A CLASSIFICATION MODEL FOR PREDICTING ACADEMIC PERFORMANCE FOR MASTER OF BUSINESS ADMINISTRATION STUDENTS/IT MICHIGAN STATE UNIVERSITY Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY ROBERT GEORGE HARRIS 1968 . .- v- _.n~.-._.——«qu-—H‘. mam LIBR A R Y ' Michigan State THEStG Univt‘rsity 7 This is to certify that the thesis entitled A CLASSIFICATION MODEL FOR PREDICTING ACADEMIC PERFORMANCE FOR MASTER OF BUSINESS ADMINISTRATION STUDENTS AT MICHIGAN STATE UNIVERSITY presented by ROBERT GEORGE HARRIS has been accepted towards fulfillment of the requirements for Ph 0 D degree in Education In mm Major pnofessor Date May 161 1968 0-169 ABSTRACT A CLASSIFICATION MODEL FOR PREDICTING ACADEMIC PERFORMANCE FOR MASTER OF BUSINESS ADMINISTRATION STUDENTS AT MICHIGAN STATE UNIVERSITY by Robert George Harris The purpose of this study was to develop a classi- fication, or probability, model for predicting academic performance in the Master of Business Administration program at Michigan State University. This model was designed to assist the admissions officer in answering two questions: (1) what is the likelihood that this applicant could successfully complete the MBA program and graduate? (2) What would be the extent of this applicant's academic success in the MBA program? The sample of students used in constructing the model was selected from the population of all students admitted and enrolling for the first time in the MBA program at Michigan State University from Fall term, 1962, to Summer term, 1966, inclusive. The "check" sample, used to validate the model's ability to correctly classify individuals, was selected from students entering MSU, under the same con- ditions in Fall term, 19660 Application of selective criteria (essentially excluding foreign students, students with incomplete records, and females), reduced the number of students eligible for selection to 51A graduates and 72 academic withdrawals° Robert George Harris From those eligible, three extreme groups were selected for analysis. These groups were : (1) Upper level graduates; (2) lower level graduates, and (3) lower level withdrawalso Each of the graduate groups contained 138 students, or 27%, of all graduates, Because of the small number of academic withdrawals, 38 students, about 50%, were included in the third groupo Two multivariate statistical procedures, both pro— grammed for machine computation, were used in the analysiso Multiple-discriminant analysis was used to investigate the significance of group differences and to determine the location of group centroids in discriminant space. Maximum likelihood classification methods were used to develop a classification model on the basis of the measurements of predictor variables for each individual in the sample. With such a model, then, unclassified individuals were assigned to a group characterized by academic performance in which the individual had the greatest likelihood--the greatest probability--of membership° By assigning members of a "check" sample--individuals not involved in the formu- lation of the model--in this manner, the model was eval- uated for its accuracy in predicting the academic perfor- mance of an "unclassified" individual, such as an applicant for the MBA programs Robert George Harris The 12 predictor variables used were those elements of information generally available for all applicants for admission from either the standard MSU application for graduate study or university record files for each indi- vidual, with the exception of a new variable which was formed in an attempt to "adjust" or "weight" the appli- cant's undergraduate GPA in respect to the "quality" of his undergraduate institution. The measure of "quality" used was the mean ATGSB Total score for the undergraduate college, as reported by ETS. In general, there is a monotone descending relationship between the mean values of the 12 predictor variables and the level of academic performance. As the mean value of the predictor variable decreases, the level of academic per- formance decreases. From the data, one observes that the more academically successful student has most of the following character- istics. He: (1) Attained a higher undergraduate GPA (both "raw" and "adjusted"); (2) scored higher on all three ATGSB tests; (3) attended an undergraduate college other than MSU that was relatively large (over 10,000) but not as large as that attended by the less successful MBA student; (A) was older, married, and delayed longer before beginning the MBA program; (5) maintained legal residence outside of Michigan, and (6) had completed a non—business undergraduate major course of study. Robert George Harris The classification model was constructed from the set of values for the predictor variables for each individual in the "main" sample. This data permitted computation of the coefficients and constant terms for a set of classifi- cation functions-—one classification function for each of the three classification groups. Evaluation of these classification functions yielded a set of three scores for each individual. The model compared the computed classi- fication scores for each unclassified individual and assigned him to the group for which he had the highest score. In this manner, the 109 unclassified individuals in the'kfluufld'sample were also assigned to the three academic performance categories. The academic performance of 53.2% of the "check" sample was accurately predicted and were termed HITS, 21.1% were MISSES, while 2A.7% were labeled NEAR HITS, since they were predicted as graduates but the level of performance was inversely predicted. If HITS were to be defined simply as predicted graduation or predicted withdrawal, the model produced 78.9% HITS and 21.1% MISSES. From the analysis of the results it was found that: (l) the three classification groups could be dis- tinguished from each other on the basis of the 12 predictor variables used in the study,_ (2) the group centroids were colinear (points on the same straight line) and ordered in the same manner as the three levels of academic per- formance, (3) (A) (5) Robert George Harris the upper level graduate groupvwm well separated from both the lower level groups, but that the lower level graduates and the academic with- drawals were relatively close together, the likelihood of individual membership in one or another such group can be predicted on the basis of the elements of information known about the individual at the time of the admissions decision, the model offers the admissions officer an objective, systematic method to assist in the selection of students for the MBA program. It establishes an order of priority in the selection of students, provides a basis for admission to eiUnn°regular or provisional status, identifies "borderline" applicants, and suggests when additional information should be considered in arriving at the admissions decision. Therefore, from these findings it was concluded that: (l) (2) (3) (A) The three groups--upper level graduates, lower level graduates, and academic withdrawals--are distinct separate groups that can be distin- guished from each other on the basis of measure- ments on the 12 predictor variables used in this study, the major problem in classification of indi- viduals results from the difficulty in discrim- inating between the lower level graduates and the academic withdrawals, the classification, or probability, model, constructed on the basis of the measurements of the 12 predictor variables for each individual in the "main" sample is a valid model for the prediction of academic performance for MBA students at MSU, the model does no worse than the less objective methods used in the admissions decision- making process during the time period encompassed in the study. Robert George Harris Admission to graduate school is a major and most serious decision. While the state of the art is not suffi— ciently advanced to entrust admissions decision-making to a computerized model, the assistance that such a model can provide should not be overlooked. By increasing the objec- tivity of the selection process, the exercise of subjective judgment, which in the final analysis is still the major element in the admissions decision, can be sharpened. A CLASSIFICATION MODEL FOR PREDICTING ACADEMIC PERFORMANCE FOR MASTER OF BUSINESS ADMINISTRATION STUDENTS AT MICHIGAN STATE UNIVERSITY By Robert George Harris A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Administration and Higher Education College of Education 1968 ACKNOWLEDGMENTS I wish to express my gratitude to the following members of the dissertation committee for their advice and counsel: Dr. Max S. Smith, Chairman; Dr. Lawrence Borosage; Dr. Max R. Raines; Dr. Winston R. Oberg, and Dr. John H. Useem. Special acknowledgment is due Dr. Smith for the guidance and encouragement provided me throughout my. doctoral program. I will always be indebted to Dr. Paul L. Dressel, for his friendship and understanding as well as guidance and assistance over the years, extending back to my freshman days at Michigan State in 19Al. I owe a debt of gratitude to Mr. Lynn H. Peltier for his assistance in data collection and processing, to Mr. Stuart Thomas, Jr., for invaluable aid in computer programming, and to Mr. James M. Peters for his critical editing and constructive comments during the writing of this thesis. I would be remiss if I did not express my deepest appreciation for the love, patience and understanding of my wife, Marie, our son, Robert, and our daughter, Patricia. R. G. H. ii TABLE OF CONTENT ACKNOWLEDGMENTS LIST OF TABLES LIST OF APPENDICES Chapter I. INTRODUCTION Need . . Purpose . . . Research Problems. . . . . . . Rationale . . . . . . . . . Overview. II. REVIEW OF THE LITERATURE Introduction . . Review of Studies of Predictions of Academic Performance . . . Review of Statistical Methods. Summary . . . III. DESIGN AND METHODOLOGY OF THE STUDY. Sample Variables Research Problems. Analysis. Summary IV. ANALYSIS OF RESULTS Description of the Sample Discriminant Analysis . Classification: The "Main" Sample Classification: The "Check" Sample Classification: Test of Significance Comparative Analysis. . . . . Summary iii Page 11 vi CD CI) \lmUl-IZ‘I-J |-‘ 21 2A 2A 31 32 3A 37 37 A5 52 53 55 Chapter V. SUMMARY AND CONCLUSIONS. Summary Conclusions. Discussion Implications for Future Research. BIBLIOGRAPHY APPENDICES. iv 0 Page Table A.l LIST OF TABLES Characteristics of MBA Students by Percentage of Each Group. Group Means for Predictor Variables Comparison of "Raw" and "Adjusted" Undergraduate GPA . . . . Projection of Group Centroids Value of Coefficients and Constant Terms for Linear Classification Functions Classification Matrix of "Main" Sample Classification Matrix for the "Check" Sample. . . . . . . . . . Comparison of Model and Actual EXperiences with "Check" and "Main" Sample MBA Students . . . . . . Page 38 39 A1 145 A8 50 S2 57 LIST OF APPENDICES Appendix Page A. Application for Admission to Graduate Study, Michigan State University . . 79 B. Classification Scores for Each Individual in "Check" Sample and Assignment to Academic Performance Group . . . . 8A vi CHAPTER I INTRODUCTION Need Problems associated with the admissions decision- making process exist in varying degrees at all educational institutions. These problems have, generally, become more acute in recent years because growing interest in educa— tion, and particularly higher education, has measurably increased therumber of applicants for admission to the nation's colleges and universities. A partial solution to these admissions problems has often been found by increasing student enrollment. However, increased enroll- ments are subject to the limitations imposed by maintenance of respectable academic standards and such factors as the (availability-~0r better, the non-availability-—of adequate /’ facilities, competent staff, and sufficient financial support. Generally, these limitations preclude unlimited or greatly increased enrollment and dictate the use of selective admissions criteria. In recent years, the Graduate School of Business Administration at Michigan State University, like many others, has been the recipient of an increasing number of applications for admission. At the same time, there has 1 been a general improvement in the qualifications of the students seeking advanced study. This has been particular- ly true in the Master of Business Administration (MBA) program at Michigan State University. In response to the increased demand, there have been larger student enroll- ments in the MBA program each year for the past several years. Because of an abundance of qualified applicants, however, and the limited (although increased) resources allocated to thegrogram, the admissions decision-making process has become more difficult. Faced with this situation, it is suggested that selective admissions criteria objectively answer two major questions about each applicant, based on the kind of information about the individual now available. First, it should be asked, "What is the likelihood that this applicant could successfully complete the MBA program and graduate?" Next, providing the first answer is in the affirmative, one should inquire, "What would be the extent of his academic success in the program?" If the answer to the first question is "no likeli- hood at all," then of course, the second question is superfluous and the admissions decision has been made. If, however, the answer is something other than that, it is important to weigh the degree of academic success that the applicant is likely to achieve. Given the answers to these two questions, it would seem reasonable to select those applicants who were most likely to attain the highest academic performance, and to reserve a decision on admis- sion of those individuals who might be expected to meet minimum graduation standards but generally perform just above the academic "borderline." "Selective admissions," as an institutional policy, is based on merit and is limited to those applicants con- sidered to be best qualified to benefit from the educa- tional Opportunity provided by the‘college. Information such as the applicant's past academic achievement, recom— mendations, reports on character, personal qualities, entrance examinations, test scores and personal interviews are often included in the selective admissions procedure. Many of the elements of information obtained about the applicant are qualitative in nature. A substantial portion of the information, however, is quantitative or is repre- sented in a quantitative form. Regardless of its form, both subjective and objective evaluation and judgments are made about the applicant, based on the available information. These elements of information are evaluated and a judgment is made about the likelihood of the applicant‘s academic success if selected for admission. Both Snookl and Stuart2 maintain that selec— tion of students must rely to a large extent upon the sub- jective judgment of the admissions officer. Nevertheless, it would seem advantageous to maximize the objective evalua- tion of all available information, both qualitative and quantitative, in the selection process. Hopefully, a more objective method would permit selection of those applicants for admission who can reasonably be expected to satisfac- torily complete a particular course of study; conversely, such a method would tend to "select out" those candidates with little likelihood of academic success. Purpose The purpose of this study is to develOp a probability model to predict the academic performance of applicants for admission to the MBA program at Michigan State University. This probability model will incorporate the recorded experience of former MBA students at the University; and, prediction of academic performance of new prospective studentsvfidl.be predicated on elements of information known about them at the time of the admissions decision. 1John L. Snook, Qualitative Admissions Factors-—Proof, Practice, and Prejudice, a paper presented to the 1966 Symposium of schools using the Admission Test for Graduate Study in Business, Princeton, N.J., May 5, 1966. 2Douglas Stuart, A Study of the Relationships of Admissions Data to Student Performance in the Graduate School of Business during the Period, 1958—1961, East Lansing, Michigan State University, Office of Institutional Research, June, 1962, p. 13. The study will use multivariate procedures to con— struct a probability model that would determine the maximum likelihood of an applicant being a member of one of three groups: top level graduates; lower level graduates, or academic withdrawals. This model will be designed to assist the admissions officer in answering the two questions previously posited: 1. What is the likelihood that this applicant could successfully complete the MBA program and graduate? What would be the extent of this applicant's academic success in the MBA program? Research Problems Two research problems are to be investigated in this study. They are as follows: Can groups of MBA students, who have either graduated or withdrawn from the program, when grouped by the overall graduate grade-point average, be distinguished from each other on the basis of the elements of information known about the students at the time of the admission decision? Can the likelihood of individual membership in each of these groups be predicted on the basis of the elements of information known about the individual at the time of the admissions decision? The elements of information known about the students at the time of the admissions decision are, for the most part limited, directly or indirectly, to the information available on the completed "Application for admission to Graduate Study" required of all persons seeking admission for advanced study at Michigan State University. Rationale Customarily, researchers concerned with predicting academic performance have used a statistical procedure known as multiple-regression analysis. This procedure requires the identification of a set of factors (indepen- dent variables), that are considered to be predictive of academic performance. The criterion for academic performance (the dependent variable), most frequently selected is the grade—point average. The rationale of the present investigation is that students grouped according to academic performance can be distinguished from each other on the basis of the measure- ments of a set of independent variables; further, that the probability of membership in each group can be determined for an unclassified individual and the most likely group membership (and therefore the associated academic perfor— mance) can be predicted for each individual. The technique of classifying individuals in this 3 manner was developed for biometric research by Rao and extended by researchers such as AndersonlI and others. In recent years, there has been increased interest in this approach in the field of educational and vocational guidance and counseling. Overview In Chapter II, the review of literature is of two kinds. A review of the literature in the general area of prediction of academic success is presented, as well as a general discussion of selected research techniques applica- ble to the problem of predicting academic performance. In Chapter III, the methodology and procedures used to conduct the study, as well as the statistical models used in the study are presented. The results of the study are analyzed in Chapter IV. The final chapter, Chapter V, presents an overall summary, the conclusions, and a discussion of the findings and their implications for future research. 3C. Radhakrishna Rao, Advanced Statistical Methods in Biometric Research (New York: John Wiley and Sons, 1952). “T. W. Anderson, An Introduction to Multivariate Statistical Analysis (New York: John Wiley and Sons, 1958), Pp- lA7—152. CHAPTER II REVIEW OF THE LITERATURE Introduction The design of this study, detailed in a later section, while not a replication of any previous research reviewed in the literature, has nevertheless evolved from experiences of earlier researchers concerned with prediction of academic performance and/or success. The literature was first reviewed to determine what research had been conducted in the general area of predic— tion of academic success of students, and more specifically, of graduate students in business administration, were examined. A second aspect of the review of literature involved focusing on statistical procedures used in the aforementioned studies and possible applications of other techniques as yet untried, or relatively unused, in the prediction of academic performance. Review of Studies of Predictions of Academic Performance The literature contains studies of performance at all educational levels, and that pertaining to undergraduates is particularly voluminous. Most studies use high school 8 grades, standard intelligence test scores or tests specifi- cally designed for the purpose, such as the scholastic Aptitide Test (SAT), as predictors of future school per- formance. In an earlier review, Cronbach reported that college level ability tests correlated about .50 to .55 with college grade-point averages.5 More recent studies have not substantially altered this finding. Typical of the many predictive studies reported in the literature is one by Glover at the University of Massa- chusetts.6 Glover used multiple-regression techniques to develop prediction equations for three groups of students: men (excluding engineers); women, and engineers. The data was collected for students admitted over a three year period. The prediction equations were cross—validated using new data collected from the most recent year group. The predictor variables used were SAT Verbal and Mathematics scores and class rank as reported by the student's secondary school. The multiple correlation ranged from .A30 for the men to .537 for the women. Attempts to improve upon the predictive efficiency of this basic variable combination by "adjusting" class rank according to the "quality" of the secondary school produced negative results. 5Lee J. Cronbach, Essentials of Psychological Test- ing (New York: Harper and Brothers, 19A9). 6Robert H. Glover, Preselection in College Admissions at the Universitygof Massachusetts, Office of Institu— tional Studies Information Series No. 9, University of Massachusetts (Amherst: University of Massachusetts, 1963). lO Linn reviews the results of several empirical studies that have used "adjusted grades" to predict academic achievement.7 His paper considers some of the possible techniques which could be used to make grade adjustments for interschool differences; it is observed, however, that most researchers have found that the improvement in pre- dictive validity due to the use of adjusted grades, as com- pared to unadjusted grades, has been "discouragingly small."8 Studies indicate that predictability of graduate school performance is generally lower than for under— graduate performance. This is probably because graduate students are a more highly selected group than under— graduates. They are of higher quality with less variation in ability-—consequently, the correlations are lower. Moreover, some of the tests used as predictors of academic success at the graduate level are often used, also, as admissions screening devices at the same level. A study conducted at Stanford University Graduate School of Business in 1957-58 provided an opportunity to observe the effect of the Admission Test for Graduate Study in Business (ATGSB) when test scores are used as a 7R. L. Linn, "Grade Adjustments for Prediction of Academic Performance: A Review," Journal of Educational Measurement, 3 (Winter, 1966), 313-329. 8 Ibid., p. 326. ll 9 Although Stanford was not using the basis for admission. ATGSB in admitting students in 1957, the School administered the test to those same MBA students in the summer of 1958 in connection with another study. By 1962, Stanford was using the ATGSB in its selection procedures. A comparison of the correlation of ATGSB scores for 1957 first year students and first year students in 1962, reflects the effect on use of the test as a predictor of first-year- average grades. The ATGSB Total score correlation for 1957 first year students was .6A, as compared to .56 for the similar group in 1962. The Admission Test for Graduate Study in Business (ATGSB) focuses on the measurement of general abilities associated with academic success rather than on specific preparation. The test provides in two broad areas of academic skill--verbal ability and quantitative ability. Scores on the ATGSB are reported on a standard normative scale. A Total score, two part-scores——a Verbal score, and a Quantitative score are obtained. The research design of many studies concerned with prediction of academic performance in graduate programs is not unlike that of the studies previously described. 9Educational Testing Service, The Admission Test for Graduate Study in Business: A Handbook for Deans and Admissions Officers, A report prepared by the Educational Testing Service (Princeton, N.J.: Educational Testing Service, 1966), pp. 5A—56. 12 For example, in a study investigating the feasibility of predicting grade-point averages for students in a graduate education course, Herbert found that the following factors were useful as predictors: undergraduate grade- point average in education courses; overall undergraduate grade—point average; type of undergraduate curriculum; grade in student teaching, and scores from the National Teacher's Examination.10 The best single predictors were undergraduate grades in education courses and overall undergraduate grade-point average. In another study of the prediction of academic success in a master's program in education, Owens and Roaden compared the grade-point averages and advisor's ratings of graduating students with their undergraduate grade-point averages, test scores, area of graduate study specialization and the enrollments of their undergraduate colleges.11 Again, it was found that undergraduate grade— point average was the best single predictor; that test scores added only slightly to predictive efficiency. Other factors were found to be of little value in pre- dicting academic performance. 10David J. Herbert, "A Predictive Study of Quality Point Averages in Graduate Education Courses," Journal of Educational Research, 60 (January, 1967), 218-220. ‘ 11Thomas R. Owens and Arlies L. Roaden, "Predicting Academic Success in Masters Degree Programs in Education," Journal of Educational Research, 60 (November, 1966), l2A-126. l3 DeGrandis evaluated the effectiveness of undergraduate grades as predictors of future academic success in the graduate study of business at the University of Southern California.12 Her finding was that undergraduate grades can be useful predictors, particularly if the student pur— sues graduate study at the same institution at which he was an undergraduate. It was also found that higher graduate grades were earned by full—time students than by part-time students, and by students who had had a signifi- cant time lapse between undergraduate study and commence- ment of graduate study. A number of studies have been conducted which in one way or another evaluate the effectiveness of the Admissions Test for Graduate Study in Business (ATGSB), either alone or in conjucntion with other measures of ability, to pre— dict academic performance in graduate study in business. In a study during 1958-1959, Pitcher showed that ATGSB Total scores are useful alone or in combination with undergraduate records in predicting first—year grade 13 averages of graduate business school students. Her study confirmed the findings of an earlier study conducted by l2Norma L. DeGrandis, "The Prediction of Scholastic Success in a Graduate School of Business Administration" (unpublished Master's thesis, University of Southern California, 1962). 13Barbara Pitcher, The Admissions Test for Graduate Stugy‘in Business as A Predictor of First-Year Grades in ‘BusineSS'School,,195841959, SR-60-3A (Princeton, N. J.: Educational Testing Service, 1960). 1A the Educational Testing Service (ETS), designers of the ATGSB, during the period 195A—1955. The multiple correla- tion coefficient of the ATGSB and undergraduate record with first-year grade averages was .50 for l95A—55, and .A9 for 1958-59. The total ATGSB score alone had a correlation of .A3 in both studies, while the undergraduate record alone was .35 in the earlier study and .28 in the 1958—59 study. It was also found that Verbal and Quantitative scores, which were not included in the l95A—55 study, predicted first-year grade averages less effectively than Total ATGSB scores. Finally, it was found that students who had undergraduate major fields of study other than business and economics did better on the ATGSB (all three scores) than did those who majored in business and economics. In 1965, Pitcher and Winterbottom confirmed the find- ings of previous ETS studies that the combination of ATGSB scores and undergraduate grade record was a useful pre— dictor of graduate school performance.lu Analyses were made of several subgroups of students defined according to undergraduate major, of time delay between college graduation and entrance to business school, of survival (drop-out vs. non-drop-out) and of national origin (foreign vs. non—foreign). The findings, while not luBarbara Pitcher and John A. Winterbottom, The Admissions Test for Graduate Study_in Business as a Pre- dictor of First—Year Grades in Business School, 1962- 1963, SR-65-2l (Princeton, N. J.: Educational Testing Service, 1965). l5 conclusive, provided sufficient evidence that these factors should be considered in the prediction of academic per- formance. The authors explained that the observed validity of both predictors had declined due to the more extensive use to which they had been put in selecting entering students. This statement further substantiated the Stan- ford study, which had found that use of the ATGSB for selection of students reduces the correlation of the ATGSB with criteria of success. In a recent handbook, ETS has reviewed the most sig- nificant studies pertaining to ATGSB.15 A University of Pennsylvania study took up the rela— tive effectiveness of various combinations of predictors with first-year grades in graduate business school.16 The three ATGSB scores and undergraduate record were validated singly and in combination against first-year grade averages. The resulting correlations were characteristic of studies of this kind. The undergraduate record, with a correla- tion coefficient of .39, was the best single predictor. The combination of all three scores and the undergraduate record, with multiple correlation coefficient of .52, was slightly higher than all other combinations. This suggests that it may be more practical to use a combination of the undergraduate record and only one score as a valid predictor, 15Educational Testing Service, op. cit., pp. Al-80. l61bid., pp. 56-58. 16 rather than going to the additional effort of including all three scores in the combination. A Harvard study also used the three ATGSB scores and the undergraduate record as predictors of the first-year grade averages at the Harvard Business School.17 In addi- tion, however, the study predicted separate course grades. The correlation of the combined predictors was found to be remarkably consistent for three different year groups of students: .A9, .A8 and .A9. There were, however, con- siderable variations among the correlation coefficients found for the various courses. It appeared that the cor- relations with the more structured courses were higher. This suggests that personality factors, which the tests do not directly measure, affect performance to a considerable degree. A study of predicting average grades at the end of an MBA program was conducted at Emory University using the three ATGSB scores, overall undergraduate grade averages, and the grade average for the junior and senior years of college.l8 The best single predictors were the ATGSB Quantitative and Total scores, with correlation coeffi- cients of .A6 and .A5, respectively. For a part of the same sample, the grade averages for the first-year of business school was also available. The correlations of 17 18 Ibid., pp. 58—59. Ibid., pp. 59-61. 17 these first-year grade averages with final MBA grade averages was .6A. It was concluded that relative to other predictors, the first-year grade average was extremely high in terms of predictive accuracy and could almost be used alone. This study shows that previous academic per- formance within the business school itself is an accurate predictor of final success in that school. It also sug- gests that a pervasive educational philos0phy within a given university may lend the undergraduate grade average in itself a degree of reliability in predicting graduate academic performance at the same institution. Stuart conducted a study of the relationships of admissions data to student performance in the Graduate School of Business at Michigan State University for the period, 1958-1961.19 The study weighed the worth of using the Miller Analogies and ATGSB tests for all stu— dents entering the school. It concluded that while both tests showed positive but low coefficients of correlation with academic performance in the graduate school, there was sufficient evidence to support use of the ATGSB in screening candidates for admission to the MBA program. Review of Statistical Methods With one exception, the statistical procedures used in all of the above studies were restricted to calculations 19Douglas Stuart, op. cit. 18 of simple or multiple correlations. (The exception was a non—parametric statistical approach employed by DeGrandis.) Consequently, before the methodology for development of the statistical model used in this study could be formulated, an extensive review of applicable statistical procedures was conducted. This study is concerned with two general problems. The first is that of discrimination, where the emphasis is on the problem of differentiating between groups of students on the basis of various elements of information about them, at the time of the admissions decision. The second is one of classification. This involves "assignment" of an individual student to one of several groups, on the basis of the same elements of information, and it also involves comparing the characteristics of an individual with that of a group. Various methods for differentiating between groups and assessing similarity are used to solve problems of discrimination and classification. Two methods available are multiple—discriminant analysis and maximum likelihood classification. These methods, however, have been put to relatively little use in the general area of prediction of academic performance, and only slightly more use in guidance and counseling. A popular and useful approach used in studies of pre— dicting academic performance is known as multiple-regression analysis or multiple correlation. Multiple regression l9 analysis provides information concerning the probable degree of success or performance of an individual in each of several groups, given data on the characteristics or past performance of that individual. Multiple discriminant analysis and maximum likelihood classification methods use the same data on the individuals in each of several groups and are designed to answer the questions, "Can one distinguish one group from another?" and "What group is the unclassified individual most like?" Multiple-regression analysis, on the other hand, is concerned with the question, "In what group would the individual perform best?" By way of clarification, it should be pointed out that the problems of "discrimination" and "classification" as defined above share a common theoretical base and historical development. They often, in fact, are viewed as simply two different aspects of discriminant analysis. Tatsuoka presents an excellent review of the develop— ment of objective methods based on the idea of profile similarity.2O He reports that a differential prediction method was first introduced in 1928, and later developed by many researchers, including R. L. Thorndike. Other approaches maximizing the overall efficiency of the method —Y 2OMaurice M. Tatsuoka, Joint-Probability of Member- ship and Success in Group: An Index which Combines the Information from Discriminant and Regression Analyses as Applied to the Guidance Problem, Harvard Studies in Career Development No. 6, Office of Naval Research, Contract Nonr- 1866 (31) (Cambridge, Mass.: 1957), pp. 1-5. 20 were later introduced by other investigators. It is ob- served that all of the early methods suffered from the major defect of forcing a multi-dimensional situation into a uni—dimensional problem. Development of a method of analysis that involved multi-dimensional space was accom- plished in recent years by researchers in both the natural and behavioral sciences. A more comprehensive review of the literature and research of discriminant analysis is presented by Tatsuoka and Tiedeman in an earlier publication.21 In one of his many significant contributions, Rao addresses the problems related to the utilization of multiple measurements in the field of biological classification.22 This is a situation where the researcher is confronted with the problem of assigning an individual to one of several groups to which he might belong. Rao presents an objective method, based on the modern theories of statistical inference, which mini— mizes errors of classification. Anderson studied the multivariate methods based on normal distribution and developed a method for classifica— tion of an individual into one of several multivariate normal distributions.23 This method is a particular 21Maurice M. Tatsuoka and David V. Tiedeman, "Dis- criminant Analysis," Review of Educational Research, XXIV (December, 195A), pp. A02-A20. 22 Rao, 0p. cit., pp. 273—378. 23Anderson, op. cit., pp. lA7-152. 21 interest in the present study since, under the proper assumptions, it provides a methodology for use in the problem at hand. In recent years, there has been increased interest in discrimination and classification methods in the field of educational and vocational guidance. A study by Cutting is typical of this new research.2u Cutting differ- entiated between groups of students on the basis of fifteen variables of self-concept of ability and occupational interest, using multiple discriminant analysis. The pur- pose of the study was to identify, through predictive variables of certain elements of the self-concept, the academic field of concentration that the college student would select. Summary Thus, the foregoing review of literature indicates that there has been extensive effort in the area of pre— dicting academic performance and success. The literature concerned with predicting success in college for graduat— ing high school students is particularly voluminous. Generally, the predictor variables and the criterion for successful performance are the same in all studies. The record of past academic performance, course grades, (or, 2“Donald J. Cutting, "Predicting the Selection of a Field of Concentration at MSU from the Personal Preference Inventory" (unpublished Ph.D. dissertation, Michigan State University, 1966). 22 more likely, overall grade averages), along with standard test scores are usually the predictors. Similarly, the college grade average, or on occasion an individual course grade, is the favorite criterion of success. A few of the studies consider other factors, such as residence, age, sex, high school or college attended, or major course of study, as contributers to academic performance. Along with the computation of simple correlations, an often used statistical technique in the studies is multiple- regression analysis. The results of this latter analysis, in Most of the studies, indicate that the past academic performance record is the best single predictor of future academic performance. The past record in combination with the scores on special predictive tests (such as SAT or ATGSB) improves this prediction. Generally, the coefficient of multiple correlation between the predictors and the criterion for academic success is between .20 and .50. There have been several attempts to improve this correla— tion by "adjusting" the past academic performance record, in accordance with the estimated "quality" of the former school. Past efforts have been generally disappointing. While the multiple regression method will continue to be a very valuable tool, it does not appear to offer any marked improvement over past studies in prediction. On the other hand, multiple—discriminant analysis, a relatively new method in educational research, offers some 23 evidence of producing useful results in future investiga- tions. Several recent studies have taken this approach in studying problems in guidance and counseling. Essen- tially, multiple-discriminant analysis uses the same predictor variables as are used in studies where multiple regression methods have been employed. Multiple discrimi- nant analysis, however, addresses the twin problems of discrimination between criterion groups and classification of an individual on the basis of these predictor variables. CHAPTER III DESIGN AND METHODOLOGY OF THE STUDY Sample The sample of students for this study was selected from a population composed of all full-time students ad— mitted and enrolling for the first time in the Master of Business Administration program at Michigan State Univer- sity during the period beginning with Fall term, 1962, and ending with Summer term, 1966. Criteria for selection of the sample were established after consideration of several problems. For example, selection of foreign students for the MBA program is quite unlike that of students from under- graduate institutions in the United States. For the most part, the records and test scores of foreign students are not available to the admissions officer. In most instances the foreign student has not taken the ATGSB, a stated requirement for all applicants. If his undergraduate record is available, it is most likely subject to considerable individual interpretation and subjective evaluation. Of a total of 1,30A students entering the MBA program during the four year period under study, 167 were foreign students. It was felt that inclusion of this group of atypical students 2A 25 in the sample would unquestionably introduce bias, and so they were excluded. For various reasons, too, there are students for which the information on file is incomplete. But the statistical design of the study did not allow for inclu- sion of individuals with missing data, and therefore, these students could not be included in the sample. Fifteen of the total number of students were females. In view of the small number, but also to avoid the possible introduction of a bias due to sex difference, these students were excluded from the sample. Finally, some students admitted during the period under study had not yet "completed" their programs--either through graduation or by withdrawal—-and for this reason were ineligible for inclusion in the sample. Of the students who had withdrawn from the program, it was known that many had done so for other than academic reasons. But since in most cases student record files did not include explicit reasons for withdrawal, it was assumed that those students--and only those students--with an overall grade—point average (GPA) pglpg_that required for graduation had withdrawn for academic reasons. It is this group that is defined herein as "academic withdrawals." A GPA of 3.00, where A=A, B=3, C=2, D=l, and F=0 is required for graduation. (All other students who withdrew were considered to have done so for other than academic reasons, but were also excluded.) 26 Consequently, the sample was selected in accordance with the following criteria: 1. 6. Student had received a baccalaureate degree from an undergraduate institution in the United States; A report of student's ATGSB test scores was on file with the University; A report of student's undergraduate grade-point average was on file with the University; Student had either completed his MBA program and received the degree, or had withdrawn from the program for academic reasons by the time data collection began; Student who had withdrawn from the MBA program had done so with an overall graduate GPA of less than 3.00; and, Student was a male. The sample was divided into three groups: (1) the upper level of degree recipients; (2) the lower level of degree recipients; and (3) the lower level academic with- drawals. After ordering the degree recipients (hereafter called the "graduates,") according to their GPAs at gradua— tion, an upper group, consisting of approximately 27 per cent of the graduates who had accumulated the highest GPAs and a lower group consisting of an equal number of gradu— ates who had accumulated the lowest GPAs, were selected. 27 The academic withdrawals, i.e., those with cumulative GPAs below 3.00, were ordered in the same way. Arbitrarily, the lower 50 per cent of this withdrawal group, rather than 27 per cent, was selected because of the relatively small number of students in this category. In this manner, a total sample consisting of 313 MBA students was selected, composed of 138 upper level graduates, 138 lower level graduates, and 37 lower level academic withdrawals. Setting of sub-sample size at 138 individuals or 27 per cent of each of the graduating groups, was determined both by certain constraints in computer programming and in light of the arguments developed by Kelley for selection of criterion groups.25 Use of a sub-sample size of 138 for both the upper and lower level graduates provided the best compro- mise between two desirable but inconsistent aims: to make the extreme groups as large as possible, and to make the extreme groups as different as possible. A "check" sample of students satisfying the same cri- teria as the "main" sample was selected from all MBA stu- dents admitted and enrolled for the first time in the Fall term, 1966. The purpose of the "check" sample was to test the validity of the model which was developed from the data obtained from the "main" or larger sample previously de- scribed. 25Truman L. Kelley, "The Selection of Upper and Lower Groups for the Validation of Test Items," Journal of Educational Psychology, XXX (1939), pp. 17—2A. 28 The "check" sample, composed of all eligible students registered Fall term, 1966, consisted of 109 students--95 graduates and 1A academic withdrawals. It was also divided into three groups. The division was made according to median GPAs of the two groups of successful graduates in the "main" sample. As a result, 57 were tagged "top level" graduates and the remaining 38 labelled "lower level." The third group included all of the 1A academic withdrawals. Variables The variables in the study were limited by the avail- able sources of data. The independent variables included all of the elements of information available to the admissions officer at the time of the admissions decision. They were, for the most part, the information available on Michigan State University's "Application for Admission to Graduate Study," a copy of which is included as Appendix A. The remainder are available in other University records held by the Registrar and/or the College of Business. In sum, the 12 independent variables, or "predictors," con- sidered in this study are: 1. Marital Status 2. Legal residence 3. Age at time of Admission A. Undergraduate major curriculum 5. Undergraduate institution attended 6. Undergraduate grade-point average 10. 11. 12. 29 Delay in beginning MBA program since last formal school attendance Size of enrollment at the undergraduate institu— tion Total ATGSB score Verbal ATGSB score Quantitative ATGSB score Product of the student's undergraduate grade— point average and the mean ATGSB score for his undergraduate institution. Data about the first eight variables were obtained either directly or indirectly from the "Application for Admission to Graduate Study" previously cited. An inter- pretation (if required) and the coding used in the study for each of these variables follows: 1. 2. Marital Status: Single = 1; Married = 2. Legal residence: Michigan = 1; Out—of—state = 2. Age at time of admission: Number of years (rounded to nearest whole year). Undergraduate major curriculum: Business = 1; Non-business = 2. Undergraduate institution: MSU = 1; other than MSU = 2. Undergraduate grade—point average: GPA for last two undergraduate years reported to two decimal points on basis of A = A.00 . . . , F = 0. 3O 7. Delay in beginning MBA program since last formal school attendance: Number of years (rounded to nearest whole year). 8. Size of enrollment at undergraduate institution: enrollment size to nearest 100 students for year of graduation.26 ETS reports to the graduate school of ATGSB scores for each individual provide the information required by variables 9 through 11. The product of the student's undergraduate GPA and the mean ATGSB score for his undergraduate institution, variable 12, is an effort to "adjust" or "weight" the individual's undergraduate GPA by a measure of the "quality" of his under— graduate college. ETS compiles a confidential publication for the use of deans and admissions officers of the graduate schools of business.27 In it, the mean ATGSB total score and the number of persons writing the test are presented in year groupings for most of the undergraduate institutions in the United States. The possibility of misinterpreting or mis- using these mean scores are acknowledged by the publisher. Nevertheless, in the spirit of experimentation and in the 26Garlan G. Parker, "Statistics of Attendance in American Universities and Colleges," Annual Reports School and Society, 83-9A (January, 1955-1967). 27Educational Testing Service, Admission Test for Graduate Study in Business, 1957-65, Statistical Summary by Undergraduate Colleges attended (Princeton, N. J.: Educational Testing Service, 1965). 31 absence of any better estimation of the "quality" of the students from a given institution seeking admission to graduate study in business, the product was used as a pre— dictor variable. To repeat, the product was formed by multiplication of each individual student's undergraduate GPA (computed to two decimal places) and the mean ATGSB Total score for his undergraduate institution. The result- ing figure was rounded to five digits for use in the com- puter program. Research Problems The two research problems investigated in this study can now be more specifically restated as follows: 1. Can the three extreme groups in the sample of MBA students, upper level graduates, lower level graduates and lower level academic with- drawals, be distinguished from one another on the basis of the measurements made on the set of predictor variables described in the pre- ceding section? 2. Can an unclassified individual be accurately classified as a probable upper or lower level graduate or as an academic withdrawal on the basis of individual measurements made on this same set of predictor variables? 32 Analysis Two statistical methods were used in the investiga— tion of the research problems. Both methods, multiple- discriminant analysis and maximum likelihood classification, are techniques for analyzing data that consist of several measures on each individual in each of several groups. Multiple-discriminant analysis requires the deter- mination of linear combinations of variates, called dis- criminant functions, such that, with respect to the dis— criminant functions, the ratio of the between-groups to within-groups dispersion is a maximum. In general, there is more than one discriminant function. The maximum number of discriminant functions is the lesser of (1) the number of predictor variables, or (2) the number of groups, mippp one. The significance of each discriminant function can be examined by means of Chi-square approximation28 or by the per cent of trace attributable to the function. The per cent of trace indicates the portion of the discriminat— ing power contained in a particular discriminant function. The location of group centroids in discriminant space is also obtainable with discriminant analysis. The deter- mination of inter-centroid distances is valuable in the examination of group separation. Tatsuoka and Tiedeman conclude that multiple dis- criminant analysis 28Rao, op. cit., pp. 372-373. 33 can be used as a unified approach in solving a research problem involving multivariate com- parison of several groups, which is likely to have as its three phases, (a) the establishment of significant group differences, (b) the study and "explanation" of these differences, and (c) the utilization of multivariate information from the samples studied in classifying a future individual known to belong to one of the groups represented.2 Phases (a) and (b), which pertain to the first of the research problems, were investigated using multiple— discriminant analysis. However, in approaching phase (0), pertaining to the problem of classification, a method based on the principle of maximum likelihood was used. Maximum}1ikelihood classification requires the compu- tation of a set of linear equations for the purpose of classifying an individual into one of several groups. One linear function is formed for each classification group. Each of these classification functions contains a constant term and a number of terms equal to the number of pre- dictor variables. Evaluation of the classification function, given a set of values for the predictor variables and the constant term for an individual, yields a classification score. Classification scores are monotone functions in relation to the probability of group membership of the individual. An individual is classified with the group for which he has the largest classification score--greatest probability of group membership. 29Tatsuoka and Tiedeman, op. cit., p. AlA. 3A The value for each of the coefficients and the constant term for each classification function were derived from the values for the set of predictor variables of each individual in the "main" sample. Using the values of the coefficients and constant terms derived from the measurements of the "main" sample, the classification score for each group was computed for each individual in the "check" sample. Each individual in the "check" sample was assigned to the group for which he had the greatest score. Since the actual classification of each individual in the "check" sample was already known, comparison of the "actual" classification with the "pre- dicted" classification indicated the validity of the model for prediction of academic success. The analysis, using both methods described above, was programmed for the Control Data Corporation "3600" Computer--CDC 3600. The programs used are: (1) Program DISCRIM--Multiple Discriminant Analysis, and (2) Program BMDO5M——Maximum Likelihood Classification.30 Summary The sample of students for the study was selected from the population of all students admitted and enrolling for the first time in the MBA program at Michigan State University from Fall term, 1962, to Summer term, 1966, 3OComputer Institute for Social Science Research, Index for Technical Reports/Program Abstracts (East Lansing: Michigan State University, 1967), p. 8. 35 inclusive. The "check" sample, used to validate the model's ability to correctly classify individuals, was selected from students entering MSU, under the same con- ditions, in Fall term, 1966. Application of selective criteria (essentially excluding foreign students, students with incomplete records, and females) reduced the number of students eligible for selection to 51A graduates and 72 academic withdrawals. From those eligible, three extreme groups were selected for analysis. These groups were: (1) upper level graduates, (2) lower level graduates, and (3) lower level withdrawals. Each of the graduate groups contained 138 students, or 27% of all graduates. This group size provided the best compromise between two desirable but inconsistent aims--to make the extreme groups as large as possible, and to make the extreme groups as different as possible. Because of the small number of academic withdrawals, 38 stu- dents, about 50%, were included in the third group. The 12 predictor variables are those elements of information generally available for all applicants for admission, either the standard MSU application for graduate study or in the college of university record files for each individual, with one major exception. A new variable was formed in an attempt to "adjust" or "weight" the applicant's undergraduate GPA in respect to the "quality" 36 of his undergraduate institution. The measure of "quality" used was the mean ATGSB Total score for the undergraduate college, as reported by ETS. The analysis is directed to the two questions posited as the research problems of this study: (1) can the three extreme groups be distinguished from one another on the basis of the set of predictor variables? (2) Can an unclassified individual be accurately classified as an upper or lower level graduate or an academic withdrawal on the basis of this same set of predictor variables? Two statistical approaches, both programmed for machine computation, were used in the analysis. Multiple- discriminant analysis was used to investigate the signifi- cance of group differences and to determine the location of group centroids in discriminant space. Maximum likelihood classification methods were used to develop a classification model on the basis of the measurements of the predictor variables for each individual in each of the extreme groups. With this model, unclassified individuals are assigned to a group characterized by academic performance in which the individual has the greatest likelihood, the greatest probability of membership. By assigning members of the "check" sample in this manner, the model can be tested for its validity in pre- dicting the academic performance classification of an "unclassified" individual. CHAPTER IV ANALYSIS OF RESULTS This chapter is presented in three parts. The first is a description of the data of the "main," or larger, sample. It is followed by a multiple-discriminant analysis~ of this data for the purpose of answering the first of the research problems posited in this study. The third part of the chapter is concerned with the classification problem, second of the research problems. Each of the parts in— cludes both a presentation and discussion of the data and a relevant analysis. The chapter concludes with a summary discussion of the three sections. Description of the Sample Because of the nature of the predictor variables, the description of the data is presented in two forms. Four of the variables are not continuous, but binary, and therefore their values provide an either-or type of knowl- edge. The mean values of these variables are presented separately in Table A.l as percentages of the group sample. About twice as many of the upper level graduates (Group I) are married compared to either the lower level graduates (Group II) or the lower level academic withdrawals (Group III). 37 38 TABLE A.1.--Characteristics of MBA students by percentage of each group. I II III Upper Level Lower Level Lower Level Graduate Graduate Academic Withdrawal Married 71.1% 36.2% 29.7% Out-of state Resident 69.6 A7.l 37.8 Non-Business Under- graduate Major 31.2 2A.6 21.6 Undergraduate College Other Than MSU 78.3 5A.A 51.A This ratio also exists in the comparison of Groups I and III in regard to out—of—state residency. However, there are only about one-half as many upper level gradu- ates from out—of-state as there are lower level graduates. The number of students in Group I who had a non- business undergraduate major field of study was also about one and one—half times greater than either of the other two groups. A similar ratio, about 3 to 2, is also found for Group I compared to either Group II or III, for students admitted from undergraduate colleges other than MSU. From Table A.1 it appears as if the more academically successful students are married, from out—of-state, had a 39 non-business undergraduate major and were admitted from an undergraduate institution other than MSU. The remaining eight predictor variables used in the study are continuous. Computed means for each of these variables are presented in Table A.2. TABLE A.2.--Group means for predictor variables. I II III Upper Level Lower Level Lower Level Graduates Graduates Academic Withdrawal N=138 N=138 N=37 Age at Admission (years) 27.0 2A.8 2A.6 Delay in Entering MBA Program (yrs) 3.9 2.5 1.3 Undergraduate GPA 3.11 2.75 2.65 Product (GPA x Mean ATGSB Total Score) 15,278 13,110 12,302 Verbal ATGSB Score 31.A 28.0 28.5 Quantitative ATGSB Score 30.8 29.0 29.1 Total ATGSB Score 52A.8 A87.2 A93.9 Enrollment at Under- graduate Institu- tion 13,560 17,317 17,8A3 The two items obviously associated with maturity of the student are age at time of admission to the MBA program A0 and the number of year's delay since last formal school attendance. The data in Table A.2 generally indicate that the older students, and/or the students who have the great- est time lapse between last school attendance and the begin- ning of graduate study, achieve greater academic success. The average age of the lower level graduate and the academic withdrawal is about the same. The number of years of delay, however, are in the approximate ratio of 3:2:1, with the top level graduates delaying almost A years between completion of their undergraduate programs and the beginning of graduate education. As one might expect, both undergraduate GPA and the "product" (individual's undergraduate GPA multiplied by the mean ATGSB Total score of the individual's undergraduate institution) are directly related to the level of academic performance. Top level graduates have attained the highest undergraduate GPAs and both the lower level graduates and lower level academic withdrawals averaged below the stated admissions requirement of a 3.00 during the last two years of undergraduate work. There is a high correlation between the undergraduate GPA and the "product" for all groups. Consequently, a similar pattern is exhibited by the product values for the three groups. A more expository presentation of the "product" relationship to undergraduate GPA and the vari- ation of the "product" between groups can be acquired by computing an "adjusted" undergraduate GPA. The national A1 mean ATGSB Total score for the time period 1957—1965 is A86.31 Therefore, if each of the "product" values is divided by A86, the quotients can be viewed as an "adjusted" undergraduate GPA. Table A.3 provides a comparison of the "raw" and "adjusted" mean GPAs of students in each of the three groups. TABLE A.3.--Comparison of "raw" and "adjusted" undergraduate GPA. I II III Upper Level Lower Level Lower Level Graduates Graduates Academic Withdrawals "Raw" UG GPA 3.11 2.75 2.65 "Adjusted" UG GPA 3.1A 2.70 2.53 The remaining variables from Table A.2 are the three ATGSB scores: Verbal, Quantitative and Total. The scores of upper level graduates were the highest in all three areas. However, all three scores were slightly higher for lower level academic withdrawals than for lower level graduates. As measure of the student's undergraduate environment, perhaps contributing to academic success at MSU, the 31Educational Testing Service, Admission Test for Graduate Study in Business:( 1957-1965, p. iii. A2 enrollment at the individual's undergraduate institution was included. From Table A.2, it is seen that the average enrollment at the undergraduate colleges of the upper level graduates is 13,560 and the enrollment of the under- graduate college for the other, less successful groups, is greater and about equal at 17,317 and 17,8A3, respectively. Discriminant Analysis The discriminant analysis of the data is addressed to the first research problem. Can the three extreme groups be distinguished from one another on the basis of the set of predictor variables? It is recalled that the discriminant functions, which in this study are linear combinations of the twelve pre— dictor variables, are determined such that the ratio of the between-groups dispersion to the within-groups disper— sion is a maximum. This ensures the optimum separation between groups in discriminant space. The dimensions of discriminant space are essentially determined by.the number of significant discriminant functions. If there are three significant discriminant functions, then there is a corresponding three-dimensional space. The maximum number of discriminant functions is the lesser of the number of predictor variables or the number of groups minus one. Therefore, in this study the maximum number of dis- criminantfmummions is two--one less than the number of “3 groups. It follows, also, that in this study, discriminant space cannot exceed two dimensions. The computer program for discriminant analysis com- puted values of the two discriminant functions: cl 0.68702 with 13 degrees of freedom ¢2 0.01867 with 11 degrees of freedom Rao's Chi-square approximations for the two dis- criminant functions were computed as:32 x: = 160.29 with 13 degrees of freedom 1 x: = 5.67 with 11 degrees of freedom 2 At the .005 level of significance with the appro- priate degrees of freedom the values from the x2 table are respectively, 2 _ 2 _ 32Rao, op. cit., pp. 372-373. AA Therefore, since in testing for significant differ- ences, 2 2 > X¢l X1.13 and, , 2 < 2 X¢2 X2,11 it can be concluded that the first discriminant function, ol, lg significant, and that the second function, ¢2, lg Egg significant. A similar conclusion can be reached by examination of the "per cent of trace" attributed to each of the two discriminant functions. It is recalled that the "per cent of trace" indicates the portion of the discriminating power contained in a particular discriminant function. The first discriminant function accounts for 97.A% of trace, while the second discriminant function (which has been shown above to be E22 significant) accounts for only 2.65% of the discriminating power. With only one significant discriminant function the discriminant space becomes uni-dimensional. This means, therefore, that projections of the group centroids (the "centers" of the multivariate distributionsh are located on one 1ine-—that they are colinear. A5 The projection of the group centroids on a line that best separates the groups are computed and translated as follows in Table A.A. TABLE A.A.--Projection of group centroids. I II III Upper Level Lower Level Lower Level Graduates Graduates Academic Withdrawals 0.3838 1.33A2 1.5761 Further, it can be seen that not only are the group centroids colinear, but that they are also located along the line in the same order as the groups vary in level of academic performance. That is, the lower level graduates, Group II, are located between Groups I and Group III. It can also be noted that the distance between Group I and either Group II or Group III is much greater than the distance between Group II and Group III. This indi- cates that the separation between upper level graduates and either of the two other groups is better defined and more easily distinguishable. Classification: The "Main" Sample The second research problem considered in this study asks, "Can an unclassified individual be accurately A6 classified as an upper or lower level graduate or an academic withdrawal on the basis of the set of predictor variables?" Analysis of the results of using the method of maximum likelihood classification is presented in this section. It is recalled that the method of maximum likelihood requires computation of a set of linear equations for the purpose of classifying an individual in one of several groups. One linear classification fucntion is formed for each group. Each of these classification functions con- tains a constant term and a number of terms equal to the number of predictor variables. Each of the terms in the function is a product of a coefficient and a variable. The evaluation of this linear classification function yields a classification score. The general form of the classifi- cation function is, ai x 13 + . . + 12L x 123 + Ki = Pij where, . th a1, . . . 11 = coefficients for i classifi- cation group le, . xl2 =set of twelve variables for 3th individual Ki = constant term for 1th classifi— cation group A7 Pij = classification score for the ith group of the 3th individual. Therefore, for each individual to be classified, three classification functions are computed. Since classification scores are monotone functions in relation to the probability of group membership, the individual is assigned to (or classified with) one of the three groups for which he has the largest score. The computer program for the maximum likelihood classification method computes the values for each of the coefficients and the constant term associated with each classification group. Values computed for this study were derived from the values for the set of predictor values of each individual in the "main" sample. Table A.5 is a table of these computed values. The reader is cautioned that the magnitude of the coefficients cannot be meaningfully interpreted individually or in comparison with the other coefficients within the same classification group. Perhaps the only valuable insight to be gained through Table A.5 is a comparative evaluation of the coefficients for the same variable between classification groups. Two observations are in order regarding the magnitude of the incremental change and the direction of this change (increasing or decreasing) from one classification group to another. Consider, for example, variable (1), the marital status of the student. A8 mmom:.mma| mmmmo.emHI ommmm.mmau zmme azaemzoo oeaoo.on mmfloo.ou mmooo.ou AcoacaoaomcH as non oaoom mmoea etc: x «no coo respond .NH oamme.mn mm:o:.mu ommae.mu onoom mmoea osaceouocnse .HH omw:z.mn mmmom.mu Hmoom.mu onoom mmoea anono> .oa ommmm.o Asmam.o maomm.o onoom mmoea Heron .m moomfi.o eoama.o Hemma.o omoaaoo o: no eccefiaoacm .m mammm.ou mflmoa.ol mmom:.ou am: onoeom aeaoo .e mmmam.mm maeaa.em omamm.:m «do on .o mmaom.mm mm:mm.mm ameom.mm coacooacmcH o: .m mmmeo.m mmoma.m mmo:m.o some: on .3 mmamm.a mHmAN.H mmmam.a coannasoa he one .m mmmfie.m mmmfie.m Hmmmm.m cocooanom flawed .m mmmma.m :HmmH.o amama.a eschew Hecate: .H manzcnoccaz anoomo< mopmsomsm mopmsomno Ho>oq nmzoq Ho>mq nozoq Ho>oq pogo: moanmflpm> HHH HH H .mCOfipocsm coauBOHmHmmmao nmocaa pom mason ucmpmsoo one mucoHOHmmooo mo osHm>II.m.: mqm 2 X0005,“ = 114.86 55 The null hypothesis is rejected at the .005 level of significance. This means that there are about five chances in 1,000 that the classification matrix in Table A.7 could have been the result of chance. Comparative Analysis Analysis of the results of this study must include a comparison of the classification model and the actual selection of MBA students at MSU during the time period covered in the study. Unfortunately, the analysis cannot be complete because there is no way of determining what "might have been"; it is impossible to determine what would have been the academic performance of those applicants who were refused admission. A certain percentage of this group of rejected applicants would probably have been successful if given the opportunity, but the number is undeterminable. Nevertheless, a compari- son of such data as are available is presented in Table A.8. In the comparative analysis, consider the classifica- tion matrices for the "main" and "check" samples presented as Tables A.6 and A.7, respectively. The definitions of Hipg and Misses provides a basis for introducing the additional terminology now used in Table A.8: 56 Hipg: Two types—- (1) "Accepted / should have been accepted" (A/A) (2) "Rejected / should have been rejected" (R/R) Misses: Two types-- (1) Type I: "Rejected / should have been accepted" (R/A) (2) Type II: "Accepted / should have been rejected" (A/R) Ideally, the analysis would compare the percentage of error attributable to each of the methods of selection-- the model and the actual system of selection used during the time period. The errors in selections occur as Misses--Type I, "Rejection / should have been accepted" (R/A), and Type II, "Accepted / should have been Rejected" (A/R). Unfortun- ately, the number of Type I Misses, (R/A), is undetermin- able from actual experience, and thus a meaningful compari— son cannot be made. However, in regard to Type II Misses, a comparison of the model and actual experience was made and the results listed in the last column of Table A.8. In Table A.8, the Type II Migg is expressed as a percentage of the total number of students. This percentage, for the model, is less than that which actually occurred in 57 .szocxcz ma mademm scans: on» now honed: oHQmLmQEoo one .mmma .Enop Hash wcfinso coammaeom condemn mpcmofiaodm Qua * aw.ma :H w w mm oHQEmm szmno: Handpo< * eo.oa m ea m as cassem exoonos "Hoooz Rw.HH em a a ham cadenm scans: ”Hashes * am.m on em AH mam oaseem sneeze "Heooz Hepoeum\< m\a <\m m\m exa oaumm momma: mpfim eccoosom ——— ——~ —-— ___ U '3 (mmcn C(II'RSICS) _; 3 | '9- _. l E 1 ii is _ _ _ _ _‘l __ _ _ -_ .5; _ I _ l____ -... >- _ __ if: _ _ ' ___ _ 4_ _ _ __ ;. " a 5 _ ._ __ l.__.____ _ __._ __ 1— ! __. - _ __ __ I _: :Mi, Credits for each grade. f AddmhcoimnwdnmmmnumbadendltfornchMJ D '5 1 Total Credit: Grade points for each grade. 1‘ '3 '3 ‘1 " H " " n H I' Muitipiv the total awash" «och ad‘- the number "div 0 ,‘ slow a. Iov mmvlo mulnpiv A cu m by . 3 ends" bv 3" ac.) TOM] Grade pom“ Total G. P. Undergraduate grade point average: mom mm mm pom bv mm arduls: = "Total CF. Grade Point Anne 3 83 MICHIGAN STATE UNIVERSITY a OIIIce of Admissions and Scholarships g. COURSES TAKEN AS A GRADUATE STUDENT Report grades as requested in No. II of Instructions. «moss AND cneons : nucanox I. ___ mamuno‘n Titan" couass 1111.: AND nouns: ssussrsn caznlrrs “__qgmrsn catnip—I A acinlrl|aiaic|nlr I |' I x _- __ ——I——I-— I-_..!_I__’__ L_ E l 5_I ___?_?__!_I___ g ! I ! I I I I__ ~ _ _ _.'_I_.’I._’_L_I_I_ j I I I I I I ' . I I 1 I .... __lwg._ L." II.____'I._.I__ ._ __ _'_I _'_ i.,|___ __'___.. _ I__I__ ,- I____ I_I__._~ J I _ ' i I 'I _-1_—— _I—i—I I a i i I I | A I '3- 3 i ‘—-I€_.Ii I 1 a :_ __ __ __ II I f E. | I I I L’s l "Er—i ‘— F E r' — ~k——~r a ' i I z. I s I i — i i I E L 1 __ i s i I I II . ‘ 2 ‘ fl —.. ' s E ' A. l I E a: I‘ , i I I L I..._ _ _ ... _ l . I‘ __ _ __ __ _ I I_ E’ I I E L I l E i | I F I“ r I "I. ' I I ——I——— I _ __;_ g_; I _I l l I E I l | I_l_ _ _ | 1; i I ' I__ I g — ___! "I“ _“ “1’7 I I I a | _ ll- ___ __ _ _ _ Credits for each grade. ‘ I. _- (Add each column to determine the number of credite [or each grade.) D | II I“ Total Credits Grade points for each grade. H ‘3 ‘3 H u H “ I! fl *1 " Multipiu the total creditrlor each {rude b! the number dirediv 0 tion; at, for example mull-Div A at as bv . 8 credits bv 3 dc.) I Total Gnde Points Graduate grade point average: mud. total ma. pom. bv ma mam.- _—. 31?:SI‘SEL Grade Point Averag 4 APPENDIX B CLASSIFICATION SCORES FOR EACH INDIVIDUAL IN "CHECK" SAMPLE AND ASSIGNMENT TO ACADEMIC PERFORMANCE GROUP Note: Group l—-Upper level graduates Group 2——Lower level graduates Group 3--Academio Withdrawals 8H RFSPONDENT iJG‘dO\IOJdRDH ~S"U9CL \:\: \: \;\) \7\t\;‘\7|4~A-Is- ‘Yd'd'eefa'd'Azh’d'fld’hu'd'd \3 \.' ‘1 \J“? \I ‘-‘—"\J\:\:\!\)'\'\)\“\3 \'\)\‘\:\;\1\)\:\:\)\'\“\: \I O.‘ Snoop SCORE 1 145.93262 1?7.64616 146.69421 1*3.09012 121.44151 117.5456? 1t5,27669 146,51911 37I.b%/z5 1)u,1"4/9 135.41666 1ue.oné17 170.74105 1“2.3i206 1'6.2‘°9¢ 1“1.’“‘18 142,615us 146.2‘290 1~l.4a9v1 1‘b.7‘261 1'u.4A21v 1'4.n"692 151.81659 1‘4,n7916 1”“.9i201 1‘9.59269 149.21608 1~$.bl4é4 191.7h/66 1"/.1Ivo5 1“1.6*251 1so.7426$ jus.15517 1‘1.54ZDI 1“b.b°6/l 141.0“6/5 1&/.41400 J‘b./“44U J<5.9“/9/ 1‘9.7~Ubb 1~4.4l§30 1‘00U50U9 1~6.22696 149,53507 1'1.5*929 3—4,¢A540 172.99004 170.5“560 1“.1*256 14102’205 1a5.bn9(6 1&4.145/5 1“O.8“/01 1*6,19999I 175.5‘212 1Iv,9uza9 1‘d.n~044 145.0560? 1‘n.7*441 |K$,2H811 85 SCORE P 145.8495‘ 1270963Q5 ‘46.“2757 115.59060 13°.3977‘ 119.4590? 155.57xbn 147.05595 127-01317 122.35fl9? 154.7368: 154.6163! 122.42fibn .153.34?7t $56.36*75 149062I41 142.2192; 156.?6541 103.7450? 138.7245h 141.14‘4S 115.5514) 1bl.i4)b° iésc'd3b7 126.3751R '31059A11 147o‘474: 1:4-*0Iés 128.92764 150.4259! ‘5?.‘464I 145.95‘9‘ 154,n7u59 1Jle“6§27 16?.ibfica 14h.§137e 1°(~.‘-babi 119.:2az1 156.4glzs IdP.”Z)7J 154.4545) 119.7(A61 153.01Q4n 14*.“6‘07 132.r445a 1530‘I75‘ 12407070“ ‘d0031il“ 155-“3954 1400;1651 ‘61.*5455 144.”C9BJ 129.‘9‘bd 136.i0/o« 135.4644‘ 130.7559‘ 16?{46“b‘ 1‘50‘I527 1J#.s734i ‘6“.‘b‘b‘ SCORE J 145.5n653 126.96062 146.37806~ 115.91611 139.69154 119.33055 135.16726 14I.99515 177.9470] 122.93015 164.6‘210 156.97005 172.9S1Z1 184.4824? 195.!5200 148.65045 141.00502 155001746 152.19256 IKBOVROOU 141.5HQOO 115.0500: 1“U./64UI 135.13915 1760,3565 1‘1060590 141.3515! 159.931b4 1?/.596~d 139.14236 152.41I12 144.01206 1s4.54)¢l 151.11176 1“104“222 ladeOQUbd 159./8$dl 1‘9063210 1‘0-39130 141.44042 1R4.o*394 119.54926 IR$.hUsUd lI’eEHQJJ 1‘2.1‘660 Lfié.e4¢na 174.cAo/o 1950/R505 1‘3.35913 1x9.57ola 160.5n025 144.51551 179.47906 1*O.;7909 1?o.17514 1€0.67110 151.17000 115054930 1‘6.4)u49 114.4%042 ASSIGNED GROUP NF*HO‘KHVHJQIOHD’RJNFUP‘NP‘NlVRJWhéui‘b"RJHO‘GUUR3MC~&WNOMJN;Q0~P0dF‘Ph‘O-UF‘PC5hHUflJMCNOHUHJN 107 106 109 5...”.-....‘JAJH'JAJJ‘JQ;JJ>A—l—A—JJ.AM§JHHIJH 4A-. A ‘J _fi J‘-_A‘.LJA-J*I.J.J‘_J 1‘0.9?997. 1:3,3nJU4 1‘9.8‘004 1*a.2"059 145.09556 1'5.15/JI 173.A7594‘ 1‘9.oso91 1‘1.0*6/6 109.2“503 1‘/.17042 159.2v694 1“9,9fl464 1*9.719/7 1‘2.oh9“d 1“J.6'149 1‘U.5“151 1'5.1”692 141.3‘563 1‘4.U‘660 3c5.aaubl 1“1.2*096 1nu,19416 1 ‘dol‘éy‘ 1‘(.deOZ 1‘9.2”SUI 1“Oo1“0“5 195.qsuflb 1‘/.hdbuo 1‘2.9“U/2 32/,5666/ 1vn.e~267 1’1.4‘U45 1‘9.5h4UH 1“\,e‘)/.’1 1"U.SQ4‘4 t“o.e‘dfid 141.h1/d1 125.4424é j“0,0“0d6 J‘l.6“év9 1*b.2~2°7 1‘9.U‘9’0 1*4.97e95 1:7.9“1U1 1"“,9‘0é6 1‘6.«711J 1‘2.54/°0 1”b.57b>2 E36 151.16331 154.64Q1s 159.36°63 151.3215‘ 145-5402I 114.9234? 171.37‘74 159e39“6* 159-“6197 *40.91123 1150‘577I 19°.RCUO7 15004436“ 169.33%01 147e§1597 ISA-79“U7 15H.Vb¢b° 155.54‘0‘ '40.91¢bfi 151-97644 I450“0“0‘ 15°0‘6?°1 159.7II4I 155.41duw 1630 Ida?" iso.auson IonJbQS’ 1g6.60059 id‘e4d-d7 13‘0‘5009 i‘He 3";[9'0 ‘47.*Z“o: 13°.Jo'z~ 15“.”5‘0‘ 161.46'64 ‘QV.Q$“U" ’55.593l” ‘910‘9'fC‘D Izd.tl’0“ 12P.q7904 149.)5}94 157.41ugu i5].41