AN ENROLLMENTFOREcAsdeMgDEL FOR A; -, * STATEWIDE SYSTEM OF’HIIGHER'EDUCATI'ON f- * Dissertation-for the Degreetlbf. Ph. D. g MICHIGAN; STATE UNIVERSITY . VERNON RUSSELL .HOFFN*ER,'JR. E 1975 , E , w TTTTTT LIBRARY Midligan State University This is to certify that the thesis entitled An Enrollmnt Forecasting Model for a Statewide System of Higher Education presented by Vernon Russell Hoffner, Jr. has been accepted towards fulfillment of the requirements for Ph.D. degree in Management flatgflmfl€ 412x Maj-sf pro ossfe fl/ / , Date [Q/ZI/7j x 0-7639 ABSTRACT AN ENROLLMENT FORECASTING MODEL FOR A STATEWIDE SYSTEM OF HIGHER EDUCATION By Vernon Russell Hoffner, Jr. This research is addressed to the problem of providing ac- curate forecasts of the statewide enrollments needed in all levels of higher education planning. Financial planning for higher education at the institutional and state levels requires some estimate of future enrollments. In addition, enrollment projections are used as a basis for the general planning and coordination of educational activities at both the institutional and state decision making levels. A Markov student flow model was used to generate the enroll- ment forecasts described in this dissertation. This method has two advantages over other types of enrollment forecasting. The first is that in addition to generating total statewide enrollment forecasts, the model can be easily expanded to differentiate between groups of institutions or classifications of students. For example, forecasts were generated for several different groups of baccalaureate institu- tions and community colleges. Forecasts of community college enroll- ments were also separated into transfer and non-transfer classifications. These classifications were used to distinguish between the group of Vernon Russell Hoffner, Jr. students most likely to transfer to a baccalaureate institution and those students enrolling in the growing area of vocational-technical programs. The second advantage is that it uses aggregate enrollment data in order to estimate the substochastic Markov transition matrix for the enrollment forecasting model. The estimation technique uti- lizes quadratic programming to obtain a least squares estimate of the transition probabilities. This is an improvement over the maximum likelihood technique which is normally used to estimate Markov transi- tion probabilities. The maximum likelihood technique requires indi- vidual student data for the estimation process, which would be very expensive and impractical-—if not impossible--to obtain when applied to a statewide system of education. The historical data required by the quadratic programming procedure has been and is currently collected by several agencies, one being the Higher Education General Information Survey. This availability of data means that no additional data col- lection activities are required in order to accurately forecast statewide enrollments. The Markov student flow enrollment forecasting model was validated by using it to generate forecasts that could be compared to actual historical enrollments. The quality of the forecasts was evalu- ated with the use of three measures of the discrepancy between the predicted and actual enrollments. The error measures were the rela- tive error, the average error in units of the forecast, and Theil's Inequality Coefficient. Vernon Russell Hoffner, Jr. The forecasting model is divided into two phases. Phase I forecasts the number of students entering the system of higher educa- tion by applying the Markov student flow concept to the statewide elementary and secondary school system. The application of the model to the forecasting of first-time entrants begins with the forecasting of statewide elementary and secondary enrollments. The flow concept is used to forecast the first-time entrants by tracing the flow of students through the elementary and secondary grade levels and into the system of higher education. The mean difference between the predicted enrollments and the actual enrollments was zero at the .05 level of significance for the two states tested, Michigan and New York. Phase II uses the forecast of the first-time entrants from Phase I as the input into the Markov flow process used to forecast the statewide higher education enrollments. This phase was also validated by comparing predicted enrollment levels with actual historical enroll- ment levels. In addition, two other methods for forecasting higher education enrollments, the population ratio method and a multiple linear regression model, were evaluated and compared with the Markov model. The forecasts generated by the population ratio method were the least accurate. The forecasts generated by the Markov student flow model and the multiple linear regression model were of comparable accuracy and both were more accurate than the population ratio forecasts. In addition to forecasting enrollments, the model can also be used to aid educational administrators in the areas of planning and policy making. The model can be used to explore and answer various kinds of "What if?" questions that might be useful in the process of develop- ing educational policies. AN ENROLLMENT FORECASTING MODEL FOR A STATEWIDE SYSTEM OF HIGHER EDUCATION By E Vernon Russell Hoffner, Jr. A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Management l975 C) Copyright by Vernon Russell Hoffner, Jr. 1975 ACKNOWLEDGMENTS It is not possible to name all of the many individuals who aided in this research effort or to state precisely what each person contributed to the final product, this dissertation. The author would like to thank his dissertation guidance committee for their valued contributions: Dr. Phillip Carter, Asso- ciate Professor of Management Science, Co-Chairman of the committee, for his suggestions and guidance in the final stages of the research and the writing of the dissertation; Dr. Patrick Toole, Assistant Professor of Engineering Design and Economic Evaluation at the Univer- sity of Colorado, Co-Chairman of the committee, for his introduction of this area of research to the author and his insight into the method- ological tools required in this research activity; and Dr; Richard Gonzalez, Chairman of the Department of Management, for his advice and interest. The author would like to express his appreciation to the staff,and particularly to Dr. James Hatcher, of the Michigan Depart- ment of Education, Higher Education Planning and Coordination for their assistance to the author in gathering and understanding the data needed for the research project. Finally, the author eXpresses his deepest appreciation to his wife, Nancy, for her patience and support during the long academic career culminating in this dissertation. ii TABLE OF CONTENTS Chapter Page I. INTRODUCTION ...................... 1 Purpose of the Study ................. 1 Limitations ..................... 5 The Need for Enrollment Forecasts .......... 8 Overview ....................... 12 II. HIGHER EDUCATION ENROLLMENT PROJECTION MODELS ..... 14 Simple Linear Models ................. l4 Ratio Models .................... l4 Linear Regression Models .............. 15 Institutional Models ................. 16 The Oliver Models ................. 17 The NCHEMS Model .................. 20 Michigan State University Model .......... 20 National Models ................... 22 The Gani Model ................... 22 Manpower Models .................. 22 Statewide Models ................... 23 The HEEP Model ................... 23 The Perkins and Paschke Model ........... 24 Summary ....................... 25 III. MODEL FORMULATION ........ . .......... 30 Model Structure ................... 32 Parameter Estimation ................. 36 Entrant Distribution ................ 36 Transition Probabilities .............. 37 Model Validation ................... 40 IV. FORECASTING FIRST-TIME ENTRANTS ............ 43 Use of the Model ................... 45 Methods of Forecasting Elementary and Secondary Enrollments ............... 45 Validation and Evaluation .............. 47 Michigan Elementary and Secondary Enrollments ................... 47 First-Time Enrollments ................ 66 iii V. FORECASTING STATEWIDE ENROLLMENTS ........... 7l Population Ratio Method ............... 72 Population Ratio--Average ............. 73 Population Ratio--Linear Regression ........ 74 Linear Regression ............. . . . . . 79 Statewide Student Flow Model ............. 83 Three Model Comparison ................ 85 Statewide Student Flow Forecasts ........... 90 Estimating P From More Recent Data ........ 94 Expanded Student Flow Model ............. 96 Separating Transfer and Terminal Enrollments ................... 99 Forecasting System Subgroups ............. 103 VI. CONCLUSIONS AND RECOMMENDATIONS ............ 118 Conclusions ..................... 120 Uses of the Model ................. 123 Planning Applications ............... 126 Recommendations ................... 128 APPENDIX AGGREGATE PUBLIC COMMUNITY COLLEGE AND BACCALAUREATE UNDERGRADUATE ENROLLMENTS FOR THE STATE OF MICHIGAN .............. l3l BIBLIOGRAPHY ......................... I33 iv Table 10. ll. LIST OF TABLES Markdv student flow model transition matrix generated from aggregate Michigan public school enrollments, 1964 to 1969 ..................... Markov student flow model forecasts of Michigan public school enrollments for the years following 1969 Markov student flow model statistical summary of Michigan public school enrollments for the years following 1969 .................... Progression ratio transition matrix generated from aggregate Michigan public school enrollments, 1964 to 1969 ....................... Progression ratio forecasts of Michigan public school enrollments for the years following 1969 ....... Progression ratio statistical summary of Michigan public school enrollments for the years following 1969 ......................... Comparison of the Markov student flow and the grade progression ratio methods for Michigan elementary and secondary enrollment forecasts .......... Markov student flow model transition matrix generated from aggregate New York public school enrollments. 1963 to I968 ..................... Markov student flow model forecasts of New York public school enrollments for the years following 1968 ......................... Markov student flow model statistical summary of New York public school enrollments for the years following 1968 .................... Progression ratio transition matrix generated from a aggregate New York public school enrollments, 1963 to 1968 ..................... Page 48 49 52 54 55 56 57 59 6O 61 62 Table Page 12. Progression ratio forecasts of New York public school enrollments for the years following 1968 ....... 63 13. Progression ratio statistical summary of New York public school enrollments for the years following 1968 ......................... 64 14. Comparison of the Markov student flow and the grade progression ratio methods for New York elementary and secondary enrollment forecasts .......... 65 15. Markov student flow model forecasts of Michigan public and private school enrollments and first- time entrants for the years following 1970 ...... 67 16. Markov student flow model statistical summary of Michigan public and private school enrollments and first-time entrants for the years following 1970 ................... . ..... 68 17. Number of live births ................. 75 18. Public community college and baccalaureate under- graduate enrollments for Michigan, 1966-1973 ..... 76 19. Enrollment forecasts based on the average ratio of enrollments to live births 18 to 21 years earlier for the years 1966 to 1970 . . . . . . . . . . . . . . 77 20. Enrollment forecasts based on the linear regression of the ratio of enrollments to live births 18 to 21 years earlier for the years 1966 to 1970 ..... 78 21. Enrollment forecasts from the Michigan Department of Education linear regression model . ......... 82 22. Markov student flow model enrollment forecasts ..... 84 23. Statistical comparison of the three model forecasts . . 86 24. List of public higher education institutions in the State of Michigan ........... . ...... . 91 25. Three state Markov enrollment forecast, 1971 to 1973 . . 93 26. Statistical summary for the three state enrollment forecast, 1971 to 1973 ................ 93 27. Three state Markov enrollment forecast, 1972 and 1973 . 95 vi Table 28. 29. 30. 31. 32. 33.' 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. Statistical summary for the three state enrollment forecast, 1972 and 1973 .............. Three state Markov enrollment forecast for 1973 Statistical summary for the three state enrollment forecast for 1973 ................. Student flow model expanded to include community college vocational enrollments with the transi- tion matrix estimated from 1968 to 1971 data . . . . Statistical summary for the expanded flow model with the transition matrix estimated from 1968 to 1971 data .................... Student flow model expanded to include community college vocational enrollments with the transi- tion matrix estimated from 1968 to 1972 data . . . . Statistical summary for the expanded flow model with the transition matrix estimated from 1968 to 1972 data .................... First institutional subgrouping ........... Enrollment forecast for the first institutional subgroups ..................... Statistical summary for enrollment forecast of the first institutional subgroups ......... Second institutional subgrouping ........... Enrollment forecast for the second institutional subgroups ..................... Statistical summary for enrollment forecast of the second institutional subgroups ........... THird institutional subgrouping ........... Enrollment forecast for the third institutional subgroups ..................... Statistical summary for enrollment forecast of the third institutional subgroups ........... vii Page 95 97 97 99 101 101 102 104 106 107 108 109 110 111 112 113 Table Page 44. Enrollment forecast for the second institutional subgroups including community college vocational enrollments ..................... 114 45. Enrollment forecast for the third institutional subgroups including community college vocational enrollments ..................... 116 viii LIST OF FIGURES Figure 1. Student flows and enrollment levels .......... 2. Application/Technique matrix of enrollment forecasting models .................. 3. The two phase Markov student flow model ........ 4. Forecasts of public community college enroll- ments for the State of Michigan ........... 5. Forecasts of public college and university enrollments for the State of Michigan ........ 6. Multiple linear regression and Markov student flow model forecasts of total public under- graduate enrollments for the State of Michigan ....................... ix 26 35 80 81 89 CHAPTER I INTRODUCTION Purpose of the Study The objective of this research is the development and vali- dation of a'Markovian student flow model, based on aggregate, histori- cal, cross-sectional enrollment data. The model can be used to trace the flow of students into and through a statewide system of higher education. A primary use of the model will be to forecast future en- rollments of the undergraduate populations of the public institutions of higher education within any given State. The enrollment forecasts can be computed by flowing the enrollment populations through the Markov process and calculating the expected.number of students in each of the states of the system at successive points in time. The structure of the model can be as detailed as the data availability permits. At one end of the scale, the system could be aggregated into only three states, one state for the community colleges, and one each for the upper and lower divisions of the baccalaureate institutions. A more detailed structure for the model is presented in Figure 1. The model in this example consists of twelve states, one state for each of the class levels at the complex universities, the state colleges, and the community colleges. The community college is separated into two parts, one for the students taking an academic .mpo>o— aces—pose» was asap» “nausea .p uczuvm zaom zmmxc ; 8-82, 8-8) ._8 £8 ._8 .18 .uo».u.u=. oavrgugao «nope: nag—c o. t: 8.: 2. as: .2 322 amuse... ._8 .18 855 LP 48:8. .1 3:3 322. a“: :2: 48 ._8 8 ~25 “:5 ~25 LL 8:5 ".222. . Eon 13¢... a fits a 558 = 526 = 553 curriculum with the option of transferring to a baccalaureate institu- tion and one for the students on a terminal, vocation or occupational program. The addition of the vocational education states to what might be considered a strictly academic model is the result of the growing number of students enrolled in these terminal programs. There are two kinds of states discussed in this dissertation. The first is the classification states of the Markov student flow model. These states refer to classification of students by class 1evel,institution of attendance, or academic program. The second is the political or geographical State, which is capitalized to differ- entiate it from the Markov state. The terms applying to institutions of higher education are defined in the following manner. A baccalaureate institution is any organization that confers a baccalaureate degree for an appropriate amount of study in a particular field of knowledge. A complex univer- sity is a baccalaureate institution that offers master's, doctoral, and professional degrees in many fields of knowledge. The primary academic emphasis at the complex university is on research and providing an academic environment for the students studying for advanced degrees. The state colleges are baccalaureate institutions that do not offer doctoral and professional degrees. They do offer some master's de- grees, but the primary academic emphasis is upon teaching at the under- graduate level and teaching at the master's level where appropriate. A community college does not offer baccalaureate degrees, only asso- ciate degrees and certificates of study. Community colleges offer programs of study requiring two years, or less, of full time study. These programs may be in the traditional fields of academic study and be acceptable toward a baccalaureate degree at any baccalaureate in- stitution, or the programs may be in the vocational areas, such as; automobile mechanics, electronics, nursing, and dental hygiene. The vocational programs provide the students with immediately salable, occupationally related skills. Conceptually, the model could contain one state for each of the class levels--freshman, sophomore, junior, and senior--at each of the institutions within the system. In addition, the community college classifications could be divided into academic or transfer programs and terminal programs for each of the two class levels at each institution. This would be an important additional classification since it would clearly identify those students which will probably transfer to a baccalaureate institution. In addition, it identifies the portion of the community college enrollment which has experienced the greatest growth during the last few years--the terminal, vocational-technical programs. The resulting model could be quite large with four state classifications for each institution. For example, Michigan has 29 public community colleges and 13 public baccalaureate institutions. Using the more finely divided classification system, the model for public higher education in Michigan would contain 168 states. Other States would have correspondingly large numbers of classification states in the Markov student flow system, depending on the number of institu- tions of higher education within the State. The flow of students into a Markov student flow model is exogenous to the model. This is indicated in Figure l with the box labeled "High School Grads" and the dashed lines represent the flow of high school graduates into the various academic entry states. In order to forecast the number of students enrolled in the system of higher education, the number of students entering the system must be known or forecasted. The flow of students through the elementary and secondary school system of a State can also be modeled with the Markov student flow process. The Markov student flow model will also be used with elementary and secondary enrollments to predict the number of students that will enter the system of higher education. The process of forecasting the number of students enrolled in the statewide system of higher education will be divided into two phases. Phase I forecasts the number of first-time entrants into the system of higher education. This is accomplished by applying the Markov student flow process to the statewide elementary and secondary enrollments. The result is a flow of students through the elementary and secondary school system, through the point of graduation and into the system of higher education. Phase II, the higher education phase, uses the forecast of students entering the higher education system with the Markov flow process to forecast the statewide enrollments. Both phases forecast enrollments on an annual basis, with forecasts being of the Fall enrollments. Limitations The development of a full scale model requires an extensive amount of data concerning the students enrolled at each institution. Due to the amount of data required in the estimation of the model parameters, only data for the State of Michigan were used in validating the higher education phase of the forecasting model. The data which were available for Michigan are quite limited and had to be gathered from various sources. There are three sources of institutional enroll- ment data for all of the institutions of higher education in Michigan. The Michigan Association of Collegiate Registrars and Admissions Officers (MACRAO), an organization of institutional administrative officers involved in the admission and registration of students at their respective institutions, has been collecting and publishing in- stitutional enrollment data for more than twelve years. The Higher Education General Information Survey (HEGIS), a comprehensive survey of institutions of higher education conducted by the United States Office of Education, has been conducted annually since its initial survey in 1966. The Michigan Bureau of Programs and Budget (BPB) has enrollment data for each of the public institutions of higher education as a part of each institution's budget request for the last four budget request cycles. The BPB institutional budget requests provide enroll- ment data beginning with the 1968-69 fiscal year. There are several problems with the data available from these three sources. This is particularly true in regard to the de- velopment of a full scale student flow model. The first problem is that each source has different definitions or methods of counting en- rolled students. This results in different enrollment figures from each source for a given Fall enrollment period. In addition, the counts had different deadlines for submittal and in the case of the MACRAO data the submission date was so early that an accurate count could not be supplied. In many instances, the count of extension stu- dents at various institutions were only estimates because of the early MACRAO reporting date. Another definitional problem results from the changes each source has made over the years in the definitions, methods, and data response forms for the collection of enrollment data. The changes in reporting methods of each information source provides for the possi- bility of error within the data that would not be detectable. The amount of detail within the data or the lack of detail is also a problem in the development of a model. The enrollment data by class level are available from the BPB for only three years. How- ever, the information is not complete for all institutions. Several institutions did not respond with the full detail requested. The HEGIS data are divided into lower division and upper division for undergrad- uates for only the last four years. Only total institutional enroll- ment data were collected by HEGIS in the first survey. The enrollment data for 1967 and 1968 were collected on the basis of undergraduate and graduate student enrollments. The MACRAO data are separated into lower division and upper division for all years since 1966. However, extension students are counted as a separate category and some portion of these students are undergraduates. The most difficult data problem concerns the number of trans- fer students and the institutional sources and destinations of these students. No data have ever been collected on a statewide basis con- cerning transfer students. The only method of obtaining these data would be to go directly to each institution.. This would be very expensive and time consuming since many institutions only have manual files on their students. As a consequence, the procedure to be followed in regard to transfer students will be to estimate the transfer rate, based on the available data. The above problems with data availability may mean that a fully developed forecasting model of student flows for a statewide sys- tem will not be possible. However, the data which are available is sufficient for the estimation of the parameters of a smaller, more aggregated model of the system. The amount of scaling down of the model will depend on the accuracy with which the model parameters can be estimated from the available data. The data limitations may also result in a problem in validat- ing the forecasting model. The underlying assumptions and the structure of the system can be examined from a logical viewpoint. However, with only eight annual observations of the system, it is unlikely that full statistical validation can be accomplished. The Need for Enrollment Forecasts The manner in which students move through the system of higher education--changing majors, leaving the system for undertermined periods of time and then returning, failing and repeating courses, continuing on for advanced and professional degrees, switching enroll- ment from one institution to another--can and does have a significant impact on the planning and management of individual institutions and on the planning and coordination of groups of institutions. Changes in student preferences, economic and social conditions, and require- ments for manpower are influencing the enrollment patterns in higher education. The emergence of community colleges as an important sector of higher education, with an increasing emphasis on occupationally related curricula, is a major influence in the changing educational patterns. In addition, many institutions are making academic changes to accommodate the changed preferences and goals of the incoming stu- dents. The projection of student enrollments has its basis in the budgetary processes used for colleges and universities. Implicitly, if not explicitly, the development of a budget involves some estimate of the enrollment for each institution, and the enrollment in each institutional subunit, during the coming fiscal year. In addition, the flow of students and enrollment projections are used as a basis for the general planning and coordination of activities at both the institutional and state decision making levels. In the past two decades, the planning and management emphasis has been in placing facilities where they are needed most, managing a continually expanding curriculum and enrollments, and obtaining the necessary qualified faculty and staff. In many instances, a signifi- cant amount of the financial support for the growth and expansion of higher education has come from the state governments. The education sector of the economy has experienced a tremendous amount of growth during the 1960's, but the 1970's have brought a termination to this exceptional rate of growth. As a result, some universities and col- leges are finding themselves with facilities which cannot be fully utilized, faculties with fewer than anticipated students, and unantici- pated gaps between resources and demands. This situation has led to 10 a more critical analysis of the system of higher education. At the state level the analysis has shifted to the development of a statewide picture and on long range needs. This is in addition to the analysis, and is used to place in perspective the operational needs and current financial requests of individual institutions. Higher education has become a complex system in which the institutions find themselves interacting at an increasing rate, both by design and by chance. This trend is not likely to be reversed. A large number of students begin their college education at a community or junior college and later transfer to a baccalaureate institution. Programs arise which must be allocated to a few institutions because enrollment is too small for the programs to be economically feasible in all colleges. In this context it is very important to consider the system as a whole, as well as the individual institutions, for to do otherwise entails great risks of creating unnecessary conflicts and ignoring potential complements within the system. Long range projections are especially important when planning capital outlay needs, but they are also useful in a broader sense. To the extent that the state can forecast its future needs in the area of higher education, it can more adequately deter- mine what priorities must be set and what policies must be implemented in order to satisfy those priorities. The structure of the system of higher education has under- gone some significant changes during the past two decades. Higher education underwent a period of slow growth during the 1950's, followed by a period of rapid growth during the 1960's, which was followed by a period of almost no growth in the early 1970's. During this time 11 the number and types of institutions grew and changed. In addition, not only did the enrollment distributions change within the system of higher education, but the goals of the students also changed. Initially, the purpose of the community college was to provide the first two years of a student's education toward his baccalaureate degree. Now the com- munity colleges are beginning to provide more terminal educational opportunities, particularly in the vocational and technical areas, rather than areas of transfer education. The pressures of these changes in the system of higher educa- tion and the greater competition for scarce financial resources have caused many institutions and state planning agencies to turn to plan- ning and management systems. These planning and management systems provide a method of organizing information, utilizing the information with planning tools and techniques for educational administrators, and maintaining a framework for economic decision making in the area of higher education. One of the key inputs in all of the planning and management systems is the projection of student enrollments. Although student enrollment forecasts are important to all planning, there has been little recent work in the area of statewide enrollment forecasting models. The primary reason appears to be that until recently the summation of the individual institutional enroll- ment projections was an accurate forecast of the enrollment within the statewide system. Typically, these projections reflected the con- straints on the capacity of the system and the individual institutions. Enrollment is more a function of capacity than it is of availability of students for most professional and graduate programs within the 12 system of higher education. Because of the capacity constraints, only a limited number of students can be enrolled in the professional and graduate programs. In most instances, there is sufficient educational demand to fill all of the vacant positions. This used to be true for the undergraduate programs, however, it is no longer true. Notwithstanding the difficulties in making accurate enroll- ment projections during this period of change in the system of higher education, there remains a need to organize the data and make as good enrollment forecasts as possible. Without reasonably good enrollment forecasts, the limited financial resources of the state governments could be budgeted and allocated to educational areas where an educa- tional demand might not materialize. The model proposed in this dis- sertation is an attempt to bring to bear new concepts and techniques on statewide enrollment forecasts in a systematic way. Overview Chapter II describes several of the techniques that have been used to analyze and forecast student enrollments. The early re- search was directed toward identifying the causitive factors influenc- ing higher education enrollments. The more recent research has been directed toward the development of institutional enrollment forecasting models. Many of these later models have a Markov structure. Chapter III discusses the application of a Markov student flow model to a statewide system of higher education. The statewide Markov student flow model is formulated. The use of quadratic program- ming for the estimation of the Markov transition probabilities is also described. 13 Chapter IV applies Phase I, the first—time entrants fore- casting phase, of the model to forecasting elementary and secondary enrollments. The results indicate that Phase I is_a valid forecasting procedure. This phase is then.used'to forecast the first-time enroll- ments for the.State of Michigan.' Chapter V completes the development of the higher education student flow model with Phase II, the higher education enrollment pro- jection phase. Forecasts of the total enrollments for the State of Michigan are generated up to 1980. These forecasts are compared to the forecasts generated by two other methods, the population ratio method and a multiple linear regression model. Following the compari- son and evaluation, the Markov model is expanded to differentiate between several different groupings of institutions. The enrollment forecasts for these institutional groups are compared with actual historical enrollments and evaluated. CHAPTER II HIGHER EDUCATION ENROLLMENT PROJECTION MODELS The importance of enrollment models and projections is the result of the need for accurate forecasts in the planning at all levels of higher education. As a consequence, many individuals and organiza- tions have constructed enrollment projection models. The Federal Gov- ernment, several educational research organizations, probably every state, and nearly every college make some attempt to forecast enroll- ments. This chapter will review the models that have been used to forecast enrollments and those models that have been proposed as enroll- ment projection models. The review will be organized into four sections: 1) simple linear models, 2) institutional models, 3) national models, and 4) statewide models. In addition, some of the forecasting models are submodels within larger resource allocation models. The remainder of the models are used to provide inputs into some resource allocation decision making process. Simple Linear Models Ratio Models The model used by the United States Office of Education to project enrollments is representative of the early models used to forecast enrollments.1 The ratio of enrollments to an age group con- sidered to be the most typical among college students, usually the 18 14 15 to 21 year old age group in the case of undergraduates, is computed for several recent years. The trend in this ratio is then assumed to con- tinue at a constant pace. This assumption, in combination with a projection of the population of the age group, is the basis for the enrollment forecasts. The resultant forecast is the product of the projected population multiplied by the projected rate of enrollment. This approach is also used by the Michigan Department of Education.2 This method has several benefits: the computation of the forecast is quite easy, the method is simple to explain to policy makers, and the data requirements are minimal. However, the simplicity of this method -does not take into consideration any factors that may cause the enroll- ment to vary other than the age group most likely to attend college. Linear Regression Models Economists have identified several factors that have had an influence on the level of enrollments.3 Linear regression analysis was the primary method used in the identification process. In many of the studies, the observations were from national data over periods of twenty or more years. The results of the various investigations were not used to forecast future college enrollments, but just to ex- plain the changes in enrollment levels. In addition to confirming the size of the relevant population group as a significant factor, it was determined that aggregate demand for higher education in the United States was positively related to disposable income per family and 4 negatively related to tuition rates. It was also verified that the discharges and changes in the size of the armed forces also affected 16 the total United States college enrollments.5 Other factors relevant to the decision to enroll in an institution of higher education that have been identified are parental characteristics, student ability, student financial aid, unemployment, and student location.6 Although the above reported studies were not oriented toward the development of enrollment projection models, their results were used in the development of a linear regression forecasting model of 7 The factors considered college enrollments for the State of Michigan. important within the forecasting model were the number of people in the 18 to 21 year old age group, the number of people discharged from the armed forces, income levels, and the unemployment rate. Institutional Models Most of the enrollment projection models are based on the analysis of cross-sectional enrollment data. The structure of the model and the values of the model parameters are the result of the observation of the number of students in particular class levels at a given point in time. The flow rates of the students from one class level to another is only observed over one time interval. No attempt is made to track the flow of students over time periods of more than one time interval. Another basis for enrollment projection models is the analysis of cohort data. The model parameters are estimated by observ- ing the paths of students in a selected cohort as they progress through their academic programs. A cohort is a group of students all having a common characteristic. Usually the characteristic that is common to 17 the group of students is the time of entry into the academic system. For example, the students entering Michigan State University as fresh- men during the Fall, 1975 term would be a cohort. The first institu- tional model described below is based on the use of cohort data; the following two institutional models use cross-sectional data for esti- mating the model parameters. The Oliver Models The cohort survival technique is used in the model developed over a period of years at the University of California by Oliver, §t_al,8 The cohort survival technique consists of determining the pass and failure probabilities for a given cohort of students as they progress through the educational system. The assumption made in using this approach is that if the cohort is selected correctly, the result- ing pass and fail probabilities will be constant from one time period to the next. The earliest form of this model used was the Grade Pro- gression Ratio method. The method generated progression ratios for each level within the university. The model has the following mathe- matical form.9 z](t+l) = a],] z](t) + y](t+l) zj+](t+l) = aj,j+l z(t) + yj+](t+l) where a1 1 is the fraction of freshmen who return to that level in the next time period, aj is the ratio of progression for the students th .j+l to the j+lst level in the next time period, and th moving from the j yj(t) is the number of new admissions to grade j during the t period. 18 th period.* yj(t) is the number of new admissions to grade j during the t These ratios are used to predict the enrollments, zj(t), for the future by the above first order difference equations. The application of the Grade Progression Ratio method to the four undergraduate class levels results in10 z(t+1) = Az(t) + y(t+l), (2.1). where z(t) is the enrollment vector, y(t) is the admissions vector, and A, the progression ratio matrix is given by 6110 o 07 a120 o o O a O 0 23 O 0 a O L. 34 ._J The original Grade Progression Ratio method does not include explicitly the possibility of a student remaining in the same class level. Oliver extended the model to include this possibility. The model structure remains the same as Equation 2.1, except that all the diagonal elements of A will be nonzero. This change in the model structure results in 6‘no 0 0 a12 at22 o o * The notation used in various equations through this disserta- tion will consist of capital letters for matrices and lower case let- ters for vectors. A matrix element will be indicated by a double subscript. Vector elements will be indicated by a single subscript. The letter t will always be used in parenthesis to signify a point in time for the associated vector or matrix. 19 " Later versions of the model have the same form as Equation 2.1, but they have been expanded from four states to five states and later to eight states. The concept of the amount of work that the student has performed toward his degree is also included in these 11 models. The student makes a decision to either attend, vacation, or drop out of school, conditioned on the amount of work completed toward his degree. As a result of this expansion, the five state model contains the four class levels of the earlier model plus one state for those students that are vacationing.12 The last formulation of this model was expanded to include 13 the graduate levels at the university. In addition, the model was formulated so it could be used for planning the resource requirements at the university. The eight cohorts identified in the eight state flow model for the whole university were:14 Students who complete studies in the lower division. Students who graduate with a bachelor's degree. Students who graduate with a master's degree. Students who graduate with a doctoral degree. Lower-division students who drop out. Upper-division students who drop out. Students working toward a master's degree who drop out. Students working toward a doctoral degree who drop out. mNOSU'I-DwN-d The output of this sector of the model was used as input to a second sector that determined the number of instructional staff re- quired in each of the following categories:15 1. Teaching assistants--these must be registered graduate students and may therefore come from cohorts 3, 4, 7, or 8. 2. Nontenured regular faculty--these may be doctoral graduates (cohort 4) or they may come from external sources. 3. Tenured regular faculty, none of whom come directly from stu- dent cohorts. 20 The NCHEMS Model The Oliver models started from the point of developing an adequate student enrollment projection model and then progressed toward the development of a resource requirements prediction model. Several groups have taken the reverse approach. They have developed resource requirements prediction models and one submodel or sector of the model predicts the level of enrollments. A widely publicized resource re- quirements prediction model has been developed at the National Center for Higher Education Management Systems (NCHEMS) at the Western Inter- 16 state Commission for Higher Education. The enrollment sector of this model separates the student flow process into two major compo- ]7 The first component produces, from historical institutional nents. data, estimates of the number of applicants for admission and the number of admitted students who enroll at the institution. The second component models the flow of the students through the institution. It uses the new enrollment provided by the admissions component and the previous enrollment to project the future enrollment by level and major. The model has the general form x(t+l) = P x(t) + y(t+l), where x(t) is the enrollment vector, y(t) is the admissions vector, and P is the Markov transition matrix. The NCHEMS student flow model was specifically designed for short term projections and to be used as a submodel within the resource requirements prediction model. Michigan State University Model The model developed at Michigan State University by Koenig, et a1. is also a resource allocation model, with the enrollment 21 projection portion as only one sector of the model.18 This particular model is a state-space systems model of the university. The student sector of the model predicts the student population by class level and major area of study at future points in time. This computation is based on past and present enrollments, available financial aids for the students, and the prediction of the incoming student population. This student forecasting model has the same general structure of x(t+l) = P x(t) + y(t+1). The specific formulation of the MSU model of the student sector of the university is: s(t) = P(t) s(t-l) + a(t) n(t) + K](t) g(t) + K2(t) h(t).]g (2.2) Where s(t) is the state vector of the student sector whose components represent the number of students in the various levels of study and areas of education. The vector s(t) depends on the enrollment in the previous year, s(t-l), the enrollment choices of the new students-- represented by the product, a(t) n(t), of the number of new arrivals entering their respective categories--the available assistantships g(t), and the available fellowships, scholarships, and the other fi- nancial aid h(t). The matrix P(t) is the transition matrix that rep- resents the proportion of students moving from one field and level classification to another during one time period and K](t) and K2(t) are the matrices that represent the effectiveness of the various fi- nancial aids in attracting and retaining students. The parameters of Equation 2.2 were estimated and verified with the use of the standard linear regression model. 22 National Models The Gani Model The earliest work on Markov student flow models was done by Gani in studying the enrollments of the universities of Australia.20 Gani's purpose was to develop a model of the university system in Australia that would accurately project the total enrollments in the system and the number of graduates of the system. The assumption for this Markov model is that the pass and repeat rates at the universities can be determined. The basic structure of the Markov model developed by Gani has the same structure as the preceding Markov models,21 x(t+l) = P x(t) + f(t). Cohort data for the years 1955 to 1960 were used to determine the transition probabilities of P. Manpower Models Two other national Markovian student flow models have been developed. A model of the British educational system was developed by Armitage, et_gl, and a model of the Norwegian system was developed by Thonstand.22 Both of these models describe more than just the school enrollments and the educational process. The Armitage model includes state classifications for teachers at all levels of the edu— cational system and a classification for people working outside of the educational system. In this respect it can be classified as a simpli— fied manpower mOdel. 'The'Thonstad model can more readily be classified as a national manpower model, with a sophisiticated educational sector. 23 The Armitage description also includes a detailed discussion of the effects of bottlenecks or constraints within the educational system.23 The bottleneck is the result of not having enough positions for the number of students applying for admission. An example of this is the situation with medical schools in the United States where there are more qualified applicants than there are position available. The effect of this problem on the model is to cause those applicants not accepted to go to the next lower opening within the system. The re- sulting model, including the bottleneck processes, can be used in the planning and decision making activities for the whole educational system. However, at this point the model is no longer Markovian in structure. Statewide Models The HEEP Model A statewide student flow model has been developed by the Office of Program Planning and Fiscal Management in the State of Washington.24 The Higher Education Enrollment Projection Model (HEEP) is purportedly a Markov model concerned with undergraduates. However, the description of the model is not that of a Markov process. It uses matrix notation to describe the status of the system at different points in time, i.e., the number of students in the various states. The use of a matrix structure does not necessarily result in a Markov student flow model. The HEEP model includes the public and private baccalaureate institutions and all the community colleges within the State. The 24 state classifications within the model are one for each of the com- munity colleges and one for each of the four undergraduate classes at the baccalaureate institutions. The model is separated into three components. The entrance component computes the number of new students in the Washington system for each of the classifications. The transfer component calculates the number of transfers from each institution to every other institution within the system and the retention component computes the number for each institution that will remain at that in- stitution. The results of the computations of these components are the number of students enrolled in each classification and for the total system for a given time period. The Perkins and Paschke Model A second model is reported by Perkins and Paschke.25 This model is oriented toward the projection of the cost of operating the statewide higher education system. The model is:26 . . concerned with the development and the use of a simulation model to estimate the costs of various policy alternatives in- volving post-secondary education in a state. The model has been used to investigate the effect upon expenditures of changes in exogenous variables . . . . The model was used to predict the operating and construction costs for each college and university in the State of Indiana. All public and private baccalaureate institutions were included. The student enrollment projection subsector of the model generates pro- jections for each institution. The enrollments are projected using one of three methods, based upon the classification of the institution. The first method uses regression analysis to identify the explanatory 25 variables for each institution in the first group. The second method uses the historical trend to predict enrollments at a second group of institutions. The projections for the remaining colleges and universi- ties were taken from an outside source.27 These enrollment predictions are then used in projecting the future costs of the system of higher education under several different policy alternatives. Summar Many techniques have been used to forecast student enroll- ments at the various levels of application. Figure 2 displays a matrix classification of the techniques and areas of application. Although work has been done on each type of model, much work remains to be done in using the models for forecasting student enrollments. Many of the models have a Markov like structure or are Markov student flow models. The algebraic structure of the Markov model appears to fit well the conceptualization of flow of students through the academic system. The application of the Markov student flow model to the forecasting of statewide enrollments still remains to be pursued. This void is evi- dent from Figure 2, where the vacant intersection of the Markov row and State application column indicates that no reported research in this area was found. The remainder of this dissertation develops and applies the Markov student flow model to forecasting statewide enroll- ments. The validity of the model is tested with enrollment data from academic institutions within Michigan. 26 APPLICATIONS JIEUHIUuu: National State Institutional 0.5. Dept. Mich. Dept. Ratio HEH of Ed. Linear References Mich. Dept. Regression 3,4,5 and 6 of Ed. Gani NCHEMS Markov Thonstad MSU Other Armitage HEEP Oliver Perkins and Paschke Figure 2. Application/Technique matrix of enrollment forecasting models. 27 Chapter II Notes 1U.S., Department of Health, Education, and Welfare, Office of Education, National Center for Educational Statistics, Projections of Educational Statistics to 1970-1980 (Washington, D.C.: Government Printing Office, 1971). 2Michigan, Department of Education, State Plan for Higher Education in Michigan (Lansing, Michigan: 1969). 3Robert Campbell and Barry N. Siegel, "The Demand for Higher Education in the United States," American Economic Review, Vol. 57 (June, 1967), pp. 482-494. Harvey Galper and Robert M. Dunn, Jr., "A Short-Run Demand Function for Higher Education in the United States," Journal of Political Economy, Vol. 77 (September-October, 1969), pp. 765-777. 4 Campbell and Siegel, "The Demand for Higher Education." 5 Function." 6A. J. Corazzini, D. Dungan, and H. G. Grabowski, "Determin- ants and Distributional Aspects of Enrollment in U.S. Higher Education," Journal of Human Resources, Vol. 7, No. 1 (Winter, 1972), pp. 39-59. Stephen A. Heenack and Paul Feldman, "Private Demand for Higher Education in the United States," The Economics and Financing of Higher Education in the United States, a Compendum of Papers sdb- mitted to the Joint Economic Committee, Congress of the United States (Washington, D.C.: Government Printing Office, 1969), pp. 375-395. R. Radner and L. S. Miller, "Demand and Supply in U.S. Higher Education: A Progress Report," American Economic Review, Vol. 60 (May, 1970), pp. 326-334. 7Michigan, Department of Education, Financial Requirements of Public Baccalaureate Institutions and Public Community Colleges (Lansing, Michigan: 1971). 8R. M. Oliver, Models for Predicting_Gross Enrollments at the University of California, Report No. 68- 3, Ford Foundation Research Program in University Administration (Berkeley, California: 1968). ‘ R. M. Oliver and K. T. Marshall, A Constant Work Model for Student Attendance and Enrollment, Report No. 69-1, Ford Foundation Research Program in University Administration (Berkeley, California: 1969 . Harvey Galper and Robert N. Dunn, Jr., "A Short-Run Demand T. K. Marshall, R. M. Oliver, and S. S. Suslow, Undergrad- uate Enrollments and Attendance Patterns, Report No. 4, Univers1ty of California Administrative Studies Project in Higher Education (Berkeley, California: 1970). 28 R. M. Oliver, 0. S. P. Hopkins, and R. Armascost, An Academic Productivity and Planning Model for a University Campus, Report No. 3, University of California Administrative Studies Project in Higher Education (Berkeley, California: 1970). R. M. Oliver and D. S. P. Hopkins, “An Equilibrium Flow Model of a University Campus," Operations Research, Vol. 20, No. 2 (March-April, 1972), pp. 249-264. 901iver, Models for Predicting, p. 5. 101bid., p. 17. 1]Oliver and Marshall, A Constant Work Model, p. l. 12 Marshall, Oliver, and Suslow, Undergraduate Enrollments, 13Oliver and Hopkins, "An Equilibrium Flow Model," p. 253. 141pm. 151b1d. 16David G. Clark, et al., Introduction to the Resource Re- quirements Prediction Model 1.6, Technical Report No. 34A, Natibnal Center for Higher Education Management Systems at Western Interstate Commission for Higher Education (Boulder, Colorado: 1973). 17C. C. Lovell, Student Flow Models: A Review and Conceptuali- zation, Technical Report No. 25, National Center for Higher Education Management Systems at Western Interstate Commission for Higher Educa- tion (Boulder, Colorado: August, 1971). 18Herman E. Koenig, M. G. Keeney, and R. Zemack, A Systems Model for Management Planning, and Resource Allocation in Institutions of Higher Education East Lansing, Michigan: Michigan State Univer- sity, 1968). 191bid., pp. 21-25. 20J. Gani, "Formulae for Projecting Enrolments and Degrees Awarded in Universities," Journal of the Royal Statistical Society, A126 (1963). 2‘Ibid., p. 406. 22Peter Armitage, Cyril Smith, and Paul Alper, Decision Models for Educational P1anning_(London: Allen Lane the Penguin Press,*1969). Tore Thonstad, Education and Manpower (Edinburgh: Oliver and Boyd, 1969)). 23Armitage, Decision Models, p. 30. 29 24Office of Program Planning and Fiscal Management, Hi her Education Enrollment Projection Model (Olympia, Washington: l970) 25William C. Perkins and Paul E. Paschke, "A Simulation Model of the Higher Education System of a State," Decision Sciences, Vol. 4, No. 2 (April, 1973), pp. 194-215. 261bid., p. 195. 271bid., p. 198. CHAPTER III MODEL FORMULATION The models described in the preceding chapter provide examples of the methods that have been used to project future levels of higher education enrollments. Each model could be used in a decision process where funds and resources are allocated to various institutions. How- ever, the decision process relies primarily on the classification of the student population within the system of higher education. This classification of the student population can be defined more specifi- cally as: the modeling of the students' movement through the educa- tional system, the determination of the distribution of the students throughout the educational system, and the description of the students within the educational system. The Markov student flow model satisfies. the needed classi- fication of the student population. The Markov process identifies the flow of students through the system. Each state indicates the de— scription of the students within it and the total number of states identifies the distribution of the students. In addition, the internal structure of the particular application can change without changing the mathematical formulation of the Markov student flow model. Within the basic formulation of the model, the states of the model can be enumerated in any way necessary to accurately and adequately describe a particular statewide system of higher education. One method of enumeration could be the classification of the students within the 30 31 state by their class levels, regardless of the institution they are attending, i.e., freshman, sophomore, junior, senior, or graduate. This would be a rather simple classification scheme. The other extreme of detail could separate the student states into the following types of classifications: (1) class level; (2) institution of attendance; (3) current credit hour load; (4) State of home residence; and (5) major area of study. The class level classifications can be freshman, sophomore, junior, senior, master, doctoral, and other. .Applying the model to the State of Michigan, and including the 47 private colleges, the 29 community colleges, and the 13 public colleges and universities, 89 different institutional classifications are obtained. The student credit hour load can be used to separate the students into part-time and full-time classifications. The residence classifications could be divided into 52 distinct elements, one for each of the United States and the District of Columbia, and one for all foreign students. The major areas of study could result in many different classification elements, a typical breakdown for areas of study could be business, engineering, science, liberal arts, education, and other. If a model containing all of the above detail were developed, it would consist of 388,752 different states.* A model that is highly simplified may not provide enough infor- mation for decision making. On the other hand, a model that is highly *This number of classification states is the product of the 7 class levels, the 89 institutions, the 2 credit hour classifications, the 52 States of home residence, and the 6 areas of study. 32 detailed may provide more information than can be meaningfully used in the decision process. The highly detailed model may not be relevant to the decision process or it may be impossible to collect all the required data. In addition, the cost of data collection for the very detailed model may be prohibitive and far outweigh the benefits that can be derived from the model. Model Structure The mathematical formulation of a statewide student flow model has the same structure as the Markov models in the preceding chap- ter. The formulation of the system of higher education can be stated as x(t) = x(t-l) P + f(t), (3.1) where x(t) is the state vector at time t, f(t) is the vector of entrants to the system at time t, and P is the transition matrix. Each ele- ment of P, pij’ is the probability of moving from state i at time t-l to state j at time t. The elements of the state vector are the numbers of students in a particular class level and institution or group of institutions at a particular point in time. The transition probabilities of P are the probability of movement from one class level/institution state at time t, to another class level/institution at time t+l. In addition, the total number of students in the system can be calculated by N(t) = Z xi(t). 1 where N(t) is the total enrollments at time t. III.IINIIT '1 (i 33 Phase II of the model is now in a form that can be used to project student enrollments at the different institutions and class levels. If the values of f(t) are known or can be forecast, then the successive values of x(t) can be easily computed, assuming that P is known. The forecasts of f(t) can be carried out with a model similar to the one just discussed for the system of higher education. The model for forecasting the first-time entrants has two stages. The first stage forecasts the number of high school graduates. This can be achieved by uSing the student flow model to flow the students through the elementary and secondary school system. The model structure con- sists of fourteen states, representing the grades kindergarten through twelve and the high school graduates. The model formulation is Y(t) = flit-1) Q + k(t) (3-2) where y(t) is the fourteen element state vector at time t, O is the 14 x 14 transition probability matrix, and k(t) is the vector of en- tries into the school system. The result of this process is the potential entrants into the system of higher education. The forecast number of high school graduates is sit) = mm. The next stage of the model projects the number of first- time entrants as a function of the high school graduates. This can be accomplished by letting f(t) = r g(t) 34 where r is the probability vector for high school graduates to enter each of the entrant states of the higher education system. Phase I of the model can now be used to forecast the first- time entrants necessary for Phase II. The structure of the two phases is illustrated in Figure 3. i There are several simplifying assumptions that could be made in the process of estimating the values of various parameters of the model. However, there are at least two assumptions that have to be made. The first is that the system is closed. The data is available only for individual States, but there will be entrants coming from outside the State, and some of the graduates will leave the State to attend college elsewhere. It will be assumed that these two flows counterbalance each other. The second assumption relates to the fact that some students do not enroll in college directly after graduating from high school. It will be assumed that this lagged portion of en- trants does not have a significant effect on the estimation of future entrants. This is due to the assumed balancing effect of the delayed entrances over succeeding time periods. These assumptions are neces- sary since the reported figures for first-time enrollments only indi- cate the total first-time enrollments at each institution. The basic assumptions of Equation 3.1 are that all flows not explicitly stated will be assumed to be either not significant or counterbalanced 'by a flow in the opposite direction. The result of this assumption will be that the transition probabilities for these flows can be assumed to be zero. For example, the flow of students dropping out of school temporarily will be balanced by those students Figure 3. 35 Phase I y(t) = y(t-l) Q + k(t) f(t) = r yum f(’0) .K\\\\\(/////V Phase II x(t) = x(t-1) P + f(t) Higher Education Enrollment Forecasts The two phase Markov student flow model. 36 resuming their studies after a temporary dropout. An example of a flow that would not be significant would be the flow of students from the community college vocational education programs to the state uni- versities. Two assumptions must also be made in the estimation of the parameters of Equation 3.2. The first assumption is that the elements of the vector k(t) are zero except for kindergarten students. The second assumption is that either the students can pass to the next grade level, or they must remain in their current grade level. This implies that O is a diagonal and off diagonal matrix, i.e., q = 0 except where j = i or 3 = 1+1. l:J These assumptions are necessary since the school enrollment numbers only indicate the number of students in each grade level. Although, the above assumptions limit the detail of the forecasts by the model, they allow the development of a forecast from the data. The assumptions and aggregation that are necessary in the development of the model result from the paucity of detailed and ac- curate data. Parameter Estimation Entrant Distribution There are three parameters in the model that have to be esti- mated from historical data. These parameters are the transition ma- trices, P and Q, and the probability vector r. The value of r can be estimated by, 37 T we T t=l t) The probability vector r is just the average of the ratio of flow of the high school graduates into the various entry states for the system of higher education. Transition Probabilities The problem with using the Markov process as a forecasting technique lies in the estimation of the transition probabilities. There are two methods of estimating the Markov transition probabili- ties, the maximum likelihood technique and the least squares method. The maximum likelihood method requires the collection of data at the level of the individual student. The formulation for the estimation procedure for the maximum likelihood technique is T yi,j(t' -l ,t) qi,j = t=2 y.(t-1) 1-1 where yi’j(t-l,t) is the number of students that move from grade i at time t-l to grade j at time t, yi(t-l) is the total number of stu- dents in grade i at time t-l, and qi,j is resulting estimate of the transition probability, averaged over the observed time periods. The use of the maximum likelihood method requires a count of the number of individuals that move from any particular grade level to any other grade level in the succeeding time period. At the statewide level, the collection of the data needed to perform a maximum likeli- hood estimate of transition probabilities would be very difficult and 38 expensive. This makes the use of the maximum likelihood technique for estimating the Markov transition probabilities highly infeasible. Using the least squares technique of estimating the Markov transition probabilities from aggregate data would appear to be the reasonable approach to take. This approach does not use the actual counts of individuals in the various states at different points in time, but rather uses the relative frequency or proportion in each state at the different points in time to estimate the transition proba- bilities.1 However, the underlying assumption is that a transition can be made from any one state to any other state. This assumption is invalid when considering student flow systems where the student remains in his current class level or passes onto the next class level. Neither the maximum likelihood method nor the least squares method is applicable for estimating the Markov transition probabilities when only aggregate data is available and there is a restriction on the possible transitions. The approach that is used in this study to estimate the transition probabilities is a modified least squares method. The actual number in each state of the system is used in the estimation procedure and restrictions on the allowed transitions are permitted. This method utilizes quadratic programming to estimate the Markov transition probabilities. Quadratic programming minimizes a least squares objective function SubjeCt to the Markov probability constraints as constraints.. The resu1ting quadratic programming formulation for Q is 39 min f... II M-i N I [xj(t) - §x1(t-l)qi5 - fjltll2 s.t. I qij 5_l for all j j qij 3_0 for all i,j, where Q is a diagonal and off diagonal matrix with each row having only nonzero elements, q and q. i,i 1,i+l' programming formulation results in the Markov transition probabilities Solving the above quadratic for the transition matrix Q. The quadratic programming formulation for P is T . 2 m1n t§2 E [xj(t) - Exi(t-1) pij — fj(t)] s.t. Z pij §_l for all j J pij 3_O for all 1,3. The structure of P will depend on the structure of the system of higher education being modeled. The quadratic programming problems for P and Q can be solved with the quadratic programming algorithm developed by Wolfe.2 3 The The resulting Markov transition matrix may be substochastic. row probabilities are not required to sum to one, but are allowed to sum to less than one.. This structure of P and Q ignores the ab- sorbing state for students dropping out of the educational system. 40 The quadratic programming method for estimating the transi- tion probabilities has two advantages over the previously described methods. The first is that aggregate data can be used to estimate the transition probabilities. This eliminates the need to collect indi- vidual transition data necessary for the maximum likelihood technique. This is a particularly important advantage when estimating transition probabilities for a large system like a statewide system of higher education. The cost of collecting the individual transition data from each of the schools in the system would be very prohibitive to the use of the model. In many cases it would be necessary to manually search a school's records to obtain the required data. The second advantage is that the transition directions can be limited by the quadratic programming constraints. The least squares method assumes transitions from any one state to any other state. However, a flow model has only a limited number of transitions and these transitions cause the individuals in the system to flow through a particular sequence of states. The maximum likelihood technique could be used to estimate the flow model transition probabilities if the individual data were available. The two advantages of the quad- ratic programming method make it a usable procedure for estimating the transition probabilities for large flow systems where only aggregate flow data are available. Model Validation The general approach to the validation of forecasting models is to use the model to forecast actual observed historical values. This is the approach that will be used in the validation of each of 41 the two phases of the forecasting model. This approach will also be used in the comparison of each phase against other forecasting tech- niques. I When comparing forecasted results against actual observa- tions, it is desirable to be able to establish the quality of the fore— cast. Three measures will be used in the evaluation of the forecasts. The first measure will be the relative discrepancy between the predicted and actual values, where Pi and Ai are the predicted and observed values, and Ri is the relative error for that pair of values. The second measure is the prediction error in units, Pi - Ai’ of the forecast for each pair of predicted and actual values. The third 4 The inequality coefficient measure is Theil's Inequality Coefficient. (U) of the pairs (Pi’ Ai) is * 2 U = (2(P1 ‘ A1) 1/2 2 1. 2A, Forecasts generated by both phases of the higher education enrollment forecasting model will be evaluated with the above three measures which provide an indication of the quality of the forecasts. The mean for each of the first two measures is computed for each fore- cast year. Each mean is tested to determine if it is significantly different from zero. Theil's Inequality Coefficient is also computed for each forecast year. The Coefficient is only used as a method of evaluating the forecasts generated by different methods for a given year. 42 Chapter III Notes 1Henri Theil and Guido Rey, "Quadratic Programming Approach to the Estimation of Transition Probabilities," Management Science, Vol. 12, No. 9 (May, 1966), PP. 714-721. T. C. Lee, G. G. Judge, and A. Zellner, Estimatipgpthe Parameters of the Markov Probability Model from A re ateiTime Series Data (Amsterdam: North-Hollanleublishing Co., 1970 . 2P. Wolfe, "The Simplex Method for Quadratic Programming," Econometrica, Vol. 27, No. 3 (July, 1959), pp. 382-398. George Hadley, Nonlinear and Dynamic Programming(Reading, Massachusetts: Addison-Wesley Publishing Co., Inc., 1964), Chapter 7. 3William Feller, An Introduction to ProbpbilitygTheory and Its Applications, Vol I (3rd ed.; New York: John Wiley & Sons, Inc., 1968), p. 400. 4Henri Theil, Applied Economic Forecasting (Amsterdam: North-Holland Publishing Co., 1966), Chapter 2. CHAPTER IV FORECASTING FIRST-TIME ENTRANTS The Markov models described in Chapter II did not consider forecasting the first-time entrants within the model. They either assumed the first-time entrants would be known, or they forecast the first-time entrants from some other source outside of the Markov stu- dent flow model. This can be a problem. Without having a valid fore- cast of first—time entrants, the use of a Markov type forecasting model, through which students would flow, would be an incomplete type of forecasting. In order to accurately forecast the enrollments for universities and community colleges on a statewide basis, knowledge of the number of students entering the system is a necessary requirement. As Oliver1 has shown, the total enrollment for the baccalaureate insti-1 tutions depends upon the number of entrants to the institutions at a time prior to that particular year. That is, the enrollment in a particular grade or class level in a given year depends upon the en- rollments in the preceding year for the preceding grade. For example, the number of juniors in a selected year is primarily dependent on the number of sophomores in the preceding year. In addition, the number of freshmen in any given year depends significantly upon the number of first-time entrants for that year. This result indicates that in order to accurately forecast the enrollments of a particular institution or group of institutions, the number of first-time entrants into those institutions is of critical importance. 43 44 When considering a full scale statewide model, there are two approaches that can be taken in forecasting the first-time entrants. One approach would be to forecast the individual first-time entrants at each institution and then to sum these to determine the number of first-time enrollments for the total statewide system. The second approach would be to forecast the total number of possible entrants into the statewide system of higher education and then to allocate those entrants amongst the schools on some prorated basis. This pro- rational basis could be determined on the basis of past history plus forecasted growth for each of the various institutions. The second approach appears to be the best when considering a statewide system of higher education. The underlying rationale is that there is a limited number of individuals available to enter the system at any given time. The number of students entering a system of higher education is a ran- dom variable. The sum of the independent estimates of the first-time entrants into the individual institutions could be greatly different from the single estimate of the number of students entering the total system of higher education. Also, we are considering here a somewhat competitive situation between the various institutions. They are all competing in the same market, i.e., they are all competing for the same students who will be entering the system of higher education. In this respect, the estimate of the number of students entering a given institution is not independent of the estimates of the number of stu- dents entering the other institutions. 45 Use of the Model The procedure used for estimating first-time enrollments is similar to the procedure for estimating the total enrollment in the system of higher education. This approach entails the estimation of a substochastic, Markov transition matrix for the flow process of the elementary and secondary school system. Using this approach, we can forecast the number of students in each grade as they progress through a statewide elementary and secondary school system. This method allows us to collect aggregate data at a very low cost which should lead to accurate forecasts of first-time enrollments. The first-time enrollments phase of the enrollment forecast- ing model is tested with enrollment data from Michigan. The statewide enrollments for public and non-public schools have been gathered in the State of Michigan for some period of time. These numbers can be used to estimate the transition probabilities needed for the Markov process in order to forecast the flow of students through the system of elementary and secondary education within the state and this will give us an estimate of the number of students that will be entering the system of higher education. Methods of Forecasting_Elementary and secondary Enrollments Phase I of the student flow model could be used to forecast the elementary and secondary enrollments for a statewide system. The process of validating Phase I of the model compares the results of the elementary and secondary forecasts with the forecasts generated by a different procedure. 46 The method that is currently used for forecasting enrollments on a statewide basis for elementary and secondary schools is the pro- gression ratio technique. This technique has the same basic mathemati- cal structure as the student flow models in Equation 2.1 and Equation 3.2. The progression ratio technique is the method used by various agencies to estimate the transition matrix for the student flow model. It has the form Egyiflgt) ,i+1 = yi(t l) is the estimated flow rate from grade i tograde i+l.“ where qi i+l This technique uses several years_of_enrollment data to estimate the ratio, q between the preceding year's enrollment for a particular i,i+l grade level yi(t-l),‘and the enrollment for the next grade level in~ the succeeding year, yi+1(t). The average of the T-l ratio observa- tions is the estimate of the transition ratio. For example, using five or six years of data for the second and third grades, the ratio of third to second grade enrollments is calculated for each pair of years. The average of these ratios for this time period is calculated and be- comes the estimate of the progression ratio for the flow of students from the second to the third grade. Using this method the progression ratios for all grades, kindergarten through the twelfth, can be esti- mated. The progression ratio technique has the advantage that it is O easy to understand, easy to implement, and provides usable forecasts. 47 An alternative method that can be used to forecast elementary and secondary enrollments is Phase I of the higher education enrollment forecasting model. Eliminating the step that forecasts the first-time entrants into the system of higher education, Phase I becbmes a fore- casting.m0del for Statewide elementary and.secondary enrollments. : These two methods of forecasting statewide elementary and secondary enrollments are compared and evaluated using the procedure outlined in Chapter III. Validation and Evaluation The two forecasting methods were tested with Michigan and New York elementary and secondary public school enrollment data. The primary reason this data is used in the testing procedure was that the data is readily accessible. The Michigan enrollment data is from the Michigan Statistical Abstract.2 The New York enrollment data is from the New York Statistical Yearbook.3 Michigan Elementary and Secondary Enrollments The first test will compare the results of the two methods using the Michigan public school data. Michigan public school enroll- ment data from the years 1964 to 1969 were used to estimate the values of Q. The results of the quadratic programming estimation process appears in Table 1. These values are used in Equation 3.2, y(t) = y(t-l) Q + k(t). (3.2) along with 1969 enrollment data to forecast the enrollments for 1970, 1971, and 1972. These forecasts appear in Table 2. 48 ooo. epm. omo. mew. mmp. mmw. m¢—. Rpm. moo. me. mno. me. moo. mmm. moo. wmo. moo. mmm. moo. Fem. mmo. oww. ooo. omm. ooo. .mom_ on womp .mucmEPFoccm Poosom ow_n=q cmmwgowz mpammcmau socw umpmcmcmm chpms cowpwmcocp _muoE zo_o p:mu=um.>oxcmz .P m4m 3oz mummmcmmm Eoce umumcmcmm x_cums cowpwmcmcp ~mvos 30pm pcwuzgm >oxcmz .m mom 3oz muommcooo Eon» oooocmoom chooe cowpwmoocu ovum; oowmmocooco .FF mom<~ 63 TABLE 12. Progression ratio forecasts of New York public school enroll- ments for the years following 1968. (thousands) YEAR K 1' “2' 3 '4 '5 6 1968 A 285 282 269 263 262 257 A 247 1969 P 276 276 266 264 262 255 A 287 281 276 267 264 261 256 R -.019 .000 -.004 -.001 .002 -.002 1970 P 278 270 273 267 263 260 A 283 284 273 273 267 264 261 R -.022 -.011 .000 -.001 -.002 -.004 1971 P 274 272 267 274 266 262 A 272 283 277 271 272 268 263 R -.O33 -.019 -.015 .007 -.006 -.005 1972 P 263 268 269 268 273 265 A 257 272 275 273 271 271 266 R -.O33 -.025 -.015 -.012 .009 -.005 YEAR 7‘ 8‘ ‘9 '10’ ‘11 12 1968 A 249 234 257 252 225 192 1969 P 255 239 266 256 228 202 A 255 242 270 258 229 200 R .002 -.013 -.014 -.007 -.003 .009 1970 P 264 245 272 265 232 205 A 264 248 275 265 231 203 R .001 -.012 -.012 .001 .005 .009 1971 P 269 253 279 271 240‘ 208 A 269 257 286 275 240 205 R -.001 -.014 -.025 -.015 .002 .016 1972 P 271 258 288 278 245 216 A 269 262 296 282 248 213 R .006 -.016 -.026 -.015 -.010 .013 P=Predicted Values A=Actual Values R=Re1ative Error P-A 64 Table 13. Progression ratio statistical summar of New York public school enrollments for the years fol owing 1968. RELATIVE ERROR STANDARD t YEAR AVERAGE DEVIATION VALUE 1969 -.004 .008 -l.769 1970 -.004 .009 -l.569 1971 -.009 .014 -2.229* 1972 -.011 .014 -2.584* ERRFDICILQN_ERRQRE STANDARD t YEAR AVERAGE DEVIATION VALUE RUNS 1969 -1.125 2.059 -1.893 8 1970 -1.166 2.351 -1.718 6 1971 -2.582 3.721 -2.404* 4 1972 -3.029 3.887 -2.699* 6 Accept the alternative hypothesis that the average forecast error is not equal to zero and that there is a negative bias in the forecast enrollments at the .05 level of significance, n = 12. Thousands Comparing the results of the two forecasting techniques the Markov student flow model does not generate significantly better results than the progression ratio technique for the first three years of the New York forecast. However, the forecasts for the fourth year are different at the .05 level of confidence. Table 14 65 displays the statistical comparisons for the New York forecasts. Theil's inequality coefficients are smaller in each of the forecast years, although the difference between the pairs of coefficients for each forecast year is small. Table 14. Comparison of the Markov student flow and the grade pro- gression ratio methods for New York elementary and second- ary enrollment forecasts. THEIL'S INEQUALITY EQUAL MEAN PREDICTION COEFFICIENT ERROR t-VALUES Year Markov Progression Relative Prediction Ratio error error 1969 .008 .009 1.246 1.204 1970 .009 .010 1.851 2.052 1971 .013 .017 1.969 2.042 1972 .015 .018 2.517* 2.437* *: Accept the alternative hypothesis that the mean prediction errors are not equal at the .05 level of significance, n = 24. The conclusion from the preceding comparison and validation process is that the Markov student flow model generates accurate enrollment projections for statewide elementary and secondary school systems. The comparison of the Markov student flow model with the progression ratio technique did not indicate a clear superiority of the Markov model. Theil's inequality coefficient was smaller for the Markov student flow model in six of the seven forecast years. The student flow forecasts contained a negative bias for only one year, compared to four years for the progression ratio technique. 66 In addition, the prediction error measures were smaller in six of the seven forecast years and there was a significant difference between the fourth year forecasts for New York. The Markov student flow model provides marginally better results than the progression ratio technique. First-Time Enrollments The final application of this chapter is the generation of the first-time enrollment predictions for Phase II.Of_the higher education enrollment projection model. This forecast will use Michi- gan public and non-public, elementary and secondary school enroll- 4 ments, with the first-time entrants to the system of higher education appended to the flow process.5 The modified flow process includes the transition from the twelfth grade to the state of enrolling in a university or community college. The total statewide elementary and secondary enrollments are used in forecasting the first-time entrants because the non-public students constitute an important factor in school enrollments. This portion of the school enrollment should be included in the prediction process when the first-time entrants are forecast. Including the non-public enrollments identifies the total potential higher education entrants. The forecast of first-time entrants and total elementary and secondary enrollments is displayed in Table 15. The forecast was generated from the base year, 1970, with the use of a transition matrix Q estimated with enrollment data from the years 1967 to 1970. Al- though the forecast can be compared with only two years of actual 67 TABLE 15. Markov student flow model forecasts of Michigan public and private school enrollments and first-time entrants for the years following 1970. YEAR K' '1 =2' 3“ 4' 5 ’6‘ 1970 A 185645 195974 192220 194606 191901 190358 194789 1971 P 192896 189492 187799 193019 189790 189499 A 173708 191512 190936 189501 193410 190330 189747 R .007 -.008 -.009 -.002 -.003 -.001 1972 P 180845 186517 185134 186354 190896 188633 A 167462 179595 186608 185740 187682 191522 188898 R .007 -.000 -.003 -.007 -.003 -.001 1973 P 174153 174900 182227 183666 184304 189606 1974 P 168409 170877 180785 181646 183538 1975 P 164536 169619 178796 180677 1976 P 163275 167753 177844 1977 P 161479 167393 1978 P 160889 1979 P 1980 P lst TIME YEAR _ 7‘ 8 ‘9 10‘ 11 '12 ENTRANTS 1970 A 194155 188125 182744 177400 158602 142758 79689 1971 P 196690 191828 186867 181951 163208 144962 83085 A 195626 192116 188544 181723 163688 144786 85697 R .005 -.002 -.009 .001 -.003 .001 -.O3O 1972 P 191927 194412 190576 186100 167395 149172 84368 A 190806 192702 192003 186374 167469 148104 83544 R .006 .009 -.007 -.001 -.000 .007 .010 1973 P 190763 190166 193213 189836 171212 152999 86818 1974 P 191588 188817 189395 192557 174650 156488 89045 1975 P 185969 189494 187907 189269 177152 159630 91076 1976 P 182875 184335 188453 187640 174128 161917 92904 1977 P 179995 181140 183656 188011 172629 159153 94236 1978 P 169999 178268 180383 183637 172970 157783 92627 1979 P 163172 168828 177501 180282 168946 158095 91830 1980 P 161902 168496 177369 165860 154417 92011 P=Predicted Values A=Actual Values R=Re1ative Error P-A 68 enrollments, the quality of this forecast can be evaluated in the same manner as the previous forecasts. However, the evaluation pro- cess is expanded to include the first-time entrants. The statistical evaluation of the predicted enrollments compared with the actual en- rollments is summarized in Table 16. Table 16. Markov student flow model statistical summary of Michigan public and private school enrollments and first-time en- trants for the years following 1970. RELATIVE ERROR STANDARD t YEAR AVERAGE DEVIATION VALUE 1971 -.004 .009 -1.522 1972 .001 .006 .647 PREDICTION‘ERRORT. THEIL'S STANDARD t INEQUALITY YEAR AVERAGE DEVIATION VALUE RUNS COEFFICIENT 1971 -502 1129 -1.604 8 .007 1972 98 1005 .351 5 .007 t Headcount Note: The null hypothesis that the average forecast error equals zero is accepted at the .05 level of significance, n = 13, for each of the forecast years. We now have a method of forecasting the first-time enroll- ments for a statewide system of universities and community colleges. As the elementary and secondary students flow through the lower school 69 system, they become potential entrants to the system of higher educa- tion. Therefore, we can apply the same flow technique to the estima- tion of first-time enrollments that we used to estimate the enrollments in the various grades in the lower school system. Now that we have the forecast first-time entrants into the system of higher education, we can use these estimates in the process of forecasting the enrollments for the system of higher education. 70 Chapter IV Notes 1R. M. Oliver, Models for Predicting_Gross Enrollments at the University of California, Report No. 68-3, Ford Foundation Re- search Program in University Administration (Berkeley, California: 1968 . 2Michigan State University, Graduate School of Business Administration, Michigan Statistical Abstract (10th ed.; East Lansing, Michigan: 1974). 3New York State, Division of the Budget, Office of Statisti- cal Coordination, New York State Statistical Yearbook (Albany, New York: 1973), p. 201 4Michigan State University, op. cit., p. 157. 5The first-time enrollment data is part of HEGIS data mentioned in Chapter I. CHAPTER V FORECASTING STATEWIDE ENROLLMENTS Forecasting statewide enrollments for higher education can now be attempted using the results of the preceding chapter. Using the forecast of first-time enrollments generated by the flow of stu- dents through elementary and secondary school system, future statewide enrollments at the undergraduate level can be projected. Before the results of the flow process are examined, let us look at the results' of two other methods. Both methods are applied to the forecasting of the total enrollments of public community colleges and baccalaureate institutions in the State of Michigan. The first is the population ratio method. The second is a multiple, linear regression model used by the Michigan Department of EducatiOn for forecasting enroll- ments. A time period of five years was selected for the estimation of P in the Markov model. This length of time was selected because there are only eight annual observations of enrollments available for estimation and validation. Using five years for the estimation of P leaves three years of actual enrollments for validation. In addition, the five year period only provides four observations for the quadratic programming estimation procedure. A shorter time period would not provide enough observations for the estimation process. A 71 72 longer time period would not allow enough time periods for the com— parison with the other forecasting methods and for validation of the Markov student flow model. The estimation of the population ratios also use a five year time period to be consistent with the Markov estimation process and to provide comparable forecasts. PopulationfiRatio Method The population ratio method is the simplest procedure for forecasting statewide enrollments in the system of higher education. However, it requires data on the 18 to 21 year old age group during the past and a forecast of the size of that age group for the future. This data is not available for the 18 to 21 year old age group or by single years, except for census years. As a result, some other method must be used to predict the size of 18 to 21 year old population. There are two possibler methods that could be employed to predict the 18 to 21 year old, college age population. One would be to use primary or secondary school enrollment by grade and year as the prediction basis for the college age group several years in the future. This method has a major disadvantaQE; only the public school enroll- ments are available for the required years. The private school en- rollments have only been gathered for a few years. This would cause no problem if the ratio of the number of students in the private schools to the number of students in the public schools was stable. This ratio has been becoming smaller since 1963 when it was 0.193.1 In 1966, the first year of usable enrollment data for the statewide system of higher education in Michigan, the ratio was 0.174. The 73 ratio drops to 0.100 in 1972. This observed drop is the result of the closing of many private schools and those students entering the public school system. The result of the student transfers has caused the public school enrollments to continue growing after the total statewide enrollments have begun to decline. Therefore, the use of only public school enrollments as a predictor of the college age group would introduce an error attributable to the transfer of students from the private schools to the public schools. The total public and pri- vate school enrollments cannot be used because the private school enrollment data is not available by grade level for all of the years needed to compare with the higher education enrollments. The second method uses the number of live births as the pre- dictor of the future college age population. Since the population ratio method only calculates a ratio from the relevant population, the live births for the time period 18 to 21 years prior to the year in question could be used in the calculation process. For example, the number of live births in the years 1945 to 1948 would be used in place of the 18 to 21 year old population for the computation of the college enrollment ratio for 1966. Population Ratio--Average There are two techniques that can be used to estimate the ratios for the population ratio method of forecasting higher education enrollments. One technique calculates the average of the ratios of the higher education enrollment to the college age population for the years of the selected historical period. The second technique 74 calculates a simple linear regression of the ratio from the ratios of the selected historical period. Both techniques are used with the live birth method to produce forecasts of the higher education enroll- ments. The live birth data used in computing the population ratios is displayed in Table 17. The corresponding higher education enroll- ments are displayed in Table 18. Using the data from these tables, the average ratio is computed for the years 1966 to 1970. The popu- lation ratio averages and the resulting enrollment forecasts for the community colleges and university undergraduates are displayed in Table 19. Population Ratio--Linear Regression The same data is used to compute a linear regression equation for the population ratio. The resulting linear equations for the community colleges and the baccalaureate institutions and the enroll- ment forecasts are displayed in Table 20. The regression parameters are all significantly different from zero. Both parameters of the community college equation and the intercept of the university equa- tion are different at the .005 level of significance. The slope para- meter of the university regression equation is different from zero at the .025 level of significance. The coefficients of determination for both equations indicate a good fit for the equations over the 1966 to 1970 historical period. The results of these forecasts are graphically presented in Figures 4 and 5 along with the results of the multiple linear regression 75 Table 17. Number of live births. 1940-1969 leer; Renter. leer. May. 1940 99,106 1955 196,294 1941 107,498 1956 206,068 1942 124,068 1957 208,488 1943 125,441 1958 202,690 1944 113,586 1959 198,301 1945 111,557 1960 195,056 1946 138,572 1961 192,825 1947 160,275 1962 182,790 1948 153,726 1963 178,871 1949 156,469 1964 175,103 1950 160,055 1965 166,464 1951 172,451 1966 165,794 1952 177,835 1967 162,756 1953 182,969 1968 159,058 1954 192,104 1969 165,760 Source: Michigan Department of Public Health, Michigan Center for Health Statistics, Michigan Health Statistics--Annual Statistical Report, (Lansing, Michigan: 1969), p. 2. 76 Table 7H3. Public community college and baccalaureate11ndergraduate enrollments for Michigan, 1966-1973. Q Community Baccalaureate Year Colleges , Undergraduates 1966 69,504 128,208 1967 79,256 135,573 1968 95,065 149,347 1969 115,791 158,130 1970 126,621 165,051 1971 132,189 169,178 1972 136,121 164,178 1973 151,652 163,327 Source: Higher Education General Information Survey, Michigan Association of Collegiate Registrars and Admissions Officers, and the Michigan Bureau of Programs and Budget as described in Chapter I. Table 19. Enrollment forecasts based on the average ratio of enroll- ments to live births 18 to 21 years earlier for the years 1966 to 1970. Year Community Baccalaureate Colleges Undergraduates 1971 107,254 163,621 1972 112,213 171,184 1973 115,901 176,811 1974 120,269 183,474 1975 124,217 189,497 1976 125,855 191,995 1977 126,165 192,469 1978 124,462 189,870 1979 122,038 186,174 1980 118,960 181,477 RATIO Standard Average Deviation Community College .1547 .0267 Baccalaureate Undergraduates .2360 .0110 78 Table 20. Enrollment forecasts based on the linear regression of the ratio of enrollments to live births 18 to 21 years earlier for the years 1966 to 1970. Year Community Baccalaureate Colleges Undergraduates 1971 145,318 176,932 1972 165,309 189,754 1973 184,453 200,786 1974 205,631 213,328 1975 227,075 225,469 1976 244,957 233,649 1977 260,486 239,445 1978 271,691 241,361 1979 280,838 241,710 1980 287,826 240,534 Community College Ratio ** ** ggfi = .0998 + .Ol83(t-1965) R2 = .962 (.0070) (.0021) Baccalaureate Ratio ** * 2 ”R = .2165 + .0064(t-1965) R = .835 (.0054) (.0016) *: The parameter below this is significantly different from zero at the .025 level of.significance, n = 5. **: The parameter below this is significantly different from zero at the .005 level of significance, n = 5. (): The number within the parenthesis is the standard deviation of the estimated parameter which appears above the parenthesis. model and the Markov student flow model. Figure 4 presents a comparison of the forecasts for the community colleges. Figure 5 presents a com- parison of the forecasts of the undergraduate enrollments for the col- leges and universities. The wide divergence of the two population ratio techniques is evident for both sectors of the system of higher education. These results are evaluated and compared with the results of a linear regression forecast and the Markov student flow forecast in a later section of this chapter. Linear Regression Statewide enrollments have also been forecast through the application of multiple linear regression. One example is a study conducted by the Michigan Department of Education to determine the effects of different variables upon higher education enrollments. The study consisted of a multiple linear regression analysis of the vari- ables affecting Michigan public higher education enrollments over the period from 1951 to 1969.2 The analysis identified four variables that strongly affected enrollments. These variables were: (1) the number of young people in the age group most likely to attend college in future years, (2) the number of people discharged from the armed services, (3) income levels, and (4) the employment rate.3 The coefficients of these four variables were determined and then used in linear regression enrollment forecasting model. The en- rollment forecasts for 1971 to 1980 are presented in Table 21 and 80 Enrollments (thousands) 3O Ratio average Ratio linear regression Multiple linear regression , 2 280) . Markov student flow // #wN—J o o o 260” /r 2401 220 200 180i 160 140w 1201 100' _ Actual __ Predicted 80w 2‘4: 5 7 74 7 7 Year Figure 4. Forecasts of public community college enrollments for the State of Michigan. Note: See Tables l6, 17, 18, 19, and 20 for numerical values. 81 Enrollments 1. Ratio average 2. Ratio linear regression (thousands) 3. Multiple linear regression 4. Markov student flow 2404 ,, ...... 2 I o’ / 22ml 7’ l l/ 200.. i” ' ‘— cm 7.5— -‘4 18a /. """"" \1 av- ------- 3 160J ____Actual 1411 _ -Predicted ‘7 t A L A l _A A. A. A A A 4 L A __- V I v v w ‘w' 66 68 7o 72 74 76 78 80 Year 1 1 . 1 Figure 5. Forecasts of public college and university enrollments for the State of Michigan. Note: See Tables 16, 17, 18, 19, and 20 for numerical values. 82 Table 21. Enrollment forecasts from the Michigan Department of Education linear regression model. Year Community V Baccalaureate Colleges Undergraduates 1971 145,800 167,700 1972 156,000 168,900 1973 166,000 169,300 1974 175,300 168,400 1975 184,300 166,800 1976 189,300 171,200 1977 191,400 173,100 1978 190,800 172,700 1979 190,300 172,200 1980 189,800 171,700 Source: Michigan Department of Education. Financial Reguirements of Public Baccalaureate Institutions and Public Community Colleges. Lansing, Michigan. 1971. p. 10. 83 graphically displayed in Figures 4 and 5. No indication was given as to the validity of the model or the goodness of fit of the linear .equations used to forecast the enrollments. Statewide Student Flow Model Forecasting statewide enrollments using the student flow model described in Chapter III, the results displayed in Table 22 are obtained. These results were obtained by using the student flow model in a highly aggregated, three state form. The three states consist of one state for the community colleges, one state for the lower division of the colleges and universities, and one state for the upper division of the colleges and universities. The community college forecast is graphically displayed in Figure 4. The total enrollment forecast for the colleges and universities is graphically displayed in Figure 5. The statewide student flow model forecasts were generated with the use of Equation 3.1. The transition matrix, P, was esti- mated with enrollment data from the years 1966 to 1970. Starting with the 1970 Fall enrollments, the Fall enrollments for the years 1971 to 1980 were iteratively generated. The annual enrollment forecasts are produced by advancing the model in steps of one year, using the previous year's forecast as the starting point for the next year's - forecast and using the first-time entrant forecasts, f(t), from Chapter IV. This procedure is indicated in the following set of equa- tions. 84 Table 22. Markov student flow model enrollment forecasts. Baccalaureate Undergraduates Year figflmflgfiig ifififi? £2213: Total 1971 137,453 87,826 82,961 170,787 1972 146,211 88,418 86,699 175,118 1973 154,704 88,792 89,570 178,362 1974 162,790 89,028 91,749 180,777 1975 170,401 89,177 93,388 182,565 1976 177,481 89,270" 94,612 183,882 1977 183,698 89,330 ' 95,521 184,851 1978 186,375 89,367 96,193 185,560 1979 187,428 89,391 96,688 186,079 1980 188,337 89,405 97,051 186,456 85 2(1971) = §(197o) P + f(1971) 2(1972) = §(1971) P + f(1972) 2(1980) = £(1979) P + f(1980), where §(t) is the forecasted enrollments for the tth year. This pro- cedure is used in all of the student flow model forecasts that follow. Three Model Comparison Each of the three models has its advantages and disadvantages. However, the primary requisite for any of the forecasting models is that it predict the future with accuracy. A secondary consideration is the ease and cost of gathering the needed data and estimating the parameter values of the model. In regard to the ease and cost of data gathering and estimating the parameter values, the models can be ranked in the following sequence: (1) population ratio, (2) student flow, and (3) linear regression. However, the important ranking is the one of accuracy. The models can be ranked on the basis of accuracy, in the following sequence: (1) student flow, (2) linear regression, and (3) population ratio. This ranking is based on the comparison of the ac- curracy of the projection of the enrollments for 1971, 1972, and 1973. Table 23 displays the forecasts for each of the models and the amount of discrepancy between each forecast and the actual enrollment in each year. In addition Theil's inequality coefficient has been computed for each forecast method and year. The coefficient for each year was computed from only two pairs of observed and predicted values, the 86 TABLE 23. Statistical comparison of the three model forecasts. 1971 1972 1973 COLL BACH COLL BACH COLL BACH Actual 132189 169178 136121 164559 151652 163327 Markov Student Flow P 137453 170787 146211 175118 154704 178362 R .040 .010 .074 .064 .020 .092 U .037 .075 .089 Multiple Linear Regression P 145800 167700 156000 168900 166000 169300 R .103 -.009 .146 .026 .095 .037 U .057 .095 .070 Population Ratio: Average P 107254 163621 112213 171184 115901 176811 R -.189 -.033 -.176 ..O40 -.236 .083 U .108 .116 .171 Population Ratio: Linear Regression P 145318 176932 165309 189754 184453 200786 R .099 .046 .214 .153 .216 .229 U .064 .180 .223 P=Predicted Value U=Thei1's Inequality Coefficient R=Re1ative‘ErrorIPEA A A=Actua1 Value 87 total community college enrollment and the total undergraduate enroll- ment at the baccalaureate institutions. This will facilitate the comparison of the results of each method on a year-to-year basis. Inspecting the results of the population ratio method, we find that both the simple average and the linear regression techniques result in relatively large forecast errors. Comparing Theil's ine- quality coefficient for each year of either population ratio technique with those of either the student flow or linear regression method, in every instance, the inequality coefficient of the population ratio technique is larger. In addition, the size of the coefficient is in- creasing much more rapidly for the population ratio techniques than the other methods. This implies that of the three methods, the popu- lation ratio method is the least accurate over the time span under consideration. The conclusion is that the population ratio method is not as accurate as other models and should not be used when other methods are available. In addition, it appears that the amount of error will increase in the more distant forecasts. This is particularly evident with the community college forecasts displayed in Figure 4. Comparing the results of the student flow model and the multiple linear regression model does not lead to a clear decision as in the preceding comparison. The amount of error measured and Theil's inequality coefficient for the different observations do not strongly indicate either method as predominately better than the other. In each of the three annual observations, the student flow method fore- casts the community college enrollments more accurately than did the linear regression method. However, the linear regression method 88 forecasts the undergraduate enrollments at the baccalaureate institu- tions more accurately. Theil's inequality coefficient measurements for each of the three years do not indicate either method as being consistently more accurate. The student flow method produces smaller Theil's inequality coefficients for the years 1971 and 1972, but the linear regression method produces the smaller coefficient for 1973. In addition, the amount of difference between the pair of coefficients for a given year is not significant relative to the magnitude of the coefficients. Neither of the two methods clearly produces more accurate forecasts than the other. Considering the results of each method over the three year period, the aggregate flow method provides marginally better re- sults. The average error is smaller, .050 vs. .069, and the average Theil's inequality coefficient is smaller, .067 vs. .074, for the student flow method. The student flow method and the linear regression method are both clearly better forecasting methods than the population ratio method. However, neither of the two is clearly superior to the other, although the student flow method did perform better than the linear regression method over the 1971 to 1973 time period. The total forecast undergraduate enrollments for both the linear regression and the student flow methods do not differ greatly over the 1971 to 1980 forecast period. This is clearly evident in the graphical presentation of the total undergraduate enrollments in Figure 6. The first seven years of both forecasts are approximately the same with very little difference in either forecast. The fore- cast totals begin to diverge after 1976. However, even at the point 89 Enrollments (thousands) -"- Markov 380,. Student flow z"."—"-‘- 360+, 34°11 Multiple linear //,4 regression / 3201, /V 1 / / / 300J -——4 280., 260if 240‘, ACtua] 220. | | 2004 ' 66 68 ‘ 7O 72 74 76 78 80 Year Figure 6. Multiple linear regression and Markov student flow model forecasts of total public undergraduate enrollments for the State of Michigan. 90 of greatest divergence in 1980, there is only a 3.6 percent difference between the two forecast totals. Considering the community college forecasts, Figure 4, and the baccalaureate undergraduate forecasts, Figure 5, the general shape of the enrollment forecast curves are the same. The linear regression method forecasts a larger number of com- munity college enrollments for each year in the 10 year forecast period. The student flow method forecasts the larger number of enrollments at the baccalaureate institutions. The shape of the 10 year enrollment forecast curves and the relationship of the individual curves reinforce the conclusion that neither method is clearly superior to the other. Statewide Student Flow Forecasts Utilizing the three state student flow model described earlier in this chapter, the statewide forecast for the years 1971 to 1980 was generated. The three state model aggregates the community college enrollments into one state. The university enrollments are aggregated into two states, one for the upper division and one for the lower di- vision students. The data used in testing the higher education Markov student flow, enrollment forecasting model is from the 29 public community colleges and 13 public colleges and universities in Michigan. Table 24 contains a list of both types of institutions. Fall enrollment data from these institutions for the years 1966 to 1970 was used to estimate the Markov transition matrix, P. Using Equation 3.1 and starting with the enrollments of a given year, the enrollments for each succeeding year was forecast. 91 Table 24. List of public higher education institutions in the State of Michigan BACCALAUREATE INSTITUTIONS Central Michigan University Eastern Michigan University Ferris State College Grand Valley State College Lake Superior State College Michigan State University Michigan Technological University Northern Michigan University Oakland University Saginaw Valley College University of Michigan Wayne State University Western Michigan University COMMUNITY COLLEGES Alpena Bay de Noc Delta C.S. Mott Glen Oaks Gogebic Grand Rapids Henry Ford Highland Park Jackson Kalamazoo Kellogg Kirtland Lake Michigan Lansing Macomb County Mid Michigan Monroe County Montcalm Muskegon North Central Northwestern Oakland St. Clair County Schoolcraft Southwestern Washtenaw Wayne County West Shore 92 The procedure followed was to initialize the model with the starting values, then iterate the model in steps of one year and use the first- time entrant values forecast in Chapter IV. The resulting forecast with a starting year of 1970 is displayed in Table 25. Although a portion of the forecasts has high error rates, the amount of error for each annual total forecast is not exceedingly large. This is indicated by the relatively small size of U, Theil's inequality coefficient, for each year where the actual and forecast values could be compared. In addition, the performance of a t-test on the prediction discrepancy and the enrollment error indicates that the difference between the predicted enrollment and the actual enrollment is zero at the five percent level of confidence for 1971 and 1973. The t-test rejects this null hypothesis at the five percent level of confidence for the predic- tions for 1972. Table 26 contains a statistical summary of this en- rollment forecast for the Michigan system of higher education. Returning to Figure 6, we see that there was no growth in enrollments between 1971 and 1972. Whatever the cause of this discon- tinuity, it is the reason for the rejection of the null hypothesis for 1972. Also looking at Figure 5, we see the enrollments for the bac- calaureate institutions dropping during the years of 1971 to 1973. In this case, the forecasts for the higher education enrollments are being made at a time of change in the particular system being forecast. Using more recent data to estimate the transition probabilities and the starting point of the forecasts should reduce the error due to the change in the system. 93 Table 25. Three state Markovenrollment forecast.- 1971 to 1973. BACCALAUREATE Community Eresh *UEnior Year Colleges Soph Senior Total 1970 A 126,621 86,888 78,163 165,051 1971 P 137,453 87,826 82,961 170,787 A 132,189 87,352 81,826 169,178 R .040 .005 .014 .010 1972 P 146,211 88,418 86,699 175,118 A 136,121 85,112 79,447 164.559 R .074 .039 .091 .064 1973 P 154,704 88,792 89,570 178,362 A 151,652 85,772 77,555 163,327 R .020 .035 .155 .092 P=Predicted Value R=Re1atiVe ErrOr P-A. A=Actual Value '7?- Table 26. Statistical summary for the three state enrollment fore- cast, 1971 to 1973. RELATIVE ERROR PREDICTION ERROR AVG STD t AVG STD t U YEAR DEV VALUE DEV VALUE 1971 .020 .018 1.904 2291 2596 1.529 . .030 1972 .068 .027 4.411*, 6883 3407 3.499*. .072 1973 .070 .074 1.644 6029 5184 2.014 .067 *: Accept the alternatiVe hypothesis that the average forecast error is different from zero at the .05 level of significance, n=3. 94 Estimating_ P From More Recent Data Applying the moving average concept, new transition proba- bilities can be estimated for P. The previous forecasts used a P estimated from enrollment data from the years 1966 to 1970. Now the transition probabilities will be estimated from enrollment data from the years 1967 to 1971. The five year historical data base for esti- mation of P has been advanced one year. The resulting enrollment forecasts starting from the base year of 1971 are displayed in Table 27. This forecast predicts the 1972 and 1973 enrollments more accu- rately than the previous forecast. The updated starting point and transition probabilities cause a more accurate forecast to be gener- ated. Statistically the forecasts for 1972 and 1973 have improved over the previous forecasts for these years. Table 28 contains the statistical summary for the forecasts in Table 27. The size of the error values have decreased. However, the forecast error for 1972 is still different from zero at the .05 level of significance. The portion of the forecast for the baccalaureate institutions is more accurate than the earlier forecast. However, the forecast for the community colleges results in a change of sign for the relative error. It appears from the data that there has been a disturbance in the general increasing enrollment pattern for the community colleges in 1972. If it is a one time occurrence, its impact will lessen over the years. However, if it is an indication of a permanent change in the enrollment pattern, then it will require a several year delay before 95 Table 27. Three state Markov enrollment forecast, 1972 and 1973. BACCALAUREATE Community Fresh Junior Year College Soph Senior Total 1971 A 132,189 87,352 81,826 169,178 1972 P 139,538 88,556 85,229 173,784 A 136,121 85,112 79,447 164,559 R .025 .040 .073 .056 1973 P 146,890 89,322 87,889 177,210 A 151,652 85,772 77,555 163,327 R -.031 .041 .133 .085 P=Predicted Value A=Actual Value R=Re1ative Error 315 A Table 28. Statistical summary for the three state enrollment fore- cast, 1972 and 1973. PREDICTION DISCREPANCY AVG STD t PREDICTION ERROR STD YEAR DEV VALUE DEV U 1972 .046 .024 3.283* 1357 5.377* .042 1973 .048 .083 1.002 7561 .062 *: Accept the alternative hypothesis that the average forecast error is different from zero at the .05 level of significance, n=3. 96 enough historical data can be collected upon which future forecasts can be made. This also applies to the college and university enroll- ments which have decreased from 1971 to 1973. Applying the moving average concept once again and estimating P with the 1968 to 1972 data, the forecast displayed in Table 29 is obtained using 1972 as the base year. The forecast of the community college enrollment deviates further than it did in the two previous forecasts for 1973. The forecast for both the upper and lower divisions of the colleges and universities improved. The size of the error measurements for the enrollment prediction decreased, as indicated in Table 30. In addition, the null hypothesis that the prediction error is zero is not rejected at the .05 level of significance. The question arises again as to whether this is a one time fluctuation in the com- munity college enrollments or whether it is the leading impulse of a new enrollment pattern. Considering each of the separate forecasts, the student flow method generated some fairly accurate forecasts in total, but some of the individual forecasts were somewhat inaccurate. The amount of error was not too large in most instances and Theil's inequality coef- ficient remained reasonable small for all the forecasts. It would appear that for the highly aggregated, three state model, the Markov student flow method generates reasonable good forecasts. Expanded Student Flow Model One of the main advantages of the student flow model is that, conceptually, the number of states can be easily expanded. Expanding the number of states, i.e., having subgroups of institutions instead 97 Table 29- 'Three state Markov enrollment forecaSt for 1973. h BACCALAUREATE Community Fresh Junior Year College Soph Senior T°t31 1972 A 136,121 85,112 79,447 164,559 1973 P 143,113 86,195 80,697 166,892 A 151,652 85,772 77,555 163,327 R -.056 .005 .041 .022 P=Predicted Value R=Relative Error P-A A=Actual Value '7?- Table 30. Statistical summary for the three state enrollment fore- cast for 1973. RELATIVE ERROR PREDICTION ERROR AVG STD t AVG STD t U YEAR DEV VALUE DEV VALUE 1973 -.004 .049 -.128 -1658 6112 -.470 .048 Note: The nUll‘hypotheSis‘that'the average forecast error is zero at the .05 level 0f signifiCance.is not rejected. 98 of a single group of each type, should improve the ability to analyze the statewide distribution of future enrollments. However, there are two problems involved with increasing the number of states of the model. The first problem concerns computational capability. As the number of states increase, the amount of computer memory required for manipulation of the data increases by about a power of 3. This prob- lem can be handled with access to a larger computer or with an addi- tional expenditure of time to employ more sophisticated data handling techniques. An example of this is the comparison of the amount of memory needed to estimate P for the three state Markov model for Phase.II--45,000 locations--and the memory needed to estimate 0 for Phase I, which contains twelve states--70,000 locations. This is one of the reasons for separating the flow model into two phases, the avoidance of excessive core memory requirements. The second problem concerns the availability of data. This is a problem with the amount of detail contained in the data. Addi- tional detail would be required concerning the number of transfers between institutions in order to greatly expand the number of states. This is particularly true with respect to the number of transfers from the community colleges and the baccalaureate institutions. Prob- lem two is evident when we separate the two groups of institutions into several classifications. This problem is also evident when we separate the community college group into the vocational-technical or non-degree oriented classification and the transfer or degree oriented classification. This will be explored in the following presentation of relevant forecasts from the Markov student flow model. 99 Separating9Transfer and Terminal Enrollments' Expanding the model to include the dichotomy between transfer and non-transfer students at the community colleges can only be accom- plished with historical data from 1968 and following years. constraint, the transition matrix, 1968 to 1971. P. With this was estimated using data from This transition probability matrix was used in Equation 3.1 to forecast the enrollments of the statewide system following the base year 1971. The results of this forecast are shown in Table 31. Table 31. Student flow model expanded to include community college vocational enrollments with the transition matrix esti- mated from 1968 to 1971 data. COMMUNITY COLLEGE BACCALAUREATE Fresh Junior Total Year Vocat Trans Total Soph Senior 1971 A 49,612 82,577 132,189 87,352 81,826 169,178 1972 P 54,125 84,750 138,875 87,595 84,590 172,185 A 59,485 76,636 136,121 85,112 79,447 164,559 R -.090 .106 .020 .029 .065 .046 1973 P 62,339 92,724 155,063 87,981 86,454 174,435 A 70,988 80,664 151,652 85,772 77,555 163,327 R -.122 .150 .022 .026 .155 .068 P=Predicted Value R=Relative Error E;A_ A=Actual Value e. A This and the following forecasts were generated in a slightly differ- ent manner than those in the preceding discussion. Instead of using the forecast values for the first-time entrants, the actual, known values were used in the student flow model forecasts. This was done 100 in order to avoid the problem of partitioning the aggregare first-time entrant forecasts into individual group forecasts for any possible grouping of institutions. The result of using the actual first-time entrants instead of forecast first-time entrants should be a more accurate forecast when compared to the actual enrollment figures. The error due to the fore- cast of first-time enrollments has been excluded from the forecast comparison of total enrollments. However, the amount of error in the community college enrollments is a -9.0 percent for the non-transfer enrollments and 10.6 percent for the transfer enrollment in the first year forecast. The amount of error increases even more in the second year of the forecast. The size of the forecast errors are also indi- cated by the increased size of Theil's inequality coefficient. The results of the t-tests, displayed in Table 32 indicate that the fore- case errors are not different from zero at the five percent level of confidence. However, the t-test results do not imply accurate fore- casts since the standard deviations of the error measures are large and the individual prediction descrepancies are large. Adding one more year of enrollment data to the estimation process for P, i.e., using enrollment data from 1968 to 1972, a new transition matrix P is obtained. Using actual first-time enrollments, as in the above, the forecast in Table 33 is obtained. The accuracy of this one year forecast is an improvement over the previous first year forecast in Table 31. The accuracy of the community college forecast has improved by a factor of two. This is evident in the comparison of the statistical summaries in Tables 32 and 34. The 101 Table 32. Statistical summary for the expanded flow model with the transition matrix estimated from 1968 to 1971 data. RELATIVE ERROR: .. PREDICTION ERROR Std t Std t Year Avg Dev Value Avg Dev Value U 1972 .027 .084 .650 2595 5781 .898 .074 1973 .042 .121 .695 3630 9158 .793 .111 Note: The null hypothesis.that the average_forecast error is zero at the .05 level of significance, n=4, is not rejected. Table 33. Student flow model expanded to include community college vocational enrollments with the transition matrix estimated from 1968 to 1972 data. COMMUNITY COLLEGE BACCALAUREATE Fresh Junior Total Year Vocat Trans Total Soph Senior 1972 A 59,485 76,636 136,121 85,112 79,447 164,559 1973 P 67,904 84,969 152,873 85,728 80,697 166,425 A 70.988 80,664 151,652 85,772 77,555 163,327 R -.O43 .053 .008 -.001 .041 .019 P=Predicted Value A=Actual Value R=Re1ative Error _P_-_A A 102 relative error,.the error in units, and their standard deViations'for the latter forecast are an improvement_over those of the preceding forecast. Table 34. Statistical summary for the expanded flow model with the transition matrix estimated from 1968 to 1972 data. PREDICTION DISCREPANCY PREDICTION ERROR Std t Std t Year Avg Dev Value Avg Dev Value U 1973 .012 .044 .057 1080 3329 .049 .039 Note: The null hypothesis that the forecast error is zero at the ' .05 level of significance, n = 4, is not rejected. The improvement in accuracy may be the result of one of several factors. The first is that the start up period of any process will contain errors in the processing activity and the reporting of the activity. This is evident in the errors in the needed breakdown of data in every year the data has been collected. (See the Appendix for the community college enrollment data.) For example, in 1971 only 25 of the 29 community colleges reported the breadkown for the first-time enrollments. In addition, only 28 of the community colleges reported the total enrollment breakdown. The liberal arts vs. voca- tional-technical data breakdown was collected in 1968, however, it was not until 1973 that all schools reported the enrollment breakdown. The second reason for poor data was that community college administra- tors were playing with the enrollment numbers in order to obtain more state funds. The extent to which this activity was practiced is 103 unknown. However, this practice was reported in two consecutive Audit_ Reports from the Office of the Auditor General for the State of Michi- gan.4 A third factor may be the result of the short period of time over which the data has been collected. When using only a few obser- vations as the basis of an estimate, the impact of a single deviation from the norm is significant. As the number of observations increases, this impact of a single deviation on the estimated value is reduced. This may be the reason that the accuracy of the second forecast was better. The number of observations had increased by one. The process of modeling the community college enrollments in two groups causes no problems in the Markov student flow model. Conceptually, it has the appealing feature of segmenting the community college enrollment population into two separate and distinct groups. The two groups require different types of educational facilities and require different faculty structures. This additional knowledge can assist the various levels of administration in planning for the two different types of students. Forecasting System Subgroups The student flow model is not limited to the forecasting of enrollments for each institutional type as a whole. The institutional classifications could be divided into groups or subgroups within the institutional type. One possible subgrouping is illustrated in Table 35. The community colleges are divided on the basis of total enroll— ment growth. Group II community colleges individually experienced a significant growth in total enrollment and Group I did not experience 104 Table 35. First institutional subgrouping. Community Colleges Baccalaureate Institutions Group I Group I Alpena Central Bay de Noc Eastern Delta Ferris C.S. Mott Grand Valley Gogebic Lake Superior Grand Rapids Michigan Tech' Jackson Northern Kellogg Oakland Lansing Saginaw Valley Macomb County Western Monroe County Montcalm North Central Northwestern Oakland St. Clair County Schoolcraft Southwestern Washtenaw Group II Group II Glen Oaks Michigan State University Henry Ford University of Michigan Highland Park Wayne State University Kalamazoo Kirtland Lake Michigan Mid Michigan Muskegon Wayne County West Shore 105 a significant amount of growth. The four year institutions are divided into the senior institutions, Group II, and the remainder of the four year schools in Group I. The transition matrix for this subgrouping and the two following were estimated with enrollment data from the years 1966 to 1970. The forecast for these institutional subgroups is displayed in Table 36. As in the preceding forecast for the vocational- technical breakdown, this forecast was generated using the actual first- time enrollments for each subgroup. The following forecasts discussed in this section are also generated using actual first-time enrollments. This should be remembered as the relative error between the forecasts and actual enrollments is discussed. These forecasts should be more accurate since no error enters the forecast results due to an error in the number of first-time entrants. The first year forecast for the baccalaureate institutions is quite good for both subgroups and the total forecast error for the baccalaureate institutions in only 1.1 percent. The community college forecasts were only partially good. The total enrollment group forecast resulted in a large forecast error. The first year forecast was off by 12.4 percent and in the third year the forecast error was 31.5 percent. The discrepancy of the total enrollment forecast, as measured by Theil's inequality coefficient was not exceedingly large for the first and second years of the forecast. The null hypothesis that the forecast error is zero is not rejected for the 1971 enrollment forecast. However, the null hypothesis is rejected for the 1972 and 1973 enrollment forecasts. The statistical summary of this forecast appears in Table 37. 106 < dam Lose” m>vmewmum mzpm> Fmapo cmpu_umgaum Nwo. mwo. mmo.- Rpm. pno. “mo. mpm. “co. m mmm.mm_ mmm.F¢ mmN.Pe umm.mm mmm.m¢ Nmm._mp mmm.¢e mum.no_ < Nm5.mnp Pmm.me ¢N~.o¢ mmm.m¢ mop.~¢ Fpm.mm_ mmm.wm mwo.mo_ a mum_ mmo. ewe. Foo.- emo. Poo. moo. mmF. omo. m mmm.¢m_.. emo.Fv mpm.o¢ mam.mm mam.¢¢ FNP.mmF oNN.F¢ mmm.¢m < mpm.¢mp mpm.e¢ mo_.o¢ mmm.Fe oem.nv mew.m¢_ omm.m¢ mno.mm a NNmF _Po. upo.- mpo.- sec. emo. soc. eup. meo. .m m~_.mo_ nom.¢e mom.o¢ mpm.m¢ m¢¢.m¢ mmF.NmF “mm.mm Now.mm < woo.pup com.m¢ m-.oe www.mm m~o.we emo._¢P mnm.¢¢ ~¢N.mm 8 _nm_ _mo.mo_ cme.~¢ www.mm www.mm oo~.N¢ _Nm.m~_ wpp.o¢ mom.ow < onm_ Punch Lowcmm snow gowcmm snow Pouch N qzogu H asogo me> gowczw smog; Lomcaq smock. N qzogu F nzogu mhth=zzou .mazogmaam _mcovpsuwpmcw pmcwm on» go; ammomcom acmEFFogcm .mm wpnm» 107 Table 37. Statistical summary for enrollment forecast of the first institutional subgroups. RELATIVE ERROR PREDICTION ERROR Std t Std t Year Avg Dev Value Avg Dev Value U 1971 .035 .052 1.649 1777 2339 1.861 .052 1972 .080 .064 3.041* 3747 2639 3.477* .083 1973 .112 .130 2.104 4683 5442 2.108 .118 * Accept the alternative hypothesis that the average forecast error is not zero at the .05 level of significance, n = 6. A second subgrouping of the institutions is indicated in Table 38. In this attempt to locate a relevant grouping of institu- tions, the community colleges were separated on the basis of geographi- cal location and the universities were separated on the basis of increasing first-time enrollments. Group II of the community colleges are all located in the southeastern third of Michigan. Group II of the baccalaureate institutions consists of those colleges that indi- vidually experienced a significant growth in first-time enrollments. The student flow forecast for this grouping is displayed in Table 39. The statistical summary appears in Table 40. The results of the t-tests for the second subgrouping indi- cate that the prediction error is not significantly different from zero. However, comparing the values of U in Table 40 with the values of U in Table 37, the conclusion is that the forecasts of the second subgrouping are less accurate than the first since the inequality co- efficients are larger for each of the three years forecast. This 108 Table 38. Second institutional subgrouping. Macomb County Monroe County Oakland St. Clair County Schoolcraft Washtenaw Wayne County Community Colleges Baccalaureate Institutions Group I Group I Alpena Central Bay de Noc Eastern Glen Oaks Ferris Gogebic Lake Superior Grand Rapids Michigan State Kalamazoo Michigan Tech Kellogg Northern Kirtland Wayne State Lake Michigan Western Mid Michigan Montcalm Muskegon North Central Northwestern Southwestern West Shore Group II Group II Delta Grand Valley C. S. Mott Oakland Henry Ford Saginaw Valley .Highland Park University of Michigan Jackson Lansing 109 < (In LOLLu m> rum meum wzpm> pmzpu umpuwumgana Nae. MAP. mwo.- mop. mmo. oo_. mm_. mmo.- x NNm.meF emp.m_ woe._m mam.mm emm.ee Nme._m_ “_e.m__ mmo.mm < mmm.mk_ _ee.N~ emm.mp mm_.me omo.me mNN.ee_ NNm.mm_ meN._m a mum_ moo. omp. ewo.- Awe. Nee. moo. mmp. eoo. x mmm.em_ NeN.m_ emm.m_ mom.Pe emu.me _N_.emp mmo.mo_ mmo.om < ,mN.mN~ ema.o~ www.mp eem.ee mem.mo meN.mep mmo.mp_ Nep.om a NNm_ m_o. emo. emo.- Noe. “_o. Nee. omo. N_o.- m mm_.mep mme.m. 8mm.mp Amm.me ep_.me mm_.NmF eme.~o_ mme.mN < No¢.PNP “_m.m_ mmN.NP eNN.me mpm.ON mko._eP mea._F_ mm_.mN a .NmF _mo.me_ mem.k_ mmo.5_ e_m.oe c_w.me _Ne.em_ Nom.wm e_N.NN < cum_ _eoee Loweem eeom eeeeem seem _eeoe N macaw _ 8:828 eee> Lowcza smog; Lowczw :mmgg N aaocw P usage meeHz=zzoo .masogmnzm _mcowpzawpmcw vacuum mgu so» ummumcom acmeppoccw .mm opaMH 110 Table 40. Statistical summary for enrollment forecast of the second institutional subgroups. RELATIVE ERROR ' PREDICTION ERROR; Std t Std t 4‘ Year Avg Dev Value Avg Dev Value U 1971 .022 .043 1.262 1852 3688 1.230 .065 1972 .064 .068 2.323* 3966 4901 1.982 .101 1973 .070 .107 1.612 5021 6730 1.828 .126 *: Accept the alternative hypothesis that the average forecast error is not zero at the .05 level of significance, n = 6. conclusion is supported by the comparison of the relative error . standard deviation; they are all larger in the second subgrouping forecasts. A third subgrouping of the institutions is indicated in Table 41. This grouping of institutions divides the community colleges such that Group II contains only institutions that admitted their first. students in 1966 or following years. The community colleges undergoing start up conditions have been segregated from the remainder of the com- munity colleges. The baccalaureate institutions are divided such that Group II contains those institutions that experienced a significant decline in first-time enrollments. The resulting forecast for this subgrouping is displayed in Table 42. The community college forecasts are interesting. The forecasts for the start up group of community colleges have a very high degree of error. The prediction discrepancy is 45 percent in the third year of the forecast. However, the fore- casts for the remaining community colleges are fairly accurate and the 111 Tab1e441. Third institutional subgrouping. Community Colleges Baccalaureate Institutions Group I Group I Alpena Central Bay de Noc Ferris Delta Grand Valley C.S. Mott Lake Superior Gogebic Michigan Tech Grand Rapids Oakland Henry Ford Saginaw Valley Highland Park University of Michigan Jackson Western Kellogg Lake Michigan Lansing Macomb County Muskegon North Central Northwestern Oakland St. Clair Shores Schoolcraft Group II Group II Glen Oaks (1967 Eastern Kalamazoo (1968 Michigan State Kirtland (1968) Northern Mid Michigan (1968) Wayne State Monroe County (1966) Montcalm (1966) Southwestern (1966) Washtenaw (1966) Wayne County (1969) West Shore (1968) 112 < wumpmmum mzpm> szao cmpuwumgaua mmo. 84,. “co. mom. omo. m__. owe. emo. m Num.me_ oem.mm www.mm m_o.mm Num.me Nmo._m_ meu.wN mmw.-~ a mmm.mk_ me_.ee Pm_.oe Nmm.ee _NN.Ne NNo.me_ mou._e e_m.NN_ a mump Nee. mwo. __o. Fmp. mes. mop. mam. “mo. x mmm.eep em“.mm mmm.oe _Pk.mm mm“.ee _N_.em_ mmN.NN mem.mo_ < okm.mnfi No_.me emN.oe m_m.ee eek.ee Pam.omp Npo.em mem.e__ a NNm_ m_o. moo.- moo.- emo. m_c. NNo. mom. .40. m mu_.me_ Nam.Pe Amm.Fe mmm.mm mmm.me mmP.Nm_ mpe.mm eNN.eoF < mee._NF mom.Pe mmN.Pe meo.~e mPN.ee eem._e_ ~_e.om Nm_.F__ a me_ Fmo.mm_ eem.mm op_.Ne “_e.wm NAN.ee _Ne.emp mNF.mN Nae.Po_ < o~m_ pmuoh gowcmm :aom gowcmm snow _muoh N azogw _ azogw gmm> Lowczn smog; gowczn smog; N azogo _ asogm , weth=zzoo .maaogmnzm _mcowuswwpm:_ news“ as» Low ammumcom ucmsppogcm .Nv mpnmp 113 relative error does not exceed 5.7 percent for any of the three years. The forecast error of the start up group is probably the re- sult of the instability of the group as individual institutions began to admit students. Part of the observed forecast error could also be the result of inaccurate data reporting by inexperienced school admin- istrators. The null hypothesis of a prediction error of zero is re- jected at the .05 level 0f significance for all three years.‘ This is indicated in the statistical summary in Table 43. Table 43. Statistical summary for enrollment forecast of the third institutional subgroups. RELATIVE ERROR PREDICTION ERROR Std t Std t Year Avg Dev Value Avg Dev Value U 1971 .051 .079 1.575 2011 2357 2.090* .052 1972 .104 .103 2.483* 4245 2870 3.623* .088 1973 .145 .168 2.125* 5433 4616 2.883 .112 *: Accept the alternative hypothesis that the average forecast error is not zero at the .05 level of significance, n = 6. Expanding the second and third groupings to include the dichotomy between the transfer and non-transfer oriented students at the community colleges, the following results are obtained. Using. enrollment data from the years 1968 to 1971, the transition matrix P was generated for the subgrouping of Table 38. The forecast generated with this transition matrix is displayed in Table 44. An interesting result is that the total enrollment for the community colleges is quite accurate, but individual forecasts for each part of the 4 1 1 NFF. —mo. eeaweweeeoo NuwpeeeeeH m._we;h Nmmp Pump < <1; coggm m>wmemmum mapm> Pmauo empowumcanm moo. Nuo. mmo.- m__. Nmo. NNo. «NF. NFF.- _mo. mm_.. m NNm.mmp mmp.mp mo¢._N mmm.mm com.¢m Nmm.PmF Noo.~m mpo.mm Noo.m_ mmm.N_ < Pmm.mnp Nmo.0N mmN.ON mmm.mo woN.¢o Nmm.¢mp eom.FN omo.Nm mum.oN omm.op a mum_ meo. Pno. eoo. mmo. mmo. ONo. mNF. mmo.- oeo. omo.- m mmm.¢o_ N¢N.NF omm.mp mON.~m omm.mm —Np.omp mm¢.mm Nom.m¢ woo.m_ om¢.op < mom.PNF mNm.mF P¢¢.mp omu.¢m m<_.mm pmn.mm_ N¢N.mm NN©.¢¢ mpm.ON Pew.m a NNmF mup.mo_ www.mp mMN.m_ Nmm.mm m__.mm mm_.Nm_ NmN.Nm Nm¢.o< qmm.m_ Pom.m < Pump _muop govcmm snow gopcmm :aom quop mcmgp pmuo> mcmch pmoo> Lmo> Lowczq smog; Lo_:=o smote . . N usogw _ azogw N nzogo F azogw mkth=zzou .macmeppogzm Pacowumuo> mom—Foo haw::EEoo mcwuzpucw maaogmazm chONuauwumcw ccoumm any Low pmmumgom unmepposcm .ee mpnMH 115 enrollment contains descrepancies that are too large in even the first year of the forecast. The inequality coefficients for each of the forecast years are larger than those of the preceding first and second year. The forecast results for the third subgrouping is displayed in Table 45. The accuracy of the total community college enrollment is better than the previous forecast. However, the accuracy of some of the individual forecasts are not as good. For example, the fore- cast error for Group II in 1973 is 35.9 percent. The reasons for the large amount of error between the forecast and the observed enroll- ments are the same as those discussed earlier. The results of the student flow model when expanded to in- clude additional states were not as good as those of the restricted model. The problems of using the expanded model and possible means of reducing the size of the forecast error will be discussed in the next chapter. 116 1 goes m>~pu m epp. mmmp < m m . P mu use. _NmF peeveweeeeu NewpezeeeH m._weee w3~m> _msgo umuuwumgaum mmo. em.. mmo. ewe. NNo. m_o. mmm. Nm_.- mmo. om_.- m “mm.mop oem.mm mam.mm m_o.mm “Km.me Nme._m_ Awe.mp 8N~.mp NN_.me NPN.Nm < mMN.NN_ mme.me m_m.oe mem.me mmw.we mam.mmp omo.~m om_.__ mem.ok __m.om a mxmp mmo. “0.. NNo. mmo. Nmo. m_o. me_. mm_.- mmo. emo.- m mmm.eep emsmom mmm.oe __N.mm mmN.ee _N_.emp me.ep NNm.o_ 8mm.mm NmN.me < NmN.mNF Fum.me mmN.Pe emm._e NmF.ee m_N.mmP mem.m_ mme.m NNm.ee o_e.ee a NNm_ mu..mez Amm._e Nam._e mmm.mm www.me mm_.mmp mum.N_ Amm.N mee.ee m__.Ne < _Nm_ Pmpoh Lowcwm :aom gowcmm snow Page» mcmch pmoo> mcmgh umoo> me> Lowcsa smote Lowcza smote N aaogo P asoco N Quota F azogw .meemm=<4eHz=zzou .mpcmsppogcm _mcowpmuo> mmmFFou harassEoo mcwczpucw mazocmnzm PacowpzpwmeP news“ as» com ammumcoe “cosy—ogcm .mw mFQmF 117 Chapter V Notes 1Michigan State University, Graduate School of Business Administration, Michigan Statistical Abstract (10th ed.; East Lansing, Michigan: 1974), p. 155. 2Michigan, Department of Education, Financial Requirements of Public Baccalaureate Institutions and Public Community C011eges (Lansing, Michigan: 1971). 31bid., p. 3. 4Office of the Auditor General, State of Michigan, Michi an Community,Colleges Enrollment Audit (Lansing, Michigan: 1971). Office of the Auditor General, State of Michigan, Michi an Community_Colleges Enrollment Audit (Lansing, Michigan: 1972). CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS Many models have been developed and used to forecast enroll- ments in higher education. Examples of several of these models were described in Chapter II. Some simplistic models have been developed for forecasting models for statewide systems, although, their use has been quite limited. Most of the developmental effort has been in the area of institutional flow models due to the accessibility and availability of good, accurate historical data at the institutions where the research was being performed. The development of a Markov student flow model requires the estimation of the Markov transition matrix. This is most easily accomplished by using the individual student data with the maximum likelihood technique. However, this is not practical for a statewide system of higher education. The purpose of this research was to develop a Markov student flow model that did not rely on the maximum likelihood technique and individual student data and which could be used to forecast statewide student enrollments from aggregate data. The Markovian flow model is the most recent type of enrollment model, with research and development continuing on its applicability to the forecasting of student enroll- ments. The application of the Markov student flow model to the fore- casting of statewide student enrollments was described. The estimation 118 119 of the Markov transition probability matrix for this application does not utilize the maximum likelihood technique. Instead, a quadratic programming, least squares technique was used on aggregate enrollment data in order to estimate the substochastic Markov transition matrix. In addition, the model utilized forecast first-time enrollments gener- ated by applying the same procedure to elementary and secondary enroll- ments. The other Markov models reviewed did not discuss the problem of first-time entrants or obtained the first-time entrants from outside the Markov student flow model. The application of the Markov student flow model to the forecasting of first-time entrants into the system of higher education begins with the forecasting of statewide elementary and secondary en- rollments. The flow concept is used to forecast the first-time entrants by tracing the flow of the elementary and secondary students into the condition or state of entering an institution of higher education. The model was compared to the progression ratio method currently used to forecast statewide elementary and secondary enrollments. The model generated slightly more accurate forecasts of the student enrollments than did the progression ratio method for statewide elementary and secondary enrollments for the states of Michigan and New York. How- ever, the difference was statistically significant only in the fourth year forecast for New York. The statewide Markov student flow model enrollment forecasts were compared with forecasts generated by two other methods. The model provides a much more accurate enrollment forecast than does the popu- lation ratio method. The comparison with a multiple linear regression 120 model did not establish the clear superiority which the comparison to the population ratio method established. The model generated slightly better results than did the multiple linear regression model when evaluated with Theil's inequality coefficient. The Markov student flow model was then used to generate forecasts of several different groups of institutions and forecasts of community college enrollments separated into liberal arts and vocational-technical enrollments. Conclusions This student flow model appears to represent the character- istics of the student flow mechanism accurately. Comparing the results generated by the model for elementary and secondary enrollments with the results for higher education, the forecasts of the elementary and secondary were more accurate. The average prediction discrepancy for the elementary and secondary statewide school systems was in the range of one to two percent, even into the third year of the enrollment fore- cast. In contrast to this, the higher education average prediction discrepancy ranged from two to seven percent for the three state model. Increasing the number of states in the model resulted in increasing the size of individual discrepancies to over 30 percent in some cases. The statewide elementary and secondary education system is basically a closed system. The students enter the system and remain in the system until their late teens. Dropouts are permitted only after a legally determined age. This results in a relatively smooth flow pattern of students from the entry level to a dropout point or 121 graduation. In addition, the educational influence for the continuing of education into the higher level appears to be relatively constant. The system of higher education is much more open, and as a result, much less stable. The environment of higher education--the economy and societal attitudes-~has a significant impact on enrollments. This is evident in the fluctuations in the enrollments when compared with the smoothed model forecasts and in the regression parameters identified by the regression model. The more open the system, the more difficult it becomes to accurately forecast the future of the system. Expanding the model to segment the community colleges and universities into several different groups did not increase the ac- curacy of the forecasts. The overall accuracy, as measured by Theil's inequality coefficients did not significantly improve. However, some of the segment forecasts developed a significant error over several time periods. There are several possible reasons for the errors in some of the forecast segments. The first reason is that the change in economic conditions and the elimination of the draft caused a significant change in en- rollment patterns. Some institutions were able to adapt quickly to these changes and maintain their historical enrollment patterns. Other schools were not able to adapt and lost enrollments as the total num- ber of students declined. The second reason is that the data collection process may not have been very accurate during its initial years. This would result in larger errors for forecasts using parameter values estimated 122 from data collected in the early years. This is indicated by the decrease in the error for the first year forecasts as the base years for estimating P were moved toward the present, dropping the initial years from the estimation base. In addition, there are start up errors inherent in the data for the community colleges. One type of start up error is in the collection of the data, just as for the baccalaureate institutions. The shift to the collection of data on the numbers of students enrolled in academic and vocational-technical programs re- sulted in inaccurate counts of the two classifications of students. A second type of start up error exists as a result of some of the community colleges not being open before the base years used in the estimation process. Of the twenty-nine community colleges currently operating, ten have opened their doors during the base years; four opened in 1966, one in 1967, four in 1968, and one in 1969. The enrollments of these schools would cause the enrollment trend to appear to be increasing at a relatively faster rate than it actually was. When sufficient data is available following the start up conditions and when the unstable economic and societal conditions can be incor- porated into the model, expansions of the model to the point of fore- casting the enrollments of individual institutions should become possible. The validity of the Markov student flow model is still somewhat open to question. The model generated good results when uSed to generate first-time enrollments. Reasonably good results were obtained with the three state application to the Michigan system of public higher education. If we apply the generally accepted 123 definition of model validation that "Validation tests whether a sim- 1 ulation model reasonably approximates a real system." or that vali- dation ". . . consists of testing the model's ability to predict the behavior of the system under study."2 then we do not have enough historical data with which to statistically validate the predicted values against actual values. The comparison of predicted enrollments with actual enrollments was attempted over a short time interval with some encouraging results from the limited comparison. Uses of the Model If we have a usable method of forecasting statewide, higher education enrollments, the question arises as to the possible uses of these forecasts in statewide educational decision making. Enrollment forecasts are the primary basis for all educational planning. Some estimate as to the nature and size of the future student population of the higher educational system is fundamental to all areas of educa- tional planning. The major areas of concern for higher education planning can be grouped under the five interdependent areas of faculty, facilities, curricula, finances, and student enrollment. Some aspect of these five areas will be affected by decisions made at the state level by coordinating or governing agencies. Only three states, Delaware, Vermont, and Nebraska, have no legal state agency for coor- dinating or controlling higher education. Of the other states, 25 have coordinating boards, and 22 have governing boards for administra- 3 tion at the state level. Each of these agencies is involved in the coordination or control of the institutions of higher education. For 124 example, the Constitution of the State of Michigan provides that the State Board of Education "shall serve as the general planning and co- ordinating body for all public education, including higher education, and shall advise the legislature as to the financial requirements in connection therewith."4 With regard to the fulfillment of this con- stitutional obligation, the Michigan State Board of Education estab- lished a set of thirty-eight goals as guides in the direction of developing an effective and efficient system of public higher education. Three goals are particularly relevant to enrollment forecasts. Goal 6. Since revisions of long-range enrollment projections are necessary in determining the need for educational pro- grams, space, and faculty, and because of the important variables affecting the college-going rate, it is the responsibility of the State Board of Education to main- tain updated long-range projections of potential and probable student enrollments.5 Goal 36. The importance of annual revision of projections for operations cannot be stressed too strongly because con- ditions constantly change. Therefore, in keeping with its constitutional mandate to advise the Legislature, the State Board of Education will carry on a continuous study of the operating needs of both the baccalaureate institutions and community and junior colleges.6 Goal 38. The projected costs of facilities in terms of future en- rollments and programs is an important undertaking if sufficient student spaces are to be available. The State Board of Education will submit updated annual capital outlay projections to the Legislature, consistent with the constitutional mandate to advise concerning the financial requirements of higher education.7 The activities implied by each of these three goals are involved with each of the planning areas identified earlier. With regard to faculty, planning must consider such factors as desired levels of teaching and research expertise, sources of teaching man- power, and the pattern of distribution of the manpower among the 125 academic fields of study for the coming years. The level of exper- tise, source, and distribution of the faculty should be considered with some knowledge of the future student population. If more stu- dents take the vocational or technical programs at the community college level, faculty with the desired background will need to be trained. The distribution of faculty among the various institutions and areas of study depends on the pattern of choices made by students regarding their fields of study. In facilities planning, total space requirements--the allocation of that space among classrooms, laboratories, faculty office space, and administration--are a function of the distribution and characteristics of the future student population. The duration of the expected population trend can influence capital budget con- siderations with regard to new academic facilities. Consideration of the major geographic distributions of future students would aid in decisions regarding the possible location of new facilities or new institutions. The decision to expand or offer new curricula is partially dependent on future student enrollments and on the societal needs for trained manpower. Although the decision to offer a new field of study at an institution depends more on the goals of the system of higher education, the size of the new program and the amount of re— sources assigned to the program depend on expected future enrollments. A large portion of the finances for the higher education institutions is from student tuition. Future financial planning is directly dependent on the forecasts of future enrollments. In 126 addition, the appropriations for the individual schools is based on two items, the predetermined activity of the institution, eg. teaching vs. research, and the number of students expected to attend the insti- tution during the next fiscal year. The forecasting of enrollments is a basic tool for the statewide planning and coordination of the previously mentioned four academic areas. However, the enrollment forecasts can also be used in the planning for future enrollments. If the enrollment forecast does not meet with the goals established for the system, then action can be planned and implemented in order to achieve specific enrollment goals. PlanningiApplications In the context of the planning of future enrollment goals or enrollment policies, the Markov student flow enrollment forecasting model can be used in a planning mode. In this mode the model can be used to answer or explore various kinds of "what if?" questions that might be asked by educational planners. One such question could be "what will be the impact of an effective, nondiscriminatory enrollment policy?" A nondiscriminatory enrollment policy is one that would re- sult in a racial mix in the higher education enrollment equal to the racial mix of the state p0pulation. If it can be assumed that the retention rate among the different racial groups is the same, than a plan that results in a balanced racial mix for the first-time entrants would result in racially balanced enrollments after four years of effective operation of the new policy. However, the question to be 127 answered is "what will be the enrollments after the implementation of _this policy?" This can be determined with the use of the forecasting model. The only aspect of the enrollment forecast that will change will be the number of first-time entrants. The number of entrants may increase to reflect the increased accesSibility 0f higher education for minority groups. The effect of this policy change can be traced with the forecasting model to determine the effects on future enroll- ments. If the policy was fully effective the first year of operation, then in four years the system of higher education would be racially balanced. However, this situation is unlikely. It would probably take several years of operation in order to become fully effective. This process of a gradual increase in minority students could also be easily simulated with the enrollment forecasting model to determine the effects of the policy on total enrollments. A second planning use of the model could be the prediction of enrollment changes due to increased financial aid. Financial aid programs have been developed to help economically disadvantaged stu- dent stay in school. Increased financial aid should cause more stu- dents to remain in school to complete their education. The effect of the aid is to increase the retention rate, which would be reflected in an increase in the transition probabilities within the student flow model. The effect of changes in future aid programs on future enrollments can be simulated with the forecasting model. This and the preceding example illustrates the use of the Markov student 128 flow model as an aid in the planning and decision making process for statewide higher education. The use of enrollment forecasts are essential in the deci- sion process necessary to control or coordinate the activities of the institutions of higher education. Considering the underlying importance of the enrollment forecast in the decision processes for higher education several suggestions are made in order to improve or expand the statewide enrollment forecasts. Recommendations At this stage of development, the student flow model still needs improvement. Additional time series data could be used for additional testing and validation of the model. The additional his- torical data would allow a longer time period over which to validate the forecasts of the model. The additional data would also allow two additional tests to be made on the transition matrix. The first test would be to determine if the transition probabilities are stationary. Any time dependency of the transition probabilities would have a sig- nificant impact on the model's forecast accuracy in the more distant future. A second test that could be carried out with the additional data would be to determine if there is any correlation between the state of the economy and any changes in the transition probabilities. The identification of a dependency relationship between the transition probabilities and the economy could lead to improved short term fore- casts. 129 The above suggestions only require the passage of time, and the research can wait until the additional data is available. The final recommendation pertains to the types of data collected by the educational agencies. The availability of transfer data for the bac- calaureate institutions would probably improve the accuracy of the model's forecasts. This data may already be available at some of these institutions, but it would be very difficult for an individual research to collect. The Markov student flow model can easily be modified to incorporate the counts of the number of transfers from the community colleges to the colleges and universities. This improve- ment could result in more accurate forecasts for the baccalaureate institutions. 130 Chapter VI Notes 1George s. Fishman and Philip J. Kiviat, "The Statistics of Discrete-Event Simulation," in John M. Dutton and William H. Starbuck, Computer Simulation of Human Behavior (New York, N. Y. John Wiley & Sons, 1971), p. 596. 2Thomas Naylor, Computer Simulation Experiments With Models of Economic Systems (New York: John Wiley & Sons, 1971), p. 158. 3Paul Wing, Statewide Planning for Postsecondary Education: Conceptualization and Analysis of Relevant Inf0rmationETBou1der, Colo- rado: National Center for Higher Education Management Systems at the Western Interstate Commission for Higher Education, March, 1972), p. l. 4Mich. Const. art. i, sec. 3 (1963). 5Michigan Department of Education, State Plan for Higher Education in Michigan (Lansing, Michigan: February, 1970), p. 20. 61bid., p. 58. 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