Journal of Educational and Behavioral Statistics

A model for retention and its counterpart, attrition, is presented. In a prototype example, students enter a program in each of k terms; some of the students complete the program, and the remainder leave. A key feature in the models proposed is that there is a dampening effect from term to term because the probability of leaving the program diminishes as the terms progress. The focus of this paper is the study of alternative models for the dampening in attrition rates. A number of alternative dampening effects are proposed that provide for different rates of attrition. Approximate maximum likelihood estimates for the underlying parameters in each model and a Bayesian analysis are provided.

After a hypothesis about some linear multivariate statistical model has been tested and rejected (e.g., in a MANOVA), many researchers employ simultaneous test procedures to locate the source(s) of the rejection. If the global test was conducted using Roy’s largest root criterion, then this procedure guarantees at least one linear combination of the model parameters relative to some linear combination of the dependent variables that is significantly different from its hypothesized value. This most significant parametric function is not always easy to find, however, because it may not manifest itself in simple or “obvious” functions. A general solution to this problem is presented along with a practical example of its application.

Ranking problems arise in setting priorities for investigations, in providing a simple summary of performance, in comparing objects in a manner robust to measurement scale, and in a wide variety of other applications. Commonly, rankings are computed from measurements that depend on the true attribute. Using the Gaussian model, we propose and compare methods for using these measurements to estimate the ranks of the underlying attributes and show that those based on an empirical Bayes model produce estimates that differ from ranking observed data. These differences result both from the effect of shrinking posterior means towards a common value by an amount that depends on the precision of individual measurements and from the Bayes processing of the posterior distribution to produce estimates that account for the uncertainty in the distribution of the ranks. We illustrate different ranking methods using data on school achievement reported byAitkin and Longford (1986). Mathematical and empirical results highlight the importance of using appropriate ranking methods and identify issues requiring further research.

Three of the potential models for the 2 × 2 contingency table are discussed: the hyper geometric Independence Trial, the double-binomial Comparative Trial, and the multinomial Double Dichotomy Trial. Then the critical regions of seven statistical tests are evaluated within each of these models. The uncorrected Pearson’s Chi-Square test and Upton’s correction are found to be overly liberal in all three models, particularly for the hypergeometric. A continuity correction, when properly applied, is shown to be an extremely good approximation to Fisher’s Exact test, which employs the hypergeometric distribution to evaluate outcome probabilities—but is an extremely conservative approximation to the double binomial and multinomial models. Haber’s test is recommended for these two cases, but the amount of work required may be prohibitive for many investigators.

An empirical sampling approach was used to assess the accuracy of a Taylor approximation for the estimation of sampling errors in the statistics involved in estimating a path model based on medium-sized samples gathered using five sample designs commonly used in educational research. The technique was found to give a reasonable approximation for the simplest of the designs but to deteriorate as clustering, stratification, and weighting became more pronounced.