ABSTRACT THE APPLICABILITY OF SELECTED RATIO AND LEAST-SQUARES REGRESSION ANALYSIS TECHNIQUES To THE PREDICTION OF FUTURE EDUCATIONAL ATTENDANCE PATTERNS BY William John Webster This study is concerned with the exploration of three major questions. 1. Can educational attendance prediction problems be formulated in such a manner that the least-squares regression model can be used in the analysis thereof? 2. Are different variables and diverse projection methodologies relevant to school districts with different growth characteristics, or is there one projection methodology that provides the best estimates, ie. estimates with the least amount of error, across all school districts. Is it worthwhile to make the necessary data transformations to estimate local population statistics for projection purposes, or will county papulation statistics, which are readily available from relatively reliable sources, provide more reliable estimates on which to base future school population projections? The population identified in this study consists of those public school districts in the State of Mdchigan that are largely coterminous with Cities of 10,000 population or more. It is divided into five strata on the basis of past community growth characteristics. Twenty-five school districts, five from each stratum, were randomly selected for study. Seventeen separateenddistinct ratio and re- gression methodologies were used to obtain estimates of elementary and secondary enrollment for each of the twenty-five districts in the sample. Pre-l960 observations on various diverse demographic variables were used as base data from which projections were made to 1965 and 1968. Actual 1965 and 1968 enrollment figures were used as the criteria against which to judge the goodness of fit of the estimates produced by each of the five ratio and twelve regression approaches examined in the study. The major findings of the study suggest that educational attendance prediction problems can in fact be formulated in such a manner that the least-squares regression model can be used to advantage in the analysis thereof; that different predictor variables and diverse ratio and regression approaches are re- levant to districts with different growth characteristics; and, that most population variables, whether they derive from local or county data, provide estimates that are inferior (in terms of good- ness of fit) to least-squares regression and ratio approaches that use time, births, and the tendencies of students to advance from one grade to another as the predictor variables. One such approach, a regression approach that is called "Transition Analysis", represents a unique approach to the projection of public school enrollment. This method provided the most accurate projections of all of the seventeen ratio and regression projection methodologies identified in this study, including those ratio approaches that are currently the most used in estimating future public school enrollment. THE APPLICABILITY OF SELECTED RATIO.AND LEAST-SQUARES REGRESSION ANALYSIS TECHNIQUES TO THE PREDICTION OF FUTURE EDUCATIONAL ATTENDANCE PATTERNS BY William John Webster A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY College of Education 1969 g4/7XE ”#2740 DEDICATION This document is dedicated to my Mother and Father in recognition of their unending support and encouragement. ii ACKNOWLEDGMENTS I wish to gratefully acknowledge the help of several individuals who contributed immensely to this study and to my general well being during my tenure as a graduate student. First, a special debt is due to Dr. James Heald, the Chairman of my Doctoral Committee, and Dr. Robert Craig. These two individuals have provided wise and stimulating council while aiding me in navigating the perils and adventures of graduate study. Their example of unselfish and encouraging interest in my progress, as well as their insightful evaluations of my work at Michigan State, will long be remembered and appreciated. Drs. Maryellen McSweeney, Joe L. Saupe, and David Smith provided invaluable aid in the formulation of my program of study and in the evaluation of the research proposal that was the genesis of this document. Each made many valuable recommendations concerning methodology and interpretation; all of them were gratefully received. Finally, the continued understanding and support provided by my wife, Linda, helped to make the completion of my graduate program and this study possible. iii Chapter I. II. III. IV. TABLE OF CONTENTS INTRODUCTION 0 O O O O 0 0 O O 0 O O O O 0 O O 0 Statement of the Problem . . . . . . . . . . . . Definition of Terms. . . . . . . . . . . . . . . Purpose of the Study . . . . . . . . . . . . . . Limitations of the Study . . . . . . . . . . . . Data Limitations. . . . . . . . . . . . . . Qualifications Inherent in the Statistical Projection Methods Utilized . . . . . . . Footnotes for Chapter I. . . . . . . . . . . . . REVIEW OF THE LITERATURE . . . . . . . . . . . . Projection Methodology Used by the United States Census Bureau. . . . . . . ... . . . . . . . . Projection Methodology Used by the State of Michigan . . . . . . . . . . . . . . . . . . . Projection Methodology Used in Public Education. Projection Methodology Used in Higher Education. Footnotes for Chapter II . . . . . . . . . . . . “THOD 0L mY O O O O O O O O 0 O O O O O O O O 0 Population and Sample . . . . . . . . . . . . . Definition of Terms . . . . . . . . . . . Classifications of Districts into Strata . Data . . . . . . . . . . . . . . . . . . . . . Data Sources and Projection Methodology . . Regression and Ratio Methods Examined , , , , , Discussion . . . . . . . . . . . . . . . . Design . . . . . . . . . . . . . . . . . . . . Footnotes for Chapter III . . . . . . . . . . . RESULTS 0 O O O O O O O 0 0 O 0 O O O 0 0 O O O Cohort-Survival Ratio Method . . . . . . . . . . Grade-Retention Ratio Method . . . . . . . . . . Enr011ment-Ratio MethOd I e o o o o o o e o e 0 iv 18 24 28 38 42 48 48 48 49 53 56 64 67 68 72 73 78 81 83 Chapter Enrollment-Ratio Method II ................................. Pupil—Home Unit Ratio Method ...................... ......... Time Analysis I... ..... . ......................... .......... Time Analysis II ...... . ................................ .... Transition Analysis... ...... . .............................. Birth Analysis... ..... ......... ..... ... ................... Enrollment Regression Analysis I... ........................ Enrollment Regression Analysis 11...... .. ........ . . . . Enrollment Regression Analysis III.. ................ ..... . Enrollment Regression Analysis IV...... ..... ... ........ .... Housing Analysis ............ ........ ..... . ................. Multi-Variable Analysis I (Regression)....... .............. Multi-Variable Analysis II (Regression)........... ......... Multi-Variable Analysis III (Regression)....... ......... . . Summary................................... ................. V. SUMMARY AND CONCLUSIONS................................ . . Results and Discussion.......................... .... . . . Recommendations for Future Research ..... ............... . BIBLIOGRAPHY........... .............. ......................... . . APPENDIX A - Assumptions Made by the University of Michigan 'Population Studies Center in Projecting Future State Population.............................. APPENDIX B — Procedures Used in Obtaining Projections............. Cohort-Survival Ratio Method... Grade-Retention Ratio Method.... Enrollment-Ratio Method II................................ Pupil-Home Unit Ratio Method.............................. Time Analysis I........................................... Time Analysis II........................................... Transition Analysis........................................ Birth Analysis............................................. Enrollment Regression Analysis I........................... Enrollment Regression Analysis II.......................... Enrollment Regression Analysis III......................... Enrollment Regression Analysis IV.......................... Housing Analysis........................................... Multi—Variable Analysis I.................................. MUlti-Variable Analysis II..... Page 85 87 89 91 94 96 98 100 102 104 106 108 111 113 115 118 120 124 128 136 138 138 141 143 144 145 146 147 148 151 153 154 155 156 158 159 161 161 9. 10. ll. 12. 13. 14. 15. 16. LIST OF TABLES Sample Districts from Exploding Communities (Stratum A) O O O O O O O O O O O O O O O O O 0 Sample Districts Drawn from Rapidly Growing Communities (Stratum B) . . . . . . . . . . . . Sample Districts Drawn from Older but Still GrOWing commities (Stratum C) e e e e e e e 0 Sample Districts Drawn from Stable Well-Established Communities (Stratum D) . . . . . . . . . . . . Sample Districts Drawn from Declining Communities (StratumE)oeeoeeeoeeeeeeeeee Factors Selected for Study and the Sources of Data Descriptive of Each . . . . . . . . . . . Projection Methodologies Examined in this Study . Number of Enrollment Projections for each District in the sample 0 O O O O 0 O O O O O 0 O 0 O O 0 Actual Enrollment of Sample Districts . . . . . . Projections by the Cohort-Survival Ratio Method . Accuracy of Projections Yielded by the Cohort- Survival Ratio Method . . . . . . . . . . . . . Projections by the Grade-Retention Ratio Method . Accuracy of Projections Yielded by the Grade Retention Ratio Method . . . . . . . . . . . . Projections by Enrollment-Ratio Method I . . . . Accuracy of Projections Yielded by Enrollment- Rat 1° MethOd I O O O O O O O O O O O O O O O O Projections by Enrollment-Ratio Method II . . . . vi Page 50 51 51 52 53 61 65 69 77 79 80 82 83 84 85 86 Table 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. Accuracy of Projections Yielded by Enrollment- RatiOMethOdIIeeeeeeeeeeoeeee Projections by Pupil-Home Unit Ratio Method . . Accuracy of Projections Yielded by the Pupil- Home Unit Ratio Method . . . . . . . . . . . Projections by Time Analysis I (Regression) . . Accuracy of Projections Yielded by Time Analysis I (Regression) . . . . . . . . . . . Projections by Time Analysis II (Regression) . Accuracy of Projections Yielded by Time Analysis II (Regression) . . . . . . . . . . Projections by Transition Analysis (Regression) Accuracy of Projections Yielded by Transition Analysis (Regr8881on) e e e o e e e e e e e Projections by Birth Analysis (Regression) . . Accuracy of Projections Yielded by Birth Analysis (Regression) . . . . . . . . . . . Projections by Enrollment Regression Analysis I Accuracy of Projection Yielded by Enrollment Regression Analysis I . . . . . . . . . . . Projectionsby'Enrollment Regression Analysis II Accuracy of Projections Yielded by Enrollment Regression Analysis II . . . . . . . . . . . Projections by Enrollment Regression AnalySiQIIIeeeeoeooooeeeeoee Accuracy of Projections Yielded by Enrollment Regression Analysis III . . . . . . . . . . vii Page 87 88 89 9O 91 92 93 95 96 97 98 99 100 101 102 103 104 Table 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. Projections by Enrollment Regression Analysis IV 0 O O O O O O O O O O O O O O 0 Accuracy of Projections Yielded by Enrollment Regre8810n AnalySiS IV e e e o e e e e e e Projections by Housing Analysis (Regression). Accuracy of Projections Yielded by Housing Analysis (Regression) . . . . .'. . . . . Projections by Multi-Variable Analysis I (Regression). . . . . . . . . . . . . . . . Accuracy of Projections Yielded by Multi- Variable Analysis I (Regression) . . . . . Projections by Multi-Variable Analysis II (Re8r33810n) e e e o o o o o e o e e o e 0 Accuracy of Projections Yielded by Multi- Variable Analysis II (Regression) . . . . . Projections by Multi-Variable Analysis III (Regression) . . . . . . . . . . . . . . . Accuracy of Projections Yielded by Multi- Variable Analysis III (Regression) . . . . Effectiveness of Projection Methodologies by Strata O I O O O O O O 0 O O O O O O O 0 viii Page 105 106 107 108 110 111 112 113 114 115 116 CHAPTER I INTRODUCTION Statement of the Problem The acute need for educational planning has long been realized. An essential facet of this planning involves the estimation of the number of pupils that are to be housed at relevant future dates. Accurate enrollment forecasts would greatly assist in the making of educational decisions affecting both educational and fiscal policies. Such decisions must be made at every educational level, although the necessary distance into the future and degree of precision may vary. A list of such decision problems would include the following. 1. Short-range District Financing. How much money will be required in the next year, and for what purposes? Are expected increases in attendance in the near future sufficient to warrant new building appropriations now? Long-range District Financing. Is such an expansion of school enrollment foreseen that funds for building and operation will need to be radically increased? Will future current taxes, if not radically revised, be able to take care of the operating budget of the school district? Given the answer to the thorny philosophical problem of how large a school unit of a certain type should be, it is from numbers of expected pupils that deductions can be made as to the numbers of sites and buildings required. State Short and Longerange Financing. How much money will be needed for public education at a state level? Is it sufficient to plan for public education collectively, or is it more feasible to consider each individual district separately? Short and Moderate-rangg District Planning. What increases in staff should be incorporated immediately into the budget in view of current and anticipated increases in attendance? How heavily must the district plan on recruiting, ie., approximately how many man hours must be devoted to re- cruiting? Longfrange District Planning. When should major physical plant changes begin? Does projected enrollment warrant major physical plant changes? From enrollment projections the community must make definite decisions with respect to space requirements, time for and placement of new buildings, length of bond issues, and even the future functions of the school plant. Moderate and Longerange State Planning. What is the state- wide demand for educational personnel going to be? Accurate enrollment projections would lend valuable insights into the question of whether or not state universities are capable of furnishing necessary personnel, and the related decisions concerning an out-of-state recruitment program. Should modifications in educational policy be imposed at a state level so that undesirable outcomes can be avoided. e.g. Should public aid to parochial education be supported in order to curtail overcrowding of facilities, or is the school-age population in the state leveling off or decreasing to the point where present facilities will be sufficient to meet future demand? In planning for the Operation of a school district, knowledge of attendance patterns alone is not sufficient, for the cost of operating a public school is not proportional to attendance alone. Expansion of a vocational education program by fifty students is not comparable to expansion of a college prep program by the same number. Other activities of the public schools may be insensitive to attendance, for example, the auditorium and gymnasium actually service the local community as well as the student body. Hence the need exists, particularly at a local level, for detailed pre-‘ dictions of attendance as well as for a knowledge of the relationships between attendance and other variables. Despite the many uses for accurate enrollment projections, the formulae currently in vogue in education are lacking in the necessary precision.1 The reasons for this lack of precision are Inany. It must be recognized that never before in the history of AAmerican Education has the estimation of future school population 1>een fraught with greater problems than at the present time. Efforts to estimate future school population must reckon with, among other things, gargantuan population and community changes. These \ changes include resurgent national growth; the increased population concentration in large metropolitan areas; great waves of in-migrant peoples of diverse races, cultural backgrounds, and school enroll- ment practices, into the large metropolitan areas of the country; accelerating decentralization within some metropolitan areas and centralization in others; and, a rapidly changing age structure in our society. Simultaneously, unprecedented community changes are under way. Included are such phenomena as changes in patterns of land use resulting from urban renewal programs, public housing projects, federal and local highway construction, increased utilization of physical planning techniques, and the ever increasing transportation flow. The problem of estimating school enrollment for local districts is further complicated by differential rates of expansion of public, private, and parochial schools, and by the need to replace obsolete school structures while, at the same time, grappling with the many problems of rapid growth. Population growth, or lack of growth, is the net result of births, deaths, and migration. Most projections that are made of papulation are based on the assumption that there will be no large— scale war, no widespread epidemic, no major economic depression, or any similar catastrophe. Within the limits of this assumption, death rates become relatively easy to estimate. For example, in the United States as a whole there has been little change in death rate since 1954.2 Available statistics indicate that the death rate in the State of Michigan has also remained largely constant during that period of time.3 Birth rates and migration present more difficult problems. Births have fluctuated widely throughout the United States during the past fifty years and could easily do so again.4 Migration to or from a given area is effected by such factors as the availability of jobs, which in turn fluctuates with industrial expansion, the availability of housing, and, the quantity and quality of available shOpping, recreational, and educational facilities. The interplay of the aforementioned factors is by no means uniform and therefore requires a separate enumeration and evaluation, district by district, to adequately estimate future school enrollment. Most American communities are faced with problems of rapid growth, however, some face actual decline in both total population and school enrollment. One projection procedure is probably not adequate for use in all situations. The best method for projecting school needs in a particular community depends to a great extent on the characteristics of the community itself, including its age, size, location, and the rate of net migration to the area. Of course, the kind and quality of available data must also be taken into account. There follows a classification of community types that may prove useful.5 A. Exploding Community. Most of the areas falling into this category are located in the suburban part or outlying fringe of large metropolitan areas. Most of them are either new communities or areas of rapidly expanding residential development in older communities with large parcels of vacant land. Projection of school needs for communities of this type is exceedingly difficult without a special survey of the local area since past enrollment figures generally cannot be used to predict future enrollment. B. Older But Still Growing Community. Areas falling into this category are characterized by parcels of vacant land that may be or are in the process of being utilized for the construction of housing units. In this type of community migration is still a definite problem in that it is difficult to predict, ie., one may encounter difficulty in attempting to predict future trends from past performance. C. Stable,_Well-established Community. Areas falling into this category are characterized by either a lack of vacant area for ex- pansion or a lack of reason to expand. Large metropolitan areas and many small rural communities would fall into this category. Difficulty in projecting enrollments for this community type stems mainly from changing population characteristics and the resultant increase or decrease in school-age youth relative to total population. D. Declining Community. Areas included in this category are characterized by a declining population. They are most likely to be rural farming and mining areas, although, they may also be communities whose rolls were filled with tuition students from "non-high school" districts that are building high schools and Siphoning students off from the host district.6 A number of methods for projecting school population have been devised. However, one mathematical model which seems to this writer to be uniquely appropriate for this type of problem, the linear, least-squares model, has not been given an adequate trial by educators.7 As early as 1932, a study of the available methods for estimating student enrollment in education revealed gross inadequacies in current methods.8 Projection methodology has largely remained the same since that time, and there is little evidence to suggest that results have become more accurate.9 It seems apparent that the time to try new projection methods in education has long since passed. It is to this task that this dissertation is addressed. Definition of Terms Throughout this manuscript several terms are used repeatedly. In the interest of clarity those terms are defined below. 1. Projection The term "projection" is used to describe the methodology involved in the forecasting of future enrollments. The term "prediction" is avoided because, considering the nature of the data and methodology, it presumes too much, implying that a statement made about enrollment at some future date does not admit of possible variation in the actual number that will be found at that date. The term "prediction" is used in the title as a retrieval term, potential readers being more familiar with the term "pre- diction" than with the term "projection." 2. Predictor The term "predictor" is used to describe the independent variable, that is the variable or variables that are manipulated in the regression and ratio equations. The predictor variable is the one whose changes or differences are associated with changes or differences in the dependent or criterion variable. Values of a predictor variable thus afford a basis for projection. 3. Criterion The term "criterion" is used to describe the dependent variable, that is the variable whose behavior is presumed to be predictable from the predictor variable. The dependent or criterion variable in this study is always some form of school enrollment. Purpose of the Study This study is concerned with the exploration of three major questions. 1. Can educational attendance prediction problems be formulated in such a manner that the least squares regression model can be used in the analysis thereof? Ultimately we are asking if simple or multiple linear regression, or a transformation thereof, will yield more accurate projections of future school attendance than will the most papular educational projection methodologies currently in use at the district level. Are different variables and diverse projection methodologies relevant to different types of school districts, or is there one projection methodology that provides the best estimates, ie. estimates with the least amount of error, across all school districts? Is it worthwhile to make the necessary data transformations to estimate local population statistics for projection purposes, or will county population statistics, which are readily available from relatively reliable sources, provide more reliable estimates on which to base future school population projections? Limitations of the Study There are two major conditions which place important limitations 1. on this study and which, therefore, must be taken into consideration when discussing any results that are obtained. These limitations are listed in their probable order of importance to the outcome of the There is a serious lack of adequate data for a study of this type. Much of the data that is used must be extrapolated or interpolated from the little data that is available. lO 2. Statements or conclusions derived from the statistical projection methods used in this study must be interpreted in the light of a number of qualifying phrases. Data Limitations This study is specifically concerned with the problems encountered in short-term enrollment projection. Generally six data points, one for each of the years 1955-1960, are used as the basis for establishing equations to project future enrollment figures to 1965 and 1968. There are definite limitations inherent in the kind and amount of data that is available. Projections based on incomplete and unreliable statistics are subject to the inadequacies and limitations of those statistics. Considered in sum, the data used in this study illustrates all too clearly that there is ample justification for the view that the develop- ment of enrollment projections of measurable accuracy and dependability is retarded by the lack of adequate and dependable data. Qualifications Inherent In The Statistical Prqjection Methods Utilized The major qualifying phrase which must be stated for every enrollment projection method utilized in this study involves the basic tenet that the environment must remain similar in the period for which projections are made to the period from which projections were drawn. Stated more succinctly, the assumption of stable conditions, or, at the very least, regular rates of change, must be made for every enrollment projection method examined in this study. To the extent that this assumption is violated, the enrollment projections will be in error. 11 This study recognizes two classifications of projection methodologies, ratio methods and regression or least-squares analysis methods. Virtually all of the projection methods des- cribed in the literature are ratio methods. The distinction between ratio and regression analysis methods is made on the basis of the measure of relationship used in the specific com- putation of projected enrollments. The classification of a projection method as a ratio method rests on the fact that the ratio, or proportion of a given predictor to enrollment, is the basis for the computation of the projected enrollment. The various ratio methods are in turn sub-divided into classifications determined by the predictor selected for study. The classification of a projection method as a regression analysis method means that the measures of relationship used as the basis for calculation are the coefficients of correlation and multiple correlation, statistics which purport to measure the degree of association between variables. The ratio methods require only one major assumption, that being the previously mentioned one that the ratios of predictor to criterion may be established historically and will remain constant or alter at a stable rate in the future. They represent a composite, or average, of past trends and occurences. The regression methods assume that the criterion variable can be expressed in the form of the equation "Y - A + BX + E," where "Y" is the criterion variable, "A" and "B" are unknown coefficients to be determined by the least squares procedure, "X" is the independent 12 or predictor variable, and "E" is the error inherent in the model. The predictor variable is generally chosen because it is a variable that is known, or can be estimated, with greater precision than the criterion variable. If more than one predictor variable is utilized, then multiple linear regression is required. In multiple linear regression a restriction is placed upon the number of predictor variables that can be included by the number of cases available for study. Generally speaking, high correlations and high multiple correlations appear as artifacts of the computational process as the number of variables approach the number of cases available for study. The regression methods also require that the ratios of pre- dictor to criterion may be established historically and will remain constant or alter at a stable rate in the future. In addition, there are several more important assumptions that must be made. The most important of these assumptions states that the line of best fit which specifies the relationship between the predictor and criterion variables be a straight line. To the extent that the data deviates from linearity, projections derived from the linear model will be in error. This assumption is not seen as a problem since most functions can be best approximated by a straight line over a short period of time or small number of data points. A second basic assumption underlying the application of regression methods is that the error terms in the regression model are independent. A great deal of use has been made in the 13 behavioral sciences of least squares regression methods in circum- stances where they are thought to be inapplicable due to the violation of this assumption.10 The classical example of autocorrelated error terms occurs when an incorrect specification of the form of a relation- ship between variables is made. If the straight forward least-squares formulae are applied directly to relationships where the preceding mistake is made, there are three major consequences. First, although the procedure will yield unbiased estimates of a and 8’ it is possible that the sampling variances of those estimates will be unduly large. Second, if the usual least squares formulae are applied for the sampling variances of the regression coefficients, serious underestimates of those variances are likely to occur. Finally, the predictions obtained from such a procedure will be inefficient in that they will have needlessly large sampling variances. Cochrane and Orcutt have demonstrated that the usual application of the method of least squares to relationships containing highly “positively autocorrelated error terms results in a marked decline of the variances of both the correlation coefficient and the re- gression coefficient as the error term becomes more random.11 Least squares methodology was applied to five different cases; a random case, a first-order Markov scheme, a limiting first order case when p - l, a second-order autoregressive scheme, and the first differencing of a random series. Even in the case of the random series there is a slight underestimate of the true error variance, but this becomes very serious in the autocorrelated cases.12 14 It should be obvious from the preceding discussion that the violation of this assumption does not affect the estimates as such, but rather the confidence intervals that can be placed around those estimates. Given the fact that there is a high probability of autocorrelated error terms in time-series data, it is extremely questionable as to whether or not conficence intervals computed in the traditional least-squares manner will add much to our knowledge about the data. This problem is not examined further in this study because, given the nature of the data, the nature of the projection problem, and the small number of data points, it is questionable as to the amount of information that the confidence intervals would supply anyway. The Standard Errors of the Regression Coefficients are merely used as relative indicants of the amount of error present in each of the various projections. The regression approach is used in an algebraic manner as a method of projecting a curve into the future. The problem of setting confidence intervals around that curve must be reserved for future study. The dther assumptions of the least—squares regression model, namely those of normality and common variance, are of little consequence since moderate variations in those entities have little affect on the results obtained from the model. 15 Footnotes for Chnpter I 1Albert Bryon Crawford and Leo Martin Chamberlain, Prediction of Population and School Enrollment in the School Survey, (Univ. of Kentucky: Bureau of School Services Bulletin, v.4, no.3, 1932 27 p.). Dean Waldfagel and Frank R. Krajewski, A Study of School Enrollment Projection Formulae, unpublished manuscript, Michigan State University, 1964. Knute C. Larson and Wallace H. Strevell, "How Reliable are Enrollment Forecasts, " School Executive, LXXI, (February, 1952), pp. 65-68. 2U.S. Bureau of the Census, Current Population Reports, Series P-25, No. 388, p.3. The present death rate (1968) is 9.5 per 1000 pOpulation and, according to the Bureau of the Census, is likely to drop somewhat to approximately 9.0 per 1000 population by 1990. 3It is certainly true for the State of Michigan. See R. Raja Indra, ed., Michigan Population Handbook, Michigan Department of Public Health, Nov., 1965, p.7. 4Current ngulation Reports, Series P-25, No. 388, op. cit., p.4. According to the U.S. Census Bureau birth rate is not likely to drop below 17 per 1000 population, but might increase and approach 25 per 1000 population within the next twenty years. Even the lower rate, being approximately twice the death rate, would guarantee a continuing increase in population. There are two reasons for believing that there will be a new spurt in births during the next few years. First, the number of women of the age of childbearing is increasing rapidly as the children of the postwar baby boom reach maturity. Second, there is evidence that the recent fall in the birth rate is partly due to delays in family formation. 5The original classification scheme was taken from: Philip M. Hauser and Evelyn M. Kitagawa, "On Estimation of School Population," American Statistical Association Proceedingg, (Washington: ASA), 1961, pp.69-70. It has been revised somewhat for purposes of clarification. 6In this category are the so-called "dead" communities. One such community of this type would be a community where there once existed a "live" mine which has since been worked out. This type of community has some unique projection needs. Provisions must be made, for example, for dropping tenured personnel as a result of declining school enrollment and the resulting drop in State Aid. 16 7At least such a trial has not been reported in the educational literature. For example, writing in 1954, and having in mind the approaches used in public school enrollment forecasting, Strevell identified four methods of enrollment projection, all of which were ratio methods. Wallace H. Strevell, "Techniques of Estimating Future Enrollment," American School Board Journal, March, 1952. This over-sight is prevalent in the educational literature, although the linear, least-squares model has been occasionally experimented with in higher education. This will be discussed at length in the review of the literature. 8A.B. Crawford and L.M. Chamberlain, op.cit., 27p. 9Chamberlain and Crawford's work has been largely ignored in educational circles. The results of an analysis of existing prediction formulae carried on at Michigan State University in an unpublished paper by Waldfogel and Krajewski, op.cit., tend to confirm Chamberlain and Crawford's hypothesis. This work was completed as recently as 1964. 10J. Durbin and G.S. Watson, "Testing for Serial Correlation in Least Squares Regression," I., Biometrika, 37, 1950, p.409. 1 1D. Cochrane and G.H. Orcutt, "Application of Least Squares Regression to Relationships Containing Autocorrelated Error Terms," Journal of the American Statistical Association, 44, 1949, pp.48-49. 12 Cochrane and Orcutt, op.cit., p.51. CHAPTER II REVIEW OF THE LITERATURE This review of the literature will begin with a presentation of the efforts of the United States Census Bureau to project both total population and student enrollment. It is felt that a review of the projection methodology utilized by the Census Bureau will serve to highlight some of the problems inherent in population projection. A consideration of the methods utilized to project total population can also be justified by the realization that many of the enrollment projection formulae currently utilized by educators rely on Census projections of total population in making their projections. Following the brief presentation of the methodology utilized by the United States Census Bureau, a review of the major projection formulae used in public education will be discussed. Each formula will be presented with a brief explanation of the assumptions involved in its use, its application, its strengths and weaknesses, and the frequency of its use. Finally, a consideration of the projection methods utilized in higher education will be presented. At this point the strengths and weaknesses, assumptions, and possible applicability of simple and multiple linear regression to educational projection problems will be discussed. l7 18 Projection Methodology Used By The United States Census Bureau The United States Census Bureau concerns itself with three types of population estimates; inter-censal, post-censal, and future. Future estimates are defined as those made for any period of time after the present moment, and, therefore, are the only type of estimate that will be considered here. Two basic procedures are used by the Census Bureau for the construction of estimates of future population. The first, and most rarely used method, involves the application of one of a variety of mathematical curves which presumably describe the manner in which the population has grown in the past and therefore is likely to grow in the future. The second involves the use of vital statistics which, in con- junction with census enumerations, permit the construction of estimates of future populations.13 The simplest mathematical procedure suggested is the use of a linear interpolation between two consecutive censuses, and subsequent extrapolation beyond the last one.14 This procedure assumes that the rate of growth over the past ten years will remain constant for the period to which the projection is being made. In the event that three or more observations are available, it is suggested that curved lines be fitted to the data and extrapolated. Such curves, specified by quadratic equations, assume no particular pattern of population growth other than that of a smooth and continuous rate of change. The quadratic approach is thought to be quite inadequate by government demographers because it l9 pre-supposes increasingly larger gains or losses to the pOpulation.15 The population estimate, if a quadratic equation is used as an approximation, will increase indefinitely into the future at a positively accelerating rate. Another mathematical procedure considered is estimation through the use of an exponential curve. In the event that a population is increasing in size, an exponential curve will predict that the absolute increment increases yearly and that population growth takes the shape of a "j curve." (The reverse holds true if the population is decreasing in size.) The further calculations are projected into the future, the greater is the absolute increment added yearly. Therefore, if one is interested in long—range estimates of population the results yielded by the use of an exponential curve might prove to be quite ludicrous.16 The logistic curve is the curve that government demographers see as being the most applicable of the mathematical approaches mentioned to the problem of long-range papulation projection. Unlike the curves referred to above, the logistic curve becomes asymptotic, indicating a plateau as far as population size is concerned. Since it is known from past observation that populations do not continue to increase indefinitely at constant, or increasing rates of growth, the Census Bureau prefers this approach.1 At the time of the publication of the Handbook of Statistical Methods For Demogrnphers, the United States Census Bureau and the 20 Office of the Coordinator of International Statistics were not extremely enthusiastic about the possible applicability of mathematical curves to population projection.18 The main criticism of mathematical curves is that they present a theo- retical pattern of growth and, therefore, cannot depict the year to year variations which result from fluctuations in the numbers of births and deaths as well as the number of immigrants and emigrants. This view is supported by another researcher who bemoans the fact that a curve may fit enrollment data for the past fifty years with a high degree of accuracy, yet fail to 19 All methods project future enrollment for the next year or two. of population projection extrapolate past trends into the future, although some make different assumptions than others. Mathematical curves perhaps do not offer the flexibility of some other methods, but, where complete and reliable vital statistics and migration data are not available, as is the case when dealing with local areas, certain varieties of mathematical curves seem quite appropriate. Perhaps the dissatisfaction of government demographers with the usefulness of mathematical curves stems from their apparent failure to utilize some appropriate models, their apparent failure to consider any variable but time as the independent variable, and the nature and complexity of the projection problem.20 The main set of methods utilized by the United States Census Bureau for the projection of the total population is based on the 21 notion that the population of any given area at a given time can be specified by subtracting the number of deaths plus the number of emigrants from the number of births plus the number of immigrants. Two methods are used in the production of projections. Projections of total population by age and sex are made through the use of the so-called "cohort-component" method, and projections of population characteristics by that same method and by the so-called "participation-rate" method. The "cohort-component" method involves dividing the population into meaningful groups based on relevant variables, and keeping track of those groups of individuals through life. Each of the components of population change - births, deaths, and interstate and international migration - is projected separately. After projecting each component of change, the projections are added to the population at a relevant base date to measure change in total population over time. The "cohort-component" method must obviously take into account the aging of the papulation over time. For example, the papulation age 20 to 24 in 1965 will be ages 25 to 29 in 1970, and, as a result will perhaps require a different mortality assumption in 1970 than it did in 1965. To keep the computations straight, the populations and the components of change are classified by the year of birth of the population. The population which was age 20 to 24 in 1965 and will be 25 to 29 in 1970, was born between 2 1940 and 1944. This pOpulation is thus called the cohort of 1940-44. 1 22 When utilizing the "cohort-component" method to make national population projections, the Census Bureau makes a number of estimates of future pOpulation. This stems from the fact that demographers have found it extremely difficult, if not impossible to make precise projections of future population. Therefore, the device of presenting a number of estimates based on a variety of assumptions about the components of population change, instead of a single estimate based on a single set of assumptions, has been adopted. Since the assignment of mathematical probabilities to the various series of projections, analogous to sampling errors of population estimates based on probability samples, is not feasible, the minimum and maximum estimates are often used as an approximation to the upper and lower bounds of a crude confidence interval, thus facilitating interpretation of the projections.22 When projecting papulation on a national scale the Census Bureau has, in the past, made four series of projections. These projections differ from one another wholly with respect to the assumptions made relative to fertility; Series A, the most liberal estimate, places the fertility rate approximately 37% higher than does Series D, the conservative estimate.23 Mertality and migration are allotted only one set of assumptions each, because it was reasoned that the possible range of those two components of population change is small.24 In summary, the "cohort-component" method, as applied to projection problems by the United States Census Bureau, involves 23 the projection of each component of change for each successive cohort on each relevant variable. Such projections involve the use of further projections of such things as age-specific life table survival rates for mortality, cumulative age-specific fertility rates for cohorts of women, and assumptions as to the total age-sex distribution of net migration. These projections involve a relatively complex procedure and require a vast amount of reliable data. In the "participation-rate" method, first, the rates at which the population participates in some activity or possesses some characteristic such as school attendance, employment, or household status, are projected, and then the rates for a given year are applied to the projections of the population for that year. As far as this paper is concerned, the most relevant use that the Census Bureau makes of the "participation-rate" method is in the projection of future national school enrollment. Projections are made of enrollment rates by age and sex, and these rates are then applied to the projections of total population in order to obtain an estimate of the number enrolled. Once again several estimates are utilized. The relevant variables that are varied are fertility rate and percent of p0pulation enrolled in school. There are two assumptions under each variable, thus contributing to a total of four series of estimates. The projections distribute enrollment among three levels of school: elementary including kindegarten, high school, and college and professional school.25 24 This procedure requires a relatively accurate estimate of total pOpulation because this is the projection to which the projected rates are applied. Therefore, one is likely to encounter data collection difficulties if he attempts to utilize a procedure similar to this in areas smaller than the state level. Projection Methodology Used By the State of Michigén States also have need for accurate population projections. These projections are made by both the United States Census Bureau and the individual states. The state population of Michigan was recently projected to 1980 by the Population Studies Center at the University of Michigan.26 The procedure used was a component procedure, similar to the methods utilized by the U.S. Census Bureau and previously discussed in this paper. In the interest of clarifying any problems that the reader may have in understanding the component procedure of population estimation, as applied to the state or national level, the assumptions and methodology of the Michigan procedure are described in detail in Appendix A. County population projections make use of all of the techniques presented in Appendix A, plus some additional procedures which adjust the county totals to equal the state total. Unique demographic characteristics of counties, as well as variations in county migration and fertility patterns, are also taken into account. Projections of total county populations in the state of Michigan are generally carried out in the following manner. First, average annual growth rates are determined for each relevant variable 25 in each county for specified time periods. Next, the share of the state total population found in a given county is projected forward by using the growth rates in the county shares obtained in the first step. Third, since it is evident that this projection procedure will not necessarily sum to 100 per cent of the state projection, the pro- jected shares of state total population are adjusted so that they do sum to 100 per cent. Finally, the adjusted shares are multiplied by the state total population to obtain the projected population for each county.28 Realizing the intense need for accurate population projections for small geographic units within the state, the Michigan Department of Commerce prepared a report entitled Preliminapy Population Projections for Small Areas in Michigan.29 The principal objective of this paper was stated as the development of an initial population projection method, suitable for statewide application on a small area basis, with sufficient flexibility for the later incorporation of additional variables and appropriate methodological refinements. It is significant that the State Resource Planning Division and their advisors chose a refinement of the least squares regression model to attack this problem. The projection technique utilized in the paper projects local units' shares of total county population. Each local unit's population is expressed as a fractional share of its county population.30 A regression line of the following form is then fitted to the shares for each local unit. P - a + b X log (B-A)31 N 26 Next, the local unit shares projected for each of the relevant years are multiplied by the projected county populations for each relevant year to arrive at the projected local unit p0pu1ations. In order to avoid negative share projections, which could result in some cases of declining shares, a constraint is incorporated that no local data unit can fall below 1/10 of 1% of its parent county.32 Since the sum of the projected shares would not necessarily total exactly 100% of the county total, the sum of the projected data units also would not necessarily exactly total the county projected population. To adjust for this difference, the projected population of each local unit was multiplied by the factor: Countnyrojected Total 33 Sum of the Local Unit Projections. The local unit projections, and the county projections to which they are tied, represent extrapolations of past trends which show, since 1940, a rapid increase in population in the southern part of Michigan and a slowly declining population in the northern part. Superimposed on this trend is the general increase of city populations, with the largest increases occurring in the metropolitan areas of southern Michigan, and the highest percentage increases occurring primarily in suburban cities and townships. The projected local unit shares, when applied to projected county populations, may yield any one of the following four results.34 27 CASE COUNTY POPULATION LOCAL UNIT SHARE PROJECTED LOCAL UNIT _____ PROJECTION PROJECTION POPULATION 1 increase increase increase 2 increase decrease increase or decrease 3 decrease increase increase or decrease 4 decrease decrease decrease As you can see, when the change in county population and the change in local unit's share of county population are opposite in sign, the projected change in the unit's population may be either an increase or decrease, depending on the relative magnitude of the rates of change of the county population and local unit's share. The most significant limitation of this type of projection comes to light particularly in suburban fringe areas where population growth is occurring in cities and townships at successively greater distances from major urban centers. Since this projection technique is based on historical trends of population change for individual minor civil divisions, it tends to show increases for units which have increased in the past, and decreases for those which have decreased in the past. This is no great problem except in the situation where a unit has shown substantial population increase in the past, but which has finally reached its "holding capacity." A model is needed that takes this "holding capacity" into consideration. 28 Projection Methodolpgy Used In Public Education Educators have modified the techniques utilized by the United States Census Bureau, apparently failing to appreciate the signifi- cance of the differences in projection problems. As a result, many of the enrollment projection methods currently used in education forfeit many of the advantages of the Census Bureau's techniques while retaining most of the disadvantages. Federal demographers working with large population and geographic areas about which large amounts of reliable data are available are certainly faced with different kinds of problems than are educators who must work with small geographic and population areas about which little reliable data is available. The most papular enrollment projection formula currently used for projecting public school enrollment is the so-called "cohort- survival" or "grade-cohort" method. This method depends upon the relationships of grade to grade enrollment throughout a school system, as indicated by grade statistics for a number of years. The method operates on the premise that by examining the enrollment data of a particular district by grade for a number of years, a trend can be established as to the preportion of students enrolled in each grade in a given year who progress to the next higher grade the following year. The essential steps involved in the procedure are as follows: 1. Begin with the actual number of births in a given school district during any six year period. (Six years is used because this is the age at which most children enter the first grade. If kindegarten enrollment is of interest, five years would be used.) 29 2. For each of the years of a given six year period, say the years "n" to "n+6", find the actual enrollment in the first grade. 3. Compute the percentage of survivorship between the number of births six years earlier, and the enrollment in the first grade for each of the known years, "n" to "n+6". Average the prOportions. Thus we have: n + n+1 + n+2 + n+3 + n+4 + n*5 Bn-6 Bn-S Bn-4 Bn-3 Bn-2 Bn-l N where: A = The average percent of survival from birth to first grade for any period of six years. (This could of course be a greater or lesser number of years.) B = The number of births in a given year. As can be seen by looking at the formula, births appear in the denominator of each ratio in the numerator, and are recorded for six years earlier than is enrollment. E = Public school enrollment in the first grade in a given year. N = The number of years for which the ratio of births to is computed. 4. Making the assumption that the average of the computed proportions applies to each future year, apply this average to the actual number of births which occurred during the years "n" to "n " to obtain estimated first grade enrollment for the years "n+6" to "n+12", the period during which those children born between the years "n" and "n+6" will enter school. Thus we have: EEn= Bn_6 (A) where: EEna Estimated first grade enrollment in the year "n". Bn—6= Births in a given year six years earlier. A = The average percent of survival from birth to the first grade. 30 5. Calculate the percentage of survivorship from grade one to grade two and so apply these to the known figures to get estimates for the years immediately ahead. Thus we have: where: ESn+1 8 Estimated enrollment in the second grade in year "n+1" . Bn-6 = The number of births in a given year, "n-6". A 8 The average percent of survival from birth to the first grade. P1,2 = The average percent of survival between the first and second grade. where: P1.2 E Sn+1 + Sn+2 + Sn+3 + Sn+4 + Sn+5 En En+1 En+2 En+3 En+4 N where: S = the enrollment in the second grade in a given year. E a The enrollment in the first grade one year earlier. N - The number of years for which ratios were computed. This procedure is carried on for each grade, then summed for a total enrollment estimate. Obviously the degree of detail depends upon the demands of the particular situation, ie., whether an estimate of how many students will be taking chemistry is desired or merely a total enrollment estimate.35 The procedure in its common form has a number of weaknesses. First, it requires an estimate of births for a number of years into 31 the future if projections of elementary enrollments of more than five or six years are desired. This is a task which demographers have not been able to accomplish with any great deal of success. Second, and perhaps of a more serious nature, birth statistics in Michigan are often not available for school districts as such. School district boundaries in Michigan do not necessarily correspond to any other govermental division. Therefore, if one is to use this approach some sort of interpolation from county or state statistics is required to estimate births by district. In an attempt to avoid estimating births by district, other formulae have been derived which are based on similar rationale but do not require birth statistics. One such formula is presented below. EEn - PSEn_5 (API x 5) where: EEn - Estimated public school enrollment in a given year. This can of course also be done by grade. PSEn_5 - Public school enrollment five years prior to the year which is being estimated. The interval could of course be greater or less than five, depending upon how far into the future you are interested in projecting, as well as the nature of the available statistics. API - The average percent increase in public school enrollment over a previous five year period. Third, this method relies quite heavily upon the averaging of percentages, a procedure that makes statisticians shudder. At least one report on the use of this approach demonstrates an awareness of 32 this problem. Liu declares that ratios may be averaged over a number of years if no clear trend is discernible. However, if a trend is discernible, Liu suggests that it be taken into definite consideration when projecting enrollments. Unfortunately, no standardized method of accomplishing this has been reported in the literature.37 Finally, the projections which the procedure yields are often inaccurate. Thomas C. Holy stated back in 1947 that the estimates were consistently somewhat high.38 No new developments since that time would lead this reviewer to believe that the formula has become any more accurate. Other weaknesses of this method will be discussed in the section that discusses common weaknesses of the various methods currently popular in education.39 Another method that has seen wide use in making educational enrollment projections is the so-called "enrollment-ratio" method. This method assumes that an observable ratio between school enroll- ment and school-age population or total population of some defined area has existed in the past and will continue to hold good for the future. The essential steps to the procedure are as follows: 1. Compute the average of the percents which total school enrollments were of total population, total school enroll- ments were of school-age population, or, grade enrollments were of grade-age population, for a relevant45eographic area for each of a number of previous years. 2. Apply that average to the future estimated total population of that relevant geographic area. 33 The formula can be expressed as follows: EEn - PSEn_10 + PSEn_9 + PSEn-8 + PSEn_7 + PSEn-6 TPn-lO TPn-9 TPn-8 TPn_7 TPn-6 where: EE - Estimated enrollment in a given year. PSE 8 Public school enrollment in a given year. In the example enrollment is being projected five years into the future. TP = The total population of a relevant geographic area in a given year N a The number of ratios involved in obtaining the average. In the example, this is five. EP - The estimated total population of a relevant geographic area in Ale year to which enrollment is being projected. It is obvious from the above presentation that projection procedures of this variety require a reliable estimation of the future population of relevant geographic areas. In Michigan, estimates of this type are made for cities, townships, and counties, but not for school districts. The three most common methods for projecting the papulation of small areas are the already discussed "cohort- survival" method, a simpler type of component method in which separate allowance is made for births, deaths, and migration without taking age into account, and the so—called "ratio" method.42 Other methods include graphic and mathematical extrapolations.43 Since the "ratio" method is going to be used quite often in this study in order to translate available data into useable form for some 34 of the formulae to be tested, there follows a discussion of it. Briefly, the "ratio" method derives population projections of the future population of an area on the basis of a series of observed historical ratios between the population of that area and the population of a larger area for which projections are already available. Such a procedure was used by the Philadelphia Planning Commission to project the pOpulation of the Philadelphia-Camden Industrial Area to the year 2000. In this case the projection was carried through the following four stages, using the already avail- able total United States projections as a starting point. 1. United States urban population projections were obtained by extrapolating the historical ratios of the U.S. urban to the U.S. total population and multiplying the extrapolated ratio by the total population projections of the U.S. 2. The Northeastern Industrial Region's urban population was then projected by extrapolating ratios of its urban population to the U.S. urban population, and applying them to the U.S. urban population projections. 3. The Philadelphia-Camden Industrial Area's urban population was next projected by extrapolating from ratios of its urban population to the urban pOpulation of the Northeastern Industrial Region, and applying them to the projected urban population of this region. 4. Projections for the total population of the Philadelphia- Camden area were obtained by extrapolating ratios of its total population to its urban population and applying them to the projected urban pOpulation of the Philadelphia-Camden Area. 4 When attempting to estimate district enrollment, the "enrollment- ratio" method encounters many of the same difficulties considered when the "cohort-survival" method was presented. There is evidence that suggests that the "cohort-survival" method yields better projections across the board than does the "enrollment-ratio" method.45 35 A relatively recent study conducted in Detroit developed a technique for projecting future enrollment based on both the "cohort- 46 survival" and the "enrollment-ratio" techniques. Different assump- tions were used to estimate future births in different types of housing areas. Elementary service areas, that is, areas from which children may attend given elementary schools, were established as the base unit for estimating future school enrollments. The number of estimated births, year by year, for each service area, in conjunction with the age distribution of the population of that service area for the latest census period, permit the determination of future age distributions. These future age distributions are then used to estimate the future school population of each service area. The results of this procedure were sufficiently good that it is currently the procedure used for projecting future school enrollment in the Detroit Schools. A third approach to the projection of school enrollments is based on the projection of dwelling units and is called the "pupil- home unit ratio" method. According to the literature, this approach is most feasible when one is projecting future school enrollment for rapidly expanding communities which are in their early period of growth and which still have relatively large parcels of unused land available for residential development.47 This approach usually involves a local survey which collects information concerning such variables as the composition of households living in various types of dwelling places at the present time; the type of housing recently 36 constructed in the area; the types of families moving into the new housing, including the number of persons per family and the number of elementary and high school students per family; the amount of land available for residential construction and the type of construction likely to be undertaken on this land; and any other information that seems relevant to the future development of the particular area.48 The information obtained from the local survey should provide a basis for: 1. Estimates of the rate of land utilization and the rate of construction of dwelling units, which can then be converted into projections of numbers of dwelling units on specified future dates. 2. Estimates of population per dwelling unit and pupil-population ratios which, when applied to the projected number of dwelling units will give the projections of total population and school enrollment. The major disadvantage of this method is the large amount of data which is necessary in order to successfully utilize it. The collection of this data involves the carrying through of a local survey for each community in which one is interested in projecting enrollment.50 A recent study examined the relationships between four characteristics of new homes and pupil yield. The four variables considered were: number of rooms, zoning ordinances, building permit valuation, and location of the home. From the cross tabulating of pupil yield with each of the four variables at each level of school, indices were derived based upon the mean yield of each category. The indices were tested by applying them to two areas of Jefferson County, 37 Colorado, where large building developments were recently completed and children were attending the Jefferson County Public Schools. The projected enrollments were compared with actual enrollments and also with enrollment projections made by the school district through the use of traditional ratios. It was found that the indices derived from the study were able to project the number and grade distribution of pupils from newly occupied homes more accurately than those methods used by the school system.51 A number of other projection methods have been tried in public education, although apparently without a great deal of success. One such method averages the percentage of population growth in the past and applies this average to the future, using time as the independent variable. The method involves plotting a curve showing how the percentage increase changed in the past and extending that curve to show future percentage increases. Experience has shown that this method does not yield very reliable projections.52 Another approach consists of studying the growth of enrollment in similar school districts from a period at which many of their demographic variables were similar to those of the district of interest, and deducing their probable future enrollment from these studies. This method, forecasting by analogy, has not proven very reliable.53 Perhaps the major disadvantage of the methods currently in use for projecting public school enrollment is the assumption inherent in their methodology that the future represents a com- posite or average description of the past. In cases of ever 38 increasing, or decreasing, total and school-age population and enroll- ment, this assumption could be invalid. The techniques to be dis- cussed shortly assume a continued linear increase in the criterion variable, perhaps a more valid assumption when dealing with problems of enrollment projection. Prpjection Methodology Used In Higher Education Researchers working in higher education have used many of the techniques that have been used in the projection of public school 54 enrollments. Writing in 1960 in a methodology handbook on the subject of enrollment projections for colleges and universities, Lins identified five methods:55 (1) curve fitting methods (2) ratio methods (3) cohort survival methods (4) combined ratio and cohort survival methods (5) correlational analysis methods. All of these methods, with the exception of curve fitting and correlational analysis, as well as practically every method des- cribed in the literature,are ratio methods. However, some re- searchers in higher education have experimented with regression analysis as a projective tool. The most comprehensive research in this area has been carried out by Savage and Brown at the University of Minnesota.56 These authors concerned themselves with two types of projection problems which are relevant to this dissertation; the projection of numbers of high school graduates in a given year, and the projection of university attendance.5 39 Several methods of projection of numbers of public high school graduates for the state of Minnesota are presented. The data utilized are compiled from births and from children's tendencies to advance from one grade to the next. The projection models consist of combinations of linear regressions of various ratios on time. The authors believe that the compound use of linear regressions in a pseudo-cohort method is a new approach to the projection of future numbers of high school graduates.58 Method I involves the estimation of numbers of public high school graduates from the ratio of number of graduates to births eighteen years earlier. A straight line is fitted to the data by the least squares method. Using this straight line and appropriate birth rates, the numbers of high school graduates can be projected eighteen years beyond the latest data available on births. An added advantage of this approach is that confidence intervals can easily be computed for the estimates. These limits are based on the assumptions that the live-birth ratio is linearly related to time over the relevant time interval, and that deviations from linearity over this interval are unbiased, independent, and normally and identically distributed variables.59 This method can also be used to project enrollment by grade through the examination of past enrollments in each grade relative to numbers of births for the appropriate earlier years. A.more sensitive and comprehensive description of changing school enrollment patterns was attained through the use of infor- mation on the changing tendencies of children to advance from one 40 grade to the next. Transition ratios were computed for each of the twelve grades and the graduating class over a period of past years. Straight lines were then fitted to the data by the least squares procedure, and extrapolated to yield future estimates. These estimates of retention ratios were then applied to the number of births of "1+6" years previously, where "i" denotes the particular grade in which one is interested in projecting the enrollment. Confidence limits were computed for the ratios, but not for the actual enrollment estimates. The problem of computing confidence limits for the enrollment estimates is a complex one, since each estimate is based on the product of thirteen dependent estimates by linear regressions. Unfortunately no attempt was made by the authors to compare the results obtained through the use of the methods described above with actual enrollment statistics. Method II, the one using the transition ratios, consistently projected about 3.3% (with standard error of t0.4%) higher than Method I, and both methods gave estimates consistently above estimates made by the Minnesota Department of Education.61 This is probably due to the assumption inherent in the regression methodology of a continued linear increase in the criterion variable rather than the customary assumption that the future resembles a composite of the past. Savage and Brown also adapted regression procedures to projecting University attendance. They used a series of regression equations which range from simple extrapolation of attendance over time to a 41 multivariate approach utilizing lags in numbers of high school 62 graduates and net changes in military personnel. As one might suspect, the multivariate procedures produced better estimates when compared with actual enrollment figures.63 In a study of enrollment projection methodology for Texas public junior colleges, multiple linear regression was tried.64 Besides the experimental use of regression, standard ratio methods of projection, including cohort-survival methods, census class projections, and total population projections were explored. It was concluded that the ratio methods vary widely in their usefulness from community to community, make very limited use of the knowledge of relationships between enrollment and community characteristics, and require assumptions that are difficult to justify except on the basis of extensive local experience. It was also concluded that multiple linear regression, as used in the study, is best justified, and implemented, at the state level.65 Linear regression, as utilized by the authors mentioned in the preceding review of literature, and as will be used in this study, is a non-analytic tool in the sense that there is no underlying theory of human behavior which suggests such linear functions. The rationale is based on the principle that, next to a static pattern of behavior, linear changes represent the simplest case. So long as dramatic changes in educational and population patterns are not anticipated, linear approximations should yield good short-term estimates of future enrollment.66 42 Footnotes for Chgpter II 13The basis for much of the discussion presented in the next few pages can be found in the Handbook of Statistical Methods For Demggpaphers by A.J. Jaffe, Demographer, Office of the Coordinator of International Statistics, (Washington: U.S. Bureau of the Census), United States Government Printing Office: 1951, 278p. 14The approach suggested here is through the use of the equation: P - A + X (2:5) where P is the papulation to be cal- N culated, A is the pOpulation enumerated at an earlier census, B is the population enumerated at a later census, X is the number of years beyond the earlier census for which the population is being calculated, and N is the number of years between censuses. Apparently, no attempt has been made to apply the least squares model to the projection problem, or to use any independent variable other than time. 15Jaffe, Handbook, 9p.cit., p.212. 16Ibid., p.213. Occasionally in the discussion the interval "ten years" is referred to. Government demographers generally work with data that is collected once every ten years, the interval between Federal Censuses. 17Jaffe, Handbook, op.cit., p.213. It seems that this objection could be gotten around simply by putting an upper bound on a poly- nomial or exponential function. For a good discussion of the logistic curve see: W.A. Spurr and D.R. Arnold, "A Short-Cut Method Of Fitting a Logistic Curve," Journal of the American Statistical Association, (March, 1948), pp. 127-34. v.43. 18The Handbook was published in 1951. Publications of the Census Bureau up through 1968 would not lead one to believe that this position has changed. 19Huan D. Kuang, "Forecasting Future Enrollments by the Curve Fitting Techniques," Journal of Experimental Education, XXIII (1955), p.273. onhe U.S. Census Bureau is making population projections for large georgraphic and demographic classifications that are specified by relatively complete and accurate data. These projections are often made far into the future. Mathematical curves are perhaps not as applicable to this type of problem. However, for moderately short estimates, say 5 to 15 years into the future, with relatively imprecise data and a relatively homogenous population, as is the situation when working at the local level, some varieties of mathematical curves might prove quite valuable for projection purposes. 43 21 U.S. Bureau of the Census, Current Pppulation Reports, Series P-25, No.388, "Summary of Demographic Projections," (Washington: U.S. Government Printing Office, 1968), p.25. 22U.S. Bureau of the Census, Current Population Reports, Series P-25, No.381, "Projections of the Population of the United States by Age, Sex, and Color, to 1990, with Extensions of Popu- lation by Age and Sex to 2015," (Washington: U.S. Government Printing Office, 1967), p.45. 23See Current Population Reports, Series P-25, No.388, OECCite , Pp.2-10. 24The presentation here is only designed to give the reader some insights into the methodology of population projection. Different assumptions are made for white and non-white populations, different age cohorts, etc. This is why assumptions are referred to as sets of assumptions in the text of this paper. For a detailed presentation of the methods and assumptions see: Jacob S. Siegel, Current Population Reports, Series P-25, No.381, op.cit., Francisco Bayo, Social Security Administration, Office of the Actuary, Actuarial Study No.62, "United States Population Projections for OASDHI Cost Estimates," 1966. When projecting populations for states, a more intensive consideration of migration is needed. See Current Population Reperts, Series P-25, No.375, "Revised Projections of the Population of States: 1970 to 1985," 1967. 25Projections of future enrollment as well as a brief dis- cussion of the projection methodology can be found in Current Pppulation Reports, Series P-25, No.365, "Revised Projections of School and College Enrollment in the United States to 1985," Jacob S. Siegel, (Washington: U.S. Government Printing Office), 1967. 26Michigan Population - 1960 to 1980. The Population Studies Center at the University of Michigan, in cooperation with the State Resource Planning Division, Michigan Office of Economic Expansion, Michigan Department of Commerce. (Working Paper No.1, January, 1966). David Goldberg and Allan Schnaiberg were the major con- tributors. 27Estimates are prepared using a ratio correlation method based on data collected during the 1950-1960 period. 28Michigan Office of Economic Expansion, Working Paper No.1, op.cit., pp.9—l4. 29Michigan Department of Commerce, Preliminaperopulation Prejections for Small Areas in Michigan, State Resource Planning Program, Office of Economic Expansion, (Working Paper No.9, November, 1966). ’ 44 30Projections of total population by counties presented in Michigan Population - 1960 to 1980, and an unpublished extension of these projections to 2000, were used as control figures. The sum of the projected populations for all local units in each county are adjusted to equal the projected county totals for each date. 31Where: P=projected population; a=the y intercept of the regression line; b-the regression coefficient; X=the independent variable; B-the year of the later census enumeration; A=the year of the earliest census enumeration; N-the interval between censuses, in this case 10. The same type of regression line was fitted to each local unit. Incorporating the log of time in the equation has the effect of reducing an indicated past rate of increase or decrease in each local unit's share of its parent county population. 32In the few cases where this constraint applied, the projected shares of other local units in the county are below the values which would have occurred without the constraint. ' 33Michigan Office of Economic Expansion, Working Paper No.9, 02.0115. , pp03‘60 34Ibid., p.7. 35The information for the preceding two pages was collected from a number of sources, too numerous to cite here. However, some of the more important sources were James D. MacConnell, Planning for School Buildinge, Prentice-Hall, Englewood Cliffs, N.J.: 1957, pp. 32-38; Alfred Liu Bangnee, EstimatingyFuture School Enrollment in Developing Countries-A Manual of Methodology, UNESCO, Population Studies, ST/SOA/Series A, No. 40, 1966; Harold J. Bowers, "Projecting School Enrollments for One State," Journal of Teacher Education, Vol. V, No.1, pp.64-66. Harold J. Boles, Step to Step To Better School Facilities, New York: Hold, Rinehart and Winston, 1965, pp.40-48; Hauser and Kitagawa, op.cit., pp.64-65. 36Waldfogel, op.cit., p.4. 37Liu Bangnee, op.cit., p.19. 38Thomas C. Holy, "What Future Needs Are Revealed by School Population Studies," Education Digest, May, 1947, V.9: pp.24-26. Even so there is some evidence to suggest that, of the various common techniques for projecting school enrollment, this method is the most reliable except in the case when a major change in the rate of increase or decrease of school enrollment occurs. Donald Lee Peterson, An Investigetion of Techniques for Predicting School District Enrollments in Florida, Ed.D. Dissertation, University of Florida, 1959. 45 39This is not to imply that there is nothing good about this projection method. It has generally proven adequate for estimating enrollments at a county or state level, where relevant data are available. 40The particular form in which school enrollment rates are computed depends upon the detail of the enrollment projections desired and the amount of age detail available in the population projections to which the enrollment rates are to be applied. For example, if enrollment projections for single grades are desired, grade-specific enrollment rates for each year of age should be computed and multiplied by the projected population in each year of age. So detailed a projection is probably only warranted when the projection is for a short span of years, no more than five years, perhaps, and if reasonably reliable pro- jections by single years of age can be obtained. When enrollment projections are desired for longer periods of time, or when population projections are not available by single years of age, projections of total elementary and high school enrollments may be obtained in the following manner. (1) compute an enrollment rate expressing the ratio of elementary enrollment in grades 1-8 to the number of persons 6-13 years of age and one of high school enrollment to the number of persons 14-17 years of age, from statistics for the last census. (2) multiply the elementary en- rollment rate by the estimated population 6-13 years of age on the projection date, and multiply the adjusted high school enrollment rate by the estimated population 14-17 years of age on the projection date. The resulting products are the estimated elementary and high school enrollments on the projection date. allnformation collected from a number of sources cited in the Bibliography. This approach is suggested in; American Association of School Administrators, 27th Yearbook, American School Buildings, School Planning Lab, Stanford University, pp.55. 42As well as some experimental methods that have already been discussed. 43For a discussion of the various methods and a citation of relevant source material see; Jacob S. Siegel, "Forecasting the Population of Small Areas," Land Economics, February, 1953, pp.72-87. In most cases population projections are obtained through the use of the "cohort-survival" method as previously described. 44Philadelphia Planning Commission, Pppulation Estimates 1950-2000,_Philadelphia-Camden Area, (April, 1948). 45 Peterson, op.cit. 46 46 Norman S. Wheeler, Techniques Used For Predicting Public School Enrollments in Detroit; Michigan, Using the Elementary School Service Area as a Base, Ed.D. Dissertation, Wayne State University, 1955. 47Abe Gottlieb, "A Planning Approach To School Enrollment Forecasts," American School Board Journal, February, 1954, p.110. See also Hauser and Kitagawa, op.cit., p.69. 48Hauser and Kitagawa, op.cit., p.69. 491a the second step, estimates of pupils per dwelling unit may be made directly and applied to the projections of dwelling units without the intermediate step of estimating total population. 50Despite the effort that must be extended in this approach, there is little evidence to suggest that it yields accurate results. It is generally felt that it yields the best results in "exploding communities." Hauser and Kitagawa, op.cit., p.69. 51William N. Driscoll, The Prediction of School Enrollment From New Home Developnents, (Research Study No.1), Ed.D. Dissertation, Colorado State University, 1965. SZMacConnell, op.cit., pp.32-38. 53Larson and Strevell, op.cit., pp.65-68. 54L.J. Linis, Methodology_of Enrollment Predictions for Colleges and Universities, American AssociatiOn of Collegiate Registrars and Admissions Officers, March, 1960. C.F. Schmid and F.J. Shanley, "Techniques of Forecasting University Enrollment," Journal of Higher Education, Vol. XXIII, No.9, 1952, pp.483-489. 55L.J. Lins, Methodolggy of Enrollment Prpjections for Colleges and Universities, American Association of Collegiate Registrars and Admissions Officers, March, 1960. 56Bryon W. Brown and I. Richard Savage, Methodological Studies in Educational Attendance Prediction, University of Minnesota, Dept. of Statistics, Sept., 1960. 57The methodological aspects of university attendance projection are of interest. 58 Brown and Savage, op.cit., p.11. 59Brown and Savage, op.cit., p.12. 47 6oIbid., p.20. 61Brown and Savage, op.cit., p.19. GZIbide , pp.23-4lo 63Ibid., p.36. 64David G. Hunt, The Develppment of Enrollment Prpjection and Prediction Methods for Texas Public Junior Colleges, The University of Texas, Ph.D. Dissertation, 1962. 65This is strictly an empirical conclusion based on data collected from Texas Public Junior Colleges and cannot be inter- preted as a condemnation of the use of multiple linear regression for projecting school district enrollment at a local level. 66Simple linear models are not appropriate for long-range projections. For example, if the birth-ratio method were pushed indefinitely into the future, more graduates from high school would be projected than babies were born eighteen years previously. Brown and Savage, op.cit., p.21. CHAPTER III METHODOLOGY Population and Sample The p0pu1ation identified in this study consists of every public school district in Michigan, the major part of which is coterminous with a city of 10,000 population or more. (According to the 1960 Census.) Building on the classification scheme proposed 67 the population has been divided into by Hauser and Kitagawa, five strata on the basis of: 1. The percent of increase of the general p0pu1ation, during the decade 1950-1960, of the urban place with which the majority of the district is coterminous. 2. The percent of increase of the number of households, during the decade 1950-1960, of the urban place with which the majority of the district is coterminous. A sample of five school districts was randomly drawn from within each of the five strata, yielding a total of twenty-five school districts. Each school district was treated as a separate case study. Definition of Terms 1. Urban Place The term "place" as used in reports of the decennial census refers to a concentration of population regardless of the 48 49 existence of legally prescribed limits, powers, or functions. Most of the places listed in the Census Reports are incor- porated as cities, towns, villages, or buroughs, however, and all of the communities in which the school districts that make up the population for this study are located are incorporated as cities. 2. City The term "city" as used in this study refers to urban places of 10,000 population or more that are incorporated as cities. 3. County The term "county" refers to the primary political divisions of the State. 4. Households Changes in definitions can have important implications for studies of population groups. In the 1950 Census, "households" did not include persons living in institutional dwelling units such as hospitals, college dormitories, or military posts. In the 1960 Census, persons living in such units were counted as households. Classification of Districts into Strata School districts were classified into strata on the basis of certain decision rules involving the population growth variables of the urban place with which the major portion of each district is coterminous. A discussion of those decision rules by stratum follows. 50 Stratum A School districts classified into this stratum are located in communities that increased by more than 100%, during the decade 1950-1960, in both general population and in the number of households. Communities that have experienced such rapid growth during the relatively brief period of ten years will be called "Exploding Communities." TABLE 1 SAMPLE DISTRICTS DRAWN FROM EXPLODING COMMUNITIES (STRATUM A) % of Pop. Increase % of Household Increase District 1950-1960 1950-1960 Allen Park 200.5 201.7 Garden City 321.8 291.1 Inkster 133.7 136.8 Livonia 280.4 252.1 Oak Park 595.5 569.3 Stratum B School districts classified in this stratum are located in communities that increased by 50% or more in either or both of the classification variables, but 100% or less in one of the classification variables, during the decade 1950-1960. Communities that can be classified into this category will be called "Rapidly Growing Communities." 51 TABLE 2 SAMPLE DISTRICTS DRAWN FROM RAPIDLY GROWING COMMUNITIES (STRATUM B) % of Pop. Increase % of Household Increase District 1950-1960 1950-1960 Ann Arbor 39.6 71.4 Birmingham 65.0 63.7 Lincoln Park 84.0 79.7 Midland 94.5 86.3 Royal oak 71.9 71.7 Stratum C School districts classified in this stratum are located in communities that have experienced an increase of 10% or more, but less than 50%, in both the number of inhabitants and the number of households during the decade 1950-1960. This type of community will be called an "Older But Still Growing Community." TABLE 3 SAMPLE DISTRICTS DRAWN FROM OLDER BUT STILL GROWING COMMUNITIES (STRATUM C) % of Pop. Increase % of Household Increase District 1950-1960 1950-1960 Alpena 11.8 11.8 Dearborn 17.9 - 27.0 Kalamazoo 42.3 40.5 Pontiac 11.6 15.8 Wyandotte 13. 1 20 .7 52 Statum D School Districts classified in this stratum are located in communities that have experienced an increase of less than 10% but equal to or greater than 0% in both the number of households and the number of inhabitants, or, an increase of less than 10% but equal to or greater than 0% in one of the classification variables, and an increase of 10% or more in the other, during the decade 1950-1960. Communities of this type will be referred to as "Stable, Well-Established Communities." TABLE 4 SAMPLE DISTRICTS DRAWN FROM STABLE WELL- ESTABLISHED COMMUNITIES (STRATUM D) % of Pop. Increase % of Household Increase District 1950-1960 1950-1960 Bay City 2.1 4.5 Grand Rapids 0.5 2.6 Ferndale 5.6 13.7 Monroe 7.0 10.2 Saginaw 5.8 7.0 Stratum E School districts classified in this stratum are located in communities that have experienced a decrease in either or both general population and the number of households during the decade 1950-1960. This type of community will be called a "Declining Community." 53 TABLE 5 SAMPLE DISTRICTS DRAWN FROM DECLINING COMMUNITIES (STRATUM E) % of Pop. Increase % of Household Increase District 1950-1960 1950-1960 Battle Creek -9.2 -3.1 Hamtramck -21.3 -9.8 Highland Park -18.0 -4.2 Ironwood -10.5 -l.6 Muskegon -4.0 -0.4 At another level of sampling, six data points were obtained on each of the predictor variables used in the study. The major reason for the small number of data points was simply the lack of availability of adequate data. The small number of data points severely limits the use of multiple linear regression, placing an upper limit of four on the number of independent or predictor variables that can justifiably be used at any given time. Data Probably the most thorough statistical description of factors related to the projection of enrollment, although admittedly junior college enrollment, was compiled by Rodgers.68 Although most of the factors that he investigated were intended merely for descriptive purposes, he did eventually select ten factors which his data suggested would form the best basis for enrollment projections. Those ten factors were: 54 1. Total population of scholastic age. 2. Public school enrollment. 3. Average daily attendance relative to school-age p0pu1ation. 4. High school enrollment. 5. High school average daily attendance. 6. High school graduates. 7. High school seniors planning to attend college. 8. College Age youth in the general population. 9. Total p0pu1ation in census years. 10. Estimated total population. Rodgers also described the relationship between enrollment and such other factors as: l. Papulation per square mile. 2. Number of farm owners. 3. Number of farm tenants. 4. Automobile registrations. 5. Poll taxes paid. 6. Number of employed persons. 7. Wages paid. 8. Income. 9. Retail sales. 10. Wholesales. 11. Value of manufactures. 12. Bank deposits. 13. County and State tax values. 55 These factors were related to enrollment by the means of ratios, however, it is evident from the data that these factors were not as highly related to enrollment and therefore would not provide as reliable an estimate as the aforementioned ten factors that were eventually used. Keeping both Rodgers' study and traditional projection methods in mind, the following independent or predictor variables were chosen as the most appropriate for use in this study. 1. Past District elementary enrollment. Past District secondary enrollment. Elementary school-age births. Secondary school-age births. Total county population. Total city population. City elementary school-age population. City secondary school-age population. Extrapolated city elementary school-age population. Extrapolated city secondary school-age population. Extrapolated county elementary school-age population. Extrapolated county secondary school-age p0pu1ation. City occupied housing. Mean number of persons per occupied unit. State Equalized Valuation of Districts. Annexations. 56 Data Sources and Projection Methodolpgy Much of the data that is used to test the various projection formulae which are dealt with in this study was extrapolated or interpolated from the little data that was available in raw form. Therefore, a major assumption of this Study must be that the methods that are utilized to obtain much of the data are valid means of doing so. There follows a discussion of the data sources and of the methods used to obtain estimates of those values that were not directly available. Much thought was given to the development and utilization of appropriate methodology for obtaining estimates of important predictor variables since the researcher is well aware that the outcome of the study is dependent in part upon the accuracy of projections of values for those predictor variables whose values were not directly available. 1. Elementary Enrollment. Elementary enrollment is defined as those full time, non- special education, students enrolled in grades K through 8 of the various school districts that comprise the sample. The K through 8 breakdown was used in order to make it possible to utilize United States Census data on the school-age population of the various cities of the sample. Non-public school children were not considered because of the extreme difficulty in obtaining data on them. The source used for elementary enrollment figures was the State Department of Education County Superintendent Summapy_Reports, State Financial and Enrollment Statistics, which records enrollment by grade. 57 2. Secondarijnrollment. Secondary enrollment is defined as those full time students in non-special education classes attending grades 9 through 12. Non-public school children were not considered. The source used for figures on secondary enrollment was the State Department of Education County Superintendent Summepy Reports, State Financial and Enrollment Statistics, which records enrollment by grade. 3. Births By City By Year. Statistics on births by city by year were obtained from the Vital Statistics of the United States, Part II,yPlace of Residence. This source was considered the most accurate for purposes of this study since births are recorded by place of residence rather than merely by place of birth. Births were recorded for each district from 1938 to 1960, and extrapolated beyond 1960 to 1963 through a linear extrapolation. 4. Births By Citngy Period. Statistics on births by city by period were computed for each district in the sample on the basis of data obtained from Vital Statistics of the United States, Part II,_Place of Residence. For example, children that were in elementary school in 1955 were born between 1942 and 1950. Therefore, by adding up the number of births in a particular city between 1942 and 1950, the total number of children born in that city during the Years that contribute to elementary enrollment in 1955 is obtained. In a like manner the number of children born between 1938 and 1942 58 should be related to secondary enrollment in 1955. Linear extrapolations beyond 1960 to 1963 were used as estimates of the number of births in the years 1961 through 1963. 5. Counpy POpulation. Population figures for Michigan Counties were obtained for the years 1950 and 1960 from reports of the United States Census Bureau. However, only estimates were available for the years 1955 through 1959, and 1965 and 1968. The county population estimates used were prepared by the University of Michigan Population Studies Center and are based on the assumptions outlined in Appendix A. These estimates were chosen because they are the most widely used estimates, as well as being the estimates utilized by the State of Michigan in projecting future population figures. 6. Citpropulation. Population figures for Michigan cities were obtained from the Center for Health Statistics, Michigan Department of Public Health. Statistics for the years 1950 and 1960 are the same as can be obtained from the U.S. Census Bureau. Estimates of city population between the years 1955 and 1960 were obtained through the use of a ratio correlation procedure which utilizes data on births, auto registration, sales tax, voter registration, and school enrollment, (See Appendix B.) Michigan Department of Health estimates were chosen because they are the most widely used estimates of city population. 59 Estimating the city population for 1965 and 1968 required a complicated process which involved the averaging of two separate estimates. The first estimate was obtained through a linear extrapolation of the ratios of city population to county population and the subsequent application of the extrapolated ratios to county population estimates for 1965 and 1968. The second estimate was obtained through averaging the ratios and applying the results to county population estimates. A subsequent averaging of the two estimates provided the needed projection of city population.69 7. Citnylementary School-Age Population. Data on city elementary school-age population was obtained for the years 1950 and 1960 from the Reports of the United States Census Bureau. Ratios of school-age to over-all population by city were established for 1950 and 1960. Linear interpolations and extrapolations were then completed in order to obtain estimates of the ratio of school-age population to total city population for each year. The resulting ratios were then applied to the estimates of total city p0pu1ation for relevant years in order to obtain estimates of city school-age population. 8. City Secondary School-Age ngulation. Estimates of city secondary school-age population were obtained in the same manner as were the estimates of city elementary school- age p0pu1ation. 9. City Extrapglated Elementapy School-Agengpulation. This represents a special type of estimate for use in one of of the regression approaches. Data on city elementary school-age 60 population was obtained for the years 1950 and 1960 from the Reports of the United States Census Bureau. Using these figures as a base, a straight line was applied to the data and extrapolated to 1965 and 1968. 10. City_Extrepolated Secondary School-Age Population. Same procedure as in "9" with data on secondary school-age population. 11. County Extrapolated Elementary School-Age Pqpulation. "9" with data at the county level. Same procedure as in 12. County Extrapolated Secondary School- e ngulation. Same procedure as in "Kr'with data at the county level. 13. Occupied Housingygnits. Data on occupied housing units per city for the years 1950 and 1960, as well as the mean number of persons per occupied unit, was obtained from the U.S. Census Bureau. Data on occupied housing units was linearly interpolated and extrapolated to arrive at estimates for relevant dates. Another estimate was obtained through the application of the mean number of persons per occupied unit in a given city to the estimated number of peOple in that city. The two estimates were then averaged. 14. State Equalized Valuation. Data on State Equalized Valuation was obtained from the Michigan State Department of Education State Aid Sheets. The data were linearly extrapolated to 1965 and 1968. 61 15. Annexations. Although districts were chosen from a p0pu1ation of districts whose boundaries were largely coterminous with city boundaries, there were some exceptions. Data on annexations between 1950 and 1968 was therefore required. This information was obtained from the State Department of Education. The following table presents a summary of the previous discussion. TABLE 6 FACTORS SELECTED FOR STUDY AND THE SOURCES OF DATA DESCRIPTIVE OF EACH Data Yeara Sourceb 1. Elementary 1950,1955-1960, Michigan State Department enrollment 1965,1968 of Education. 2. Secondary 1950,1955-1960, Michigan State Department enrollment 1965,1968 of Education- 3. Births by City 1938-1960 Vital Statistics of the United States, Part II, Place of Residence. 1961-1963 Linear extrapolation of pre-1960 data. 4. Births by City by 1938-1963 Births per year added Period over relevant periods. Utilize data on Births by City. 5. County Population 1950,1960 U.S. Census Bureau. 1955-1959,1965, The University of Michigan 1968 Population Studies Center, 62 TABLE 6 (cont'd.) Data Year8 6. City Population 1950,1960 1955-1959 1965,1968 7. School-Age 1950,1960 Population 1955-1959 1965,1968 8. Extrapolated City 1950,1960 and County School- Age Population 1955-1959 Sourceb U.S. Census Bureau. Michigan Department of Public Health. Linear extrapolation of the ratios of city population to county population and the sub- sequent application of the resulting ratios to county p0pu1ation esti— mates for 1965 and 1968. In addition, the averaging of the same ratios and subsequent application to estimates of county population. Results are then averaged. U.S. Census Bureau, Linear interpolation of ratios of school-age population to total city population, and their subsequent application to estimates of city. population Linear extrapolation of ratios of school-age population to total city population and their subsequent application to estimates of city p0pu1ation, U.S. Census Bureau. Linear interpolation of 1950 and 1960 school-age population figures. 63 TABLE 6 (cont'd.) Data Yeara 1965,1968 9. Occupied Housing 1950,1960 Units 1955-1959 1965,1968 10. State Equalized Valuation 1950,1955-1960 1965,1968 ll. Annexations 1950-1968 Sourceb Linear extrapolation of 1950 and 1960 school- age population figures. U.S. Census Bureau. Linear interpolation of 1950 and 1960 occupied housing figures and mean number of persons per occupied unit figures, application of the latter to population figures, and the subsequent averaging of the two estimates. Linear extrapolation of 1950 and 1960 occupied housing figures and mean number of persons per occupied unit figures, application of the latter to projected population figures, and the subsequent averaging of the two esti- mates. FMchigan State Department of Education. Linear extrapolation of pre-1960 figures. Michigan State Department of Education. ‘The symbol "-" is to be interpreted as the word "through". Thus 1955-1960 is to be read, "1955 through 1960". bA detailed citing of each source appears in the part of the Bibliography that deals with data sources. 64 Regression and Ratio Methods Examined This study examines the applicability of seventeen separate and distinct ratio and regression methodologies to the projection of future public school enrollment. The following criteria were used in choosing and developing projection methods: 1. The method should be uncomplicated enough so that it can be easily and quickly explained and simple enough so that computations can be made on a desk calculator or with standard electronic-computer programs. 2. There should be grounds for believing that the model will in fact approximate events. 3. The data necessary for making projections as well as for testing the model should be available As you recall, the distinction between ra n. d regression analysis methods is made on the basis of the measure of relationships used in the specific computation of projected enrollments. Ratio methods utilize the ratio or prOportion of a given predictor to enrollment, while regression methods use coefficients of correlation and multiple correlation, statistics which purport to measure degree of association. Table 7 presents the seventeen projection methods that are considered in this study in terms of wehther each is a ratio or regression method, the predicor variable or variables used in each, and the relationship on which each projection is based. This Table should be read.while keeping the previous discussions of assumptions and data limitations in mind. 65 TABLE 7 PROJECTION METHODOLOGIES EXAMINED IN THIS STUDY -..:L. Name Type Cohort-Survival Ratio Grade-Retention Enrollment- Ratio Method I Enrollment- Ratio Method II Pupil-Home Unit Time Analysis I Time Analysis II Transition Analysis Ratio Ratio Ratio Ratio Regression Regression Regression Predictor (e) 1. Births by City 2. Past District Enrollment Past District Enrollment Total City Population Total County Population Home Units (City) Time Time 1. Births by City 2. Past District Enrollment Relationship Past tendencies of students to advance from one grade to the next. Past trends in public school enrollment. Past ratio of public school enrollment to total city population. Past ratio of public school enrollment to total county population. Past ratio of public school enrollment to city occupied housing units. Past relationship between public school enrollment and time. Past relationship between the log of public school en- 'rollment and time. Past tendencies of students to advance from one grade to the next. (Also: from birth to kindergarten) TABLE 7 (cont'd.) Name 9. Birth Analysis 10. ll. 12. 13. 14. Enrollment Regression Analysis I Enrollment Regression Analysis II Enrollment Regression Analysis III Enrollment Regression Analysis IV Housing Analysis I123. Regression Regression Regression Regression Regression Regression Predictor(e) Births by City City School- Age Popula- tion 1. City School-Age Population 2. Total City Population 1. City School-Age P0pulation 2. Total County Population 1. Extra- polated City School-Age Papulation 2. Extra- polated County School-Age Population 1. Housing 2. State Equalized Valuation Relationship Past relationship between births and public school enrollment. Past relationship between city school- age p0pu1ation and public school en- rollment. Past relationship between city school- age population, city total population, and public school enroll- ment. Past relationship between city school- age population, county total population, and public school enroll- ment. Past relationship be- tween extrapolated city school-age population, extra- polated county school- age population, and public school enrollment. Past relationship between occupied housing, SEV, and public school enrollment. TABLE 7 (cont'd.) Name 15. Multi-Variable Analysis I 16. Multi-Variable Analysis II 17. Multi-Variable Analysis III TIE Regression Regression Regression 67 Predictor(s) 1. Births by City 2. City School-Age Population 3. Total City Population 4. Total County Population Same variables as in 15 Logs of variables used in 15 Relationship Past relationship between the predictor variables and public school enrollments. Past relationship between the predictor variables and the logs of public school enrollment. Past relationship between the logs of the predictor variables and the logs of public school enrollment. Discussion It is one of the basic tenets of this study that most relation- ships can best be described by a linear approximation over a short period of time. This assumption is tested through the use of logarithmic transformations in three of the regression equations. These transformations provide varying approximations of positively accelerating curves. If these methods do in fact provide accurate PrOJections of future enrollment, it will suggest that there should be greater experimentation with non-linear projection Emiliodologies. 68 Design The seventeen ratio and regression analysis projection methods presented in the preceding section were used to develop projections of elementary and secondary enrollment for the twenty-five school districts in the sample for the years 1965 and 1968. These projections were based on data which were available during or prior to the year 1960. In this manner it was possible to use actual enrollment in the years 1965 and 1968 as the criterion against which to judge the relative performance of the various projection methods. As was previously mentioned, the sample school districts were divided into five strata on the basis of past community growth characteristics. Each of the seventeen projections methods was evaluated on the basis of its applicability to districts within each of the five stratum. The criterion of goodness of fit between estimated and actual district enrollment in 1965 and 1968 formed the basis for evaluation. Goodness of fit was determined in the following manner. Each of the twenty-five districts represented in the sample has sixty-eight separate estimates of different aspects of its enrollment. Table 8 presents a breakdown of those estimates. 69 TABLE 8 NUMBER OF ENROLLMENT PROJECTIONS FOR EACH DISTRICT IN THE SAMPLE 1965 1968 Level Ratio Regression Ratio Regression Elementary 5 12 5 12 Secondary 5 12 5 12 This means that for any given school district in any given stratum for any given method there are four separate estimates of enrollment; one for elementary enrollment in 1965, one for secondary enrollment in 1965, one for elementary enrollment in 1968, and one for secondary enrollment in 1968. Needless to say there are also four actual enrollment figures. Goodness of fit is determined by substracting the actual enrollment figure from the projected enrollment figure for each of the four aforementioned estimates, dividing each by its actual enrollment figure, and reporting each for its respective estimate. The resulting statistic, a measure of the percentage of error in each estimate, is called the Webster Coefficient and is labeled "W" whenever it appears. The smaller the Webster Coefficient, the better the estimate. In other words, goodness of fit is determined by the Webster Coefficient (W). For any given school district within any given method there are four W Coefficients reported. 70 W1 - (Projected Elementary - Actual Elementary) (Enrollment1965 Enrollment1965 ) Actual Elementary Enrollment1965 W2 8 (Projected Secondary - Actual Secondary) (Enrollment1965 Enrollment1965 ) Actual Secondary Enrollment1965 W3 - (Projected Elementary -' Actual Elementary) (Enrollment1968 Enrollment1968 ) Actual Elementary Enrollment1963 W4 - (Projected Secondary - Actual Secondary) (Enrollment1968 Enrollment1968 Actual Secondary Enrollment1963 In addition, the Webster Coefficients are ranked within each district. As previously mentioned, each district has seventeen separate estimates of each of the four aforementioned enrollment figures. These seventeen separate estimates are ranked within each of the four sub-categories by district according to the size of the W Coefficient. The W Coefficients are then summed across the four sub-category estimates in order to obtain a composite W Coefficient for a given projection approach within a given district across the four estimates. The resulting composite W Coefficient is then ranked and is used as one index of the effectiveness of a given projection approach for a given district relative to other Projection approaches for that district. A second index of the effectiveness of a given projection a'Pproach is a frequency count by stratum of the number of projections 7l yielded by each methodology that are within a given degree of accuracy of the actual enrollment figures. These observations are grouped by elementary and secondary projections, but collapsed over years. Although the observations certainly are not independent of one another, they nevertheless serve as indicants of the degree of success that a given methodology enjoys in a given stratum. The two indicants discussed above are used, finally, to determine those projection formulae that are best suited for districts with growth characteristics similar to those described in each of the five strata that comprise the sample. Although at first glance it may appear as if the study utilizes a very small sample of districts, the reader is reminded that the entire popu- lation to which the study wishes to generalize consists of only some eighty Michigan school districts. Although there are many generalizations that may probably be extrapolated to other populations, the researcher would hesitate to recommend the unqualified application of the results of this study to states with growth patterns different from those of Michigan. 72 Footnotes for Chapter III 67Hauser and Kitagawa, op.cit., pp.69-70. 68Jack Rodgers, Criteria For the Establishment of Local Public Junior Colleges in Texas, unpublished Ph.D. Dissertation, University of Texas, June, 1956. 69This procedure was followed for every district in the sample except Livonia. In this one case it was obvious that the population estimates emanating from the method outlined in the text was com- pletely unrealistic. Therefore, only the estimates obtained through a linear extrapolation of the ratios of city population to county population and the subsequent application of the extrapolated ratios to county population estimates for 1965 and 1968 were used. CHAPTER IV RESULTS The results of the study are presented in seventeen separate sections, each section dealing with one of the seventeen projection methodologies examined in the study. In order to reorient the reader to the salient details of each of the methodologies, a short description of each is presented prior to the display of the results of the application of that methodology to the projection problem. The display of the results of each methodology is facilitated through the use of two tables. The major table presents the following data for each District. ( EE ) . Projected Elementary Enrollment, 1965.(1965) The W Coefficient for the 1965 elementary projection. (W) The rank of the 1965 elementary projection relative to projections yielded by the other sixteen projection methodologies examined in the study. (R) (SE) Projected Secondary Enrollment, 1965. (1965) The W Coefficient for the 1965 secondary projection. (W) . The rank of the 1965 secondary projection relative to projections yielded by the other sixteen projection methodologies examined in the study. (R) (EE) . Projected Elementary Enrollment, 1968. (1968) . The W Coefficient for the 1968 elementary projection. (W) The rank of the 1968 elementary projection relative to projections yielded by the other sixteen projection methodologies examined in the study. 73 74 ( SE ) 10. Projected Secondary Enrollment, 1968. (1968) 11. The W Coefficient for the 1968 secondary projection. (W) 12. The rank of the 1968 secondary projection relative to projections yielded by the other sixteen projection methodologies examined in the study. (R) 13. A rank of the composite W Coefficients (R) The W Coefficients are summed within districts and projection methodologies across the four separate enrollment esti- mates. The resulting composite W Coefficient is then ranked across projection methodologies in order to provide one basis for assessing the relative performance of the seventeen separate projection methodologies. The composite ranks range from 1 to 17. The Districts are organized by stratum in order to further facilitate comparison. Another way of looking at the data is to look at the number of projections yielded by each methodology that are within a given degree of accuracy of the actual enrollment figure. The second table that displays information about each of the methodologies presents the number of estimates yielded by each methodology that were within a W Coefficient of .1000 of actual enrollment figures. Besides presenting this information by stratum, the second table also presents a composite picture of the data, that is, the percentage of the time by stratum that all seventeen approaches examined in this study yielded projections that were within a W Coefficient of .1000 of actual enrollment figures. The information supplied by this table thus provides another basis for an assessment of the performance <>f a given methodology relative to the performance of the other 75 sixteen projection methodologies examined in this study. Pro- jections that appear in this table are collapsed over years while maintaining the elementary and secondary breakdown. Multiple correlation coefficients, regression coefficients, standard errors of regression coefficients, and F Statistics assessing the usefulness of the regression lines as predictors, are all available for each of the regression approaches. They are not reported because of a number of reasons: 1. The criterion by which the usefulness of a particular methodology is assessed is not the degree to which the regression line fits past data, but rather the accuracy of the projections yielded by that line as determined by the degree of discrepancy between those projections and actual data. That the regression line fit past data is a necessary but not sufficient requirement of the methodology. 2. Due to the nature of the projection problem and the rather unorthodox use made of the regression model in this study, the statistics mentioned above do not supply a great deal of useful information. This point will be discussed in greater detail shortly. 3. There are some eight-hundred and fifty separate and distinct regression equations used in this study, requiring some eight-hundred and fifty pages of data to describe them all. The above-mentioned data is of only limited usefulness because of the following considerations. As one might suspect, due to the nature of the data and the small number of data points, less than five percent of the correlation and multiple correlation coefficients computed for the eight-hundred and fifty regression equations used 111 this study are less than .9000. It is therefore rather obvious tirat most of the F Statistics assessing the usefulness of the various 76 regression lines as predictors are significant at p23 .05. This is not suprising since it is one of the basic tenets of this study that, over a relatively short period of time or small number of data points, any function of the type examined here can best be approximated by a straight line. The predictor variables,as was suspected, are all highly inter- correlated. This fact of course throws some doubt on the usefulness of multiple linear regression as opposed to simple linear regression as a projection technique. As the number of predictor variables in the regression equations are increased from one to four, the multiple correlations increase to the point where, with four predictor variables, there is not a single multiple correlation coefficient below .9800. However, the fact that the high multiple correlation coefficients are artifacts of the computational process in many instances is attested to by greatly increasing standard errors of regression coefficients as the number of predictor variables increase. There are cases when, with four predictor variables, the standard errors of the regression coefficients are larger than the regression coefficients themselves, suggesting the presence of a great deal of error. From an examination of the standard errors of the regression coefficients, one must conclude that, given the nature of the projection problem as outlined in this study, one or two carefully chosen predictor variables are sufficient. This of course may not be the case with an increased number of data points. However, it is relevant at this point to remind the reader of the difficulty of Plrocuring data. At the local level in Michigan, ten years is about tble most one can expect to obtain data for. 77 Table 9 presents the actual enrollment figures for each of the twenty-five districts that comprise the sample used in the study. These are the figures that provide the criterion against which the accuracy of the various methods is judged. Notice that the districts are grouped by stratum, as they will appear throughout the remainder of this manuscript. TABLE 9 ACTUAL ENROLLMENT OF SAMPLE DISTRICTS EE SE EE SE District 1965 1965 1968 1968 Allen Park 4629 2135 4677 2301 Garden City 10223 2981 10841 3124 Inkster 3652 1140 3687 1231 Livonia 22868 7019 25151 9936 Oak Park 5120 1956 4987 2110 Ann Arbor 11947 3881 14039 4746 Birmingham 11462 4557 11882 5633 Lincoln Park 9192 3965 9507 4063 Midland 8340 2970 8721 3396 Royal Oak 13883 5680 13305 6026 Alpena 5548 1882 5961 2256 Dearborn 13374 8395 13290 8998 Kalamazoo 13018 4750 13226 5084 Pontiac 16865 5529 17617 5798 Wyandotte 5797 2655 5722 2654 Bay City 8817 5202 9063 5452 Grand Rapids 23887 7814 24250 8774 Ferndale 5585 2463 5531 2528 Munroe 3688 2123 6065 2458 Saginaw 15526 6148 16079 6368 Battle Creek 8361 2958 8111 2952 Hamtramck 2375 1271 2207 1004 Highland Park 4258 1827 4140 2071 Ironwood 1930 748 1916 780 Muskegon 6397 2694 6125 2684 78 Cohort-Survival Ratio Method The Cohort-Survival Ratio approach was used in the form presented in the Review of the Literature on pages 28 through 30. This means that nine separate ratio equations are used to obtain each of the elementary projections and that an additional four ratio equations are used to obtain each of the projections of secondary enrollment. The mean number of births per year for the relevant period of time is applied to the ratio equations to obtain the projection. Table 10 presents the projections yielded by the Cohort-Survival Ratio Approach. The key to the symbols in this table and the other sixteen tables like it that appear in this chapter is on pages 73 and 74. Table 11 displays the number of projections yielded by the Cohort-Survival Ratio Method, within and across strata, that are within a W Coefficient of .1000 of the actual enrollment figure. Please observe the footnotes to this table that explain what the headings stand for, since there are sixteen more tables in this Chapter that are set up exactly like this one. Appendix B-l outlines in detail the procedures used to obtain projections from the Cohort-Survival Ratio method. In a like manner, procedures used in obtaining projections from each of the sixteen other Inethodologies examined in this study are outlined in detail in Appendices B-—2 through B-l6. 79 MWQ‘MIN [\HHO‘IN H H HOOP-‘0 [\NHNl-n MI mm—nn—a MNHNl-n Hv-l Ml oqmo. homo. mmoo. obno. meo. haqH. «woo. muoH. ohHH. mnHH. omNH. mcHo. omuo. HNoo. omNo. mmoo. NHHN. mmOH. mono. muwN. memo. HomH. mmmo. “H00. NHNH. 3| + +»+ +.+.+ + + ouwu own whom cam uaHm Oman mmmm wnNN comm oumq wmmm omwn omHm mmqw qoum HH¢o mHHq mono wemn owoc noou hqow ooHH NHmm ome womH mm «man—cm ... HOO‘WI‘ H .16 \OMCDHd’ uwp4 H OOON v-i ‘0“qu Ml ammo. QOmH. mHoH. ammo. Hqu. ammo. mneo. mmqo. wooo. oqmo. «0H0. mumo. HNmH. ammo. ommo. mooo. man. womH. ammo. wmbH. nmoo. NwHH. smoo. omwo. MHno. 31 . + +-+ + + + . + + + + + + MHon wooH mHhm nmou shun meaH ohmm thn ashNN oaww amen NmooH wehmH wNNMH .aHmo ommmH ¢¢¢OH HmnoH #quH wommH mmom @NHwN hmmm cthH hHaq momH um NH wH #H mH «waN. ooHo. mhmo. 05mm. mono. mhoo. thH. @mMH. mmmH. wwmo. Hqu. quo. mHoo. . qqu. mooc. cNoH. wMHN. wmoo. Nmeo. Name. Hhoc. HmHH. mnqo. homo. HmmH. 3| + + + + + + + oowH ooh ooou mom con MOHo Nomm mHHN own» wwcm mwNN moum wmhm meHh «wwH muse moon wHuq coma Noun osmH mNNo omoH comm ~0¢N mooH mm \Ov-lex'l’ cad-once) H Hh «H mH 0H 0H mnmno H Ml «Hmo. unho. mmuo. oooo. mowo. oeoo. mnoo. mono. QNHO. mane. wmho. tho. wau. NmNH. memo. HNoo. mme. NoHH. mHoo. hmmH. mnmn. mono. mwoo. mono. nqao. 31 + + +»+ +»+ + + + + + + + + $000 wNNH omH¢ mumw macs moqu swam homn oo¢m~ hHHw nNNo momoH thoH NNOmH moon oqth «coo «#NOH onNH wome Hows N~H¢~ «mom onHH omwu nomH mm dowexmnz moosdouH swam edoaamam moamuuaum Russo oHuuam suckum oouuoz onvmuom muHmwm compo huge sum ouooeauhz ouHuaom oonqsuHuM muonuuon umoaH< xmo Huhom uawHqu Mumm uHoouHH amnmcHauHm uonu< uu< atom ado uHaobaH youmqu some ooeuuo xuum uoHH< DUHuumHn nomemz OHHH>MDwnHmomoo HmH Nm monHUMHOMm OH HHmHH Houume moss oooooo sham doaa< ooapooan anHMZ OHHHH «H m HHmo. - moHH HH oHam. + moon m ammo. - HmoH mH mon. + ammo HoooaoH a o oooH. - oomm o momH. + HmmmH a omom. - momm a mooo. + ooooH muHo doouoe oH mH ommH. - momH oH oHom. + omoo oH HooH. - ommH o mooo. + oaom anus doHH< m m m 83 m. m mooH m m 32 m m 32 Hora-Ho . mm mm mm mm H ocean: oHeom-ezmzHHoezm mm moneommomm ¢H HHS 85 presents the projections yielded by Enrollment-Ratio 172 Table 15 displays the number of projections yielded by Enrollment-Ratio Method I, within and across strata, that are within a W Coefficient of .1000 of the actual enrollment figures. TABLE 15 ACCURACY OF PROJECTIONS YIELDED BY ENROLLMENT-RATIO METHOD I"l EE SE ._Stra.._.tum £1. E 2 3 E E E E A 4 6 .40 .38 2 8 .20 .29 B 6 4 .60 .50 3 7 .30 .49 C 4 6 .40 .52 O 10 .OO .39 D 5 5 .50 .61 8 2 .80 .52 E .1. .6. .39 .39. _3 .7. .29 .22 Total 23 27 .46 .50 16 34 .32 .44 aFor an explanation of the table headings, see the footnotes to Table 11, page 80. The results suggest that Enrollment-Ratio Method I, provides generally inadequate projections of public school enrollment for Districts across all Strata. Enrollment-Ratio Method II This method uses the same procedures as Enrollment-Ratio Method 1, except that it uses total county population as the figure to which the inatios are applied. The methodology is described on pages 32 and 33 HH NH oH omoN. - Hoo o mNoo. - omom m ooHN. mom oH oNoo. - ooom HoooHdH o «H omom. - oNoH m ooNH. - omoo oH mmNm. oomH m oomH. - Noom NoHo coon-o NH oH NNoN. - mmoH o mooo. + oomo NH ooHN. NNoH o mooo. + oomo aura ooHH< H H N 83 m m 83 m N 23 m m GS 3233 . um um no on HH ocean: oHNHH mH o oomo. NoHH mH oomm. + HHom N mNNo. - NmoH oH omoN. + oomo Hoo.HoH o HH NmNN. mNoN m oooo. + NmNHH HH NmoN. - omHN m HNmo. - memo NDHo ooouoo NH o omoo. mNHN NH NmNo. + oHoo NH Nooo. - mNoH NH NNoN. + mooo Hoom ooHH< m m m mooH m. m mooH m m mooH m m mooH olooHo . mm mm mm mm oomemz oHeom NHzo mzom-HHmoN mm oonNommomo OH Humans 89 Table 19 presents the number of projections yielded by the Pupil-Home Unit Ratio Method, within and across strata, that are within a W Coefficient of .1000 of the actual enrollment figures. TABLE 19 ACCURACY OF PROJECTIONS YIELDED BE THE PUPIL-HOME UNIT RATIO METHOD EE SE __so.-.mm E M 2. .1: E H E E A 2 8 .20 .38 4 6 .40 .29 B 5 5 .50 .50 1 9 .10 .49 C 5 5 .50 .52 0 10 .00 .39 D 2 8 .20 .61 4 6 .40 .52 E .2 ..5. as .22 .2 .3 .99 .5_2 Total 19 31 .38 .50 15 35 .30 .44 aFor an explanation of the table headings, see the footnotes to Table 11, page 80. The results suggest that the Pupil-Home Unit Ratio Method provides generally inadequate estimates of public school enrollment for Districts across all Strata. _Iime Analysis I Time Analysis I represents the first of the regression approaches Aexamined in this study. The approach bases its projections on past Irelationships between enrollment and time. Table 20 presents the lorojections yielded by Time Analysis 1.75 90 «3.4qu H \‘I’hlnow O‘QOOI’N v—l Otnmtnm Hw‘Dhfi' Ml ONNO. oomH. NNOH. OOOO. ONOO. OOOO. OOOH. OHNO. NOOO. NOOO. OHOH. ONON. OONO. ONNO. NONH. NONO. ONNO. OONO. NOHH. HOOH. NOOH. NNON. OOOH. NNON. NNNO. 31 OHON NOO «NNH OOOH ONON NOOO OOHN ONON HONO NNHN HOHN HNNO NNOO OONO ONOH mONO ONON ONON HOOO ONOO OONN NNHO NNOH NOON OOON HmNm mm v—‘hlnw MHMMH mm NH N HH NH HH m. OHOH. NONO. NONO. NHOH. OOON. Nooo. NNNN. NNHO. NHoo. ONNO. ONHH. HOOO. OHON. NNNH. mNOH. OONN. OOOH. NOHN. OOOO. quo. + +.+.+ + + + + + + + + + +~+ + + + NONO.H+ NoHH. OOOH. NNNH. NOON. _m .+. + + + ONNN NOON NONO OHmN OOOO NOOOH HHOO OOON OONON NOOO NONO OHOOH ONOOH NONHH NOOO ONONH OOOO OONHH NHONH mONmH ONOOH NNOON ONOO OOmmH NNON OOOH um HH NH HH NH mH mN ”€001” ONONN? H OOON. OOOO. NOOO. OHOH. HOOO. NONO. ONNO. NOHO. Hoqo. NOOO. OOOO. NNON. Oomo. OONO. NHOO. OOno. OOOO. oNHH. NOOO. NNOO. NmNO. OOON. NNNH. OOON. ONNO. m. + NOON NOO NNNH NNoH HNNN + ONOO NOON NNON OOON ONOO + OOON ONHO Oomo + ONOO OHNH NNNN + oNHN OHON OONO OOmN + ONON HOOO OOO «HHN + «OHN NOOH MO OQNO‘ H OHNHq moo OH NH HH OH aim HH NH NH Ml OOOH. Nomo. ONOO. OOOO. NOHH. ONOO. NNoH ONHo. ONOO. HONO. ONNO. NOOO. NONH. NHOO. OOno. NOHH. NOOO. NONH. OOHO. OOOO. NONO. NOOO. NNmo. OONN. NOOH. 31 . + + + + + . + + + + + + +.+ + + + + + + + + NOON NNON ONNO OHON ONNO NNoNH ONOO OONm OONNN ONOO OONO NHONH OHNNH OOHNH ONON OOONH NOOO ONOOH NOOHH ONHNH ONNO ONONN NOON NNONH OONN NOOH mu uomoxnsz oooauouH Hoom odoHemHm Josuuusum Howuo mHouum somHOum ooumoz OHoocuom moHamO ocmuo homo mom ouuouuwhz OuHuuom coaustmM uuonueon euomH< ado Hahom OudHOHz xuam aHoouHH anomoHauHm uonu< um< swam goo OHOO>HH noumqu NuHo ooouoo Huuo ooHH< uOHHuan Aonmmmmommo H mHmNHHH noumxcH NuHo cuckoo xuum :OHH< ooHaoaHo NN MHOHH Houmqu NOHO ooouoo xnsm ooHH< DOHuumHO AZOHOOMOUMMV OHOVHHH neuaHuH NHHO dooumo HNom OOHHO ooHauoHo Honmmmmommo mHmeHH m mH HomN. mNm m mmHo. + mmNm mH NmoN. - mom m NHHo. - HHom HouaHaH m NH oomN. oooN o mNoo. + mNmHH m mNmH. - omoN H HNoo. - omHoH NHHO ooouuo o N momo. onN m oNoH. + NmHm oH oomo. - mooH m HHoo. + NHoo Huuo ooHH< m m : mooH N m mooH N m mooH m N mooH ooHuuoHo . mm mm mm mm H mHmNHO' N4 MNQNTN HI—l O‘flNNv-i MN OH OH OOON. ONOH. ONHO. NOOH. NHOO. OONH. NNNH. NOOO. NONO. NHHO. HOOH. OOOH. NNOH. OOOO. OOOO. HNOO. OOOO. HOHO. OHOO. HNNO. OONO. ONOH. HONN. OHON. NONO. _m OHNN HNO NOOH OOOH OOON HOOO NOOH OONN NOON HOHO NNON NOOO ONON OONO NONH NOOO HONN OOON NNOO OOHO NNON NHOO NNO HONN HOON OOOH MO ”NNOO IOVONInln NH OH NH NH O OH NH H OH OH O. OOOH. OOOO. ONNO. OOOO. NHOO. ONHO. HOOH. NNHO. OHNO. NOHO. NOOO. OONO. NONO. OOOO. HOOO. NOOH. OOOO. OOOH. NHOH. OHOO. OOOO. NONN. OHOO. NNNN. NOOH. 3| +-+-+ + + + + + + + + + + + + HOON OHON OOHO HOON ONHO NONOH OONO OOOO OONNN ONOO OONO OOONH ONONH NHNNH ONOO HOHOH OOHO NHNOH HHNNH OOONH OONO OONON OOON OOONH NONO OOOH mm domoxonz ooozuouH atom odoHomHe Husduusom Macao oHuuoO suuHOum ooumoz OHOOOHON mOHOom Oauuo NHHO Nam uuuooasmz uuHuaom oouosunM auonuuom adomH< Huo HuNom OumHOHz HHOO mHooaHH aanaHauHO HOHH< mm< spam moo oHdo>HH Houmde NuHo mevuuo xuum ueHH< ooHuuuHo HH mHmmHHH Houmqu NDHU auouuo swam OOHH< ooHuuoHn NN HHmHHN O N OHOH. - NNOH oH OONO. + NOON O OOHN. - HOO O NHOO. + ONON NOOOHOH NH O OOON. - OOON OH OONO. + HOOOH NH ONON. - ONON NH OOON. + OOONH NOHO aOONOo NH OH OONO. + ONON OH NOON. + OhOO H oHNo. + OOHN OH NOOH. + OHOO HNON OOHH< m. m m OOOH m m OOOH m m OOOH m m OOOH OOHNOOE . NO mm mm mm >H OHONHOzO onOOmNONN HzmzHHomzu um OonHouOONN ON mqmflw 106 TABLE 35 ACCURACY OF PROJECTIONS YIELDED BY ENROLLMENT REGRESSION ANALYSIS IVa EE SE __Stratum £1 E F. I E E .13. E A 4 6 .40 .38 2 8 .20 .29 B 7 3 .70 .50 6 4 .60 .49 C 6 4 .60 .52 6 4 .60 .39 D 8 2 .80 .61 6 4 .60 .52 E _6 .3 .60 .49 _é .6. .39 .22. Total 31 19 .62 .50 24 26 .48 .44 8For an explanation of the table headings, see the footnotes to Table 11, page 80. The results suggest that Enrollment Regression Analysis IV provides an adequate projection methodology for Districts having growth characteristics similar to those in Strata B and D, although the secondary projections leave something to be desired. It should be pointed out, however, that the projections yielded by this methodo- logy, though adequate, are not the best for Strata B and D of any of the estimates examined in this study. (Also Stratum C). HousingiAnalysis The predictor variables used in this approach were occupied housing and state equalized valuation. Table 36 presents the projections yielded by Housing Analysis.83 Table 37 presents the number of projections yielded by Housing Analysis, within and across Strata, that are within a W Coefficient of .1000 of the actual enrollment figures. 107 00th r-| OH .pq NH OH ON OOH NONH. OHON. OOOH. ONOO. OwOO. OOHO. OONH. NHOO. NOOH. mmNo. NONO. OHNN. NONH. ONNH. HNNN. NOOH. HHOO. ONNO. NONH. OHOH. ONHN. NONO. NNNN. ONOH. OONO. 3| NNON Nam HNhH NOOH HOON NOOO OOON NNNN OOON ONNN OONN ONNN OONO ONON ONNH NONN OOON OONN HNmO ONNN NONN HOON OOO OHON HNNN mama NO \o~O\0l~\o Ha F4 H QONGQ InN‘OGJh coco OH MO‘OHG) ml ONHH. HOOH. NNOO. HONN. NNOO. NNOO. NONN. NONO. NNOO. NHNO. ONNO. NNOO. ONHH. ONOO. NNNO. NNOH. NONH. ONNH. NNNH. NNNO. NOON. NOOO. OHNO. NNON° ONNN. m. +,+ + +.+ + + + + + + + + +-+ + + NNOO NONN OHNO ONNN NONN OmOOH NONN HONN NNHNN ONNN ONOO OOHOH NNNOH NONNH omOn NOONH NNOO OHOOH mNmNH HOONH NONO OONNN NONN OOONH NNON mama mm Q'OQIDN m HNOH. NONO. NOOO. NONH. NOOO. HOOO. OOOO. NNOH. ONNO. NNNO. HOOO. NOON. HNOH. NOHH. NHNH. NONH. NONO. NOOO. ONNO. ONHO. ONHN. ONNN. HNNN. HNNN. ONNO. m. NNON mHh NNNH OOOH HOON NNOO ONON NONN NOHN OOON NOON HONN ONOO NNON NNOH ONOO HNHN ONON NHNO HNNN NNNN NNOO HON OONN OOON NOOH mm OONNN MI NOOO. OmOH. NOHO. NONH. OHOO. mmoo. ONOO. ONNO. NNNO. HNOO. OONO. NHOO. HONO. NOOO. ONHO. ONOO. ONOO. NOHH. NONO. mmoo. NNNO. mmoo. NNOO. NNOO. OONH. 3| + +-+ + + + + + + + + + n + + + + + + HHOO OHNN HNNO OONN NNNO NmOOH OHNN ONNN NOHNN Oman NONN NNONH ONHOH snONH NnOm ONNNH NONN NmNOH HOONH ONONH OOON NONHN NNON OmOHH OOON nomH MM domoxnsz vooadouH xuum vaanme xuauuuaum xmouu oHuuuN suanwm mouse: mHuvauum mvHamm wanna NOHO Nam ouuovuu%3 odHuaom ooudauHuM auonuwun uaumH< xuo Huhom OHHOHOHHH xumm dHooaHH aunwaHEHm uonn< aq< NNON sue «Hao>HH HouuxaH NOHO uowuuu NNON doHH< uOHuumHn HonNNuNONNO NHNNHHA Houaqu Nufiu dovumo xumm aOHH< NOHNOOHN Aonmmwmwmmv H NHNWHnHHHDZ Mm mZOHHUMOOMm NN mqmHA Houmde NOHO auuuuo xuum :OHH< uuHuumHn AonmmmMQMMv HH NHNMHaHHHOS Wm NZOHHUNHOMm OO MHmHA HouuxaH spam dovumo xuum HHOHHO NOHHOOHH HonNNmNumNO HHH NO NHNOH NHNMHIHHHDZ Mm NZOHHUMOOMm A|||| 115 TABLE 43 ACCURACY OF PROJECTIONS YIELDED BY MULTI-VARIABLE ANALYSIS III (REGRESSION)a EE SE Stratum H 11 g g H M g _l: A 2 8 °20 2 8 .20 B 3 7 .30 7 3 .70 C 1 9 .10 5 5 .50 D l 9 .10 l 9 .10 E .2 .1 .29 .2 .3 .99. Total 10 40 .20 21 29 .42 8For an explanation of the table headings, see the footnotes to Table 11, page 80. Summary Table 44 presents a breakdown by Strata of the effectiveness of the seventeen projection methodologies examined in this study. In order to be judged as adequate for a given Strata a projection methodology must meet the following criteria. 1. The methodology must receive a composite W Coefficient rank of 10 or better on four of the five Districts classified in that Strata. 2. The methodology must yield projections that are within a W Coefficient of .1000 of actual enrollment figures at least six out of ten observations on both elementary and secondary enrollment estimates. Table 44 also identifies those formulae that appear to be best suited to the problems of projection relevant to Districts in each of the Strata identified in this study. Only one formula per Stratum is identified as best. Criteriazirecomposite W Coefficient rank and number of projections yielded that are within a W Coefficient of .1000. 116 TABLE 44 EFFECTIVENESS OF PROJECTION METHODOLOGIES BY STRATA Stratum Projection Methodofigy A B 9 D E l. Cohort-Survival Ratio X 0 X 0 X 2. Grade Retention Ratio 0 0 O 0 O 3. Enrollment Ratio Method I 0 O 0 0 0 4. Enrollment Ratio Method II 0 0 O 0 O 5. Pupil-Home Unit (Ratio) 0 0 0 0 0 6. Time Analysis I (Regression) 0 O 0 b 0 7. Time Analysis II (Regression) 0 0 0 X 0 8. Transition Analysis (Regression) b b b X b 9. Birth Analysis (Regression) 0 0 0 0 X 10. Enrollment Regression Analysis I 0 0 0 X X 11. Enrollment Regression Analysis II 0 0 0 0 0 12. Enrollment Regression Analysis III 0 X 0 X 0 l3. Enrollment Regression Analysis IV 0 X X X 0 14. Housing Analysis (Regression) O O O X 0 15. Multi-Variable Analysis I O O 0 0 0 l6. Multi-Variable Analysis II 0 0 0 O 0 17. Multi-Variable Analysis III ‘9 .Q ‘Q .9 .9 Total 2 3 3 7 4 X - projection methodology is adequate as determined by previously outlined criteria. b = projection methodology yields best estimates for Districts in this Strata. O = projection methodology is not adequate as determined by previously outlined criteria. 117 Footnotes for Chapter IV 0In order to be judged as adequate for a given Strata a projection methodology must meet the following criteria. a. It must receive a composite W Coefficient rank of 10 or better on four of the five Districts classified in that Strata. b. It must yield projections that are within a W Coefficient of .1000 of actual enrollment figures at least six out of ten observations on both elementary and secondary enrollment estimates. 71The 72The 73The 74The 75The 76The 77The 78The 79The 80The 81The 82The 83The 84The 85The 86The methodology is outlined methodology methodology methodology methodology methodology methodology methodology methodology methodology methodology methodology methodology methodology methodology methodology is is is is is is is is is is is is is is is outlined outlined outlined outlined outlined outlined outlined outlined outlined outlined outlined outlined outlined outlined outlined in in in in in in in in in in in in in in in in Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix B—2. B-3. B-4. B-S. B—6. B-7. B-9 0 B-lO. B—ll. B-12. CHAPTER V SUMMARY AND CONCLUSIONS This study explores three major research questions. 1. Can educational attendance prediction problems he T formulated in such a manner that the least squares f regression model can be used in the analysis thereof? ; Ultimately we are asking if simple or multiple linear 2 w E; regression, or a transformation thereof, will yield more accurate than will the methodologies 2. Are different methodologies districts, or projections of future school attendance most popular educational projection currently in use at the district level. variables and diverse projection relevant to different types of school is there one projection methodology that provides the best estimates, ie. estimates with the least amount of error, across all school districts? 3. Is it worthwhile to make the necessary data transfor- mations to estimate local population statistics for projection purposes, or will county population statistics, which are readily available from relatively reliable sources, provide more reliable estimates on which to base future school population projections? The population identified in this study consists of those public school districts in the State of Michigan that are largely 118 119 coterminous with cities of 10,000 population or more. It is divided into five strata on the basis of past community growth characteristics. Twenty-five school districts, five from each stratum, were randomly drawn from the population. Seventeen separate and distinct ratio and regression projection methodologies, using pre-196O observations as base data, were used in order to ‘ F obtain estimates of elementary and secondary enrollment for each of the twenty-five districts in the sample for the years 1965 and 1968. Actual 1965 and 1968 enrollment figures were then used as! the criteria against which to judge the relative accuracy of the various ratio and regression approaches. The ratio and regression analysis methods examined in this study all have one thing in common, they are all relatively simple to apply. Every one of the twelve regression approaches can be run on canned computer programs currently available at Michigan State University. This is their major strength, but also their major weakness. It is a strength in that, given the applicability of a given regression approach to enrollment projection, educators can apply that approach to their educational projection problems with relatively little difficulty. With some modifications of current computer programs, a canned program could be made available that, given the values of relevant predictor variables, would project elementary and secondary enrollment at given intervals into the future. It is a weakness in that, given districts with certain kinds of growth characteristics, particularly districts similar in growth 120 patterns to those classified in Stratum A, the approaches examined in this study tend to oversimplify the estimation problem. It is probable that in districts that have unusual growth characteristics, a local survey that considers a large number of demographic variables is needed in order to obtain consistently adequate projections of future enrollment. It seems crucial that at least the approximate number of people that a given area can support be determined so that an upper limit can be established beyond which projected population values cannot increase. Results and Discussion A number of major conclusions that are pregnant with implications for education can be drawn from the results of this study. Of primary importance, the results suggest that educational attendance prediction problems can be formulated in such a manner that the least squares regression model can be used in the analysis thereof. One need only point to the relative accuracy of the projections produced by Transition Analysis, a regression approach, to support the contention that regression analysis offers a viable methodology that can be used to great advantage in the estimation of future public school enrollment. Adding further support is the fact that all of the regression methodologies examined in this study, with the exception of the abortive attempts to use four predictor variables with six observations, outperform all of the comparable 121 ratio methods in terms of accuracy of projections yielded. Also of major interest is the fact that it appears as if different projection methodologies perform differentially across districts that are located in communities with diverse past population growth characteristics. For example, Time Analysis I, a regression approach, appears to provide the most accurate projections for districts with past growth characteristics similar to those classified in Stratum D. Other projection methodologies also predict with differential success across strata. Adding additional support to this position is the differential ease with which all of the projection methodologies, taken as a group, predict across the five strata. For example, while it appears to be relatively difficult to obtain accurate estimates for those districts classified in Stratum A, there are numerous accurate projections of various aspects of enrollment of those districts classified in Statum D. On the other hand, Transition Analysis, a regression approach, appears to supply adequate projections of future school enrollment across all districts, actually supplying the best estimates in four of the five strata. Therefore, it appears that one could not go too far wrong in using Transition Analysis, regardless of the past growth characteristics of the community in which the district of interest is located. At another level of differentiation, many of the projection methodologies appear to have enjoyed differential success in the Projection of elementary and secondary enrollment. In general, the 122 various methodologies experienced more success in projecting elementary enrollment than in attempting to project secondary enrollment. This seems to be particularly true of Birth Analysis, Enrollment Regression Analysis III, Enrollment Regression Analysis IV, and Housing Analysis, all regression methodologies. The question concerning whether it is more efficient to use population estimates derived from local or county data, appears to be an insignificant one. Births, used in conjunction with past tendencies of students to advance from one grade to the next, appear to form the most adequate basis from which to project future school enrollment. Both Transition Analysis, a regression approach, and Cohort-Survival Ratio Analysis, approaches that use the afore- mentioned predictors, far outperform all of the other methodologies examined in this study. Generally speaking, neither total city population estimates nor total county population estimates, when taken singularly, correlated exceptionally highly with enrollment across all districts. However, it is significant to note that correlations were increased by using school-age population, a variable that was computed in the manner outlined on page 62. This explains the relatively high multiple correlations that were observed in the cases of Enrollment Regression Analysis II and Enrollment Regression Analysis III, as well as casting serious doubt on the validity of using total city or total county p0pu- 1ation in conjunction with school-age population as predictors, since school-age population by itself appears to produce results 123 that are at least comparable to the muhi‘variate approaches. All things considered, this researcher has grave doubts as to the viability of either city or county population estimates to the projection of future school enrollment. Another interesting result, considered in the context of the question of the viability of using estimates of p0pu1ation variables as predictors of future school enrollment, is the performance of a relatively unique regression approach, Enrollment Regression Analysis IV. While the inter-censal p0pu1ation estimates produced by the University of Michigan Population Studies Center, and used as the basis for most of the population estimates used in this study, are based on such demographic variables as births, auto registration, school census count, voters, and sales tax, the inter- censal estimates used as predictor variables in Enrollment Regression Analysis IV were computed in strictly an algebraic manner. Census counts for the years 1950 and 1960 were obtained, and a straight line was simply drawn between those two points to obtain approxi- mations of the base data on which the regression equations are formulated. The line was then extrapolated to obtain estimates of relevant population parameters for 1965 and 1968. Despite the relative simplicity of this method, as opposed to the complicated techniques used to obtain population estimates for all of the other methodologies examined in this study, Enrollment Regression Analysis IV outperformed the more complex methods in terms of accuracy of projections. 124 Finally, the results of the study suggest that the accuracy of projections is inversely related to the distance into the future that one wishes to make those projections. That is, the further into the future a projection is made, the more inaccurate it is likely to be; This is generally true of all projection methodo- logies across all strata. Recommendations For Future Research As the study progressed, an exponentially increasing number of unanswered questions made themselves apparent. These questions are incorporated into recommendations for future research. One principle that is suggested by the results of the study is tempered, at this point, by far too much uncertainty. Of major concern for future research is the apparent inapplicability of regression equations with more than two predictor variables to projection problems of the type outlined in this study. Unless predictor variables can be located that are characterized by low inter-correlations, yet maintain a high correlation with the criterion variable, it is extremely doubtful that more than two such variables are useful. However, before any conclusions can be drawn as to the validity of this hypothesis, a great deal of future research must be perpetrated. Of primary importance is a specification of the optimum length of past time series observations to be used in estimating the constants in the regression model, that is, providing such an optimum length exists. If, in fact, a greater number of observations provide more ICCUrate projections, assuming that it is possible to obtain 125 the data, can more predictor variables efficiently be added? As soon as an increase in the number of observations is contemplated, a new research question arises, that being a determination of the length of time, or number of data points, that are required before the linear model no longer furnishes an adequate projection methodology. This question calls for experimentation with trans- formed scales and higher order curves as non—linear projection techniques, in the context of varying the number of data points. The small amount of experimentation done with transformed scales in this study suggesusthat, for a relatively small number of data points, the linear model provides the most adequate projection technique. An empirical determination-of the point at which this is no longer the case would be extremely valuable. The reader will recall some earlier concern with the implications and effects of autocorrelated error terms on the regression model. The effects of autocorrelated error terms lie entirely in the computation of standard errors and confidence intervals for the regression coefficients. While the estimates are admittedly un- biased, their sampling variances may be unduly large. The problem arises when the usual least-squares formulae are applied to estimate the sampling variances of the regression coefficients. Serious underestimates of these variances are likely to result. Although this state of affairs did not effect the outcome of this study, it would be helpful at some time in the future to be able to compute realistic confidence intervals for population projections. This 126 of course would also involve the development of methods designed to combine the error in the projection of the predictor variable with the error in the projection of the criterion variable into a meaningful statement of the amount of error inherent in the projection. A theoretical study of the applicability of the regression model to time series analysis, in the context of its use in this study, would provide invaluable assistance to the science of demography. Also of major interest is a replication of the results of this study on districts other than those included in the sample and the population. Transition Analysis appears to Offer a viable new approach to the projection of future public school enrollment. It would be extremely interesting and valuable to determine if these results hold across different populations and samples. Finally, experimentation should be conducted with a greater diversity of projection methodologies utilizing a number of different demographic variables. The models investigated in this study might be improved in a number of ways. First, greater experimentation with methods of projecting various demographic variables, such as future city and county population, should be encouraged. If more adequate estimates of relevant demographic variables were available, it is probable that more reliable estimates of future school enroll- ment could be obtained. Second, non-public school data, if it were made available, could be used as additional variables in the pro- jection equations. Lastly, a study of the feasibility of using 127 various indicants of economic status should be encouraged, especially if it becomes possible to use several decades of data in projecting future public school enrollment. This study only serves as a starting point in the investigation of the applicability of least-squares regression to the projection of future school enrollment. Educators have relied on largely inadequate techniques for far too long. The time for the appli- cation of some of the techniques of the behavioral sciences to the projection of future enrollment has long since passed. It is hoped that the results of this study, in conjunction with the recommendations flx-future research, will encourage such a course of action. BIBLIOGRAPHY rt . United United United United United United United United 128 BIBLIOGRAPHY Public Documents States Bureau of the Census, Current Population Reports, Series P-25, No. 362, "Illustrative Projections of the Population of States 1970 to 1985 (Revised)", March 7, 1967. States Bureau of the Census, Current Population Rgports, Series P-25, No. 339, "Methods of Population Estimation: Part I Illustrative Procedure of the Census Bureau's Component Method II", June 6, 1966. States Bureau of the Census, Current Population Reports, Series P-25, No. 390, "Projection of Educational Attainment 1970 to 1985", March 29, 1968. States Bureau of the Census, Current Pepulation Repprts, Series P-25, No. 360, "Projections of the Number of Households and Families 1967 to 1985", February 20, 1967. States Bureau of the Census, Current Population Reports, Series P-25, No. 394, "Projections of the Number of Households and Families 1967 to 1985", June 6, 1968. States Bureau of the Census, Current Pppulation Reports, Series P-25, No. 359, "Projections of the Population of the United States By Age, Sex, and Color to 1990, with Extensions of Total Population to 2015", February 20, 1967. States Bureau of the Census, Current Population Reports, Series P-25, No. 381, "Projections of the Population of The United States, By Age, Sex, and Color to 1990, with Extensions of Papulation By Age and Sex to 2015", December 18, 1967. States Bureau of the Census, Current Population Reports, Series P-25, No. 85, "Projections of School Enrollment in the United States, 1953 to 1965", December, 1953. R1 129 United States Bureau of the Census, Current Population Reports, Series P-25, No. 345, "Projections of the White and Non White Population of the United States, By Age and Sex, to 1985", July 29, 1966. United States Bureau of the Census, Current Population Reports, Series P-25, No. 375, "Revised Projections of the Population of States 1970 to 1985", October 3, 1967. United States Bureau of the Census, Current Population Repprts, Series P-25, No. 365, "Revised Projections of School and College Enrollment in the United States to 1985", May 5, 1967. United States Bureau of the Census, Current Papulation Repprts, Series P-25, No. 388, "Summary of Demographic Projections", March 14, 1968. United States Bureau of the Census, Whelpton, P.K., (ed.), "Forecasts of the Population of the United States, 1945-1975", Washington: 1947. Books Boles, Harold W., Step to Stgp to Better School Facilities. New York: Holt, Rinehart and Winston, 1965. Draper, N.R. and Smith, H., Applied Regression Analysis. New York: John Wiley and Sons, 1966. Jacobs, E.G., Methods of School Enrollment Prpjection. New York: Columbia University Press, 1959. Jaffe, A.J., Demographer, Office of the Coordinator of International Statistics, Handbook of Statistical Methods for Demographers. Washington: United States Government Printing Office, 1951. MacConnell, James D., Plannipg for School Building . Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1957. 130 Articles and Periodicals Bernert, E.H. and Narn, C.B., "Demographic Factors Affecting American Education," NSSE Yearbook, 1960. Bowers, Harold J., "Projecting School Enrollments For One State," Journal of Teacher Education, Vol. V, No. 1, March, 1954. Campbell, George C., "The Forecasting Problem in School Administration," The American School Board Journal, November, 1950. Cochrane, D. and Orcutt, G.H., "Application of Least Squares Regression to Relationships Containing Auto-correlated Error Terms," Journal of the American Statistical Association, Vol. 44, 1949. Coleman, John, "The Shifting Seas of Enrollment," School and Society, April 15, 1968. Dorn, Harold F., "Pitfalls in Population Forecasts and Projections," Journal of the American Statistical Association, September, 1950. Durbin, J. and Watson, G.S., "Testing for Serial Correlation in Least Squares Regression," Biometrika, I, II, Vol. 37 and 38, 1950-51. Edmission, R.W., "How to Predict Your District's Enrollment," School Executive, May, 1954. Gottlieb, Abe, "A Planning Approach to School Enrollment Forecasts," American School Board Journal, February, 1954. Hauser, Philip M. and Kitagawa, Evelyn M., "On Estimation of School Population," American Statistical Association Proceedings, Washington: ASA, 1961. Herrick, John H., "Estimating Future School Enrollments in Rapidly Growing Communities," Educational Research Bulletin, 31, April 16, 1952. Holy, Thomas C., "What Future Needs Are Revealed by School Population Studies?" Education Digest, Vol. 9, May, 1947. \ 131 Kuang, Huan D., "Forecasting Future Enrollments by the Curve Fitting Techniques," Journal of Experimental Education, Vol. XXIII, 1955. Larson, Knute G. and Strevell, Wallace H., "How Reliable Are Enrollment Forecasts," School Executive, Vol. LXXI, February, 1952. Mitchell, Donald P. and Turell, John E., "Prediction of Enrollments Includes In-Migration of Pre-School Children," Nations Schools, August, 1955. Odell, Charles W., "Estimating Future School Enrollments," American School Board Journal, January, 1951. Richey, Herman C., "Population Change and its Implications for Education," Journal of Teacher Education, Vol. V, No. 1, March, 1954. Schmid, C.F. and Shanley, F.J., "Techniques of Forecasting University Enrollment," Journal of Higher Education, Vol. XXIII, No. 9, 1952. Siegel, Jacob 8., "Forecasting the Population of Small Areas," Land Economics, February, 1953. Spurr, W.A. and Arnold, D.R., "A Short-Cut Method of Fitting A Logistic Curve," Journal of the American Statistical Association, March, 1948. Strevell, Wallace H., "Techniques of Estimating Future Enrollment," American School Board Journal, March, 1952. Tarwin, B.C. and de la Torre, Jaime, "Educational Trends for Specific Populations in the United States by Cohort Analysis," Rocky Mountain Social Science Journal, Vol. 4:1, April, 1967. Whelpton, P.K., "An Empirical Method of Calculating Future Popu- lation," Journal of the American Statistical Association, Vol. 31, 1936. 132 Reports American Association of School Administrators, 27th Yearbook, American School Buildings, School Planning Lab, Stanford University, 1955. Armstrong, Charles M. and Harris, Mary S., A Method Predictipg School-Agg Population, Division of Research, State Education Dept., The University of the State of New York, Albany, New York, 1949. Bangnee, Alfred Liu, EstimatingyFuture School Enrollment in Develppipg Countries - A Manual of Methodology, UNESCO/United Nations Statistical Reports and Studies, Population Studies No. 40 ST/SOA/Series A, 1966. Bayo, Francisco, United States Population Projgctions for OASHI Cost Estimates, Social Security Administration, Office of the Actuary; Actuaril Study No. 62, 1966. Brown Jr., Byron W. and Savage, I. Richard, Methodological Studies in Educational Attendance Prediction, University of Minnesota, Dept. of Statistics, September, 1960. Crawford, Albert Bryon and Chamberlain, Leo Martin, Prediction of Population and School Enrollment in the School Survey, University of Kentucky: Bureau of School Services Bulletin, Vol. 4, No. 3, 1932. Click, Paul C., Estimates of NUmber of Families in the United States: 1940 to 1960, Series P-46, No. 4, Population - Special Reports, United States Department of Commerce, Washington: 1946. Goldberg, David and Schnaiberg, Allan, Michigan POpulation - 1960 to 1980, State Resource Planning Division, Office of Economic Expansion, Michigan Department of Commerce, Working Paper No. 1, January, 1966. Linis, L.J., Methodology of Enrollment Predictions for Colleggg and Universities, American Association of Collegiate Registrars and Admissions Officers, March, 1960. 133 Michigan Department of Commerce, Preliminapy Population Projections for Small Areas in Michigan, State Resource Planning Division, Office of Economic Expansion, Working Paper No. 9, November, 1966. Philadelphia Planning Commission, Population Estimates 1950-2000, Philadelphia-Camden Area, April, 1948. Schmidt, Calvin F. and Miller, Vincent A., Enrollment Forecasts for the State of Washingpon, 1957-1965, Seattle: Washington State Census Board, 1957. Thoden, John F. and Taylor, C.L.. Potential Michigan Public School Enrollments, Quarterly Bulletin, of the Michigan Agricultural Experimental Station, Michigan State College, East Lansing, Michigan, Vol. 34, No. 4, 1950. Thomas, J. Allen, School Finance and Educational Opportunityyin Michigan, Michigan School Finance Study, Michigan Department of Public Instruction, 1968. Thompson, Ronald B., Enrollment Projections for Higher Education, 1961-1978, The American Association of Collegiate Registrars and Admissions Officers, September, 1961. Werdelin, 1., Statistics for Educational Plannipg and Administration Part I: Dempgraphy, No. 22, 1967. Unpublished Materials Driscoll, William N., The Prediction of School Enrollment From New Home Developments, Research Study No. 1, Ed.D. Dissertation, Colorado State University, 1965. Hunt, David G., The Develgpment of Enrollment Projection and Prediction Methods For Texas Public Junior Colle es, The University of Texas, Ph.D. Dissertation, 1962. Peterson, Donald Lee, An Investigption of Techniqus For Predictipg School District Enrollments In Florida, Ed.D. Dissertation, University of Florida, 1959. 134 Ploughman, Theodore L., Darnton, William T., Heuser, William A., An Assignment Progpgm to Establish School Attendance Boundaries and to Determine School Construction Require- ments, Oakland Intermediate School District, July, 1968. Rodgers, Jack, Criteria For the Establishment of Local Public Junior Colleges in Texas, Unpublished Ph.D. Dissertation, University of Texas, June, 1956. Waldfogel, Dean and Krajewski, Frank R., A Study of School Enrollment Projection Formulae, unpublished seminar paper completed at Michigan State University, 1964. Wheeler, Norman 5., Techniques Used For Predicting Public School Enrollments in Detroit, Michigan, Usipg:§he Elementagy School Service Area As a Base, Ed.D. Dissertation, Wayne State University, 1955. Data Sources Michigan Department of Public Health, Annual Statistical Report, Michigan Health Statistics, Health Statistics and Evaluation Center, 1950, 1955-1960. Michigan Department of Public Health, Michigan Population Handbook, R. Raja Indra (Ed.) Health Statistics and Evaluation Center, November, 1965. Michigan Department of Public Instruction, Computer Print-out of 1968 SEV by School District, Available at State Department. Michigan Department of Public Instruction, County Superintendent Summary Reports; State Financial and Enrollment Statistics. Lot 180, Michigan State Bureau of Records. Michigan Department of Public Instruction, File showing annexations in the State of Michigan. Michigan Department of Public Instruction, Individual forms showing 1968 public school enrollment by grade. Available at State Department. 135 Michigan Department of Public Instruction, Ranking of Michigan Public High School Districts By Selected Financial Data, Bulletin 1012, Available from 1961-62 School Year through the 1967-68 School Year, with the exception of 1963-64. Michigan Department of Public Instruction, State Aid Sheets, Lot 180, Michigan State Bureau of Records. United States Bureau of the Census, Census of Pppulation: 1950, Vol. II, Characteristics of the Population, Part 22, Michigan. United States Bureau of the Census, Census of Pppulation: 1960, Vol. I, Characteristics of the Population, Part 24, Michigan. United States Department of Health, Education, and Welfare, Public Health Service, Vital Statistics of the United States, Part II, Place of Residence, National Center for Health Statistics, 1938-1960. University of Michigan Population Studies Center, Michigan Counties: Pppulation 1940 to 1980. Dr. David Goldberg, December, 1963. APPENDICES APPENDIX A ASSUMPTIONS MADE BY THE UNIVERSITY OF MICHIGAN POPULATION STUDIES CENTER IN PROJECTING FUTURE STATE POPULATION APPENDIX A ASSUMPTIONS MADE BY THE UNIVERSITY OF MICHIGAN POPULATION STUDIES CENTER IN PROJECTING FUTURE STATE POPULATION The state population was projected by using a component procedure. That is, births, deaths, and migration were projected individually. The assumptions about these components are described below. Mortality A life table was constructed based on mortality in the Michigan population for the period 1959—1961. It was assumed that age specific rates of mortality would remain constant throughout the projection period. Although this assumption is unrealistic it does not seriously affect the p0pu1ation projections. Gains in longevity are likely to be small and the impact on p0pu1ation growth is trivial by comparison to a 10 or 15 per cent change in fertility. The present population is "aged forward" by using survival rates for that age level constructed from the life table. The procedure is straight forward. If there are say, 1000 people at a certain age level in a given year, we take that 1000 times the survival rates for that age level in order to obtain a projection of the number of survivors. When this process is completed for all age groups, we obtain a projection of population by age for all groups 5—9 or older. Two elements are lacking - net migrants by age and the population 0-4. These components are handled by the assumptions on migration and fertility. Migration Census data from 1960 indicate that the state lost approximately 180.000 net migrants for the period 1955-1960. Estimates of net migration by age for the period 1960-65 were prepared by obtaining estimates of the state p0pu1ation by age in 1965 and noting the discrepancy between expected survivors of the 1960 population and estimated 1965 p0pu1ation: Est. Pop. (1965) — Expected pop (1965) = Estimated Net Migrants These estimates imply an out migration from the state about equal in magnitude to the out migration during the preceding five years. Two sources give us confidence in the migration estimates: 136 137 1. The school census shows a consistent year to year out migration of the population age 7-16 for the period 1956- 1964. 2. A recent release of the Census Bureau using different procedures obtains similar results. For the period 1965-1980, we have assumed that net migration would gradually diminish toward zero by 1980, but with the age pattern of migration being fairly similar from period to period. If migration outward were to continue at the same pace experienced by the state in 1955-60 or 1960-65, the state total population would be about 350-375 thousand less than the figures shown in the projections for 1980. Fertility We have made fertility projections by using cohort procedures. This method of projecting fertility focuses on the size of completed family and the spacing of births, rather than on age specific birth rates (the Census Bureau has recently shifted over to this method in their national projections). Several national studies conducted at the University of Michigan indicate that married women now in the childbearing ages are likely to complete their families with three children. Given the fact that about 95 per cent of all women eventually marry, this means that each woman will have approximately 2.85 children. The projections used in this report are based on the results ofv these national surveys and are consistent with them. Projections of births are obtained as follows: If there are say 1000 women age 15-19 in 1960 and an expected 950 women age 20-24 in 1965 (based on the mortality and migration assumptions) then based on the birth assumptions it is relatively simple to determine their prediction. Calculations are carried out for all women in the childbearing ages. This procedure generated a prediction of births for the State of Michigan for the Period 1960-1965 that was less than one per cent difference from the actual number. Once births have been projected for each five year period, they are "survived" to age 0-4, migrants added or subtracted and we are ready to repeat the entire operation for the next five year period. APPENDIX B PROCEDURES USED IN OBTAINING PROJECTIONS APPENDIX B-l PROCEDURES USED IN OBTAINING PROJECTIONS FROM THE COHORT-SURVIVAL RATIO METHOD 1. Obtain the actual number of births by place of residence by year for the period 1950 through 1955. 2. Obtain actual enrollment figures by grade for kindergarten through twelfth grade for the years 1955 through 1960. 3. Compute the average percentage of survivorship between the number of births five years earlier and kindergarten enrollment thusly: AB-K - K1955 + K1956 + K1957 + K1958 + K1959 + K1960 B1950 B1951 B1952 B1953 B1954 B1955 N where: AB-K = Average percentage of survivorship from birth to kindergarten. K = Kindergarten enrollment in a given year. B = Births in a given year. N = 6 4. Compute the average percentage of survivorship between successive grades. The computation of the average percentage of survivorship from kindergarten to grade one is illustrated below. All other needed average percentages of survivorship are computed in a simi— lar manner, except obviously using different input variables. AK-l = F1956 + F1957 + F1958 + F1959 + FI960 K1955 K1956 K1957 K1958 K1959 N where: AK-l = Average percentage of survivorship from kindergarten to grade one. F = First grade enrollment in a given year. K = Kindergarten enrollment in a given year. 138 E'-‘ 5. 7. N = 5 139 The procedures outlined in steps 3 and 4 yield thirteen separate ratios. AB-K AK—l = " A1-2 = " A2—3 = " 3-4 = " 4-5 = " 5-6 = " 6-7 = " 7-8 = " >>|>>> A8-9 = " A9-10 " A10-11 " A11-12 " Average percentage of survivorship from birth to kindergarten. " kindergarten to first grade first to second grade second to third grade " third to fourth grade " fourth to fifth grade " fifth to sixth grade sixth to seventh grace " seventh to eighth grade " eighth to ninth grade " ninth to tenth grade " tenth to eleventh grade " eleventh to twelfth grade Determine the mean number of births by place of residence for the years 1952 through 1960, 1948 through 1951, 1955 through 1963, and 1951 through 1954. Those dates correspond to the birth dates of the children in elementary school in 1965, in secondary school in 1965, in elementary school in 1968, and in secondary school in 1968, respectively. Symbolically: EB1965 Mean number of births by place of residence for the years 1952 through 1960. SBI965 Mean number of births by place of residence for the years 1948 through 1951. E31968 Mean number of births by place of residence for the years 1955 through 1963. SB1968 - Mean number of births by place of residence for the years 1951 through 1954. An example of the procedures used in projecting elementary and 140 secondary enrollment to 1965 follows. a. b. 'EB1965 x AB-K = K1965 K = kindergarten K1965 x AK-l = F1965 F = first grade F1965 x A1-2 = 81965 S = second grade 81965 x A2_3 = T1965 T = third grade T1965 x A3_4 = Fo1965 Fo = fourth grade Fo1965 x A4-5 = F11965 Fi = fifth grade F11965 x A5-6 = 811965 Si - sixth grade 811965 x A6-7 = Se1965 Se = seventh grade Se1965 x A7_8 = E1965 E = eighth grade Sum the nine separate estimates to obtain a projection of total elementary enrollment for 1965. .Ip order to obtain a projection of secondary enrollment, take SB and recurse through the identical process outlined in steps (a) through (1). Once an estimate of eighth grade en- rollment has been obtained in this manner, use that estimate as a basis for obtaining estimates of the various components of spcondary enrollment thusly: (E1965 = eighth grade enroll— ment 1. N = ninth grade E1965 x A8-9 = N1965 2. Te N1965 x A9_10 = Te1965 tenth grade 3. Te E1 1965 x A10_1l= E11965 eleventh grade 4' E11965 x A11-12= Tw1965 5. Sum the four separate estimates to obtain a projection of total secondary enrollment in 1965. Tw twelfth grade 8. Follow the same procedure, but input EB' and SB. respectively, to obtain projections of elementary and lgggondary enrollment for 1968. APPENDIX B-2 PROCEDURES USED IN OBTAINING PROJECTIONS FROM THE GRADE-RETENTION RATIO METHOD Obtain the actual enrollment figures by grade for kindergarten through twelfth grade for the years 1955 through 1960. Combine the enrollment figures into a sum of kindergarten through grade eight and grade nine through grade twelve figures. Compute the average percent increase in elementary school enroll— ment. APIE =%1955+ $331956 + LEE-1957 + %%1958 + $31959 1956 1957 1958 1959 1960 N where: APIE = Average percent of increase in elementary enroll- ment by year for the years 1955 through 1960. EE = Elementary enrollment in a given year. N = 5 Compute the average percent increase in secondary school enrollment in a like manner. a. PEE _ 1965 - (5 x APIE) x EE1960 b. PSE1965 = (5 x APIS) x SE1960 c. PEE1968 = (8 x APIE) x EE1960 d. PSE1968 = (8 x APIS) x SE1960 where: EE = Elementary enrollment in a given year. APIE= Average percent increase in elementary enrollment by year for the years 1955 through 1960. SE = Secondary enrollment in a given year. APIS= Average percent increase in secondary enrollment 141 142 by year for the years 1955 through 1960. P = When a P appears in front of EE or SE, it signifies projected enrollment. APPENDIX B-3 PROCEDURES USED IN OBTAINING PROJECTIONS FROM ENROLLMENT-RATIO METHOD I. Obtain population figures for total city population for the years 1955 through 1960. Obtain actual elementary and secondary enrollment figures for the years 1955 through 1960. Compute the average of the percents which elementary and secondary enrollment are of total city p0pu1ation. Apply those average percentages to the future estimated total city population for the years 1965 and 1968 to obtain projections of e1- ementary and secondary enrollment for 1965 and 1968. An example of the formula, projecting elementary enrollment for 1965, looks as follows: PEE1965 = EE1955 EE1956 EE1957 EE1958 EE1959 EE1960 TP1955 + TP1956 + TP1957 + TP1958 + TP1959 + TP1960 N x ETP1965 Where: EE = Elementary enrollment in a given year. TP = Total city population in a given year. N = 6 ETP = Estimated total city p0p81ation in a given year (1965). P = When a P appears in front of EB or SE, it signi- fies projected enrollment. 143 APPENDIX B-4 PROCEDURES USED IN OBTAINING PROJECTIONS FROM ENROLLMENT-RATIO METHOD II Obtain population figures for total county population for the years 1955 through 1960. Obtain actual elementary and secondary enrollment figures for the years 1955 through 1960. Compute the average of the percents which elementary and secondary enrollment are of total county population. Apply those average percentages to the future estimated total county population for the years 1965 and 1968 to obtain projections of elementary and secondary enrollment for 1965 and 1968. An example of the formula, projecting secondary enrollment for 1968, looks as follows: PSE1968= SE1955 SEI956 SE1957 SE1958 SE1959 SE1960 ————+ '———- +-———- +———— +—— +—— TP1955 TP1956 TP1957 TP1958 TP1959 TP1960 N x ETP1968 where: SE = Secondary enrollment in a given year. TP = Total county population in a given year. N = 6 ETP = Estimated total county population in a given year. (1968) P = When a P appears in front of EB or SE, it signifies projected enrollment. 144 APPENDIX B—5 PROCEDURES USED IN OBTAINING PROJECTIONS FROM THE PUPIL-HOME UNIT RATIO METHOD Obtain figures on the number of home units by city for the years 1955 through 1960. Obtain actual elementary and secondary enrollment figures for the years 1955 through 1960. Compute the average of the percents which elementary and secondary enrollment are of the number of home units. Apply those average percentages to the future estimated total number of home units for the years 1965 and 1968 to obtain projections of elementary and secondary enrollment for 1965 and 1968. An example of the formula, projecting secondary enrollment for 1965, looks as follows: PSE1965 = SE1955 SE1956 SE1957 SE1958 SE1959 SE1960 —————+————+———+———+-—-——+——— HU1955 HU1956 HU1957 HU1958 HU1959 HU1960 N X EHU1965 where: PSE = Projected secondary enrollment in a given year. (1965) SE = Secondary enrollment in a given year. HU = Home units in a give year. N = 6 EHU = Estimated housing units in a given year. (1965) 145 APPENDIX B-6 PROCEDURES USED IN OBTAINING PROJECTIONS FROM TIME ANALYSIS I Obtain actual elementary and secondary enrollment figures for the years 1955 through 1960. Utilize the values 1955 through 1960 as values of the predictor variable in establishing each of the two regression equations re— quired. Use the values corresponding to elementary enrollment in the years 1955 through 1960 as the values of the criterion variable in one of the equations, and the values corresponding to secondary enrollment in those years as values of the criterion variable in the other. Each year on the x axis is paired with the correspond- ing value of elementary or secondary enrollment on the y axis in defining the needed parameters of the regression equations. Utilizing the least-squares criterion, two equations of the form y = a + bx are obtained. (One equation to be used in projecting future elementary enrollment, and one to be used in projecting future secondary enrollment.) For example, in projecting elemen- tary enrollment for the year 1965, the least-square equation would be of the following form: PEE1965 = a + b (1965) where: PEE = Projected elementary enrollment in a given year. (1965) a = The y intercept of the line defined by the relationship between time and elementary enrollment for the years 1955 through 1960. b = The slope of the line defined by the relationship between time and elementary enrollment for the years 1955 through 1960. 1965 to which it is desired to project enrollment. 146 "2| The value of the predictor variable as defined by the year APPENDIX B-7 PROCEDURES USED IN OBTAINING PROJECTIONS FROM TIME ANALYSIS II 1. Obtain actual elementary and secondary enrollment figures for the years 1955 through 1960. Transform those figures to logarithms to the base ten. 2. Utilize the values 1955 through 1960 as the values of the predictor variable in establishing each of the two regression equations re- quired. Use the values corresponding to the logarithms of elemen- tary enrollment in the years 1955 through 1960 as the values of the criterion variable in one of the equations, and the values corresponding to the logarithms of secondary enrollment in those years as the value of the criterion variable in the other. Each year on the x axis is paired with the corresponding value of the logarithm of elementary or secondary enrollment on the y axis in defining the needed parameters of the regression equations. 3. Utilizing the least-squares criterion, two equations of the form Y - a+bx are obtained. (One equation to be used in projecting future elementary enrollment, and one to be used in projecting future secondary enrollment.) For example, in projecting secondary enrollment for the year 1965, the least-squares equation would be of the following form: PSE1965 = a + b (1965) where: PSE - The logarithm of projected secondary enrollment in a given year. (1965) a = The y intercept of the line defined by the relationship between time and the logarithms of secondary enrollment for the years 1955 through 1960. b = The slope of the line defined by the relationship between time and the logarithms of secondary enrollment for the years 1955 through 1960. 1965 The Value of the predictor variable as defined by the year to which it is desired to project enrollment. 4. Convert the logarithm of projected enrollment for a given year into the projected enrollment figure for that year. 147 . ". AA 13 -5"! VA; 1 APPENDIX B—8 PROCEDURES USED IN OBTAINING PROJECTIONS FROM TRANSITION ANALYSIS Obtain the actual number of births by place of residence by year for the period 1950 through 1955. Obtain actual enrollment figures by grade for kindergarten through EEK twelfth grade for the years 1955 through 1960. * Utilizing the least-squares criterion, compute the following thir- teen regression equations with predictor and criterion variables as specified. Predictor (enrollment) Criterion (enrollment) rfi a. Births, 1950-1955 Kindergarten, 1955-1960 b. Kindergarten, 1955-1959 First Grade, 1956-1960 c. First Grade, 1955-1959 Second Grade, 1956-1960 d. Second Grade, 1955-1959 Third Grade, 1956-1960 e. Third Graed, 1955-1959 Fourth Grade, 1956-1960 f. Fourth Grade, 1955-1959 Fifth Grade, 1956-1960 g. Fifth Grade, 1955-1959 Sixth Grade, 1956-1960 h. Sixth Grade, 1955-1959 Seventh Grade, 1956-1960 1. Seventh Grade, 1955-1959 Eighth Grade, 1956-1960 j. Eighth Grade, 1955-1959 Ninth Grade, 1956-1960 k. Ninth Grade, 1955-1959 Tenth Grade, 1956-1960 1. Tenth Grade, 1955-1959 Eleventh Grade, 1956-1960 m. Eleventh Grade, 1955-1959 Twelfth Grade, 1956-1960 The procedures outlined in step 3 yield thirteen separate regres- sion equations of the form Y = a + bx. For example, transition from birth to kindergarten is defined as: Y = a + bx where: Y = Projected enrollment in kindergarten in a given year. a = The Y intercept of the line defined by the relationship between births for the years 1950-1955 and kindergarten enrollment for the years 1955-1960. b = The slope of the line defined by the relationship between births for the years 1950—1955 and kindergarten enrollment for the years 1955-1960. x = The value of the predictor variable, in this case births during a given period. (See step 5.) 148 149 5. Determine the mean number of births by place of residence for the years 1952 through 1960, 1948 through 1951, 1955 through 1963, and 1951 through 1954. Those dates correspond to the birth dates of children in elementary school in 1965, in secondary school in 1965, in elementary school in 1968, and in secondary school in 1968, respectively. Symbolically: .Efil965 = Mean number of births by place of residence for the years 1952 through 1960. lug SB1965 = Mean number of births by place of residence for the years 1948 through 1951. EB1968 = Mean number of births by place of residence for the years 1955 through 1963. f-* SB1968 = Mean number of births by place of residence for the years 1951 through 1954. 6. Define T = a + b (the regression line) for each of the thirteen cases, projected elementary and secondary enrollment for the year 1965 would be computed in the following manner. a. TB-K(EBl965) = K1965 K = kindergarten b. TK-1(K1965) - F1965 F = first grade T1-2(F1965) S1965 S = second grade d. T2-3(Sl965) = T1965 T = third grade e. T3-4(T1965) = Fo1965 Fo - fourth grade f. T4-5(F°1965) = F11965 Fi - fifth grade Si g. T5—6(Fil965— = 811965 sixth grade h. T6_7(Se Se seventh grade 1965) = se1965 i. T7_8(Se E - eighth grade 1965) ’ E1965 Note: Every one of the above equations is of the form Y = a + bx. For example: ( TK—l K1965) = F1965 150 a = The Y intercept of the line defined by the relationship between kindergarten enrollment in the years 1955-1959 and first grade enrollment in the years 1956-1960. b = The slope of the line defined by the relationship be- tween kindergarten enrollment in the years 1955-1959 and first grade enrollment in the years 1956-1960. K1965 - Projected kindergarten enrollment in 1965. F1956 = Projected first grade enrollment in 1965. j. Sum the nine separate estimates to obtain a projection of total elementary enrollment for 1965. k. _Ip order to obtain a projection of secondary enrollment, take SB and recurse through the identical steps outlined in steps (a) through (i). Once an estimate of eighth grade enroll- ment has been obtained in this manner, use that estimate as a basis for obtaining estimates of the various components of secondary enrollment thusly: T8-9(E1965) = N1965 N = ninth grade 2. T9-10(N1965)= Te1965 Te = tenth grade 3. T10_11(Te1965) = E11965 E1 = eleventh grade 4. T11-12(E11965) = Tw1965 Tw = twelfth grade 5. Sum the four separate estimates to obtain a projection of secondary enrollment for 1965. 7. Follow the same procedure, but input E31968 and SB respectively, to obtain projections of elementary and secondary enrollment for 1968 O 1. APPENDIX B-9 PROCEDURES USED IN OBTAINING PROJECTIONS FROM BIRTH ANALYSIS Obtain actual elementary and secondary enrollment figures for the years 1955 through 1960. Obtain the number of births by place of residence for the years E‘T 1938 through 1960. Determine the total number of births during each of the six nine year periods that correspond to the period 5 during which children who were enrolled in elementary school in each of the six years, 1955 through 1960, would have been born. Do likewise for the total number of births during each of the six four year periods that correspond to the period during which chil- r‘ dren who were enrolled in secondary school in each of the six years, 1955 through 1960 would have been born. Utilize the summated births that correspond to the years 1955 through 1960 as the predictor variables, one set for projecting elementary enrollment and the other for projecting secondary en- rollment. Use elementary and secondary entollment, respectively, as the criterion variables. Using the least-squares criterion, two equations of the form Y = a + bx are obtained. (One equation to be used in projecting future elementary enrollment, and one to be used in projecting future secondary enrollment.) For example, in projecting elem- entary enrollment for the year 1965, the following procedure would be followed: a. Define the predictor and criterion variables. Predictor Criterion 1. births 1942 through 1950 1. EE1955 2. births 1943 through 1951 2. EE1956 3. births 1944 through 1952 3. EE1957 4. births 1945 through 1953 4. EE1958 5. births 1946 through 1954 5. EE1959 6. births 1947 through 1955 6. EE 1960 b' PEE1965 = a + b EBI952-1960) where: PEE = Projected elementary enrollment in a given year. (1965) a:= The Y intercept of the line defined by the relation- ship between births and enrollment for the years 1955 through 1960. (as specified above) 151 152 b = The slope of the line defined by the relationship between births and enrollment for the years 1955 through 1960. (as specified above) = The sum of the number of births, 1952 through EB 1952—1960 1960. APPENDIX B-lO PROCEDURES USED IN OBTAINING PROJECTIONS FROM ENROLLMENT REGRESSION ANALYSIS I Obtain actual elementary and secondary enrollment figures for the years 1955 through 1960. Obtain city school-age population figures for elementary and secon- dary school-age p0pu1ation for the years 1955 through 1960. Utilize the values corresponding to elementary or secondary school- age population in the years 1955 through 1960 as the values of the predictor variable, and the values corresponding to elementary and secondary enrollment, respectively, in the years 1955 through 1960, as the values of the criterion variable in formulating the regres- sion equations. Utilizing the least-squares criterion, two equations of the form Y = a + bx are obtained. (One equation to be used in projecting future elementary enrollment, and one to be used in projecting future secondary enrollment.) For example, in projecting secon- dary enrollment for the year 1965, the least-squares equation would be of the following form: = a + b (ESA PSE1965 1965) where: PSE = Projected secondary enrollment in a given year. (1965) a = The Y intercept of the line defined by the relationship between city elementary school-age population and elem! entary enrollment for the years 1955 through 1960. b = The SlOpe of the line defined by the relationship between city elementary school-age population and elementary en— rollment for the years 1955 through 1960. ESA = Estimated city elementary school-age population in a given year. (1965) 153 APPENDIX B-ll PROCEDURES USED IN OBTAINING PROJECTIONS FROM ENROLLMENT REGRESSION ANALYSIS II Obtain the actual enrollment figures for elementary and secondary enrollment for the years 1955 through 1960. Elementary enrollment is the criterion variable in one of the regression equations while secondary enrollment is the criterion variable in the other. Obtain city school-age p0pu1ation figures for elementary and secon— dary school—age population for the years 1955 through 1960. Ele- mentary city school-age population represents one of the predictor variables in one of the regression equations while secondary city school-age population represents one of the predictor variables in the other. Obtain total city p0pu1ation figures for the years 1955 through 1960. Total city p0pu1ation represents a second predictor variable used in both regression equations. Utilizing the least-squares criterion, two equations of the form Y = a + bl x + b x are obtained. (One equation to be used in projecting future elementary enrollment, and one to be used in projecting future secondary enrollment.) For example, in pro- jecting elementary enrollment for the year 1968, the least—squares equation would be of the following form: = a + bl (ESA 8) + b2 (TCP PEE1968 1968) where: 196 PEE = Projected elementary enrollment in a given year. (1968) a = The Y intercept of the line defined by the relationship between elementary enrollment, and city elementary school-age p0pu1ation and total city population for the years 1955 through 1960. b = The weight that city elementary school-age p0pu1ation contributes to the definition of the slope of the regression line. ESA = Estimated elementary school-age p0pu1ation in a given year. (1968) b = The weight that total city population contributes to the definition of the slope of the regression line. TCP = Estimated total city p0pu1ation in a given year. (1968) 154 APPENDIX B-12 PROCEDURES USED IN OBTAINING PROJECTIONS FROM ENROLLMENT REGRESSION ANALYSIS III Obtain the actual enrollment figures for elementary and secondary enrollment for the years 1955 through 1960. Elementary enrollment is the criterion variable in one of the regression equations while secondary enrollment is the criterion variable in the other. Obtain city school-age population figures for elementary and secon- dary school-age population for the years 1955 through 1960. Elem- entary city school-age population represents one of the predictor variables in one of the regression equations while secondary city school—age p0pu1ation represents one of the predictor variables in the other. Obtain total county population figures for the years 1955 through 1960. Total county p0pulation represents a second predictor variable used in both regression equations. Utilizing the least-squares criterion, two equations of the form 1 + b2 x are obtained. (One equation to be used in projecting future elementary enrollment, and one to be used in pro— jecting future secondary enrollment.) For example, in projecting secondary enrollment for the year 1968, the least-squares equation would be of the following form: Y = a + b x PSE1968 where: PSE a TP 3 + b1 (SSA ) + b2 (TP 1968 1968) Projected secondary enrollment in a given year. (1968) The Y intercept of the line defined by the relation- ship between secondary enrollment, and city secondary school-age population and total county p0pu1ation for the years 1955 through 1960. The weight that city secondary sChool-age p0pu1ation contributes to the definition of the SlOpe of the regression line. Estimated secondary school—age population in a given year. (1968) The weight that total county population contributes to the definition of the slope of the regression line. Estimated total county population in a given year. (1968) 155 APPENDIX B—l3 PROCEDURES USED IN OBTAINING PROJECTIONS FROM ENROLLMENT REGRESSION ANALYSIS IV Obtain the actual elementary and secondary enrollment figures for the years 1955 through 1960. Elementary enrollment is the criterion variable in one of the regression equations while secondary enroll— ment is the criterion ovariable in the other. Obtain extrapolated city school-age population figures for elemen— tary and secondary school-age population for the years 1955 through 1960. City elementary school-age population represents one of the predictor variables in one of the regression equations while city extrapolated secondary school-age population represents one of the predictor variables in the other. Obtain extrapolated county school-age population figures for elemen- tary and secondary school—age population for the years 1955 through 1960. County extrapolated elementary elementary school—age popula— tion represents one of the predictor variables in one of the re— gression equations while county extrapolated secondary school-age population represents one of the predictor variables in the other. Utilizing the least—squares criterion, two equations of the form Y = a + b x + b x2 are obtained. (One equation to be used in pro- jecting future eIementary enrollment, and one to be used in project- ing future secondary enrollment.) For example, in projecting elem- entary enrollment for the year 1965, the least squares equation would be of the following form: PEE = a + bl (EECP1965) + b2 (EETP1965) 1965 where: PEE = Projected elementary enrollment in a given year. (1965) a = The Y intercept of the line defined by the relationship between elementary enrollment, and city and county extrap- olated elementary school-age population for the years 1955 through 1960. b = The weight that city extrapolated elementary school-age 1 population contributes to the definition of the slope of the regression line. EECP = City extrapolated elementary school—age p0pu1ation in a given year. (1965) b = The weight that county extrapolated elementary school-age population contributes to the definition of the slope of the regression line. 156 EETP - County extrapolated elementary school—age population in a given year. (1965) 157 APPENDIX B-l4 PROCEDURES USED IN OBTAINING PROJECTIONS FROM HOUSING ANALYSIS Obtain the actual elementary and secondary enrollment figures for the years 1955 through 1960. Elementary enrollment is the crite— rion variable in one of the regression equations while secondary enrollment is the criterion variable in the other. Obtain figures on the number of home units by city for the years 1955 through 1960. Home units represents one of the predictor variables in both regression equations. Obtain figures on State Equalized Valuation by district for the yean31955 through 1960. State Equalized Valuation represents the second predictor variable in both regression equations. Utilizing the least-squares criterion, two equations of the form Y = a + b x + b x are obtained. (One equation to be used in projecting future elementary enrollment, and one to be used in projecting future secondary enrollment.) For example, in project— ing secondary enrollment for the year 1965, the least-squares equation would be of the following form: = W' a + bl CH} ) + b2 (S PSE1965 EV196S) where: 1965 PSE = Projected secondary enrollment in a given year. (1965) a = The Y intercept of the line defined by the relationship between secondary enrollment, and occupied housing and State Equalized Valuation for the years 1955 through 1960. b = The weight that occupied housing contributes to the definition of the slope of the regression line. HV = The number of estimated occupied housing units in a given year. (1965) b = The weight that State Equalized Valuation contributes to the definition of the slope of the regression line. SEV = Estimated State Equalized Valuation in a given year.(1965) 158 APPENDIX B-15 PROCEDURES USED IN OBTAINING PROJECTIONS FROM MULTI-VARIABLE ANALYSIS I Obtain the actual elementary and secondary enrollment figures for the years 1955 through 1960. Elementary enrollment is the criterion variable in one of the regression equations while secondary enrollment is the criterion in the other. Obtain figures on total city population for the years 1955 through 1960. Total city p0pu1ation is one of the predictor variables used in both regression equations. Obtain figures on total county p0pu1ation for the years 1955 through 1960. Total county p0pu1ation is the second predictor variable used in both regression equations. Obtain city school—age population figures for elementary and secondary school-age population for the years 1955 through 1960. Elementary school—age city p0pu1ation represents a third predictor variable in one of the regression equations while secondary city school-age population represents a third predictor variable in the other. Obtain birth statistics for elementary and secondary school-age population for the years 1955 through 1960. (For procedures, see pp. 156-157 of Procedures Used in Obtaining Projections from Birth Analysis.) Elementary births represents a fourth predictor variable in one of the regression equations while secondary births represents a fourth predictor variable in the other. Utilizing the least-squares criterion, two equations of the form Y = a + blx + bzxz + b3x3 + b4x4 are obtained.(0ne equation to be used in projecting future elementary enrollment, and one to be used in projecting future secondary enrollment.) For example, in projecting elementary enrollment for the year 1968, the least-squares equation would be of the following form: a + b1(TCP ) + b2(TP ) + b3(ESA ) + b4(EB PEE1968= 1968 1968 1968) where: 1968 PEE = Projected elementary enrollment in a given year. (1968) a a The Y intercept of the line defined by the relationship between elementary enrollment, and total city p0pu1ation, total county population, city elementary school-age p0pu- 1ation, and city elementary birth statistics for the years 1955 through 1960. 159 EB 160 The weight that total city population contributes to the slope of the regression line. Estimated total city population in a given year. (1968) The weight that total county population contributes to the slope of the regression line. Estimated total county population in a given year. (1968) The weight that city elementary school-age p0pulation contri- butes to the slope of the regression line. Estimated elementary school-age population in a given year. (1968) The weight that city elementary births contributes to the SlOpe of the regression line. Total number of births during the years that contributed students to elementary school in a given year. (1968) APPENDIX B-l6 PROCEDURES USED IN OBTAINING PROJECTIONS FROM MULTI-VARIABLE ANALYSIS II AND MMLTI- VARIABLE ANALYSIS III Multi-Variable Analysis II uses the same procedures as Multi- Variable Analysis II, except that the criterion variables, elementary and secondary enrollment, are transformed into logarithms to the base ten. This of course means that the projections are in logarithmic form and must be transformed into raw numbers. Multi-Variable Analysis III uses the same procedures as Multi- Variable Analysis I, except that all of the criterion and predictor variables are transformed into logarithms to the base ten. Once again the projections are in logarithmic form and must be transformed into raw numbers. 161