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The estimation of transition probabilities in a student flow model

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  • This study was concerned with the estimation of transition probabilities in a finite state Markov-type student flow model. Goals setting for this study were three in number: (1) Study of the methods which will provide conclusions that (a) the transition probabilities are stationary or nonstationary and (b) the process is of order one. (2) Study of the methods to estimate the transition probabilities under various stationarity conditions and data availability conditions, and (3) Formulate the Markov-type student flow model for the Oregon State University system and then perform actual enrollment projections for this system. In testing the statistical hypotheses about the transition matrices, the study started with the investigation of the probability distributions and the limiting distributions of the numbers of transfer students. Several x²-tests were developed for various testing purposes. The works done by Anderson and Goodman (1957) and Halperin (1966) were cited. Actual tests performed on OSU data indicated that they were nonstationary and order one transition matrices. The techniques of estimating transition probabilities were presented according to the stationarity property of the transition matrices. In the stationary case, emphasis was made on the estimation based on the aggregate data. A restricted least square technique was used. Hypothetical data were created to test the method. It was found that the estimates based on aggregate data closely resemble the actual transition probabilities. In the estimation of nonstationary transition probabilities three approaches were presented: (1) the causative matrix method first used in the market research then analytically studied by Harary, et al., (2) the linear regression method which formulated the transition probability as a function of a set of exogenous variables, and (3) the composite chain method which further divided a state into several homogeneous groups, where each group was assumed to possess its own stationary transition pattern. Extensive discussions were made on the linear regression method. This study provides the theoretical support and points out the method of estimating regression equations by using the aggregate data. Each method was followed with an actual projection of transition probabilities based upon the first five years of OSU student data for the last four years. Comparisons were made according to the numerical results and some other factors. In performing the actual student flow projection for the OSU system, the linear regression method was selected for estimating the future transition matrices. In the model verification it was found that the projection results were very reliable at the university total level, while on the school level, an average absolute error of 5.75% was reported. The final enrollment projection covered five year period starting 1971-72 school year. Problems encountered in this study were analyzed. Suggestions made include: the setup of student related data bank for research use and the study of state selection technique in the model. The applicability of this model to the OSU system was discussed. Possible extensions of the model in economic anaylsis, marketing system, accounting system and demographic system was offered.
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