTOhI OF PROCL:JUI)1‘J’S IXIT.) PBZ—fbrnvaTs 0 O O O O O O O O O C O O O O O C O O O O O I 11:]— Evaluation of Ecperimental Des irns...................... lll Evaluation of EXperirn- ental P'rocedures................... llh Evaluation of Analytical Procedures..................... 116 The Continuous Function Analysis..................... llé Utilization of Soil Test Heasures.................... 121 ‘Economic Interpretation and.Evaluation of Results....... 12- Concluding Remarks...................................... 12L BIBLJIOQmPEIYOOCOOOOOOOOOOOOOCOOOOOOQOOOOOOCOOOOOOOOOOOOOOOOOOOOOO. 127 ix 10. ll. 12. 13. 1h. 15. 16. 17. LIST OF TABLES United States Fertilizer Consumption 1910-1955.............. Michigan Fertilizer Consumption 1939-1955................... Experimental Design for the Oats, Wheat, Alfalfa and Corn Rotation.................................................... Experimental Design for the Continuous Corn Experiment...... Experimental Design for the Beans, 'heat and Corn Rotation.. Experimental Design for the l95h Potato Experiment.......... Experimental Desipn for the I956 Potato Experiment.......... Observed and Estimated Oat Yields, l956..................... Changes in Oats Yields Resulting from Unit Changes in Nitrogen fipplj-cati0nSOOOO0.00.0.0...OOOOOOOOOOOOOO0.00.00... Changes in Cats Yields Resulting from Unit Changes in P205 ApplicationSoOOOOOOOOOOOOOOO0..OOOOOOOOOOOOOOOOOOOOOOOO Changes in Cats Yields Resulting from Unit Changes in K20 ApplicationSOO0.0.0.0....OOOOOOOOOOOIOCCOOOOOOOOOOOOOOOOO... Observed and Estimated Wheat Yields, 1956................... Changes in Wheat Yields Resulting from Unit Changes in Nitrogen Iipplj.cati0n800000..OOOOOOOOOOOOOOOOOOOOOOOOO0...... Changes in Wheat Yields Resulting from Unit Changes in P205 AppliLZa-tionSOO.OOOOOOOOOOOOOO...0.0.0.0...0.0.0.0000... Changes in Wheat Yields Resulting from Unit Changes in K20 Applications.OOOOOOOOOOOOO0.00.00.00.000.0.0....0.00.0000... Comparison of Observed and Predicted Corn Yields on a Kalamazoo Sandy Loam Soil, l95§............................. Observed and Estimated Bean Yields, l956.................... 61 63 67 ‘3 \J.) C37) pf LIST OF TRBLES - continued TABLE 18. 19. 20. 21. 22. Changes in Bean Yields Resulting from Unit Changes in Ap131j-ed hIitrO£jeHOOOOOOOOOIIOOOOOOOOOOOOOOOOOOOOOIOOOOOOOOOOO Changes in Bean Yields Resulting from Unit Changes in Applied F‘IIOSC’IfilCI'iC [LeidOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOO. High Profit Fertilizer Inputs for Field Beans with Varying Bean Pricesoo0000000000000.000.00.00000000000coco-0.00.00... Incidence of weed Infestation and Plant Lodging on Oat Plots as Related to Nitrogen Applications......................... 1 Comparison of Amounts of Yield Variance Associated with Alternative Production Function Formulations................ 1 Estimated High-Profit Plant Nutrient Applications for 90 Cl If? 4.x) Varj.0us CropSOOOOOOOOOOOOOCOOC...0......OOOOOOOOOOOOIOOOOOOO 1—23 xi FIGURE 1. \‘l o O\ 0 LIST OF FIGURES Partial derivatives of polynomial and exponential functions for oats with reSpect to nitrogen........................... Partial derivatives of the polynomial and exponential functions for oats with respect to phOSphoric acid.......... Partial derivatives of polynomial and exponential functions for oats Pfi'tkl reSpeCt t0 pOtaShoooooooo0000.00.00.00.0000000 Partial derivatives functions for wheat Partial derivatives functions for wheat Partial derivatives functions for wheat Partial derivatives functions for beans Partial derivatives functions for beans of the polynomial and exponential with respect to nitrogen................ of the polynomial and exponential with reSpect to phOSphoric acid......... of the polynomial and exponential with respect to of a polynomial with reSpect to of a polynomial with a 4- espect to potaShOOOCOOOOOOOOOOOOOO and two exponential IlitrOSGHOOOOOOOOOOOOOOOO and two eXponential phosphoric acid......... Pap (D 70 72 CHAPTER I THE NATURE AND MAGNITUDE OF FERTILIZER USE PROBLEMS Current Research Problems in Fertilizer Use Recently much attention has been devoted to the economics of ferti- lizer use in the United States. Research receiving increased emphasis includes various attempts to determine the most efficient forms and carriers of the three primary plant nutrients: nitrogen, phOSphorus and potassium. A second important area of research is that of attempting to determine the relative effectiveness of alternative methods of fertilizer appli- cation. One such alternative is broadcasting the fertilizer and plowing it down prior to planting the crop. A second alternative is placing all or a portion of the fertilizer in bands of varying depths and dis- tances from the seed. A third method is that of applying all or a portion of the fertilizer by tOp dressing the growing crop. Other alternatives include combinations of the above listed procedures. A third major area of fertilizer research, which is interrelated with the previously mentioned two, is that of deriving fertilizer input- crop output ratios and relationships. Current research includes deriving such input-output relationships for the three primary plant nutrients to for a number of crops on a variety of soil types and for differing management practices. This by no means exhausts the list of fertil work currently being conducted. iowever, it in‘icates three of the major areas in which fertilizer research is being conducted and illustrates N I the diversity of current fertilizer research. The latter researca area, that of der 1% Vin: input-output relationships plus an economic interpreta- 1* tion of these relationships, s the primary concern in ‘his thesis. Derivation of physical input-output ratios or physical production functions is only the first step in an economic analysis designed to determine optimal fertilizer use. Once such physical relationships have been empirically established, profit maximization principles can be employed to determine Optimal fertilizer use with a given set of crop and fertilizer prices and given the earning power or marginal value productiv- ity of other farm expenditure or investment categories. The Importance of Fertilizer as an Aoricultural Productux1Factor The eXpanded interest and resources currently being allocated to obtaining more detailed and reliable information about the economics of fertilizer use appears to be warranted by (l) the importance of fertilizer as production factor in United States and Michigan agriculture and (2) the need for greater production from.American agriculture in the years ahead. The latter can be obtained only by the use of more production resources and/or a more efficient combination of production factors. Fertilizer consumption in the United States has increased rapidly over the past several decades as indicated by the data shown in Table 1. Consumption of the primary plant nutrients in 1910 totaled h6,000 tons of nitrogen, h99,000 tons of P205, the common fertilizer form of phos- phorus, and 211,000 tons of K20, the common fertilizer form of potassium. By 195h these totals had increased to 1,868,000 tons of nitrogen, 2,228,000 tons of P205 and 1,868,000 tons of KéO. Recently particularly large increases have occurred in the consumption of nitrogen and potassium with nitrogen consumption more than doubling from 19h9 to 195h. Preliminary estimates indicate further substantial increases for 1955 with a slight decline in 1956. The decline in consumption in 1956 was accompanied by a decrease in total crOp acreage for the United States as a whole during that year . Increases in fertilizer consumption have occurred in Michigan with even greater relative increases in recent years than for the United StateS‘ as a whole. In contrast to total United States consumption which declined slightly in 1956, Michigan consumption increased slightly over that of 1955. The annual consumption of the primary plant nutrients for Michigan during the period 1939 to 1955 is indicated in Table 2. During this period total nitrogen consumption increased over 11 fold from 3,31h to 37,18h tons while consumption of P205 increased from 18,016 to 88,228 tons. Consumption of K20, which was only 9,97h tons in 1939, increased to 85,3h3 tons in 1955. Assuming a price of $.15 per pound for elemental . l . nitrogen, $.10 per pound for P205 and $.11 per pound for £20, the total 1These are the prices currently being used by fertilizer experts as being typical of prices paid by Michigan farmers. TABLE 1 UNITED scares FERTILIZER CONSUNPTION 1910-1955 :— Year Prhmry Plant I‘éutrients in Thousands of Tons N P205 th 1910 us u99 211 1920 228 660 257 1925 279 680 282 1930 377 793 35h l9h0 h19 912 h35 19hl hSB 993 h67 19h2 399 1,131 5&6 19u3 508 1,238 6&3 l9hh 63S 1,h05 6h9 19h5 6&1 1,h35 7S3 l9u6 759 1,671 85h 19h? 835 1,775 878 19h8 8hl 1,8h2 956 19h9 911 1,88h 1,065 1950 1,126 2,073 1,215 1951 1,265 2,091 1,h13 1952 1,h8h 2,218 1,607 1953 1,6u8 2,209 1,720 195h 1,868 2,228 1,868 1Source: agricultural Statistics 1955, U. S. Department of.Agri- culture (Washington: U. 3. Government Printing Office, 1956). TABLE 2 MICHIGAN FERTILIZER CONSUMPTION 1939-1955 Year Primary Plant Nutrients in Tonsl N P205 K20 1939 3,318 18,016 9,9'8 1980 3,931 19,672 11,078 1981 8,588 21,328 12,175 1982 8,991 38,730 19,303 1983 8,651 39,967 19,280 1988 7,223 36,687 19,628 1985 7,995 37,078 28,909 1986 9,235 51,291 26,096 1987 9,821 87,823 27,986 1988 9,898 56,361 32,186 1989 12,078 59,923 37,898 1950 18,898 66,786 85,171 1951 16,981 70,002 56,272 1952 21,798 75,937 66,513 1953 23,887 75,117 70,253 1958 30,190 76,27? 78,172 1955 37,188 88,228 85,383 1Source: Michigan Agricultural Statistics, (Michigan Department of Agriculture, July, 1956). These estimates were made by the Soil Science Department at Michigan State University. cash expenditure for Michigan would have been $11,155,200 for nitrogen, $17,685,600 for phosphorus, and $18,775,860 for potassium in 1955. The total cost for all three of the primary plant nutrients would have been $80,630,280 in 1958 and $87,576,260 in 1955. Although not all fertilizer is used in production of agricultural crOps, non-agricultural uses in Michigan were estimated1 to be only about 5.3 percent of the total nitrogen, 2.1 percent of the total P205 and 0.9 percent of the total K20 consumed. Estimates made in The 1958 Census 2 of.Agricu1ture indicate the total expenditure for fertilizer for farm use in Michigan was only'$31,l63,000 in 1958. Consequently, at least a portion of the plant nutrients were purchased at prices less than those listed as typical. The estimated cost for the total on-farm consumption of the three primary plant nutrients for the entire United States was $1,028,105,000 in 1958.3 Thus, farm expenditures on fertilizer exceeded a billion dollars in 1958-and was still increasing. Reasons for Increased Fertilizer Use Several reasons exist for increased use of commercial fertilizer by farmers. Plant nutrients have become much cheaper relative to most other farm inputs due primarily to a reduction in bulk and utilization lEstimates made by'W. H. Heneberry, Department of.Agricu1tura1 Economics, Michigan State University. 2"Use and Expenditures for Fertilizer and Lime," adapted from The 1958 Census of.Agriculture, (washington: U. S. Government Printing Office, 1956). 3Ibid. of more efficient manufacturing processes. Excluding transportation costs, the 1958-55 price of a unit of nitrogen was only about one- third of the adjusted 1920 price.1 A unit of K20 was only one-fifth of the adjusted 1920 price in 1958-55 while the adjusted price of a unit of P205 decreased about 27 percent during this 35-year period. A second important reason for increased fertilizer use is the availability of more information concerning the yield benefits realized by various crops from application of the primary plant nutrients. This information has been forthcoming in increasing quantities from numerous sources. Experimental results from Agricultural Experiment Stations and private fertilizer companies have been.utilized by farmers. .Agencies such as the Federal Extension Service, the Tennessee Valley.Authority and others have aided in providing farmers with educational materials and demonstrations of the affects of fertilizer on crop yields. In ad- dition, farmers personal experiences with plant nutrients together with those of their neighbors are the basis for increased fertilizer use by many farmers. It seems that we can validly conclude that commercial fertilizer is an important agricultural production factor as indicated by the fact that the value of the three primary plant nutrients used exceeded a billion dollars in 1958. It is a productive input used by a great number 1T. P. Hignett, "Our Changing Technology," Methodological Procedures in the Economic Analyses of Fertilizer Data, Edited by E. L. Baum, Earl O. Heady and John Blackmore7fhmes: Iowa State College Press, 1956) p. 205. of farmers producing a variety of crops. Farmers have greatly expanded fertilizer use in the past decade. They need additional information as to what expenditures for fertilizer are yielding in dollar returns. Such information is ne essary if farmers are to allocate optimally their capital resources between alternative farm investments and expenditures. Lonngun Agricultural Production Needs One of the major problems currently facing.American.Agriculture is that of surpluses for some of the major farm crops. In view of this problem, a question arises as to the logic of engaging in research which could result in recommendations indicating greater use of commercial fertilizer, larger crOp yields and greater total production. Several studies have been made in which attempts have been made to forecast future needs for farm products in the United States. Predictions of future potential demands for agricultural products are all considerably higher than quantities supplied by current production. Two factors seem to be of primary importance in these higher predictions. First, large population increases have been predicted. Using the period 1951-53 1 as a base, predictions made by the Bureau of the Census in 1955 are for a pOpulation increase of 11 percent by 960 and an increase of more than one-third of the base period population by 1975. U) ca 0 O E p: (*1 U “s (5 [—1 o ncreases in consumer income accompanying an -xpand- l ' nave been predicoed. Estimates made by the United States 4 ing econom 1Bureau of the Census, Current POpulation Reports, Series P-25 No. 123 (washington: U. S. Government Printing Office, October 20, 1955). 'Department of Agriculture indicate an increase of real per capital con- sumer income of almost two-thirds greater than the 1951-53 base period by'1975.l Estimates of total crop production needs for 1975 are about 25 percent above actual 1951-53 production.2 This overall increase is not uniformly distributed over all crops, however. For example, more than preportional increases are predicted for pasture and feed grain crOps since a needed increase of h5 percent in livestock production is forecast. The needed average yearly increase in production of feed grains from the 1951-53 base period to 1975 is 5 1/2 times the historical average annual long-term increase. No attempt will be made here to provide a comprehensive analysis of future agricultural production needs. Rather, the point being made here is that agricultural production needs will be much higher in the years ahead. This greater production must come from use of more resources, 3 more productive resources and/or a more productive combination of resources. 1H. H. Wboten and J. R..Anderson, “Agricultural.Land Resources in the United States-~with Specia1.Reference to Present and Potential Crop- land and.Pasture," Agricultural Information Bulletin IhO (washington: U. S. Department of.Agriculture, June, 1955). 2G. T. Barton and R. 0. Rogers, "Farm Output, Projected Changes and Projected Needs," Agricultural Information Bulletin No. 162 (washington: ‘Agricultural Research Service, August, 1956). 3A particularly critical problem currently faced by farmers and by ‘farm management researchersis that of finding combinations and quantities of other resources which will increase the marginal value productivity of labor. Numerous farm management studies have indicated an extremely low marginal value product for this extremely important farm resource. A discussion of the low'marginal value productivity of labor as well as a bibliography of other work on this subject may be found in, E. I. Fuller, "Michigan Dairy Farm Organizations Designed to Use Labor ‘Efficiently," Unpublished Masters Thesis, Department of.Agricultural Economics, Michigan State University, 1957. 10 Research workers in the Agricultural Research Service1 have made pro- jections of probable increases in pasture and cropland acreage of 25 million acres by 1975 or an increase of about one million acres per year. If the projected increase in cropland occurs, the necessary annual increase in crop production per acre will still be about 50 percent larger than that occurring in the post world war II period. In view of the long-run needs for farm products it is apparent that there will be a need for improved or increased use of farm resources in the next two decades. Improved information about the productivity of various resources, including fertilizer, will help farmers make the necessary production adjustments on an economical basis. The_Iype of Information Needed by Farmers In order to make economically sound decisions regarding how much and what analysis of fertilizer to use, farmers need rather Specialized information. First, they need information on yield reSponse to the three primary plant nutrients of the various crOps which they produce. They need information about the affects of different forms of fertilizer and different application methods. In addition, this information must be applicable to their particular type of soil, the soil management practices which they use or should use, and the weather conditions which they encounter. Differences in the fertility level of the soil will influence the yields obtained by various amounts of applied plant nutrients; thus, ca. 1H. H. Wboten and J. R..Anderson, gp, cit. 11 .effects of residual fertility need to be known. Finally, the price of fertilizer and the price of the crop produced will influence the high profit combination of plant nutrients to apply. Specification of the type of information needed by farmers is a guide in determining what research is needed and what research procedures may be followed in obtaining this information. For example, the effects of different variables such as the effects of the various plant nutrients on crop yields, the affects of weather on yield reSponses and the signifi- cance of crop and fertilizer prices on Optimal fertilizer use will be treated quite differently in the analysis. Applied plant nutrients can be measured and controlled and their effect on crop yields determined by statistical estimation. weather cannot be controlled but if experi- mentation is carried out over a number of years and a variety of weather conditions, yield reSponses for several sets of weather conditions and a probability distribution of responses with respect to weather can be acquired. In the case of crops and fertilizers, various prices may be applied to the physical input-output relations to correSpond with expected farm conditions. Currently, researchers are attempting to devise methods for incorporat- ing information about soil fertility acquired by chemical soil tests into their predictions of yield reSponses to fertilization. An attempt will be made in this thesis to reduce unexplained variances in crOp yields by taking into account soil test data. l2 Succeeding chapters of the thesis will pertain to alternative analytical procedures, specification of the experimental work being carried on at the Michigan Agricultural Experiment Station, analysis of the experimental data and, finally, evaluation of the results. CHAPTER II METHODS OF.ANALYSIS It is the generally recognized task of scientific endeavor to establish and verify relationships which are universal to some popula- tion.1 When relating various phenomena in the real world we find two dimensions of such relationships subject to variance. First, the relationships may vary with respect to the reliability of the empirical estimates which we can derive or establish for them, i.e., variance in the reliability dimension. Secondly, the size of the population to which such relationships are universal may vary considerably, i.e., variance in the application dimension. One would not expect, for example, to establish relationships between plant nutrients and crop yields as accurate or as general as those which have been established between the volume and pressure of gas as Boyle's law. However, if we believe that there are logical, systematic and describable relationships existing between plant nutrients and crop yields, it seems to be our task as scientists to attempt to quantify such relationships to the best of our ability. This is true particularly in view of the need for such infor- mation indicated in Chapter I. In so doing, an optimum level of 1Most of these relationships will of course be probability state- ments about relationships. Thus the universality referred to here does not imply absoluteness of the relationships specified, but rather implies universal applicability to some population of the deductions and infer- ences made. 1h accuracy of quantitative estimates can be defined by equating the cost of additional accuracy with its value. Failure to structure and quantify relationships systematically, when such action is possible, is likely to result in a failure to make Optimum use of scientific procedure in deveIOping a body of interpersonal information useful to researchers working on this and related problems of soil fertility and/or farm management. Methods of Collecting Data Two methods of securing data for use in determining the relation- ships existing between variables are generally recognized as being valid forms of scientific methodologg. These are (1) controlled experimentation and measurement of relationships and (2) collection of non-controlled observations, as in astronomy, which typify the population being studied and to which relational inferences are to be made. Both of these two methods have advantages as well as some disadvantages which vary some- what with the nature of the Specific problem being studied. The discussion which follows is an attempt to evaluate the two procedures in the context where determination of plant nutrient input—crOp yield output relation~ ships is the problem being investigated. The former method, controlled experimentation, has the relative advantage of lending itself to more precise estimation of rel tionships between relevant variables. Greater accuracy is usually obtained in controlled experimentation for two reasons. First, variables can be measured more accurately. Fertilizer applications and crop yields, 15 for example, can be measured quite accurately on experimental plots. Secondly, controls can be enforced quite rigorously; for example, tillage practices, insect infestations, soil characteristics,etc.,can be controlled better on experimental plots than under farm conditions. . Such controls, though facilitating accurate estimation of relationships between studied variables have an accompanying disadvantage. This dis- advantage is that there is a possibility that no pOpulation other than the experimental one may have exactly the same combination of controlled and uncontrolled variables interacting in the production processes being studied. It follows that one may not be able to draw inferences from the experimental results and apply them validly to any given farm popu- lation. The alternative method, that of collecting non-controlled observations by a sample survey procedure, has proven effective in numerous types of research. It is difficult, however, to utilize this method when acquiring fertilizer reSponse information because (1) dif- ferences in numerous uncontrollable factors such as insect damage, weather, tillage and harvesting methods are apt to bias the results or introduce excessive unexplained variance and (2) studied inputs are difficult to measure accurately. Another shortcoming of using the sample survey method in estimating fertilizer reSponse surfaces is the difficulty of acquiring observations dispersed over the range and combination of plant nutrients necessary to obtain a statistically reliable estimate of the yield reSponse surface. These and other problems have been 16 encountered by researchers working with non-experimental data. It is the opinion of most soil scientists and other researchers that the controlled experiment method of obtaining data is not only the more scientific method but the only one producing reliable estimates of fertilizer input-output functions. As experimental input-output data .1. become available a logical follow-up stage of analysis would be CO V , L J. L.) a test the applicability of these results under farm conditions. This procedure should indicate whether or not results obtained from the experimental sample may be validly inferred to some farm population. The Concept of Functional Relationsnips The principles utilized by economists in determining various Optimal conditions of resource use and production output are stated in numerous publications by numerous authors. However, it seems desirable to outline briefly some of the principles of economic theory which can readily be applied to the production relationships of interest in agronomic-economic work. In order to apply effectively the deductive principles of economic theory, the relevant production relationships need to be specified rather systematically or formally. 1For a discussion of problems encountered and results obtained using non-experimental data in fertilizer input-crOp output determinations see E. W} Kehrberg, "Some Problems Involved in Fitting Production Functions to Data Recorded by Soil-Testing Laboratories," Methodological Procedures in the Economic.Analy§es of Fertilizer Data, Edited by E. L. Baum, Earl O. Heady and John Blackmore (Ames: Iowa State College Press, 1956) pp. lBh-lhO and H. H. Yeh, "Estimating Input-Output Relationships for Wheat in Michigan Using Sampling Data, l952-5h, Unpublished Masters Thesis, Michigan State University, 1955. 17 Agronomists have hypothesized for years that plant nutrients and crOp yields are functionally related. Numerous attempts have been made to Specify these relationships in equation form for various creps and plant nutrients. In its simplest form, this functional relationship may be written Y - f(X) where Y is the crop yield and X the plant nutrient, in this example nitrogen. Recognizing that other factors interact with nitrogen, X1, and are necessary for crop production, we write: Y . f(X1,X2,oooo,Xi,oooo,Xn) where X1 represents nitrogen andX2 to Xn are other factors such as P205, K20, water, temperature etc. To symbolize that all factors except nitrogen are fixed at some constant level, we write Y . f(Xl :2,oooo,Xi’oooo,Xn)o Furthermore, if all factors affecting crOp yields cannot be isolated and specified, we say Y I f(X1/X2,oooo’Xi,oooo,Xn)+ U A where U is an error term representing the unexplained variance of Y 1 (predicted yield) from Y (observed yield). If it can be validly 11f unexplained variance is to be validly attributed solely to components of the error term, U, the specified functional relationship must be the right one, i.e., it must be the real world functional relationship. 18 assumed that: (1) factors which contribute to U, i.e., unspecified factors, are normally and randomly distributed with respect to the measured variables (in this case X1) and (2) that the expected value of U is zero, the existence of this unSpecified source of yield variance does not bias statistical estimates of the influence of the observed variables on Y. The specification of the functional relationship between plant nutrients and crop yields, commonly called a production function, has taken different forms over a period of years. Justice Von.Liebig's "Law of the Minimum" was an early attempt to Specify the form of fertili- zer production functions. This formulation postulated that crop yields increased in direct proportion to additions of the nutrient which was limiting plant growth. Thus, other production factors were assumed to be perfect complements of the limiting factor. This formulation of the fertilizer-crOp yield production function has been rejected because researchers have observed that: (1) production factors are not perfect complements, i.e., a given crop yield may be produced with varying quantities and combinations of applied N, P205, K20, water etc. and (2) additional inputs of a factor limiting crop yields does not typically result in linear additions to crOp yields but rather it results in diminishing additions to crOp yields for a time and eventually further additions of the factor cause an actual decrease in total yield. Since Von Liebig's early formulation, numerous attempts have been made to use different forms of production functions to describe these input-output relationships. although numerous types of functions have been formulated, none has been accepted as "best." These various func- 1 tions have received adequate discussion in other literature and will not be analyzed here. There are, however, several criteria which must be satisfied by a particular function if it is to provide a re: fornulation of the input-output relationships between fertilizer inputs and crop outputs. The function should be cap able of reflecting suc- cessively the following yield responses to added inputs of plant nutrients: (l) vields increasing at a diminisuinw rate and (2) decreasi Lng total yields. If the soil is relatively low in initial fertility, an earlier state of input-output relationship may be present. This is the stage C. "I 1 where yieics increase at an incram sin: rate in response to additional 3 input 3 of plant nutrients. In addition, if interaction between plant nutrients is expected, L-e formulation should incluce equma Wior 1 variables '3 to Specify this interaction. If the characteristics of plant growth and crOp yields could be formulated theoretically to the extent that a prOper equational form lHistorical descr ription of use of production functions in estimat— t ilizcr-CTOP yield relations may be fornd in the following 7ubli- ing fer ‘ cations: John C. Redman ands at eph 1en Q. allen "Some Int+3 rolationsutns f acoromnc and.A:ronomic Concepts ," Journal of Farm BC or emits, vol. KIIVI (nu :ust, l9:~d), pp. L5 2 UGS and T r‘l C. deady, John T. Pese: and William Br wn, Crop i": orte Surf? '0'}: and Esono C (Ah a 1:1 Ft.ert?'l__.i7-;-er Use, Research Bulletin l¢., Agricultural ixperiment Station, Iowa tate College, lQSS). 2Such interaction may be incorporated into the functional relation~ ship in several ways. It is in a sense automatically included in a production function of product form such as an exponential. Special cross product terms may be included in a polynomial type equation. The point of importance is that it be included so that partial derivatives of yield with reSpect to individual plant nutrients reflects the level at which other interacting nutrients are considered. 20 could be deduced, statistical estimation of the production function would be greatly singlified. The statistical task would then be only that of estimating parameters for the variables in the functional relationship and obtaining reliability measures for these parameters. However, lack- ing a precise theory as to the proper functional form, it is necessary to compare various equations to see which "best" describes the observed relationships. Various problems of design and alternative analysis necessitated by lack of knowledge about the apprOpriate functional form will be developed later in this chapter. Detrnnhioijvt'fleormuvic (hitira After obtaining an estima,e of the production function for plant fi nutrients, various optimal combinations of phant nutrients may be determine". If for example, the following equation: Y - + b " + h ” 3 + b T + b V 9 + b " + b V 2 ‘ a 1&1. “2“1 see ar~2 the see describes the relation of yield to the three plant nutrients Kl, KB, and X3, then the following proeoiuce is used to find the combination of plant nutrients producing the naximua yield. the *artial derivatives of the three nutrients with respec to yield gives: 0 ’ - 1 FA 1.- (l) -—+- = 01 + aogal A h) v n 0" m + 2134:: 2 TT' + 2004-3. G] t. _ U) u) c1 ('1‘ r ,4 (I) as ,0 ’t‘ {11 ’ S C - t... 9) p4 CL 6) *3 t... f- U: pvacive equal to zero and olving the three equations simultaneously gives the combiaation of plant nutrients pro- ducing the maximum crop yield. To obtain the economically optir O . u ‘fl -- V '7 -0 J-’ J. . w’ .0 _ Q _. “'1 ~~ > ‘ ‘_-¢ '1: 0 1‘ ‘ r _| '§ 3. have sen or.s1 ied primarily with the OOjective Oi pIOvLLlRH ahuQHdhC data for estiration of the reSpective crop j/ie ld surfaces icr c..l;1-ut #3 A] ‘1. nsx'» . ‘\r\- . . ,‘A‘. -'~“ mmmmm ,. .\ ‘I 1‘, mvzral int er ested ajenciss LaVC aLuJQ the e: p “ilenta 20" by .L -. ,. .-.. A. - J.‘ - .. , . ,. .L‘. ,.. .' m1, .1. ' , .- contrisuting funds or OUJ’F r; {urCUu, ULCQC include: ;.3 lLa at the l Ilant fl r a '..-‘ m“, ' -‘l‘ 9v. " ‘45 o ‘r (\J“ . h ,1, r“ \ "if“ '3'! 1. '1" reed COdnCil, The navison Che teal Covworetion and The iw.vess3) Ja'iq -‘ J-‘ .3 J- all CEO 1‘4. by . ’3’“ it“. .1 33 crops. This objective, together with the restriction Of limited funds for experimentation, provide the main restrictions for the framework within which the experimental designs were developed. In conforming to these restrictions the designs have the following general characteris- tics: (1) individual observations cover those portions of the production surfaces of interest to researchers, (2) the experiments contain a minimum number of replicated plots since the objective is to estimate the entire surface over the range in which it is Of economic importance thus minimizing the need for establishing accurate measurements of individual surface points (3) the designs involve numerous check plots (plots to which no fertilizer is applied) to establish the origin of fitted functions, i.e., the yield value with no plant nutrients applied and (h) to the extent possible, intercorrelations among the amounts of nutrients applied have been minimized to facilitate estimation of the equational parameters with greater reliability than would be the case if such intercorrelations were high. The designs vary somewhat for different experiments but may be broadly classified as incomplete factorials. The Cats, Wheat,gglfalfa and Corn Rotation In the Spring of 1955 an experiment was initiated for a rotation of oats, wheat, alfalfa and corn. This experiment is located at two sites in Kalamazoo and Calhoun counties on a Kalamazoo sandy loam soil. This is a light upland soil having a tendency to be somewhat drouthy and of relatively low natural fertility. Each crop of the rotation is 3h grown each year; thus there are four fields each having the same experi— mental design. The experiment includes all three Of the primary plant nutrients, nitrogen, phosphoric acid and potash in varying combinations. Six treatment levels, including the zero application level, are included in the experiment for each of the plant nutrients. These treatment levels measured in pounds per acre are: N - o 20 to 80 160 2&0 P.o5 - o to 80 160 320 hBO K20 - o 20 to 80 160 2ho Ninety-one individual surface points are sampled, twenty-seven of which are replicated twice in a 3 x 3 x 3 factorial at the 2nd, hth and 6th treatment levels. There are eleven replications of the check (0,0,0,) treatment. There are one hundred and thirty plots in each of the four fields in the experiment. Individual plots are SO x 1h feet in size, making a total area per plot of about l/62.S of an acre. The lh foot width facilitates use of a 7 foot grain drill for fertilizer and seed application and a 7 foot self propelled combine for harvesting Operations. Almost all fertilizer applications are made by broadcasting the fertilizer, either mechanically or by hand, prior to plowing the ground and preparatory to planting the crop. Two notable exceptions are: (l) the first level of applied P205 (ho pounds per acre) is applied in the row at planting time as a starter 35 fertilizer and (2) the alfalfa crop is fertilized by top-dressing in the Spring. The design for this experiment is shown in detail in Table 3. Continuous Corn An experiment in which corn is grown in continuous culture was initiated in Tuscola County in 1956. This experiment is located on a Wisner clay loam soil which is one of the heavier,more productive soils occurring in the state. The experiment contains 20h individual plots representing 139 surface points. Included in the design is a 3 x 3 x 3 factorial replicated three times including observations at the 2nd, hth, and 6th treatment levels. In addition, there are eight check plots. Inclusion of the triplicated factorial allows limited study of yields and other experimental data by analysis of variance techniques. The seven treatment levels in pounds per acre for the three plant nutrients in this experiment are as follows: N - 0 20 &0 80 160 2&0 320 19.05 - 0 &0 80 160 320 &80 6&0 6 K20 - o 20 &0 80 160 2&0 320 Individual plots are 55 x lh feet in size allowing h rows Of corn Spaced h2 inches apart to be grown on each plot. The design for this experiment is shown in detail in Table h. 36 TABLE 3 EXPERIMENTAL DESIGN FOR THE OATS, WHEAT, ALFALFA AND CORN ROTATION Plant Nutrients NO. Plant Nutrients NO. (Pounds Per.Acre) of (Pounds Per.Acre) of N P205 K20 Plots N P205 K20 Plots 0 O O l 80 160 80 O O O 80 160 2&0 O hO 20 80 320 ‘ hO 0 160 O 80 320 160 O 160 80 80 320 ZhO O 160 2h0 80 NBC 0 O hBO 80 80 h80 20 O hBO 2hO 8O hBO 80 2O 0 20 80 b80 160 20 &0 O 80 hBO 2hO 20 hO 20 160 hO to 20 DO 80 160 to 160 20 hO ZhO 160 80 20 20 80 hO 160 80 DO 20 80 160 160 80 80 20 160 20 160 80 2hO 160 160 hO 160 160 160 160 160 2&0 20 160 80 20 160 ZhO 20 320 hO n>FJADRJFJFJFJF’FJRDRJRDFJRJRJRJFJF4FJFJFJFJFJRDFJFJFJFJP‘F‘F4FJFJPJFJRJFJAJRDFJF‘F‘FJAJAJ 20 320 160 160 320 20 20 &80 20 160 320 80 20 &80 80 160 320 160 20 &80 2&0 160 320 2&0 &0 &0 &0 160 320 &0 &0 &0 160 160 &80 80 &0 80 20 160 h80 160 &0 80 &0 160 &80 2&0 ho 80 80 QhO O 80 &0 80 2&0 2&0 0 2A0 &0 160 &0 2&0 &0 20 &0 160 160 2b0 &0 80 &0 320 20 2&0 ho 2h0 &0 320 80 2&0 80 160 &0 320 2&0 2&0 160 20 &0 &80 he 2&0 160 80 &0 &60 160 2&0 160 2&0 8O 0 O ZhO 320 O 80 0 80 2&0 320 &0 80 0 2&0 2&0 320 160 80 &0 20 2h0 320 2h0 80 &0 80 2&0 &80 0 80 &0 2&0 2&0 use 20 80 80 hO 2h0 hBO BO 80 80 160 2&0 &80 160 80 160 0 2&0 &80 2&0 nDF‘FJFJAJRDRDFJFJFJFJFJF‘F’F’F’F‘F’F’RDFJFJFJnDRJAJFJF’AJAJRJFJF4AJADAJFJPJF‘F‘F‘F’F‘F‘F‘P‘ 80 160 20 ‘1 1 J 313 h tn: _ Li- EKPeRILEKIAL ssszen FOR THE csnumxtsccmzssustuir 37 Plant I‘éutrients I~.(:. Plant I‘Iutrients I20. (Pounds Per Acre) of (Pounds Per Acre) of N P205 K20 Plots H P20r K20 Plots 0 0 c a hC 320 to 1 C C 11.0 1 LC! 22C 1 :11) l 0 &3 2&0 1 &o 32‘ 320 1 0 80 0 1 &0 &80 &0 1 0 80 &0 1 &0 &80 80 1 0 160 320 1 &0 &80 2&0 1 0 320 160 1 &0 6&0 20 1 0 &80 20 1 &0 6&0 160 1 0 6&0 80 1 &0 6&0 320 1 0 6&0 320 1 80 0 0 1 20 &0 20 3 80 0 160 1 20 &o 80 3 80 &0 20 3 20 &0 160 1 80 &0 80 3 20 &0 2&0 3 80 &0 2&0 3 20 8O 20 1 80 80 &0 1 20 80 80 1 80 80 160 1 20 80 2&0 1 80 80 2&0 1 20 160 20 3 80 160 20 3 20 160 &0 1 80 160 80 3 20 160 80 3 80 160 2&0 3 20 160 2&0 3 80 160 320 1 20 320 20 1 80 320 0 1 20 320 160 1 80 320 &0 1 20 320 320 1 80 320 160 1 20 &80 20 3 80 &80 20 3 20 &80 &0 1 80 &80 &0 1 20 &80 80 3 80 &80 80 3 20 &80 2&0 3 80 &80 2&0 3 20 6&0 160 1 80 6&0 80 1 20 6&0 320 1 80 6&0 320 1 &0 o 0 1 160 0 20 1 &0 0 &0 1 160 0 80 1 &0 &0 20 1 160 &0 0 1 &0 &0 &0 1 160 &0 80 1 &0 &0 80 1 160 &0 2&0 1 &0 &0 160 1 160 80 &0 1 &0 &o 320 1 160 80 160 1 &o 80 0 1 160 160 0 1 &0 80 &0 2 160 160 20 1 &0 80 2&0 1 160 160 80 1 to 160 20 1 160 160 160 1 &0 160 80 1 160 160 2&0 1 no 160 160 1 160 160 320 1 &0 160 2&0 1 160 320 &0 1 &0 320 20 1 160 320 80 1 &0 320 &0 1 160 320 160 2 Continued 38 Table h concluded Plant Nutrients NO. Plant Nutrients No. (Pounds Per Acre) of (Pounds Per.Acre) of N P205 K20 Plots N ’ P205 K20 Plots 160 h80 20 160 hBO 80 160 hBO 320 160 6hO 80 160 6hO 2hO 2&0 6h0 160 2&0 6&0 320 320 0 80 320 &0 20 320 to 2&0 2&0 0 0 320 80 to 2&0 0 &0 320 80 80 2&0 &0 20 320 80 160 2&0 &0 80 320 80 2&0 2&0 &0 2&0 320 80 320 2&0 80 0 320 160 0 2hO 80 &0 2&0 80 160 2hO 80 320 2&0 160 20 2hO 160 80 2h0 160 2h0 2h0 320 to 2h0 320 160 2h0 320 320 2h0 h80 20 2hO hBO 8O 2hO h8O 2h0 2hO OhO &0 320 160 &0 320 160 320 320 320 &0 320 '320 80 320 320 160 320 320 2&0 320 &80 20 320 &80 160 320 &80 320 320 6&0 &0 320 6&0 80 320 6&0 2&0 320 6&0 320 FJbJUJU)F‘FJFJuJquJFJFJF‘FJUJbJUJF‘F‘F’F’F’F’F’ nJF‘F’F‘F‘F’FJFJFJFJF‘F‘F’F‘F‘F’F‘k’k‘k‘k‘k’k‘F‘ Field Beans,_Wheat and Corn Rotation An intensive rotation of field beans, wheat and corn was initiated in Gratiot county in 1955. Corn was produced on these plots in 1955 and field beans in 1956. The experiment is located on a Simms loam soil, a heavy productive soil which can be crOpped.quite intensively without hazard of erosion damage. The seven treatment levels for the three plant nutrients are identical to those in the continuous corn experiment. 39 The treatments in pounds per acre of applied plant nutrients are: N - O 20 hO 80 160 2hO 320 PnO5 - 0 ho 80 160 320 hBO 6hO ~ K20 - O 20 &0 80 160 2hO 320 The experiment is an incomplete factorial consisting of 193 individual surface points of which twenty-seven are replicated twice in a 3 x 3 x 3 factorial at the lst, hth, and 6th treatment levels. There are ll check plots in the basic experimental design which contains a total of 233 individual plots. Extra plots were included in the experiment for purposes of other analyses bringing the total number of plots to 258. Individual plots in this experiment are SO x lh feet in size. The design for this experiment includes a more complete Specification of the production surface than any of the other experiments with the total of 193 different surface points exceeding that of any other experiment currently being conducted. The experimental design for this experiment is shown in Table 5. Potatoes Two experiments have been established to measure the response of potatoes to variable quantities of applied P205 and K20. The first of these experiments was initiated in l95h and the second in 1956. Both experiments are being conducted on a Houghton muck soil on the Experiment Station muck farm near East Lansing. Only two plant nutrients, P205 and K20 are treated as variables in this experiment. The muck soil is high in organic matter content and consequently high in nitrogen. Available TABLE 5 EXPERIMENTAL DESIGN FOR THE BEANS, WHEAT AND CCBN ROTATION hO I Plant Nutrients NO. Plant Nutrients No. (Pounds Per.Acre) of (Pounds Per Acre) of N P305 K20 Plots N P205 K20 Plots 0 O 0 ll 20 320 160 2 O 0 2O 1 20 320 320 2 O 0 ho l 20 hBO 2O 1 O O 80 1 2O hBO hO l O O 160 l 20 htO 160 l O O 2hO 1 2O hBO 320 l O O 320 l 20 OhO 2O 2 0 ho O l 20 GhO hO l O &0 2hO l 20 ého to l 0 to 0 1 20 6&0 160 2 O to &0 l 20 6hO 2hO l O 160 O l 20 OhO 320 2 O 160 320 l &0 O O l 0 320 0 1 &0 0 &0 1 O 320 160 l &0 hO 2O 1 O hSO O 1 ho hO to l O th 2O 1 hO hO 160 l 0 6&0 0 1 &0 &0 320 1 O 6hO 60 1 ho to O l 0 6&0 320 1 &0 80 &0 2 2O 0 O l hO BO 2hO 1 2O &0 2O 2 ho to 320 l 20 hO hO 1 ho 160 20 l 20 &0 60 1 &0 160 60 1 2O hO 160 2 hO 160 160 1 2O hO 2hO 1 ho 160 320 l 20 hO 320 2 hO 320 20 l 20 8 2O 1 hO 320 80 l 20 SO 80 l hO 320 160 l 20 80 160 l hO 32 320 l 20 to 2hO l &0 hCO LO 1 2O . 60 32C 1 hO hOO to l 20 ISO 20 l &0 th 2hO 1 20 ISO &o 1 ho 6&0 2O 1 20 160 150 l hO ehO 60 1 2C 160 2&0 l hO éhO 160 l 20 160 32 l &o 6&0 320 l 20 32C 20 2 OC O O l 20 320 LO 1 CO 0 320 l 20 320 tO I SO no 20 l _‘ . - I Continued Table 5 continued Plant Nutrients NO . Plant Nutrients NO . (Pounds Per.Acre) of (Pounds Per Acre) of N P 205 K 20 Plots N P 205 K 20 Plots 80 to 80 8O &0 160 80 &0 320 80 80 &o 80 80 2&0 80 80 320 160 320 20 160 320 &0 160 320 80 160 320 160 160 320 2&0 160 320 320 80 160 20 160 &80 20 80 160 80 160 &80 &0 80 160 160 160 &80 160 80 320 20 160 &80 320 80 320 80 80 320 160 80 320 320 80 &80 &0 80 &80 2&0 80 &80 320 80 6&0 o 80 6u0 20 80 6&0 &0 80 6&0 160 80 6&0 2h0 80 6&0 320 160 6&0 20 160 6&0 &0 160 6&0 80 160 6&0 160 160 6&0 2&0 160 6&0 320 2&0 0 0 2&0 &0 20 2&0 hO 80 2&0 &0 160 2&0 &0 320 2h0 80 NO 160 O O 2h0 80 320 160 O 160 2hO 160 20 160 hO 20 2hO 160 80 2h0 l60 160 2h0 160 ZhO 2h0 320 20 2ND 320 80 2ND hBO 160 2&0 320 6&0 2hO hBO hO 2h0 hBO 80 2hO hBO 2h0 2h0 6h0 20 2110 611,0 80 2&0 6ND 160 2h0 6&0 320 320 O 0 320 O 80 160 &0 &0 160 &0 80 160 &0 160 160 &0 2&0 160 &0 320 160 80 20 160 80 80 160 80 160 160 80 2&0 160 80 320 160 160 20 160 160 &0 160 160 160 160 160 320 160 320 0 I—‘l—‘l—‘i—JHl-‘Hl-‘HHmHmHHml-‘l—‘HI-‘HHHI—‘I—‘HHHHHHHNHl—‘Hl-‘l—‘HH HHHHHHNHHI—JHHHHHHHHHHHHHHNHNHHml—‘HHHNHNHHM Continued h2 Table 5 concluded Plant Nutrients No. Plant Nutrients NO. (Pounds Per Acre) of (Pounds Per Acre) of N P205 K20 Plots N P205 K30 Plots 320 0 320 1 320 320 &0 1 320 &0 20 2 320 320 80 1 320 &0 &0 1 320 320 160 2 320 &0 80 1 320 320 2&0 1 320 &0 160 2 320 320 320 2 320 &0 2&0 1 320 &80 20 1 320 &0 320 2 320 &80 &0 1 320 80 20 1 320 &80 160 1 320 80 80 1 320 &80 320 1 320 80 160 1 320 6&0 0 1 320 80 2&0 1 320 6&0 20 2 320 160 0 1 320 6&0 &0 1 320 160 20 1 320 6&0 80 1 320 160 &0 1 320 6&0 160 2 320 160 160 1 320 6&0 2&0 1 320 160 320 1 320 6&0 320 2 320 320 20 2 nitrogen may be in short supply at a given time but largely because Of weather conditions not conducive to sufficiently rapid nitrification. Thus, it usually does not pay to apply commercial nitrogen fertilizers unless temporary nitrogen deficiencies are evident. Furthermore, if nitrogen applications are made, the amount of applied nitrogen is, at best, a poor indicator of the amount of nitrogen available for plant use. The l95h Potato Experiment 1 The experiment initiated in l95h consists of an incomplete factorial 1The experimental design and treatment rates used in this experiment were established by Professors G. L. Johnson and J. F. Davis of the Michigan.Agricultural Experiment Station. b3 of seven levels of P205 and nine levels of K20. The treatment rates in pounds per acre are as follows: P205 - l 25 50 100 200 300 hSO K20 - l 25 50 100 200 350 550 750 900 The design includes forty-seven surface points, twenty-nine of which are replicated twice for a total of seventy-six plots in the experiment. Individual plots are h9 by ll feet in size. Application of a single pound of a plant nutrient to some plots constituted a substitution for the zero treatment level. Utilizing a one pound treatment alleviated the problem of having to use negative logarithms when fitting exponential functions to zero treatments. The design for this eXperiment is presented in detail in Table 6. After the first potato crop was produced on these plots, the plots were Split and half of the plot continued to receive the original fertilizer treatment while the other half of the plot received no ferti- lizer in subsequent years. This procedure was practiced because of complications due to the high fertility level of the land at the time the experiment was initiated. This original high fertility level resulted in negligible yield changes with additional applications of plant nutrients. Continuous soil testing Of these experimental plots in succeeding years should provide valuable information with reSpect to residual fertility values Since over time a wide range of fertility levels should deveIOp on these plots. *J 1.4) '._l 6 assumes. Deena»: FOR "we 19% POTATO Hemmer? 1 P205 Treatment (pounds per acre) 1 25 50 100 200 300 hSO F" N N >4 (R p; “i. j :3 {>4 s per acre) g. N ii 1 'C‘; X XX £301 71 H O O E: ii (~. + U 200 X XX XX X :3; he use >4 £2 350 550 750 "5 K20 Treatmen ii ii MN :23 {>4 ii >5 :2” g; XX XX 1 1 r~ :- 1’, AA A p4 900 X Y lEach X represents one experimental plot. m1 A r 3 -1,.:. 1-..... .' M. .1. nr3135e Iotam3.iperinenu A new potato experiment was inititted in l956. This eXperiment was established on a newly cleared muck soil which had not been previously farmed and which had not received previous applications of plant nutrients. The experimental design includes at different surface points. neplications and check plots bring the total number of plots to 114. All surface points are replicated twice with the exception of a h x h triplicated factorial amw.h.check plots. The design utilized in this experiment is basically an incomplete factorial but it also includes several additional features. Included in the design are: (l) A h x h triplicated factorial. The treatment levels included in this factorial are the following in pounds of plant nutrients per acre: P205 - 100 200 300 hOO K20 - 200 &00 600 800 Inclusion of this triplicated factorial allows limited study by analysis of variance procedures. The experiment produces a large amount of useful agronomic data as a by-prOduct of the basic input-output study. Such data includes information on the quality and chemical compositions of the product, plant characteristics, residual fertility,etc. Analysis of these data by continuous function analysis may not be feasible or appropriate.1 By including a triplicated factorial in the experimental design, analysis of variance treatment Of such data is facilitated at a small additional cost. 2 A 3 x 3 composite design is included in the experiment for the 1For example, protein content Of wheat may increase linearly or curvilinearly with additional nitrogen inputs up to some maximum value and then remain unchanged with additional nitrogen inputs. In such an event, continuous function analysis might not be the appropriate means of analyzing data to acquire determinations of quality differences. 2This design is described by R. L. Anderson in "A Comparison of ‘Discrete and Continuous Models in.Agricultural Production.Analysis,“ Methodological Procedures in the Economic Analyses of Fertilizer Data, Edited by E. L. Baum, Earl O. Heady and John Blackmore (Ames: Iowa State College Press, 1956) p. h9. A6 purpose of comparing it with a 3 x 3 factorial design as to its effective- ness as a basis for estimating the reSponse surface by least squares techniques. The treatments used for this comparison are shown in Table 7. TABLE 7 EXPERIMENTAL DESIGN FOR THE 1956 POTATO EXPERIMENT r ‘1- Plant Nutrients No. Plant Nutrients NO. (Pounds Per Acre) of (Pounds Per Acre) of 13,305 K20 Plots 13205 K20 Plots 0 O h 200 600 3 O 200 2 200 700 2 25 25 2 200 800 3 25 75 2 250 250 2 50 50 2 250 hOO 2 50 100 2 250 500 2 50 150 2 250 600 2 50 hOO 2 250 750 2 75 225 2 300 200 3 100 O 2 300 300 2 100 100 2 300 hOO 3 100 200 3 300 600 3 100 300 2 300 700 2 100 hOO 3 300 800 3 100 600 3 300 900 2 100 800 3 350 350 2 150 150 2 350 hOO 2 150 300 2 350 500 2 150 hSO 2 350 700 2 ISO 600 2 hOO 200 3 200 100 2 hOO hOO 3 200 200 3 hOO 600 3 200 hOO 3 hOO 800 3 200 500 2 hOO 900 2 1:7 (3) Points on three constant~proportion_PgOs-KQO diagonals are sufficiently sampled to permit estimation of these diagonals individually. These are the diagonals in the experimental design in which P205 and K20 are applied in 1:3, 1:2 and 1:1 preportions reSpectively. Estimation of yield reSponse along these constant proportion of P205 and K20 diagonals allows comparison of these estimates with those derived from estimates Of the entire surface. The Total Experimental Prggram Exclusive of the sugar beet experiment initiated in 1957, the nutrient level experiments described in this chapter contain about 1150 individual plots.l Relative to experimental work undertaken elsewhere, this is an elaborate project. Over 20 acres of land are required for the experimental work. Acquiring soil samples, plant tissue samples, crop yields and quality determinations for the various crOps are tasks entailing large labor inputs. In addition, chemical analysis as well as tabulation and computational analysis of these data are time consuming undertakings. In addition to the basic input-output determinations, the experi- Inents produce a large amount of by-product data of interest to agronomists éxnd economists. For example, data acquired from these experiments are 1Deingutilized to compare alternative methods of testing soil for 1Numerous other experiments are conducted by the Michigan.Agri- Ctfiltural Experiment Station, many Of which also provide data for fertilizer input-output determinations. AB residual quantities Of the three plant nutrients. The influence of various plant nutrient treatments on the quality of crops produced is being studied utilizing data from these experiments. A comparison of experimental results from field and greenhouse experiments is being conducted in conjunction with the basic input-output studies. In brief, the experiments described in this chapter produce a wealth of data which is being used for a diversity Of research projects. Data produced from the experiments described in this chapter from l95h-56 were used for the analysis conducted in Chapter IV. CHAPTER IV ANALYSIS OF THE DATA The Oats,_Wheat,_Alfalfa and Corn Rotation The oats, wheat, alfalfa and corn rotation experiment was initiated in 1955 and data have been collected for two years. Only two harvested crOpS were produced in 1955 as alfalfa and wheat stands could not be established in time for harvest during the 1955 crop year. field data were acquired for both corn and cats in 1955 and all four crops were produced in 1956. Due to a heterogeneous stand of alfalfa, no data were acquired for that crop in 1956. Analysis of the Oats Data Oats were produced on two of the experimental sites in Calhoun and Kalamazoo counties in 1955. Preliminary graphic analysis of these data indicated that the variance present in the yield data was not associated with different quantities of applied plant nutrients. This hypothesis was further substantiated by fitting a polynomial equation to the data. The equation fitted was of the type I = a + bl N + ‘02 N2 + b:3 P + b4 P:3 + be Ii + by, K2 + b7 'J‘TP + b8 1;}; + 6, P1:. The variables N, P and H represent per acre applications of N, P305 Le O‘ and K20, respectively. None of the variables in this equation had estimated parameters significantly different from zero. Apparently weather conditions were the main determinants limiting crOp yields during the 1955 crop growing season. Unfavorable weather conditions, largely the result of a late summer drouth, prevented crop yield increases which might have occurred with increased applications of plant nutrients under more favorable weather conditions. Yield data for cats were acquired again in 1956. Preliminary graphic analysis of these data indicated that positive relationships existed between oat yields and applied N and P205. Furthermore, these relationships appeared to be curvilinear, reflecting diminishing returns to plant nutrient inputs. The first formulation of the fUnctional relationship which was attempted for the 1956 data was a polynominal equation identical to the one fitted to the 1955 data. This polynomial contains first and second degree terms for N, P205 and K20 and first degree, cross-product terms for all nutrients taken two at a time. This formulation containing the estimated parameters is shown in equation I. Values listed below the estimated parameters and included in parentheses are standard errors of the respective parameters. N, P and K represent per acre applications of N, P205 and K20 reSpectively as is the case in all equations unless otherwise indicated. Equation (I): i0 - h3.326378 + .u0112190 N - .00130761 N2 - .00650205 P (.05115313) (.0019075) (.02579697) + .0000053h P2 + .06186818 K - .00010387 K2 +.00000068 NP - .COOlC905 NK (.oooou775) (.051965u8) (.000191h8) (.ooooéh95) (.00013020) + .OOOO75h2 PK (.00006h30) 51 The adjusted coefficient of multiple correlation for this equation was .690. The coefficient of multiple determination indicated that about h8 per cent of the variance in crop yields was associated with variance explained by the regression equation. Estimated coefficients for the nitrogen variables were significant at the one per cent prob- ability level. None of the coefficients for other variables were sig- nificant at the ten per cent level of probability. Because of the large amount of variance not associated with re- gression and the non-significant coefficients which were estimated for several variables, a second formulation of the production function relationship was attempted. This formulation was an exponential equation of the Carter-Halter1 type. This exponential equation is quite flexible depending on the magnitude of parameters estimated for the variables. In addition to retaining the curvilinear properties postulated to exist in fertilizer input-crop output relationships, use of this equation facilitates estimation of input-output relationships ranging over all three stages of production, i.e., returns to additional plant nutrients which (1) increase at an increasing rate (2) increase at a decreasing rate and (3) become negative. This formulation in equational form is: N bgci K Y = aNbl c1 P b3c§ By taking the logarithm of this equation, we can acquire an equational form of this relationship for which the parameters can be estimated by 1The usefulness of this equation as a production function formu- lation was first noted by H. 0. Carter and A. N. Halter. 52 the technique of least squares. The form in which this equation is fitted statistically is: LogY== loga+bl logN-I-N 10g Cl+b2 logP+P log c24- b3 log K +‘K leg cs. The equation with estimated parameters is shown in Equation II. Egation £11): Log 1?, = 157315152 + .16L175’028 log N - .00057687 N - (.02022815) (.000150h6) .ozuu1092 log P + .000096095838 P + .ooé3u3hh87 log K + .000217567362 K (.016u801o) (.0000669u) (.02017332) (.OOClHYlh) The coefficient of multiple correlation for this equation was .760. The coefficient of multiple determination indicated that about 58 per cent of the variance in oat yields was associated with regression. In this equation, coefficients for nitrogen variables were signifi- cant at the five per cent probability level as was the coefficient for the first phOSphorus variable. Testing the significance of coefficients for individual variables in an equation which contains more than one variable for a given plant nutrient is a practice of limited usefulness. The related variables in an equation such as N, N2, log N, etc., are obviously highly correlated. Estimates of individual parameters may be subject to large standard errors reflecting these high intercorrelations. One might conclude that since individual parameters are not statistically significant, no sig- nificant effects are present. This conclusion might well be fallacious. If the aggregate effect of all variables representing a particular plant nutrient could be tested for significance, the test might indicate. a significant aggregate effect. This situation illustrates an inadequacy of current statistical testing procedures. In cases where (1) two or more independent variables in a production function occur in product form or (2) more than one variable is used to measure the effects of a particular plant nutrient, it would be desirable to obtain a reliability measure on the derivative of crep yield with respect to individual plant nutrients. Such derivatives are necessarily utilized in determining marginal nutrient effects and consequently Optimal applications of plant nutrients. A satisfactory procedure for computing reliability measures for such derivatives has not yet been develOped but is a critical need in much analytical production economics work. Interpretation of the Sta istical Results It appears desirable to investigate several aSpects of the two alternative production function formulations presented here. A comparison of the production surfaces generated ey the two functions is of particular J interest. In addition, it is interesting to compare the combinations of plant nutrients which (1) maximize yields and (2) maximize profits under various plant nutrient and crop prices. A comparison of predicted eat «.1 . the two functions an‘ ye ‘ yields usin; selected combinations of applied plant L nutrients is shown in Table 8. Observations from twenty-eight combinations n a o ‘ .‘ .' 7_ fl ;- T, , A .o ' . q p-. _ f5 ‘ “va r _,', - ._‘ 01 plant nutrients are included in Taole o. fuss: ineiuce UUSei‘aolOLS from all 2" ‘lots in the l x 7 x 3 replicated factorial in addition to .3 2 .J ._ a w yield from all check plots. (I) the averag« TABLE 8 OBSERVED AND ESTIMATED OAT YIELDS, 1956 Treatment Observed Residual2 (Pounds per Acre) Predicted Yield Yieldl v § LBu. per Acre) Lgu._per.Acre) (‘1’ i) N P205 K20 Exp.3 Poly. Exp. Poly 0 O O 37 01$ )4303 38-7 103 -1416 20 80 20 56.7 51.8 67.5 10.8 15.7 20 80 80 58.9 58.9 55.1 -“.8 0.2 20 80 280 68.3 59.6 63.5 -0.8 3.9 20 160 20 56.3 51.3 51.0 ~5.3 ~0.3 20 160 80 58.5 58.9 70.8 11.9 15.5 20 160 280 63.8 61.2 57.9 -5.9 -3.3 20 880 20 58.8 50.8 56.8 -2.8 5.6 20 880 80 61.1 55.9 60.0 -1.1 0.1 20 880 280 6617 66.0 60.6 -6.1 -5.8 80 80 20 65.7 67.9 75.8 10.1 7.9 80 80 80 68.8 70.6 72.1 3.7 1.5 80 80 280 78.6 78.3 88.2 9.6 9.9 80 160 20 65.3 67.8 76.9 11.6 9.5 80 160 80 67.9 70.7 89.8 ~18.5 ~21.3 80 160 280 78.0 75.8 71.0 -1.8 -“.0 80 880 20 68.2 66.9 61.2 -7.0 -5.7 80 880 80 70.9 71.7 72.3 1.8 0.6 80 880 280 77.8 80.6 88.3 6.9 3.7 280 80 20 63.7 68.8 71.7 8.0 6.9 280 80 80 66.2 66.5 66.6 0.8 0.1 280 80 280 72.3 67.3 61.7 -10.6 55.6 280 160 20 63.2 68.3 57.2 -6.0 -7.1 280 160 80 65.8 66.6 66.2 0.8 -0.8 280 160 280 71.7 68.9 69.2 -2.5 0.3 280 880 20 66.1 63.9 76.2 10.1 12.3 280 880 80 68.7 67.6 72.3 3.6 8.7 280 880 280 75.0 73.7 80.6 5.6 6.9 1The observed yield for the 0-0-0 treatment is an average of yields from 11 plots, all other observed yields are averages of two plots. 2Residuals are deviations of predicted yields from average observed yields. 3In computing Ii for zero treatments of plant nutrients using the exponential equation, inputs of a single pound of N, P205 and K20 were used. This introduces a slight upward bias in the predicted yield but overcomes the problem of having Yi = 0 when any of the treatments is zero. This procedure is utilized throughout this chapter when computing Yi from exponential equations. Statistical measures derived for the equations, including the co- efficient of multiple correlation and standard errors of the regression coefficients, indicate that the exponential is a slightly, but not significantly, more appropriate formulation than the polynomial. Measures such as correlation coefficients and standard errors of regres- sion coefficients and equations are not without some limitations in comparing these two functions. The observations, and hence the variances, of the variables are not readily comparable since in one instance they are in real numbers and in the other in logarithms. The real numbers and logarithms, although bearing a consistent monotonic relationship to each other, do not maintain a relationship of equivalence or of constant ratios, Hence, the listed statistical measures should not be given an absolute interpretation for comparative purposes, i.e., they should, instead, serve as a basis for a rough comparison. InSpection of the residual values, (Ii - Ii) for both functions provides little basis for choice between functions since the individual residual values about the two functions are about equally dispersed with reSpect to magnitude and direction. Some additional insight into the appropriateness of the two altern- ative functions may be gained by comparing the derivatives of these functions with reSpect to their correSpendence to input-output relation- ships postulated to exist in accordance with currently held theory. In addition, the derivatives are used to calculate plant nutrient combi- nations which preduce (1) maximum yields and (2) maximum profits. lThese residuals are shown in columns 7 and 8 in Table 8. 56 Maximum yields occur where the first order partial derivatives of the functions are equal to zero. Maximum profits occur where the partial derivatives with reSpect to individual plant nutrients are equal to the plant nutrient-crOp price ratios. Since most of the variance explained by regression is associated with the nitrogen variable, the derivatives of the functions with reSpect to nitrogen are of particular interest. The partial derivatives of the two functions with reSpect to nitrogen,’:%%fi-, are represented by the following equations III and IV. All derivatives are taken for a unit (one pound) change in plant nutrients. E9 Egg-ation ETD - polynomial: 5% a 81 - 282 N + 1»7 P + be K Substituting in the estimated parameters from equation (I) gives 21:1 = .80112190 - 2(.00130761)N + .00000068 P - .00010905 K 49 N , , . ey- Nbl Nlnc N b1_1 Equation (IV) - exponential. éfifl a R(I cl ,1 + 01 b1 N ) Where R a antileg (a + b2 log P + + log 02 + b3 log K + K log Cs). The expression of the partial derivative of the exponential may be simplified by factoring out YO which leaves flD'Y b A: 4K ,2 N YO (ln cl«+ N ). Substituting in the estimated parameters from equation (II) gives 2 Y , 16875028 —._-—Q 2: — , 1” —'———.—'.-i——— The partial derivatives of the two functions with reSpect to N are shown in Table 9 with P205 and K30 fixed at three different levels, 20-hO, 80-160 and 2h0-880 pounds per acre respectively. These derivatives are also shown in Figure I. Derivatives of the exponential function are larger at small nitrogen inputs than is the case for the polynomial function. It is the Opinion of the author that the exponential generates a production surface rising too rapidly with small nitrogen inputs. If this is true, the derivatives are probably too reSponsive to input changes, i.e., they probably are too large with small inputs and change rapidly to become too small with larger inputs. This phenomenon is due in part to the fact that when Xi = O, Y a O. The function may still be quite reliable ever the range of moderate inputs. The derivative of the exponential function is l.h56 bushels per pound of nitrogen with a nitrogen input of 5 pounds and decreases to .77? bushels when 10 pounds are applied. These values of the derivative seem excessively high from a viewpoint of plant physiology, i.e., it is difficult to visualize how one pound of additional nitrogen could result in the production of l.h56 bushels of additional wheat. However, the derivatives of the exponential type function are not restricted to a linear function of plant nutrient inputs as is the case with a polynomial with only first and second degree terms. This linear restriction on the derivatives of a polynomial can be overcome by modifying the formulation to include variables raised to fractional powers, e.g., powers such as 3/2, l/2 etc. and/or by adding variables involving powers hig TABLE 9 CHANGES IN OATS YIELDS RESULTING FROM UNIT CHANGES IN NITROGEN APPLICATIONS Treatment Level Nitrogen Treatment Derivative of Derivative of of P205 &zK20l Level Polynomial2 Exponential2 (pounds per acre) (bu. per acre) (bu. per acre) 1 20 .387 .393 l 80 .298 .17h l 80 .190 .050 1 120 .085 .005 l 160 -.019 -.018 1 200 -.128 -.031 1 2hO -.229 -.O39 2 2O .3hl .h06 2 80 .288 .180 2 80 .188 .051 2 120 .079 .005 2 160 -.013 -.018 2 200 -.130 -.032 2 280 -.235 -.080 3 20 .323 .h62 3 hO .270 .205 3 80 .166 .059 3 120 .061 .006 3 160 -.Oh3 -.021 3 200 -.188 -.037 3 280 -.253 -.086 1Nitrogen is varied with P305 and K20 fixed at three levels: (1) 80-20, (2) 160-80 and (3) 880-280 pounds per acre reSpectively. 2The derivatives are those resulting from an additional pound of nitrogen. The statistical fit might not be improved by such a modification but derivatives would be allowed to become a curvilinear function of additional plant nutrients. Further experimentation with the use of fractional powered and more complex polynomials and additional derivative of exponential derivative of polynomial ‘ “L di------ 1.0 0.75 13 $4 (J CO I. Q) 8* 0.5 3 U) ,4 c0 0 3 0.25 '0 r" (1) H >~. .5 fig 0 [1 :13 £1 L) 0.25 0.5 “-- —- —-—*... .. .- "‘ --«X hO 80 1’0 160 ‘6 Nitrogen Applied (pounds per acre) 200 Applications of P205 and K30 are fixed at 160 and 80 pounds per acre respectively. Fig. 1. Partial derivatives of polynomial and exponentiai functions for cats with respect to nitrogen. O\ O inSpection of the derivatives of these f1nct ions is neeee The partial derivatives of yield with respect to P205 and K20 are y.‘ shown in Tables 10 and 11 and Figures 2 and 3 respectively. As in tle .1 case for the derivatives witn reSpect to nitrogen, the de1vatiV.es f O the expenentials take more extreme values than those of the polynomial. However, the derivatives of be oh functions are small and may be non- significant and the absolute value of the difference between the two derivatives is not large. High Profit Cem'inatidns of Pl_ant lutrients The Optimal amount of plant nutrients to apply, as has been previously stated, is a function not only of the productivity of applied nutrients but also of plant nutrient and crep prices. To solve for the combination of applied plant nutrients which will maximize yields, the partial derivatives of yield with respect to all plant nutrients are set equal to zero and solved simultaneously. For the polynomial equation, the maxi- mum estimated yield is obtained with 153 pounds of N, a slightljf neg;at1ve quantity of Png and .l p aid of NBC. The estimated amounts of P205 and K20 resulting in maximum yields are neither statistically nor economically significant, 1. e., they are not significantly different from zero. It becomes profitable to a13ply plant nutrients to oats only when the price of cats is in excess of 1 CO per bushel and even then only nitreg en 1 appl lications are profitable. ¥Plant nutrient prices used in computing the high profit inputs were $0.15 per pound for nitrogen, $0.10 for phOSphoric acid and $0.11 for potash. 61 TABLE 10 CHANGES IN OATS 111133 RBSULTBJG FROM UNIT Gimmes IN P205 APPLICATIONS Treatment Level of N and K201 Derivative of Exponential3 (bu. per acre) P205 Treatment Derivative of Level Polynomial2 (pounds per acre) (bu. per acre) 1 20 -.005 -.022 1 ho -.00u -.00h 1 80 -.003 .003 1 160 -.002 .007 1 2&0 -.002 .008 1 320 -.001 .009 1 u80 .000 .010 2 20 .000 -.027 2 no .000 -.006 2 80 .001 .005 2 160 .002 .008 2 2u0 .003 .010 2 320 .oou .011 2 use .005 .012 3 20 .012 -.028 3 to .013 -.006 3 80 .013 .005 3 160 .011 .009 3 ZhO .OlS .011 3 320 .016 .012 3 uso .017 .013 IPhosphoric acid is varied with N and K20 fixed at three levels: (1) 20-20 (2) 80-80 and (3) 2hO-2h0 pounds 2Regression coefficients for phOSphoric acid variables were not significant. per acre reSpectively. 3The regression coefficient for only one phosphoric acid variable was significant. . per acre) (bu Change in yield of oats O -O.?S 70.75 -l.25 -1.S oz -A‘t-_---_‘_‘----- ”””””” =—-’"’P’-”’—-—'-.' ."7"' ‘t__ f derivative of polynomial E derivative of exponential I I I a I I I I I I ’I I I I I I I I I I I I I I I I I l I 80 160 2DO 320 L00 &80 PhOSphoric Acid Applied (pounds per acre) Applications of N and K30 are fixed at 80 pounds per acre. ‘ Fig. 2. Partial derivatives of the polynomial and exponential functions for oats with respect to phosphoric acid. TABLE 11 CHANGES IN OATS YIELDS RESULTING FROM UNIT CHANGES IN K20 APPLICATIONS Treatment Level K20 Treatment Derivative of Derivative of of N and P2051 Level Polynomial Exponential (pounds per acre) (bu. per acre) (bu. per acre) 1 20 .059 .0h6 1 b0 .Oéh .038 l 00 .0h6 .03h 1 120 .037 .033 1 160 .029 .033 1 200 .021 .031 1 2u0 .012 .03A 2 20 .061 .05 2 hO .057 .Ohh 2 80 .0L9 .039 2 120 .oto .038 2 100 .032 .030 2 200 .024 .039 2 2&0 .015 .C39 3 20 .C07 .tjh 3 MO .06h .Uhh 3 00 .CjS .OHO 3 120 .0h? .039 3 180 .039 .ce9 3 200 .030 .039 3 2h0 .022 .OhO 1Potash is varied with N and P205 fixed at three levels: (1) 20-hO (2) 80-160 and (3) 2hO-h80. 0.33 0.3 Ci: cl—u‘-.- '- . derivative of exponential \ derivative of polynomial h0 80 120 160 200 iin L A:(-/ Potash Applied (pounds per acre) Apptications of N and P205 are fixed at 80 and 100 pounds per acre respectively. Fig. 3. Pirtial derivatives of polynomial and exponential funCLitns for oats with respect to potash. On the basis of yield reSponse measured for 1955 and 1956, icrtiliz- ing of oats was not a profitable practice. The only possible justifi- cations for application of plant nutrients to the cats crop appear to be when a seeding is being established with the cats and/or the benefits derived from residual plant nutrients by other crops in the rotation. Analysis of the Wheat Data Wheat was produced on the Kalamazoo County experimental site in 1956. The yield data produced in this experiment were analyzed in the same manner as the oats data. The original function fitted to the wheat data was a nine variable polynomial. This formulation with estimated parameters is shown in Equation V. Equation (v): §¥,= 28.538730321 + .08598h69h1 N - .0002208h56 N2 + (.016959906h) (.0000632u18) '0163750688 P ' -000035115h P3 + .0085708071 K + .000021323n K2 + (.0085530282) (.0000158325) (.0172292t2u) (.000063h867) .00001902h6 P - .0000799h31 NK + .0000151210 PK (.0000215352) (.0000hh5667) (.709260110) The adjusted coefficient of multiple correlation for this equation was .66 and the coefficient of multiple determination indicated that about hh.per cent of the variance in yield was associated with variance in applied plant nutrients. As was the case for cats, only the estimated parameters for the nitrogen variables were statistically Significant at the one per cent probability level. However, the phosphoric acid vari- ables, P and P%.were significant at the five per cent probability level. 66 A Carter-Halter type exponential function was also fitted to the wheat data. The results of this fit are shown in Equation VI. Equation (VI): Log §w = l.hh50h7179 + .0226358023 log N + .0001726882h8 N + (.0117u51331) (.0000873600h8) .0165763652 log P - .0000173738hl.P4-.0012796889 log K + .000108232188 K (.0095688935) (.000038867869) (.0117132969) (.000085h36998) The adjusted coefficient of multiple correlation for this equation was .65. The adjusted coefficient of multiple determination indicated ‘that about h3 per cent of the variance in crOp yields was associated with variance in the amounts of applied plant nutrients. The first three estimated coefficients in this equation were significant at the l0 per cent probability level and were almost significant at the five per cent probability level. The last three coefficients in the equation were not statistically significant. A comparison of observed yields with yields estimated by using the two functions is shown in Table 12. As was the case for the cats data, the tabular comparison includes observations and predictions for 28 combinations of applied N, P205 and K20. The observed yield values are averages of two replications for all treatments except the check (0,0,0) treatment which is an average of eleven replications. The coefficients of multiple correlation and determination indicated that the two functions were about equally effective in explaining variance in wheat yields. Inspection of the residuals for the two functions, (Yi - Ti), further substantiates the conclusion that the two functions produce about equally good fits. These residuals are shown in columns 7 and 8 of Table 12. 67 TABLE 12 OBSERVED AND ESTIMATED WHEAT YIELDS, 1956 ‘ _._‘-—_——-: _- I H r *- Treatment Predicted Yield IObserved Yield1 Residual2 (pounds per acre) (bu. per acre) (bu. per acre) (Yi ~ Ti) N P205 K20 Exp. Poly. Exp. Poly. 0 o o 27.9 28.5 28.2 0.3 -o.3 20 to 20 32.2 30.9 29.5 -2.7 -1.u 20 to 80 32.7 31.5 29.3 -3.h -2.2 20 to 2t0 3h.l 33.8 3u.6 0.5 0.8 20 160 20 32.8 32.1 31.2 -1.6 -{L9 20 160 80 33.3 32.8 3o.t -2.9 -2.t 20 160 2uo 32.7 35.u 35.1 o.u -o.3 2o t8o 20 33.0 3o.t 32.1 -o.9 1.7 20 ubo 80 33.5 31.u 31.9 -1.6 0.5 20 t80 2u0 3h.9 3h.8 3t.9 0.0 0.1 80 to 20 3h.0 3h.7 37.5 3.5 2.8 80 to 80 3u.6 35.0 3u.9 0.3 -o.1 80 to 2to 36.1 36.6 37.2 1.1 0.6 80 160 20 3u.6 36.1 no.9 6.3 h.8 80 160 80 35.2 36.5 36.7 1.5 0.2 80 160 2&0 36.7 38.3 36.5 -o.2 -1.8 80 u80 2o 3u.8 3u.7 28.7 ~6.1 -6.0 80 hbo 80 35.b 3S.h 39.0 3.6 3.6 80 u80 2uo 36.9 38.0 38.6 1.7 0.6 2uo to 20 37.2 37.0 36.6 -o.6 -o.h 2t0 to 80 37.8 36.6 35.1 -2.7 -1.5 2to to 2to 39.1 36.1 35.1 -“.3 ~1.o 2to 160 20 37.8 38.8 t2.1 t.3 3.3 2uo 160 80 38.5 38.h 39.3 0.8 0.9 2to 160 2to no.1 38.2 38.h -1.7 0.2 2u0 A60 20 38.1 38.h t2.9 t.8 h.5 2uo t8o 80 38.7 38.3 38.2 -0.5 -0.1 2&0 hBO 2&0 no.3 38.8 38.8 1.5 0.0 1The observed yield is the average of two replications except for the check (0,0,0) treatment which is the average of 11 replications. estimated yields. gfiesiduals are the difference between average observed yields and 68 Derivatives of the two functions with respect to all three of the plant nutrients are presented in Tables 13-15 and in Figures h-6. The derivatives of the two functions produce different estimates of the productivity of the various plant nutrients. For example, the derivative of the polynomial indicates that the marginal productivity of nitrogen over the range of 30 to 100 pounds, which is a common range of application, is almost double the marginal productivity schedule generated by the derivative of the exponential. Derivatives of the two functions with reSpect to P205 also exhibit substantial differences over the range of usual applications. However, the marginal productivity of phOSphorus is low and the absolute value of the differences between the two deriva- tives is small as is shown in Table lb and Figure 5. Estimated derivatives of the two functions with respect to K20 also differ widely as indicated in Table 15 and Figure 6. The derivative of the polynomial with reSpect to K20 exhibits increasing returns to addi- tional applications of K20 which is not a logical phenomenon. The deriva- tive of the exponential exhibits only slightly diminishing returns. As previously indicated, however, the K20 variables in both equations lack statistical significance at any acceptable probability level. Inferences made about the productivity of all three plant nutrients will vary considerably depending on which function is chosen as "best". InSpection of residuals of the two functions does not provide any satisfactory basis for choosing between the two functions. MP‘W. 7“ Diana 13 69 CHANG'S IN WHEAT YIELDS RESULTING FROM UICIT CHANGES .LN TTT 110/” 71". APTTICATICIS UULM Treatme nt Level of ‘v 01 IIitrogen Tr atment Level (pounds per Derivative of Polynomial acre) (bu. per acre) Derivative of Exponential (bu. per acre) lwmit (l) hC- '20 F’FJFJFJFJFJFJ TORDRJNNMM \nwwwwwxn en is varied with P305 PO C," C: :ufi' .‘\I\)'\ J C) CL" V\ I r“; [UNI—1H \ J 4_ ) C.“ 1:" f\ C. O C) 120 150 200 ZQC F)“ (.U bu 120 160 200 2210 .076 (2) L30-1’0 and (3) hEG-2SIC pounds per a01e .0h9 .032 . c2 3 . L 21 . 019 .019 o L]-8 . 0;1 .033 .62' and K 0 iixed at three level '33 reSpectively. Change in yield of wheat (bu. per acre), ‘ya— deriVative of exponential derivative of polynomial .05 \ -.035 to 80 120 160 T\J O C) r E: Nitrogen Applied (pounis per acre) Applications of P205 and K20 are fixed at 160 and 80 pounds per acre respectively. Fig. h. Partial derivatives of the polynomial and exponential functions for wheat with respect to nitrogen. TLBLE 1h 012151150513 I“? L'JIZBLIT 11:11:13 :‘tESLlTl’i‘l-CI E“. 101'. UL ITL " ._.3.I=.7<_}.ZS Treatment Level P205 Treatment Derivative of Derivative of of 1 Level Polynomial Exponential N and K20 (pounds per acre) (bu per acre) (bu. per acre) 1 ho .Olh .012 1 to .011 .006 1 130 .006 .002 l 2h0 .000 .001 1 320 -.005 .001 1 800 -.011 .000 l h80 -.017 .000 2 hO .016 .013 2 80 .013 .006 2 160 .008 .003 2 2h0 .002 .001 2 320 -.003 .001 2 hOO -.009 .000 2 h80 -.015 .000 3 hO .022 .015 3 80 .019 .007 3 160 .013 .003 3 2h0 .008 .002 3 320 .002 .001 3 800 -.008 .000 3 h80 -.009 .000 1P2 05 is varied with N and K30 fixed at three levels: (1) 20-20 (2) 80-80 and (3) 2h0-2h0 pounds per acre respectively. 00 ’15. .05 | I I I ‘ u I I | I | I | ./ darivat ive of exponential ‘ derivative of poiynomiai 80 160 ELO 3‘0 Phosphoric Acid Applied (pounds per acre) N and Ké0 are fixed at 80 pounds per acre Fig. 5. Partill derivatives of the ptlynonial anl exponential functions for wheat with respect to phosphoric acid. 73 TABLE 15 CHANGES IN WHEAT YIELDS RESULTING FROM UNIT CHANGES IN K20.APPLICATIONS —-_. T _._‘_ Treatment Level Potash Treatment Derivative of Derivative of of Level Polynomial Exponential N and P2051 (pounds per acre) (bu. per acre) (bu. per acre) 1 20 .008 .010 1 h0 .009 .009 l 80 .011 .009 1 120 .013 .009 1 160 .018 .009 1 200 .016 .009 1 2h0 .018 .009 2 20 .005 .010 2 hO .006 .009 2 80 .008 .009 2 120 .010 .009 2 160 .011 .009 2 200 .013 .009 2 2h0 .015 .009 3 20 —.003 .011 3 80 -.002 .011 3 80 .000 .010 3 120 .002 .010 3 160 .003 .010 3 200 .005 .010 3 2h0 .007 .010 tPotash is varied with N and P205 fixed at three levels: (1) 20-h0 (2) 80-160 and (3) 2h0-h80 pounds per acre reSpectively. ‘F ii ulv loll}. Change in yield of wheat (bu. per acre) C-.-...-..—. C' -CD' ‘- derIVative of exponential b0 80 120 160 300 Ad? PotaSh applied (pounds per acre) Applications of N and P205 are fixed at 80 and I60 pounds per acre respectively. Fig. 6. Partial derivatives of the polynomial and exponential functions for wheat with respect to potash. 75 Maximum Yields and High Profit Plant Nutrient Applications Maximum yields of about 39 bushels per acre are predicted using the polynomialequation. This yield occurs with plant nutrient appli- cations of about 196 pounds of N, 300 pounds of P205 and 61 pounds of K20. The maximum yield predicted using the eXponential is in excess of any yield observed in the experiment and requires plant nutrient applications in excess of any quantities applied in the experiment. Since the predicted maximum yield and the plant nutrient inputs producing this yield lie beyond the range of observed values, they are probably invalid inferences. Both functions generated reSponse surfaces which illustrated sub- stantial positive yield reSponse to N and P205. However, because of the moderate slopes of the reSponse surfaces, the value of additional production was less than the cost of plant nutrients necessary to obtain the increases in yields.1 Applications of nitrogen and phOSphoric acid would.have been profit- able only at wheat prices of about $3.00 per bushel. Such wheat prices appear extremely unlikely. It is important to note that although the derivatives of the two functions differ considerably, the same conclusion, that no fertilizer applications were profitable at typical prices, would be reached using either function as a basis for computing high~profit fertilizer inputs. 1No credit was given for possible residual fertility. With large fertilizer applications some carr"over fertility would be expected, however, the magnitude and value of this carryover can only be determined over time. . 76 Analysis of the Corn Data Two corn crOps have been produced and harvested in the rotation experiment. The corn plots were located at the Calhoun county site in 1955. A severe summer drouth reduced corn yields in this area, particularly on the lighter upland soils. An extensive analysis of the 1955 corn data was conducted by Jack Knetsch and has previously been reported.1 Knetsch found that a Carter-Halter type exponential provided the best statistical fit to the data. Significant reSponse was found to exist only for applied nitrogen. The fitted function was .18627 (N + 0.1) YC - 39.?1(N + .01) .96230 , where N was measured in 20-pound units. The addition of .1 of a unit alleviated the problem of forcing the function to have a value of zero when any one of the plant nutrient inputs was zero. The coefficient of multiple correlation for this equation was .69. The high profit nitrogen application varied from 29 to Sh pounds per acre as the price of corn was varied from $.80 to $2.00 per bushel with nitrogen priced at $.15 per pound. A comparison of observed and predicted yields is shown in Table 16. 1The results of this analysis are contained in: Jack L. Knetsch, "Methodological Procedures and Applications for Incorporating Economic Considerations into Fertilizer Recommendations," Unpublished Master's Thesis, Michigan State University, 1956, and Jack L. Knetsch, L. S. Robertson, Jr., and w. B. Sundquist, "Economic Considerations in Soil Fertility Research," Michigan Agricultural Experiment Station, Quarterly Bulletin, August, 1956, pp. 10-16. TABLE 16 COMPARISON OF OBSERVED AND PREDICTED CORN YIELDS ON A KALAMAZOO SANDY LOAM SOIL, 1955 Average Predicted Marginal Product Number of Actual Yield of of 20-pound N Per Acre of Yields, Corn Units of N (pounds) Plots (bu. per acre) (bu. per acre) (bu. per acre) 0 18 26.3 25.8 0 20 2h 80.6 38.2 12.8 to 18 83.5 81.8 3.6 80 29 h3oh hh.l 1.15:L 160 18 82.8 83.0 -0.501 2h0 27 h0.7 39.8 -O.8Sl 1Average marginal product of 20-pound units of nitrogen for the hO-pound incremental intervals shown in column 1. Corn was produced on the Kalamazoo county site in 1956. Once again the crop was damaged by a severe late summer drouth. Check plot yields were not significantly dif erent from those receiving applied plant nutrients. Preliminary tabulations indicated very little association of yield variance with variance in any of the three applied nutrients. This lack of relationship was further substantiated by functional analysis. A.nine—termgpolynomial was fitted to the data with the estimated para- meters shown in Equation VII. Equation (VII): Y0 = 58.8ouc809 - .0085761891 N + .000113767680 N2 - (.0391267523 (.000185895858) .0181889261 P + .0000093102508 P2 + .0269173892 K - .000092531u50 K2 + (.019683161) (.0000365288729) (.039750uu01) (.00186857332) .0000669778119 NP + .0000357228950 Nx;¢ .000056027392 PK (.0000u96773588) (.0000995h8385) (.0000891638629) ‘- l None of the {ma ineters in thi seqLation are sL _nificaatly diiferent from zero. This lack of significance is not surpri31ng since the adjusted coefficient of multiple correlation for the ecuation is only .23 and the adjusted coefficient of mul‘iple det ruina ion is only .Oj a value which is not signifi Lcantlv cil' rent from zero. A Carter-D alter trLe equation which was fitted to the oata is snown in Equation VIII. S“ A ~\“e+“on (Vlll)= L02 Y0 = 1.719957583 + .012753199 103 N - (0((l43. I + CO :7 W977 31L~ P - .080110811 P ~ .005317996 103 K + .00c18587o K (. 0131:;326) (.0Loc55c33) (.t16278 , 3 203) (.LL8118L;6) Only the fourth tern in this equation, F, is Statistically Sivni-ica 8. Since the phOSphoric ac (1 variable is repres-x rted by two terms, orie of which is not significant, no very valid inierenccs can be made about the aggregate influence of phi 'Lhoric acid. None of as terms repie- r‘ v- “I ’L r “. ‘ r‘ o r“ ‘ '9‘. rs r‘ “L 7' : O ' ' “fir/I rv “ ‘1 Senting nitrogen 01 pOCaSJ are signiiioantly oiiierent 110m zero. about the same amount of total variance in yield is associated with regression for t11e polunom -al. The adgusted coeiliciro nts of multiple U] c l" P a h—J C} O 9.} U] U) a S 1'58. ‘ y a n K" J... - .‘ "~‘-~ ~~ ‘a' ~'-~ '- t-‘i 3 -~.-»-. ,\ 1"!ch '. *F multiple determ1haoion are .24 ano .co SQyUCULVClJo O *5 *3 L.J h) CE ‘rJo O :5 E E. n " .L_° . --_. 1, °,,._-..‘ - .1. rs -. - _. .-. -9 - -, - ,3 i .. ‘-.. ..'. oisticalif SILUIIICano. ihe only inierence alien Sdphn o) 0 O ! - _:’_ _f‘ Till-11.081113 IC‘LLI‘C 01 V8.1" 1-611103 rieid w‘s associa tad w th applied plan t nutiiints. ' \ \—‘ O‘I’IC‘ \“n TY» I‘T‘( "-“fi’l '3".,_ 1".C)“1)1(‘.u - LA.) (1.L 1.... J'. )1 .141, JL. L” La. wax) (I A. - - it. , In sunrary, onlv moo2rLte apuiica per acre, were profitable on corn on Kalamazoo sa.id/ loan In 1933 and no .‘1 \VL.) 7w 1.3.3 .r' . x . ° n, - r" J. - A, D 1. . -2 : 3,4. , no aliaiia was frosn in tdd iirsc year 01 tJG experiment o v. .' n...- -4. 1 11 ' 1. r . ‘. -. » 2. .2-1 a - ' a; c the inaullloy to esca31153 any harvestaole grouti 1n CA5 first year 01 1 . ~.-~ fr. ,- .L -—.. -' d’” .L' J... -1 n , h, ‘ f‘r. ,- 7 _ .. - ~ .1- ._ i .i tile 1:}C}2231"l3:1:;l‘1u. in .L‘," O om? 83.3arlo- Ol d11dlh-d. ILLS JCI‘J 112-3 oOI‘O_’-~IIJI’J_';Ollb. w- «— ’ —'— -. .. A ‘ ' o -'_ 1. . ‘-o .. 3 r‘ .~‘ "a. f " J- . . .4. t \ - u .. LarLe contiguois areas in use i11ld had hO‘uTao31y good SoanS while 7 other areas had almost no alfalia growing on them. to attempt was made U to collect and analyze yield data because yield difchGnCVS were obviously ( ”I Q a function of difiercnces in stand not associated with applied plant J._, -‘ , . J- Analgsis o” the Continuous Corn Data The initial corn crOp in a continuous corn rotation was produced on a Wismer clay—loam soil in Tuscola county in 1956. Preliminary inspection of the data indicated small and heterOgeneous yield reSponses to applied plant nutrients. The eight check plots in this experiment had an average yield of 100.6 bushels per acre, while the average of all 210 plots in the experiment was 109.7 bushels per acre. A nine variable polynomial was fitted to the data and the results of this formulation are shown in Equation IX. Equation gig); it . 10h.565510278 + .06991h3u6 N + .05075haso p - (.03u70b916) (.017279507) .001629512 K - .000356932 N2 - .000068956 P2 - .000053579 K2 - (.OBhBSSSB? (.000108282) (.0000287h3) (.000116055) .000039695 NP + .000112058 NK + .000060759 PK (.000039782) (.000079363) (.oooou383o) 80 Coefficients of four of the variables N, N2, P, and P2 were sig- nificant at the one per cent probability level, whereas none of the potash variables were statistically significant. Only a small portion of yield variance was associated with applied plant nutrients as the coefficients of multiple correlation and multiple determination were only .hO and .16 respectively; Since none of the independent variables containing a potash term were statistically significant, the polynomial was reformulated, drOpping the variables containing a potash term. The shortened polynominal is shown in Equation.X. A Egpation X: Y + 10h.082698823 + .07370h5h6 N + .050022736 P - ° (.O3h298683) (.017116212) .000331599 N2 - .000056021 P2 - .OOOOZShéO NP (.000107266) (.00002733h) (.000038963) In equation X the first four coefficients are significant at the one per cent probability level. The fifth term, a cross product, was not significant at any acceptable significance level. The coefficients of multiple correlation and multiple determination for the shortened polynomial were .39 and .16 reSpectively. Due to the small portion of yield variance associated with applied plant nutrients as indicated by inspection and the fitted polynomials, no attempt was made to fit an exponential type equation to the data. Maximum Yield and High Profit Combinations of Plant Nutrients Coefficients for the nitrOgen and phOSphoric acid variables were similar for the two polynomials fitted to the data. Since the potash coefficients were not significant, the plant nutrient combination providing maximum yields was restricted to N and P205 and was calculated from Equation X. The maximum predicted yield, 123.h bushels per acre, was obtained using 95 pounds of N and h25 pounds of P205. The cost of using any amount of applied plant nutrients exceeded the returns unless corn prices exceeded $2.00 per bushel. The latter corn price situation is, of course, an unlikely phenomenon. The high check plot yields, in excess of 100 bushels per acre, probably indicates the soil was quite fertile prior to additional appli- cations of plant nutrients although soil tests indicate only a moderate fertility level. Other possible sources of yield variance were present in the experimental field, including differences in previous crepping history; Although yields from the plot areas with different crOpping histories were not statistically different, this factor of heterogenity may have contributed some variance to crop yields. Analysis of the Bean Data from the Corn, Beans and Wheat_fiotation Field beans were produced on a Simms loam soil in Gratiot county in 1956. The bean crOp is part of an intensive cash crop rotation of corn, beans and wheat. Experimental plots had received plant nutrient treatments in 1955 identical to the 1956 treatments. Thus, some residual fertility might have been expected to be present in 1956, particularly on plots receiving heavy fertilizer applications the previous year. Preliminary tabulation of the data indicated a substantial response to 82 nitrogen applications, a smaller reSponse to phOSphoric acid, and no appreciable response to applied potash. Three functions were fitted to the bean data. The first two func- tions are exponential type formulations and the third a five variable polynomial. The original production function formulation is a six variable exponential of the Carter-Halter type. Although preliminary analysis had indicated no reSponse to potash, variables containing potash terms were included in this original exponential which is shown in Equation XI. Equation 131); Log Y5 = 1.203h797 + .032812261 10g N + .000398971 N + (.017529035) (.001c69u) .019527h3h log P + .000062271 P + .001880612 log K + .000050911 K (.015569387) (.oooougsoh) (.018591118) (.000068525) The adjusted coefficient of multiple correlation for this equation was .605 and the coefficient of multiple determination was .366. This indicates that about 37 per cent of the variance in bean yields was associated with regression. Because of the large standard errors for fine potash coefficients, a second formulation of the exponential was made dropping the potash terms. This exponential is shown in Equation XII. Equation (X11): Log Y5 = 1.207h135791 + .O3h7393520 leg N + .000396596h N (.016076657) (.00010650195) + .021h607700 log P + .0000597327 P (.01u609617) (.0000L8352231) The adjusted coefficient of multiple correlation for the shortened exponential was .607 and the coefficient of multiple determination was b.) .369. Coefficients of the nitrogen and phOSphoric acid variables were not changed appreciably by omitting the non—significant potash terms. Phosphoric acid terms were not significant at the 10 per cent probability level as the size of the estimated coefficients for these terms exceeded their reSpective standard errors. Finally, a five variable polynomial was fitted to the bean data. The results of this fit are shown in Equation XIII. .1: Equation (XIII): fit = 17.60231tu + .0636878985 N - .OOC10708hh1 r2 + (.011h222) (.000035h159) .0127h99698 P - .0000105617 P2 + .OOOOO63h92 NP (.00580265) (.00000373620) (.000013095?) The adjusted coefficients of multiple correlation and determination for this equation were .6h6 and .hl7, respectively. A comparison of observed and predicted yields using the three functions fitted to the data are presented in Table 17. As in previous cases, inSpection of the residual quantities, i.e., differences between predicted and observed values, of the three functions provides little basis for choosing any one function over the others. This is true because of the relative uniformity of the magnitude and direction of the residuals. Partial derivatives of the three functions with reSpect to nitrogen are shown in Table 18 and Figure 7. Partial derivatives with reSpect to phOSphoric acid are presented in Table 19 and Figure 8. Kaximum Yields and Optimum Inputs of Plant Nutrients Derivatives of the two exponential equations with respect to nitrogen are characterized by preperties which are unusual for marginal product OBSERVED AND ESTIMATED BEAN YIELDS, 1956 h Treatment Observed Residual3 gpounds per acr§)_ Predicted Yieldl Yieldg (r1 - ii)» N P205 K20 Poly 325(1) EJMZ) Poly mm m7?) 0 o 0 17.6 16.1 16.0 17.h -0.2 1.3 1.8 20 no 20 19.3 19.8 19.6 25.h 6.1 5.6 5.8 20 no 160 19.3 19.8 20.0 25.9 6.1 6.1 5.9 20 no 320 19.3 19.8 20.h 19.6 0.3 -0.2 -0.8 20 320 20 21.9 21.5 21.2 15.h r6.5 -6.1 -5.8 20 320 160 21.9 21.5 21.6 2h.5 2.6 3.0 2.9 20 320 320 21.9 21.5 22.1 21.h -0.5 -o.1 -0.7 20 6&0 20 22.7 22.9 22.5 25.8 3.1 2.9 3.3 20 6ho 160 22.7 22.9 23.0 21.0 -1.7 -1.9 -2.0 20 6u0 320 22.7 22.9 23.u 18.8 -7.9 -8.1 -8.6 no 80 to 21.0 21.1 20.9 21.9 0.9 0.8 1.0 80 160 80 23.9 23.0 22.8 31.1 7.2 8.1 8.3 160 no 20 25.6 28.2 23.8 23.8 -1.8 -O.h 0.0 160 no 160 25.6 2u.2 2h.3 26.9 1.3 2.7 2.6 160 to 320 25.6 2h.2 2h.8 27.8 2.2 3.6 3.0 160 320 20 28.8 26.3 25.8 31.6 3.2 5.3 5.8 160 320 160 28.8 26.3 26.h 33.8 5.h 7.5 7.h 160 320 320 28.11 26.3 26.9 25.6 -2.8 -0.7 -1.3 160 6h0 20 29.5 27.9 27.h 2u.7 -h.8 -3.2 -2.7 160 6h0 160 29.5 27.9 28.0 33.5 h.0 5.6 5.5 160 6h0 320 29.5 27.9 28.5 29.3 -0.2 1.h 0.8 2uo hBO 2u0 31.1 29.6 29.9 29.5 -1.6 -0.1 -0.u 320 no 20 27.6 28.7 28.2 32.8 u.8 3.7 8.2 320 no 160 27.6 28.7 28.8 27.5 -0.1 -1.2 -1.3 320 to 320 27.6 28.7 29.h 26.u -1.2 -2.3 -3.0 320 320 20 30.7 31.2 30.6 311.1 3.1; 2.9 3.5 320 320 160 30.7 31.2 31.2 29.7 -1.0 -1.5 -1.5 320 320 320 30.7 31.2 31.9 27.8 -2.9 -3.8 ~h.1 320 6h0 20 32.2 33.1 32.5 3u.8 2.6 1.7 2.3 320 6u0 160 32.2 33.1 33.1 33.7 1.5 0.6 0.6 320 6u0 320 32.2 33.1 33.8 30.6 -1.6 -2.5 -3.2 l 1Exp (1) is the four term exponential and Exp (2) is the six term exponential. 2The observed yield for the 0-0-0 treatment is an average of yields from ll plots, all other observed yields are averages of two plots. 3Residuals are deviations of predicted yields from average observed yields. 85 TABLE 18 CHANGES IN BEAN YIELDS RESULTING FROM UNIT CHANGES IN APPLIED NITROGEN Treatment Nitrogen Treatment Derivativecfl? DerivativecflTDerivative of Level of Level Polynomial Exp. (1) Exp. (2) P205 and K201 (pounds per acre) (bu.per acre) (bu.per acre) (bu.per acre) 1 20 .060 .052 .050 1 to .055 .037 _ .035 1 80 .0h7 .030 .029 l 120 .038 .028 .027 1 160 .030 .027 .027 l 200 .021 .027 .027 l 2&0 .013 .027 .027 l 320 .005 .029 .029 2 20 .060 .055 .052 2 hO .056 .039 .037 2 80 .0h8 .031 .030 2 120 .039 .029 .029 2 160 .030 .029 .028 2 200 .022 .029 .028 2 2h0 .013 .029 .029 2 320 .OOh .031 .031 3 20 .062 .057 .055 3 hO .058 .0h0 .039 3 80 .050 .032 .032 3 120 .0h1 .030 .030 3 160 .032 .030 .030 3 200 .02h .030 .030 3 2uo .015 .030 .030 3 320 .002 .032 .032 h 20 .063 .060 .060 u to .059 .0t2 .0h2 h 80 .051 .03h .03h h 120 .0h2 .032 .033 h 160 .033 .031 .032 h 200 .025 .032 .032 h 2h0 .016 .032 .033 u 320 .001 .033 .035 lNitrogen is varied with P205 and K20 fixed at: (1) h0-20 (2) 160- 80 (3) 320-160 and (h) 6h0-320 reSpectively. Derivatives of the poly— nomial and Exp. (1) are independent of applied K20 since there were no K20 variables in the functions for which these derivatives were taken. Y‘"" 54.. Z’VII LV 5 131 6) v::r;;xf i? ‘3Y7N=f‘7:1?..;i O r K ._ - -- K‘g-‘ianttt V 9 . - dau'lvat :vr'cwl li‘Jar I e/ 131519. ‘,'):‘;)._,r[fi_ nfjiul I I II I 'K '( 10 “ O |‘ X .I I '1 ‘l \ .1 l'\ 0.18 " “ ‘K .‘ “ derivative of po.ynom.al k -_- “ ,“~‘.‘¢‘-*.‘-. .‘-‘-&- !_x-<_x -X-x’x-x'! SO TOO 150 ECO (X; 3'. n ‘_ v) A4 Nitrogen applied {pounds per acre) PpOS and KdO are fixed at 160 and &0 pounds per acre respectively. Fig. /. Partial derivatives of a polynomial and (we expunentiui Tenetions for beans with respect to nitrogen. ("h \, —4 TABLE 19 CHAN ES IND KIN YILLDS IflNULIILa FROh UIIT CEaIGES IN APPLIED PIL‘SPH IC ACID Treatment P205 Treatment Derivative of Derivative of Derivative of Level of Level Polynomial Exp. (1) Exp. (2) N and K20 (pounds per acre) (bu.per acre) (bu.per acre) (bu.per acre) to .012 .013 .012 80 .011 .008 .008 160 .009 .006 .005 2ho .C08 .005 .C05 320 .006 .00h .00h hOO .OOh .CCh. .OCh u80 .003 .00h .00h 6u0 .001 .00h .00h FJFJFJF‘F‘FJFJF‘ 2 L0 .012 .015 - .015 2 80 .012 .CC9 .009 2 130 .010 .tts .006 2 2nc .008 .kk .005 2 32‘ .006 .005 .005 2 hIO .C05 .C05 .005 2 h80 .003 .00; .oou 2 6h0 .000 .eoh .005 3 hC .013 .01” .015 3 80 .c13 .tio .010 3 160 .011 .CC? .C;7 3 233, .CC9 .CChS .(X,6 3 320 .000 .C05 .CC5 3 LOO .006 .005 .CC5 3 . L80 .CCa .005 .C 3 6LO .OCd .005 .CL: 9‘11“ C‘C’ PC 0 0 CC) l#}4 ‘JJL" o o C‘C I_‘l—- [0N m .JL—J J: h 160 .011 .00) .uLu h 230 .010 .0’7 ..L( N 30‘ .CCS .LL) .LCS h nCC .CC3 .ct3 .cL7 h hLC .CC5 .Cxfii .Lcé . 1 04C .CCI .CCG .LCG ngOr is varied with N and K30 fixed at (1)2 -20 (2) tC-JO (f) 170- 10v and (V) 320- 320 reSpectively. Jen'vaulv7r of the polvnom a1 and h.;p. (l) are Independent of a7plied p70 since there wece no ‘70 variaoles Change in yield of beans (bu. per acre) .09 Kflfilx 0) L0 |¥ O 0 U1 statue-fin eff-##1##?- f. .03 ' * . \ 00..) \ derivative of pcivnomiai *-¥-1‘°*—-—*— 100 200 300 hOO 500 600 PhOSphoric Acid Applied (pounis per acre) N and K30 are fixed at 80 pounds per acre Fig. 8. Partial derivatives of a polynomial and two exponentia; functions for beans with respect to phOSphoric acid. d) v A 1 1 _ ’ 3 ,_ _o_ J_ o y, 1 ___ _0 7'1, 1. g r) r. , j n: _, r; if - !, seneoules. Tnese derivauives, shown in ladle lo and Figure ;, fired (‘3‘,"(\ I.“ '. J‘ n ‘V‘f‘, V] .—() n " ': r-v': 1'] .1 r". . I I" c '7'“ u \l" '\ v" -‘ 3“ “‘fi ’) ‘3 ‘ " 7 ': ‘1 "3 ’ '-_‘ I" ' (I suing/it; d. lanky OI L- ........ inL_L_.'x-;J tawtl‘eg “LU. 0.511 “CULHL; ‘9 Jaiuio Oi . r . ’- ’ 7.1-. r 1 I - r H l‘ncrUa‘S'L‘Ilc; r ~( -4. Ultra 39 .4‘ -XJCIUJ) v .2 U _ 1 '9. U‘ ./ 1,1) U1. LI'I ) Y 'K).l(\.41‘1-{J IGNLL “ 1--” .. U .r-‘- -'- ,. 1.1. _"\" ._'., ,._ _ -.. .1--- n 13,: 1' 33".,” .1. .r‘ " . 1 ‘ ,, ' , this rapier 1110,1ca1 UfOquuJ oi Qifliniohldw returns lolioued 0v increts— ing returns to successive nitrogen inputs, the polynomial equation is probably a more appropriate approximation to the fertilizer response surface. Because of the phenomenon of increasing returns to nitrogen inputs exhibited by the exponential functions, maximum yields and high profit plant nutrient inputs lie beyond the range of experimental inputs. The maximum yield as calculated from the polynomial equation is 32.2 bushels per acre. This maximum is achieved using slightly less than 318 pounds of nitrOgen and about 629 pounds of P205. The quantities of N and P205 producing the maximum bean yield are almost identical with those of the highest treatment level in the experiment. Despite the large phosphoric acid inputs which produced maximum yields, applications of phosphoric acid were not profitable at typical crop and fertilizer prices. Assuming a price of $0.10 per pound for P205, use of P205 became profitable only with bean prices in excess of $7.00 per bushel. Nitrogen inputs, on the other hand, were profitable over a wide range of been and nitrogen prices. Assuming a price of $0.15 per pound for nitrogen, the high profit quantity of nitrOgen ranged from about 76 pounds with bean prices at $3.50 per bushel, to 205 pounds at $7.50 per bushel. Estimated high profit nutrient inputs for various bean prices are shown in Table 20. 90 TABLE 20 HIGH PROFIT FERTILIZER INPUTS FOR FIELD BEANS wITH VARYING BEAN PRICES High Profit Plant Bean Price Nutrient Inputl Predicted Yield (per bushel) N P205 (bu.per acre) $3.00 35 o 19.70~ 3.50 76 0 21.82 h.00 106 0 23.15 8.50 130 0 2h.07 5.00 lh8 0 2h.68 5.50 16h 0 25.16 6.00 177 0 25.52 6.50 188 O 25.79 7.00 197 O 25.99 7.50 205 35 26.60 1N and P205 were priced at 20.15 and 0.10 per pound reSpectively. Analysis of the Potato Data V M The original potato experiment was initiated in 195h. Data have been collected for three successive years. Only'P20s and K20 were varied in this experiment. The response surface estimated for the 195h data was one of diminishing absolute yields with additional inputs of P205 and K20. The pre-treatment fertility level of the plots was such that, given the weather conditions existing in 195b, the portion of the reSponse 91 surface characterized by the experiment was that of stage three in the input-output dimension, i.e., negative marginal returns to additional plant nutrient inputs. Preliminary analysis of data collected from the two succeeding years, 1955 and 1956, indicated no significant change in potato yields associated with applied plant nutrients. Further analysis of the soil test data and associated changes in yields over time may provide useful information as to depletion rates and residual fertility as well as yield response to applied plant nutrients. However, data collected to date from the original potato experiment do not indicate P205 and K20 reSponses of economic consequence. Data were also collected in 1956 for the lit plot potato experiment initiated on a previously unfarmed parcel of muck soil. Preliminary analysis of these data indicated that much of the variance in potato yields was not associated with variance in applied plant nutrients. However, as some discernible relationships were evident in the data, a functional analysis was conducted and is presented in Equations XIV and XV. The first formulation attempted was a five-variable polynomial which is shown in Equation XIV. A l Equation (XIV) : Yb = 38.u8031h37 - .06692653 P + .Izobosoo K + (.OL313US9) (.019792e?) .00003556 P2 - .00013183 K2 + .00022502 PK (.00009859) (.00002250) (.00003975) 1Yields expressed in Equations XIV and.XV are pounds per plot. IMultiplying pounds per plot by a conversion factor of 6.8062 gives the potato yield in bushels per acre. \O I’D The adjusted coefficients of mul iple correlation and multiple de termin- ation for this equation are .501 and .251 re ectively. The second, fourth and fifth variables of this equation have coefficients which are statistically Sig ii" Icant at the one per cent probability level. The second formulation of the functional relationship was an emponenti ial type equation as shown in Equation XV. quatlcn (XV)? L08 Y = 1-h213095; - .OJSGPPYC log P - .00013065 P + p (-03h07L68) (.0001h319) .20502059 log K - .ooo21112 K (.0289h7h2) (.oooeo317) The adjusted coefficients of multiple correlation and multiple determin- ation for Equation XV are .585 and .3h2 respectively. In the latter ormulaiion -, coefficients estimated for both K20 variables are sitnifican+ H) at the one percent probability level. r—o 0 f" '- ro o _ \_ _ ~ -L- _o r v r. ‘ _o c_ 0 .axrr 3m YiCl as and Tjrh Profit Plant huurient applI aLlOfiS' Because of the complex nature of the PROS-K20 yield relationship, it is extremely difficult to determine the amounts of plant nutrient inputs (1) whi h maximize yields or (2) which maximize profits. The complexity of these relations hips is further exemplified by the derivatives of the functions. For example, he part al derivative of yield with reSpect to phosphoric acid for the polynomial is neca tive for almost any quantity of P205 unless K20 is fixed at a level of at leas 250 pounds per acre. In the case of the exponential, the partial ee Ivative ofy wlo with ream) ct to phos sphoric acid is always negative. Because of these unusual 93 phenomena, at ordinary potato prices the calculated highdprofit quantity of P205 is negative and consequently outside of the range of observed values. It is the expressed Opinion of soil scientists that the interaction between P205 and K20 is an important complementary relationship for potato production on muck soils. This interaction effect may exceed in importance the individual effects of either plant nutrient. There is, in particular, a commonly held belief that applied P205 will cause significant increases in potato yields only if adequate amounts of K20 are concurrently present in the soil. In view of this, it may not be illogical to assume that P205 applications had a non-significant effect on yields at lOW'KQO treatment levels. Both equations contain at least one nonsignificant P205 coefficient which may bias the estimate of the production surface and consequently the derivatives of the function. Further detailed analysis of these data is needed; however, it appears that, given the weather conditions of l956,ru>substantial appli- cations of plant nutrients were profitable. If any plant nutrient applications were profitable at all in 1956 they were only moderate applications of potash. CHAPTER V SOURCES OF UNEXPLAINED VARIANCE IN YIELDS.AND BIAS OF REGRESSION COEFFICIENTS The analysis presented in Chapter IV was designed prinarily to explain variance in crop yields with logically formulated functional relationships between quantities of applied plant nutrients and crop yields. Statistical estimates of the parameters of the plant nutrient variables were made using alternative production function formulations. On the basis of these estimates, inferences were made as to the shape of the plant nutrient-crop yield production surface, plant nutrient combinations producing maximum crop yields, plant nutrient combinations producing maximum dollar profits, etc. Variance in crop yields was not solely a function of variance in the quantities of applied plant nutrients. The adjusted coefficients of multiple determination ranged from a high of .58 to a low of .05 for the crops analyzed. Lacking knowledge of the exact form of the functional relationship between applied plant nutrients and crop yields and, furthermore, lacking completely effective control over unstudied variables, one should not expect 100 per cent of the variance in crop yields to be associated with regression. One might, however, expect a greater proportion of yield variance to be associated with regression than was found to be the case in the analysis of the preceding chapter. 9h Failure to characterize the major portion of yield variance by functional analysis raises questions as to whether or not experimental controls were rigidly enforced. This chapter will be directed first towards an explanation of variance in crOp yields not explained by the regression of applied plant nutrients. An additional problem deals with whether or not unspecified variables were randomly and normally distributed with respect to the independent variables studied. Sources of Unexplained.Variance in Yields Sources of unexplained variance in yield can be broadly classified as being due to (1) experimental error with reSpect to variables Specified and measured and (2) inadequate control over unspecified and unmeasured variables. Since these two sources of yield variance should be normally and randomly distributed with reSpect to treatment variables, they may be viewed as being sources of within treatment yield variance. Experimental Error Some portion of the unexplained variance in crop yields is undoubt— edly due to experimental error. Such errors are made by not applying the Specified amounts of plant nutrients on individual plots or errors made in acquiring yield measurements from the plots. ther sources of experimental error are uneven seed and fertilizer distribution on plots to mention only a few. In general, however, these errors are expected to be somewhat normally and randomly distributed with reSpect to 96 treatments and should be averaged out in the statistical estimating process. Researchers should recognize that this component of variance is present even in rigorously controlled experiments. Competent researchers should attempt to minimize such errors subject to the con- dition that the cost of reducing the errors is not in excess of the value of the gain in accuracy resulting from their reduction. For example, mechanization of controlled experiments may introduce experi- mental error in excess of that occurring with the use of hand—labor methods. However, minor increases in experimental error may be more than offset by the acquisition of additional information and better functional analysis resulting from additional plots and/or larger plots. Thus, reduction of experimental error should not be established as an absolute goal but rather one subject to economic considerations. It is the opinion of the author that the data analyzed in the preceding chapter did not, in general, have excessive experimental error. Some experimental error, however, was present. In particular, the con- tinuous corn experiment was characterized by a considerable amount of such error. Due to unfavorable weather conditions it was necessary to harvest the continuous corn plots by hand. Only a subsample from each plot was harvested; consequently, due to the smaller harvested sample a larger experimental error would be expected. Furthermore, the previous crOpping history varied for some of the plots in this experiment. Although corn yields from plots on the two areas with different cropping histories were not significantly different statistically, this hetero~ geneity of previous land use probably contributed to a minor amount of 97 variance in yields. Since the total yield variance was small originally, the existence of experimental error made it difficult to isolate the effects on yield variance due to variance in the quantity of applied plant nutrients. Uncontrolled and Unmeasured Variables Numerous factors such as weather, insects, bacterial action in the soil, etc.,are possible sources of variance in crOp yields not explained by the functional analysis in Chapter IV. The field bean input—output experiment was duplicated in the greenhouse. Results of the greenhouse experimentation are presented here to substantiate the hypothesis that yield variance could be explained by functional analysis given adequate control of unmeasured variables affecting yield and/or specification and measurement of these variables. A nine variable polynomial was fitted to data produced in the green- house. The soil contained in individual greenhouse pots was acquired from the correSponding field plots. The same number of observations were acquired using the same treatment levels as in the field experiment. Yields acquired in the greenhouse were for bean numbers per pot since the 1 beans could not be allowed to mature under greenhouse conditions. The results of th's regression analysis are presented in the following equation: 1Bean count and bean yields are not perfectly correlated, however, the two measures should be sufficiently correlated to allow valid infer- ences to be made from one to the other quantity-(yield) wise. Bean count might, however, be a considerably less valid measure of the quality of the crop. 98 Igh = 9.22679560 + .hShooioB N + .0203075 P + .07h73918 K - .00081362 n2 - (.o378h008) (.01919871) (.03779oho) (.00011092) .00001986 P2 - .000258338 K2 + .000197867 NP + .00012933 HK - .00003616 PK (.000019855) (.00011181) (.00003765) (.00007785) (.00OO3762) i‘ = .91 P? = .828 These results indicate that about 83 per cent of the variance in bean count for the greenhouse pots was associated with regression. In the functional analysis of the field data, however, only h2 per cent or about one-half as much of the variance in bean yields was associated. with regression. The inference suggested by this comparison of analyses is that explanation of more of the variance in yield under field condi- tions would be possible if variables affecting yield could be better controlled and/or measured and Specified in the functional relationship. Effects of Within Treatment Variance on Statistical Estimates The presence of within treatment variance should be noted when evalu- ating the relative success of particular functional forms in characterizing input-output relationships. If there is a difference in the yields from plots receiving the same plant nutrient applications, any function fitted to these data by least squares techniques, or any valid estimating procedure, will miss one or both yield observations. The greater the difference in yields between replicated plots, the greater will be the variance which cannot be explained by the function. Failure to explain this within treatment variance is not therefore a valid criticism of a particular functional form. 99 The effects of within treatment variance on the amount of total variance explained by regression may be exemplified by use of data from the 1956 wheat experiment. Data from all 130 plots were used in acquir- ing the statistical estimates made for the nine variable polynomial function presented in Equation V. A second polymial equation was fitted to the average yields of the plots which had a minimum of two replications for a given treatment. These observations include yields from the 3 x 3 x 3 factorial which was replicated twice and the ll check plots for a total of 65 plots averaged into 28 observations. Statistical results for the function fitted to the average yields from replicated treatments is shown in Equation XVII. A Eiuation XVII: Yfi a 27.87287 + .11222667 N - .ooo28877 N2 + .02207660 P - (.0292398h) (.00010520) (.01h61992) .oochoos P2 - .01h07013 K + .OOOllOll K2 + .00003065 NP - .00013398 NK + (.oooo2o3o) (.29239Ch) (.0001c32o) (.oooo273o (.ooooSM72) .00001889 PK (.oooo2736) The adjusted coefficients of multiple correlation and multiple determin- ation for this equation were .79 and .62 respectively as compared to .66 and .hh for the function fitted to all 130 individual observations. This sizeable increase in the amount of yield variance explained by regression illustrates that within treatment variance was an important component of total yield variance. The parameters of a function fitted to the average value of replicated plots, if all plots are replicated an equal number of 100 times, should be the same as those for a function fitted to the non- averaged observations, however, avera jng values for replicate Wb ations discards part of the information prov: ded oy the experime ental data. rectors R.lated to the Independent Variables O [Jsfixi III-I;:1*:s;,IoIIIInasflgsIIs LIL Visual obse ervation oft he experimental plots indicates that there were yield variance creating comg>onents which were associated with plant nutrients and which therefore were either (1) sources of biases in the estimat ed mf etc of plant nutrients on yields or (2) sources of yield riance which should be considered when evaluating the aggreg'te ef ects of applied plant nutrients. Incidence of Needs, LOdging and Plant Disease Observational data collected for oats in 1956 ind cated mignificant differences in weed growth and plant lodging which were as wsociat d With nitrogen applications. Prior to harves gthe oats crop, individual plots were ranked as to the degree of weed infesta oion and plant lodging and then these ranks were tabulated against nitrogen applications. The results of this classification are shown in Table 21. The incidence of weeds in plots was ranked from O to 3 with an increase in number rank indicating an increase in weed infestation. Lodging was ranked similarly from 0 to 8. ‘Weeds and lodging not only affected the absolute crop yields produced on some plots but the harvestability of the crop as well. The ratio of the amount of grain produced to the amount of grain lOl TMEE21 INCIDENCE OF hfiBD INFESTATIUN AND PLANT LODGIKG ON CAT PLOTS AS RELATED TO NITROGEN APPLICATIONS it Standard Standard Nitrogen Number ‘Average Deviation Average Deviation Application of Lodging of Lodging Weed2 of Need (pounds per acre) Plots Scorel Score Score Score 0 18 .889 .7hl O O 20 2h 2.250 1.561 .167 .3hO ho 1h 2.lh3 1.187 .lh} .32h 80 29 b.690 l.Sll .828 .hS? 160 18 6.hhh 1.257 1.556 .889 iPlots were ranked from O to 8 according to the extent of lodging present. 2Weed incidence was ranked from O to 3. harvested was probably significantly different for badly lodged plots as compared to non-lodged plots. No statistical measures of these differences were made, however. Incidence of plant disease as well as weed growth varied with plant nutrient applications on the field bean plots. Particularly, quack grass infestations were more pronounced on high nitrogen plots than on plots receiving smaller applications of nitrOgen. Plots with a large amount of plant foliage tended to have more shading of lower leaves and bean pods and consequently more disease infestation. The quantity of foliage on plots was, in turn, associated with the quantity of applied nitrogen. 102 These and other variance generating factors which are not inde- pendent of quantities of applied plant nutrients, but which may influ— ence yields in a manner other than that Specified in the production function formulation are sources of bias, i.e., they distort the absolute crop yield producing effects of plant nutrients. If such distortion or bias is a necessary consequence of applying plant nutrients it should be measured and considered when evaluating the effects of plant nutrient applications. In some instances, however, utilization of improved crOp management practices may eliminate such effects. For example, if weeds could be adequately controlled and if a sufficiently strong strawed variety of wheat were available for planting, the potential effects of n eercts (D applied plant nutrients on crOp yields might be realized. Th of factors which are sour es of bias in plant nutrient input-crOp yield output estimates as well as a discussion of other factors interacting with plant nutrients in the production of crOps are discussed adequately i in other literature and will not be enlarged upon here. Relationships Between Residual Fertility and Crop Yields L All plots in the two rotation experiments received the same plant a. nutrient applications in 1955 and in 1956. It seemed logical to expect 1For a discussion of these factors see L. S. Robertson Jr., G. L. Johnson and J. F. Davis, "Problems Involved in the Integration of Agrono- mic and Economic Methodologies in Economic Optima Experiments," Fertiliser Innovations and Resource Use, Edited by E. L. Baum, E. O. Heady, J. T. Peach and C. G. fiildreth (Ames: Iowa State College Press, 1956) pp. 226— 2h2, and L. S. Robertson Jr., W} B. Sundquist and L. N. Shepherd "A Frog- ress Report of the Studies on the Economics of Fertilizer Use on Beans and Potatoes," himeographed Report presented at a T.V.A. sponsored sym- posium on the economics of fertilizer use at Knoxville, Tennessee, Harch 1957. 103 some carryover or residual effects in 1956 from plant nutrient appli- cations made in 1955. This was particularly true for the plots receiving heavy plant nutrient applications in the preceding year. Soil tests for P205 and K20 were taken preceding and following every crOp produced. Consequently, differences in fertility between plots prior to plant nutrient applications made in 1956 would be expected to be related to these soil test measures. The method utilized in attempting to relate variance in crop yields to soil fertility as measured by soil tests will be summarized briefly. Soil test measures were first correlated with the applied amounts of the same nutrient for the individual plots. If this correlation was very high it would indicate that (l) variance in yield could probably be explained as well by the original functional analysis using only applied plant nutrients and/or (2) it would be diffi- cult to include both soil test and applied plant nutrient measures in a 1 functional analysis since the presence of high intercorrelations would reduce the reliability of estimated parameters of a function containing both measures as variables. If, on the other hand, the correlation between quantities of applied plant nutrients and soil tests was low, indicating some independence of the two measures, soil tests might successfully be used to explain a portion of the variance not associated with regression. The procedure used in relating soil test data to un- explained variances was to correlate the soil test data with the 1The distinction between high and low correlations is quite arbi- trary, however, as intercorrelations approach .70 the reliability of estimated parameters probably begins to decrease quite rapidly. th residuals computed from the original functional analysis. The results obtained by this method of analysis are shown for one crop from each of the rotation experiments, namely, the wheat and bean crOps produced in 1956. ,Effects of Residual Fertility;on wheat Yields Soil test data were collected for P205 and K20 preceding the wheat crop grown in 1956. No analysis of residual nitrogen has been completed to date.1 ‘The first analysis conducted was that of correlating pre-l956 crop soil test measures2 with the applications of P205 and K20 made in 1955. In the following discussion, soil test values of P205 and K20 are designated as Pst and.KSt reSpectivelyy Applied P205 and K20 are designated Pa and Ka' A regression analysis was conducted using Pst as the dependent variable and Pa as the independent variable. The resulting regression equation is as follows: 13340 a memo + .166659 Pa (.016565) The coefficient of correlation for this equation was .662 and the co- efficient of determination .h38. A similar regression analysis was 1Several nitrogen tests determinations have been made for soil samples from these plots. No nitrogen soil tests have as yet been generally accepted as satisfactory. A statistical comparison of the effectiveness of alternative nitrogen soil tests for residual nitrogen is currently in process using soil samples from this experiment. zThese soil samples were actually acquired in September of 1955 immediately preceding seeding of the wheat crop which was harvested in 1956. 105 conducted relating soil test measures and applied quantities of K20. The resulting regression equation is as follows: A Kst = 75.108u + .276689 Ka (.039190) The coefficients of correlation and determination for this equation were .526 and .277 reSpectively. A preliminary inspection of the residuals of these functions indicates that little, if any, improvement could be made by changing the formulation, i.e., by fitting a curvilinear form such as Pst = a + Pa + P: to the P205 variables. As evidenced by the preceding analysis, applied and residual plant nutrients Show a moderate amount of interdependence or correlation. A correlation as high as .66, as was found between Pst and Pa, might indicate that the effects of residual and applied P205 could not be easily separated. The correlation between Kst and Ka, .53, does not, however, appear to be prohibitively high. On the basis of the preceding exploratory analysis it was decided that some reduction in unexplained yield variance might be accomplished by incorporating the residual fertility measures into the analysis. Simple correlation analysis was conducted using Pst and Kst as separate independent variables and the residuals from the nine variable polynomial (Equation V) as the dependent variable. Designating the residuals or deviations from the polynomial (Ii - ii) as D, the results of these simple correlation analyses are as follows: 1C6 U) = -.l§9696 + .OO’ZZ‘D'E) Psi) (.006331) ”I r2 I: .OCDCS D = -1.L58§83 + .01h232 Kst (.ooohto) F” = .190 Eb = .0362 Phosphoric acid soil test measures appear to bear no relation to the unexplained residuals of the original functional analysis, whereas, potash soil tests are slightly, but not significantly, related to these residuals. The inference suggested by this analysis appears to be that soil test measures do not provide an aid in reducing unexplained yield variance in the case of wheat, at least not in the simple relational form analyzed here. It should be remembered, however, that soil test measures were correlated with quantities of applied plant nutrients and that most of their effects on yields are probably incorporated in the 1 original functional analysis. Additional work is currently in progress evaluating soil test procedures and relating these measures to quanti- ties of applied plant nutrients. lGordonhnderson of the Department of Agricultural Economics and Arthur welcott of the Department of Soil Science of the hichigan.Agri- cultural Experiment Station are COOperating on this phase of research work. 107 Effects of Residual Fertility n Dean Yielcs As in the case of wheat, soil test data were collected prior to fertilization and planting of the 1956 bean crop. P205 soil test observations ranged in values from h8 to 50h, however, only one observ- ation was in excess of hOO. K20 soil tests ranged from a low of 8h to a high of hOO. Application rates in the bean experiments ranged from O to 320 for K20 and from O to 6hO for P205. A regression analysis was conducted using Pst as the dependent variable and Pa as the independent variable. The results of this regression were as follows: -97.9670078 + 2.253587 P (.11616h) "U > U) (+- II a F2 = .593 The same analysis was conducted using potash soil test measures and treatment rates as variables. The results of this regression are shown in the following equation: Kst = ~7h.970613 + 1.2h59oo Ka (.091887) E7 a .6h5 F2 = .hlé As in the case of the wheat experiment, a greater portion of the variance in P205 soil tests was associated with variance in P205 appli- cations of the preceding ear than was the case for K20. Correlations as large as these, .770 and .6h5 reSpectively, indicate that the effects 108 of residual fertility on bean yields migit well have been explained at least in part by the original functional analysis in which yield vari- ance was formulated as a function of variance in applied plant nu.trients alone. Despite the high correlation between soil test measures and quanti- ties of applied P205 and K20, a multiple regression analysis was conducted . A using Pet and Kst as dependent variables and the residuals, (Ti - Ti), from Equation XI, the six variable enponen al equation, as the dependent variable. The results of this multiple regression analysis are as follows: E = 2.0157 - .07973 Pst - .020h2 Hot (.C'OET9) (.Oflhhg) o The parameters of this equat ion are highly SLfniiicant however, the adjusted coefficient of mul_tiple correlation is only .068. This value of §.is not significantly different from zero. As in the wheat experiment, no significant amount of the variance in yields not explained by the original regression analysis with applied plant nutrients as independent variables can be attribute d to residual fertility as measured by K20 and P205 soil tests. (D p) ‘1 O m 0 <1 0 *1 p H '1 C.) :1, £0 0 E Ther- why failure to relate unexplained yield variance to soil test measures should not be interpreted as meaning that crOp yields are not a function of residual fertility. P305 and K20 soil tests were found to be significantly related to amounts of applied nutrients, hence, a portion of their effect on yield variance would be expected to be characterized by the original functional analysis. In addition, soil test measures for nitrogen, which in almost all experiments had the predominant effect on crop yields, were not included in the analysis. Soil test measures are themselves subject to consider- able variance beCause of errors in sampling and in testing the samples. Additional research needs to be undertaken in calculating sampling and testing variances for soil test procedures. Such research would provide an aid in evaluating the accuracy and adequacy of Soil testing procedures currently being used. Another possible explanation of the low correlation between soil tests and residuals is that a more complex formulation of the relationship between soil est measures and unex- plained residuals would have been more appropriate, i.e., the linear relationship assumed in simple correlation analysis may be an over- simplification of the relationship between these variables. Because of the importance of soil test data in making current fertilizer recommendations, additional work needs to be done relating alternative soil test measures to: (l) variance in crOp yields (2) quanti- ties of applied plant nutrients to establish substitution ratios between applied and resi‘ual plant nutrients and (3) other soil testing methods to determine the most effective soil test procedures available. The plant nutrient input—output experiments described earlier should provide data well adapted to an analysis of soil esting procedures. The experiments contain extremely high and extremely low levels of plant nutrient applications and consequently a wide range of residual fertility values is develOping on the plots. As individual plots become extremely depleted of plant nutrients or extremely fertile they will provide a 110 wide range of soil test observations. If plant nutrient applications were to be rerandomized on the plots, a wide range of residual and applied nutrient combinations could be observed. Thus the effects of residual fertility might be studied without the complicating influence of highly correlated plant nutrient applications. ,Furthermore, substi- tution ratios between residual and applied nutrients could be estimated from a wide range in the combinations of the two. Conclusions Several sub-inferences may be drawn from the analysis presented in this chapter. The important conclusion, however, seems to be simply this: Given (1) adequate control over Specified factors affecting crOp yields, and (2) a random and normal distribution of other factors affect- ing crop yields, functional analysis should provide an adequate repre- sentation of plant nutrient input-crop yield output relationships. The relatively small amount of total variance in crop yields explained by functional analysis is not an inherent characteristic of the analysis and/or the functional forms used but rather is largely a function of the uncontrolled factors enumerated in this chapter. EVALUATION OF PROCEDURES AND RESU TS Evaluation of Experimental Desirns The experimental designs used in the several experiments described in this work were formulated with several restrictions and objectives in view. Prior to designing the experiments, it was decided that continuous function analysis of the experimental data would provide a better basis for (l) estimating plant nutrient input-crop yield output coefficients and (2) facilitating an economic analysis to determine optimal plant nutrient applications, than would alternative methods of analysis. Thus the experiments were designed to provide data suitable for continuous function analysis. Restrictions on funds, labor and equipment limited the number and/or size of the experimental plots. Individual treatments or cells in the eXperimental designs were selected to: (1) describe the economically relevant portion of the production surface sufficiently to obtain reliable estimates of parameters of the production functions (2) establish with adequacy input-output measures for critical points on the production surfaces, e.g., origin of the functions and their inflec- tion points and (3) minimize intercorrelations among treatment variables. It is the Opinion of the author that the experimental designs were quite satisfactory as a basis for providing data for continuous functions lll H }-J [‘0 analysis. The experimental designs utilized in the two original rota- tion experiments are not highly efficient in providing data which 1 reatnent yield variance cf- readily facilitates estimation of (1) within (2) crop quality differences associated with treatments, (3) differences in plant nutrient content of plant tissue and (h) differences in other plant and soil characteristics associated with plant nutrients but not in the manner prescribed for the basic input-output relationship. Once committed to an incomplete factorial design with a minimum number of replications, analysis of such factors as those listed above 2 may be quite difficult. However, he designs which were used are adequate for these determinations if (1) the determinations can be made by corre— lation analysis or (2) if the determinations for one plant nutrient can be assumed to be independent of the treatment level of other plant nutrients. In the latter case this means that all observations for which the treatment level of the studied variable are constant can be considered as replications of that treatment. A modification of the incomplete factorial--minimum replication design used in the rotation experiments was incorporated into the con- tinuous corn, the 1956 potato and the new sugar beet experiments. These designs include a triplicated factorial in addition to other treatments which were replicated twice. This modification was incorporated into 1Inability to Specify within treatment variance is not considered to be an important criticism of the experimental design. 2The inference being made here is that some of the determinations listed above can best be acquired by analysis of variance techniques. 113 the designs to facilitate analysis of by-product data produced in the experiment. The experimental designs, as modified, still provide numerous non-replicated treatments in order to Specify the production surface adequately for continuous function analysis. Inclusion of a factorial into the experimental design facilitaoes utilization of analysis of variance techniques on a limited basis at little additional cost. A possible criticism of the experimental designs which were used might be the large Spacing between treatment levels of the various plant nutrients. Obviously, it would be desirable to have observations at treatment levels intermediate to those contained in the experiment, however, the experiments already were large and required a considerable amount of land, labor, machinery, equipment and supervision. Larger experiments would have created problems in conducting experimental work, such as seeding, harvesting etc., with apprOpriate timeliness. The primary consideration in not enlarging the experiments by including intermediate treatment levels was that of the additional time and cost which would be necessitated by such an expansion. The correlation between applied and residual plant nutrients is relatively high in these exgeriments since individual plots receive the same treatment in successive years. A more comprehensive analysis of residual and applied plant nutrient relationships would be facilitated by rerandomizing treatments on the experimental fields. Such a modifi- cation of the experimental design would provide observations over a much wider range of combinations of residual and applied nutrients. This is a modification of the experimental design currently being contemplated. 11h Evaluation of Experimental Procedures To the extent feasible, the experimental work was conducted utiliz- ing mechanized procedures. When soil conditions allowed, plant nutrient applications were made with a 7 ft. ‘ractor drawn drill. Small grain seedings were also made with a 7 ft. drill which required one round on the plots which were 1h feet wide. Wheat and oats crOps were harvested with a 7 ft. self propelled combine. A portion of the corn crop was harvested by using an eSpecially constructed single-row corn picker. In instances where weather conditions prevented fertilizer application and corn harvest by machine, this work was accomplished by use of hand labor. Some amount of additional experimental error undoubtedly occurs due to use of machinery as compared to hand labor; for example, plant nutrient applications are not precisely weighed out and delivered in exact amounts to individual plots. Small amounts of grain remain in the combine from one plot to another when harvesting etc. and introduce some small experimental error. These errors should, for the most part, however, average out and not bias the plant nutrient input-crOp yield output esti— mates made. Mechanization of experimental work provides some interesting and important implications particularly with reSpect to the number and size of individual plots which can be satisfactorily included in an experiment. Two objectives of plant nutrient input-crop yield output research appear to be of relevance here. First, we want research results to be validly inferrable to some farm population. Farmers typically operate as units fields of a minimum of several acres in size. The larger the experi- mental plots, the more nearly they represent the conditions actually existing on farms. Farmers, and consequently researchers whose objective is to make input-output estimates applicable to farm conditions, are not particularly interested in measuring within treatment yield variance. Rather, they are interested in determining the variance in yield which can be attributed to variance in plant nutrient applications under farm conditions e.g., the change in yield resulting from application of an additional 20 pounds of nitrogen etc. Researchers are interested, however, in having some assurance that within treatment yield variance is not prohibitively large so as to constitute a large portion of total yield variance. Within treatment variance is reduced by increasing the size of individual experimental plots and the harvested portion of these plots. Increases in plot size are facilitated by mechanizing the experi- mental procedures used. Errors of inference due to excessive within treatment yield variance can be eliminated alternatively by replicating a given treatment several times and averaging the yields of the several replications. Additional replications of a treatment require more labor and have a higher cost than is true for a comparable enlargement of a given plot. It is the opinion of the author that when the main objective of experimentation is to estimate plant nutrient reSponse surfaces, increasing plot size is a more efficient alternative. A second objective of our research, that of estimating input—output coefficients to which we can attach acceptable reliability measures, 116 is aided by increasing the number of individual plots in an experiment. .9 standard error of estimate for parameters in a functional equation diminishes as the number of observations increases. .Attainment of both accurate and applicable research results is, therefore, enhanced by increasing the size and number of experimental plots. It is the opinion of the author that within treatment variarce in the experiments was probably higher than was necessary. Use of larger plots and/or harvesting a larger portion of individual plots would probably have provided results the additional accuracy of which would have been worth the cost of obtaining this accuracy. Y"! 0 v o ' - evallation of Analytical Procedures The Continuous Function Analysis A brief justification for utilizing continuous function analysis was presented in Chapter II and will not be repeated or expands here. Rather, a brief a posteriori evaluation of the effectiveness of the con- tinuous function analysis used will be attempted here. Both polynomial and exponential type formulations of the reSpective production functions were fitted for all creps for which preliminary analysis indicated that an appreciable amount of variance in yield was associated with variance in applied plant nutrients. No criteria are available which provide a basis for saying one formulation is "absolutely" more appropriate than the other; however, some measures which provide somewhat of a quantitative basis for comparis n are available. rthermore, logic and theory provide ll? a basis for selecting one formulation in preference to the other in at least two instances. As previously mentioned, comparison of the coefficients of multiple correlation for the two functions provides a guide as to the relative amount of yield variance associated with regression. This comparison is somewhat subjective, however, since: (1) in the case of the eXponentials, variance is measured in logarithms and in the polynomials it is measured in real numerical values. Although the logarithms and real numbers bear a consistent monotonic relationship to each other over the range of the values which they take in the data, they do not retain a relationship of constant ratios. (2) The two formulations differ as to the number of variables in the respective equations, hence there is a small difference in the number of degrees of freedom used in the two analyses. The latter difficulty is not an important one, however, because of the large number of observations and, hence, degrees of freedom, present in the analysis. A comparison of the coefficients of multiple correlation and determin- ation for the functions fitted is shown in Table 22. In three of the six comparisons a larger amount of yield variance is explained by regression for the exponential equations than for the polynomials. In one case, that of the field beans, the polynomial equation has larger values of h and fig, whereas, in the remaining two comparisons values of E and i2 for the two equations are almost identical. This comparison provides no very conclusive indication as to the superiority of either type of formulation. TM1322 COMPARISON OF AMOUNTS OF YIELD VARIANCE ASSOCIATED WITH .ALTERHATIVE PRODUCTION FUECTICN FCRLULATIOKS CrOp Function Number of Variables R R3 Oats, I956 Polynomial 9 .69 .h8 Exponential 6 .76 .58 Wheat, 1956 Polynomial 9 .66 .bb Exponential 6 .65 .h2 Corn, 1955 Exponential 6 .70 .h? Polynomiall 9 .6h .hl Corn, 1956 Polynomial 9 .23 .CS Exponential 6 .2h .06 Cont. Corn, I956 Polynomial 9 .hO .16 Polynomial S .39 .16 Beans, I956 Polynomial S .65 .h2 Exponential 6 .61 .37 Exponential h .61 .3 Potatoes, I956 Polynomial S .50 .25 Exponential h .59 .)h P7 1The polynomial used on the polynomial of the form Y = a + b1 + b9 [EF— 1955 N + Corn data was a square root 2 b /fi. + b3P + b4 /P" + b5K + b6 /i— + 119 A second comparison of the two types of functions was included in A the analysis. Residual measures, (Ii - Ii), were computed for both types of functions. These residuals are measures of the deviation of predicted yields from observed yields. The residuals are almost identi- cal for both t; es of functions for all crops. This is true for the magnitude of residuals as well as for their sign or direction. In summary, inSpection and measurement of the residuals provides no discern— able basis for choosing one function in preference to the other. A third comparison of the polynomial and exponential functions which might provide some basis for choosing the more apprOpriate one is an inspection of the derivatives of these functions. InSpection of the partial derivatives of the exponential functions with respect to individual plant nutrients shows that the derivatives are usually of extreme magnitude (negative or positive) for small inputs of the plant nutrients 1 and then become extremely small quite rapidly. Extremely large deriva- Y O 0 fl 9 ., With small inputs or the Xi are a consequence of he yield tives ~—:~ ’ Z>Al being zero when any of the Xi = O. Derivatives of the polynomials in comparison usually take less extreme values. It is the opinion of the author that over moderate plant nutrient input ranges for most crOps, generally in the range of 20 lbs. to 200 lbs., the exponential is probably a satisfactory formulation of most of the 1There are exceptions to this statement. For example, the partial derivatives of bean yields with reSpect to nitrogen decreases at firs and then increase with additional nitrogen inputs. here are other exceptions to this statement as well. input-output relationships. Derivatives of the exponentials for field beans and potatoes, however, are contradictory to the usually accepted concept of diminishing returns. Maximum yields predicted using the exponential functions were outside of the range of observed inputs for 1 the bean and potato crops. However, the maximum potato yield predicted was secured using quantities of plant nutrients outside of the range of observed inputs using the polynomial as well. Calculation of the quant it es of plant nutrients which result in maximum profits is a much more complex procedure using the exponential type formulation than using a polynomial. Solving the exponential for optimal inputs requires use of a series of successive approximations 2 known as Newton‘s method. This method requL was in part a graphic approximation refined by solving a series of equations. Statistical estimates of the parameters of both types of equations are rather easily acquired by methods of least squares. The primary advantap e of the exponential type formulation as com- pared to the particular polynomial used is that it permits derivatives, Q X.) 2’ l to takecninon-linear forms. Deeratives of a polynomial containi ng " 1 range of observed inputs is a criticism 01 the inn reality the maximum yield does occur within t1:e ranfe of ooserved inputs 1The phenomena of maximum predicted yields being outside of the c and is fallaciously pr mt d to be outside. I1, indeed, the true maxi- mum yield exists in ziond the range of observed inputs it is the experi- mental desi;.ii, not the function, v1111c1i s1‘lould oe critici ed. 3a complete ex,1anation of t 3 method of solving a Carter—halter t pe e: :ponent ial is explained in a forthcoming article in the Journal of Farm Economics by A. h. haltei E. C. circa“ and J. G. hocking. u.) .. . -‘ <- O “’._1“ V. a A)". ,'.‘._' r‘ nr‘n-“ur n Tne authors also discuss in some cecail the cronnroic) of this ianiiy oi "unctions. ._ D. L 1 r an “ “‘1 — a. >7 a P 1. - only llfSu ant Szcuvu 1333:; c»rm3 xiii necessarily be restrict-c to g «f V. 5 "q- _. fl -3‘.> _ 5c , .. _ . f‘ -‘. ' P '. - s ' O _ 0 ~..- . linear iorm. Iv .Lb L113 opinion Oi {1:123 author brat until a 31:1“ 001111.111 ._ , ,-.1 1H - '1 I,” , ,1 , ,., 7__'1I.-.. ‘ I "I I‘l. ' 113 proceeuies for soiVan a Carter—naiter ty] e .ponkn iai icr OptiMal plant nutrient inputs are available that modifications of the polynomial 3 "““j '7‘ I, ~r"'—-.l -' '1‘ '- 1.3 '13 :firu'."v|r~"-‘ - ~— ‘.-4-.,.. 3-}. - _ ’7‘ Byte: iOl I1 Lei 1.; on 1111,3311; Du {7.011, C in: 1-1; :A'Jl *3 . .LIlCUl pol a biflf5‘ \fal'lL-J 1.13:? ()l v. '1 ’3 WW) . pa . 7~........ . A .- -.—‘ . .7’1 .-...'I - 9 degree i/e or nxa or oi a CULECB 2< c dLU not equal to l will resait in non—linear derivatives. uliiuaercn of Soil Test Measures No significant amount of variance not explained by regression was explained by use of P205 and K20 soil test measures. Soil tests for residual quantities of P205 and K20 were not significantly correlated A with residuals (Vi - Yi) from the functions fitted for wheat and beans. The analysis presented here does not provide a very comprehensive exploitation of the possibilities of using soil test measures in supple- menting functional analysis of applied plant nutrients. The Federal Extension Service, as well as Agricultural Experiment Stations and private fertilizer companies rely heavily on soil test data as a basis for making fertilizer recommendations. Because of the wideSpread use of these soil test procedures, any additional information relating variance in soil test measures to crOp yields would be a very valuable contribution. Economic Interpretation and Ev alu a ion of Results The most profitable amounts of plant nutrients to apply we*e co puted for all crops except alfalfa. The analysis presented in Chapter IV 122 indicated a significant reSponse to nitrogen for corn produced on a Kalamazoo sandy loam soil in 1955. Significant yield responses to applied nitrogen were recorded for oats, wheat, field beans and corn produced on a Wismer clay loam soil in 1956. The only crop not showing a significant reSponse to nitrogen in 1956 was the corn produced on a Kalamazoo sandy loam soil. Statistically significant reSponse to applied phosphoric acid was recorded for wheat, field beans and the corn produced on a Wismer clay loam soil in 1956. Oats and corn produced on a Kalamazoo sandy loam soil in 1956 did not Show significant yield response to applied phosphoric acid. The only crOp showing significant yield reSponse to applied potash was the potato crop produced in 1956 on a Houghton muck soil. DeSpite the several significant reSponses recorded, only small amounts of plant nutrient applications were indicated to be profitable. Predicted high~profit plant nutrient inputs for the various crOps are shown in Table 23. No applications of P205 and K20 were indicated to be profitable for any of the crOps produced at typical crop and fertilizer prices. Nitrogen applications were profitable for five of the crOps produced if crOp prices were sufficiently high. Assuming typical prices, however, nitrogen applications were profitable only for corn produced in 1955 and field beans produced in 1956. Some qualification of these results seems to be warranted. First, the 1955 and 1956 growing seasons were characterized by severe summer drouths. Thus the reSponses recorded.may not typify the long-run expected responses to applied plant nutrients. Additional data collected TABL 3 23 ESTIMATED HIGHéPROFIT PLANT NUTRIENT APPLICATIONS FOR VaRIOUS CROPS Estimated High Profit Plant Nutrient Tintitsl Crop V e i A v 5 N PROS 320 Oats, 1955 None None None Oats, 1956 Only if the price of oats None None > €211.00 Wheat, 1956 Only if the price of wheat None None > 253 .00 Corn, 1955 About hO lbs. (on Kalamazoo Sandy loam soil) at Corn None None prices of Corn, 1956 (on Kalamazoo Sandy loam soil) None None None Corn, 1956 Only if the (on Wiener Clay loam soil) price of corn None None > 2.00 Field Beans, 1956 75 lbs. with Only if None beans 33.50, bean prices 150 lbs. with are >ifi7.00 beans at $5.CO and 200 lbs. with beans at $7.00 Potatoes, 1956 Not varied in None None at experiment ordinary prices 1Computed with N at $0.15 per 1b., P205 at $0.10 per lb. and K20 at $0.11 per 1b. over time are needed to obtain a probability distribution of yield reSponses over the range of existing weather conditions. As a further qualification, it should be noted that the experimental results reported in the preceding analysis were obtained from soils either (1) relatively unproductive, as in the case of the Kalaaazoo sandy loam soil or (2) relatively heavy and productive in the cases of the Simms loam and Wiener clay loam soils. One might expect, a priori, to obtain the greatest yield reSponse to applied plant nutrients from soils with a high productive potential but with low fertility levels. Greater yield reSponse may be noted in future years on low nutrient level ‘ 0 plots as residual fertility is depleted. Concludinv Remarks The analysis of experimental work presented here is rather limited in sc0pe with respect to nunber of soils, crops and growing seasons. Additional work is needed before the optimal plant nutrient treatments estimated here can be substantiated or invalidated as long-run Optimal applications. The distribution of yield responses over time is likely to be characterized by wide diSpersions, particularly in the case of the lighter soils which are frequently subject to damaging drouth periods. However, some interesting questions and implications are posed by the results of the analyses presented here. J P w No significant response was obtained from applied potash for t several crops grown on mineral soils durin* a two-year period of I I L.) experimentation. This lack of reSponse poses a question as to the validity of recommending a program of "ba lanced” plant nutrient appli- cations. Rather, the general reSponses recorded from thes experiments indicate that nitrogen was the rite ary source of crOp yi as r Sponse. On the basis of these results it appea‘s that a plant nutrient combination weighted more heavily with nitr05:en relative to potash might be Optimal at least until residual potash is depleted somewhat. A second general implication posed by the experimental results is, "despite statistically significant yield responses, in most cases the cost of pplying additional plant nutrients exceeded the value of the additional crOp produced." This general result would indicate that analysis which only detects significant yield differences which are associated with plant nutrient applications is not an adequate procedure for determinin~ the most profi able application rates. This result in itself would seem to validate or at least vindicate the general type of anal uiS used in this dissertation, i.e., that of continuous function analysis to which economizing principles may be applied. In conclusion, at the farm management application level of fertili- zation practices, these practices cannot be considered independent of other alternative farm business expenditures nor can they be considered independent of the numerous factors with which they interact. For examp_e , a livestock farmer may find it profitable to fertilize oats, not for the oaty ield benefits, but in ordeu to establish a clever or grass seeding which is essential to his livestock ente rplise However, if a farm manager is to intelligently and economically Synth ethe costs and benefits of the numerous components of his farm business he needs infor- mation as to the productivity of expenditures made for plant nutrients for the various crops he produces. Additional plant nutrient input-crOp yield output estimates will help to provide this information. BIBLIOGRAPHY Barton, G. T. and Rogers, R. 0. "Farm Output, Projected Changes and Projected Needs," Agricultural Information Bulletin No. 162, washington: ‘Agricultural Research Service, August, 1956. Baum, E. L., Heady, Earl O. and Blackmore, John. Economic Analysis of Fertilizer Use Data, Ames: Iowa State College Press, 1956. Baum, E. L., Heady, Earl 0., Pesek, John T. and Hildreth, Clifford G. Fertilizer Innovations and Resource Use, Ames: Iowa State College Press, 1957. Bureau of the Census. Current Population Reports, Series P—2S No. 123, washington: U. 3. Government Printing Office, October 20, 1955. Bureau of the Census. "Use and Expenditures for Fertilizer and Lime," Adapted from the l9§h Census of Asriculture, washington: U. 5. Government Printing Office, 1956. Fuller, E. I. "Michigan Dairy Farm Organizations Designed to Use Labor Efficiently," Unpublished hasters Thesis, Department of.Agricultural Economics, Michigan State University, 1957. Heady, Earl 0., Pesek, John and Brown, William. "CrOp Response Surfaces and Economic Optima In Fertilizer Use," Research Bulletin LEh, Ames: Agricultural Experiment Station, Iowa State College, 1955. Hildreth, C. "Point Estimates of Ordinates of Concave Functions," Journal of the American Statistical Association, Vol. h9, September, 195u, pp. 598-019. Hutton, Robert F. "An Appraisal of Research on the Economics of Ferti- lizer Use." Agricultu~al Economics Branch, Division of Agricultural Relations, Tennessee Authority Report No. T SS—l, Knoxville: Tennessee Valley authority, 1933} Johnson, Glenn L. "A Critical Evaluation of Fertilization Research," Farm Hanagpment in the west Problems In Resource Use, Report No. l, The Economics of Fertilizer Application. Conference Proceedings of the Farm Management Research Committee of the western.hgricultural \ r—l Economics Research Council, Corvallis: 1956, pp. 33-up. Johnson, Paul R. "Alternative Functions for Analyzing a Fertilizer Yield Relationship," Journal of Farm Economics XXIV November, 1953. pp 0 519 “929 o 127 Ibach, D. B., and Hendum, S. N. "Determining Profitable Use of Ferti- lizer," U. S. Department of fisriculturc,_F. r. 1C5 naslinston. U. S. Government Priiting tiiice, l9;:. Hnetsch, Jack L. "Lethodological Proceduress and App nlicati ons for Inc corporating Economic Considerations into Fertilizer Recommend— ations," Unpublished Lasters Thesis, Departmm nt of.Agricultural Economics, hichifan State University, 1956. Knetsch, Jack L., Robertson Lynn. 8., and Sundquist, W". B. "E'conomic Considerations In Soil Fertility Research, Fichigan Aggrieultural hVT\Qr’IT‘mJY‘-+ STEM—L ‘Ofi QhJcir-l: 91‘]. V‘ hFMl—letln, I'L‘Llflle, l//\). 11p. lb-{LSO Michigan Department of.Agriculture. Hie h ,an Agricultural Statistics, July, 1956. Redman, John C. and Allen, Stephen Q. "Some Interrelationships of Economic and Agronomic Concepts, " J'tiurra l of Farm Economics Vol. W“J'I,Au{:ust,1951;.pp. L53 h ' M“) o Robertson, L. 8., Sundquist, W. B. and Shepherd, L. N. "A Progres 3 Report of the Studies on the Economics of Fertilizer Use on Beans and Potatoes," Himeoxranhed Report of the U‘onwcin Loriculir a1 Experiment Station, harch, 1957. U. S. Department of.Agriculture. .figri cultural Staei miies, 1955, 'Washington: U. S. Government P unblfl Oiiice, l9po. Hooten, H. H. and Anderson, J. R. "- facultural Land Res ources in The United States with dossial Reference to Present and Potential Cropland and Pasture, ricultural Information Bulletin lhO, washington: U. S. Deparomwrt of A*ri culture, June, 10;S. Yeh, H. H. "‘“tlflatlfifl Input- Output Relationships for Wheat in Michigan Using Sampling Dana, 1952-5h," npublished Kasters Thesis, Department of Agricultural Economics, Michigan State University, 1955.