Melvin R. Novick

Multivariate Generalized Beta Distributions with Applications to Utility Assessment

Two multivariate probability distributions, namely a generalized beta and a generalizedF, that appear to be useful in utility modeling are derived. They reduce to the standard beta andF distributions, respectively, in special cases. Reproduction of distributional form is demonstrated for marginal and conditional distributions. Formulas for the moments of these distributions are given. The usefulness of these distributions in utility modeling derives from the fact that they generally do not demand increasing risk aversion as do most standard forms. An example of the use of the bivariate generalized beta distribution in utility modeling is presented. This distribution compares favorably in an example given here to both a normal model and an unstructured model.

The estimation of proportions inm groups

In many applications, it is desirable to estimate binomial proportions inm groups where it is anticipated that these proportions are similar but not identical. Following a general approach due to Lindley, a Bayesian Model II aposteriori modal estimate is derived that estimates the inverse sine transform of each proportion by a weighted average of the inverse sine transform of the observed proportion in the individual group and the average of the estimated values. Comparison with a classical method due to Jackson spotlights some desirable features of Model II analyses. The simplicity of the present formulation makes it possible to study the behavior of the Bayesian Model II approach more closely than in more complex formulations. Also, it is possible to estimate the amount of gain afforded by the Model II analyses.

A Bayesian Indifference Procedure

Abstract In a logical probability approach to inference, distributions on a parameter space are interpretable as representing states of knowledge, and any prevailing state of knowledge may be taken to have been arrived at from a previous state of ignorance (indifference) followed by an accumulation of prior data. In this paper an indifference procedure is introduced that requires postulating what size and what kind of samples will and will not (in a special sense) permit statistical inference and prediction—e.g., one observation from a two-parameter normal model is not (in our special sense) sufficient to permit inference about the variance but two observations are. In essence, the procedure stipulates that prior indifference distributions be improper but become proper after an appropriate minimal sample. With some limitation on the family of priors considered, this procedure permits unique specification of indifference for the more commonly encountered statistical models. Furthermore, these specification...

Bayesian Full Rank Marginalization for Two-Way Contingency Tables.

A general approach is proposed for modeling the structure of anr ×s contingency table and for drawing marginal inferences about all parameters (e.g., interaction effects) in the model. The main approach is relevant wheneverrs −r −s + 1≥5. The approach may also be used to check the adequacy of Rasch’s multiplicative Poisson model. In general, the posterior estimates of the cell probabilities compromise between the cell frequencies and the fitted values obtained under the reduced model in the spirit ofLeonard (1975). It is also possible to compute reasonable approximations to the full posterior densities of many parameters of interest, followingLeonard (1982) andTierney and Kadane (1984). All prior parameters are evaluated with the assistance of the data via a hierarchical Bayes procedure, thus reducing the subjectivity involved in the analysis. Anr ×s cross classification of 5,648 Marine Corps clerical students by school and test grade is analyzed in detail, and the posterior densities of the 96 possible interactions are used to suggest a simplified structure partitioning and collapsing the table into a meaningful 3 × 2 table.

Bayesian Analysis for Binomial Models with Generalized Beta Prior Distributions

The Libby-Novick class of three-parameter generalized beta distributions is shown to provide a rich class of prior distributions for the binomial model that removes some of the restrictions of the standard beta class. The posterior distribution defines a new class of four-parameter generalized beta distributions for which numerical posterior analysis is easily done. A numerical example indicates the desirability of using these wider classes of densities for binomial models, particularly in an interactive computing environment.

PLU Robust Bayesian Decision Theory: Point Estimation

Abstract The development of data analysis techniques that are robust with respect to wild or extreme observations is now a major concern. From a Bayesian point of view, the concept of robustness also pertains to the choice of a prior density (P robustness) and a utility function (U robustness), as well as the likelihood (L robustness). A technique for converting commonly used nonrobust density and utility functions to robust versions is described that provides convenient solutions for point estimates. Applications of this procedure to the robust Bayesian analysis of the linear model are provided.

Fixed-State Assessment of Utility Functions

Abstract This article presents a fixed-state method of assessing utility functions that requires the statement of probabilities that equate certain gambles, emphasizes the importance of coherence checking, and provides a least squares fit to help resolve revealed incoherence. This approach may be a useful alternative to fixed probability methods, but only when made available in an interactive environment in which the resolution of incoherence is encouraged and facilitated.

Marginal distributions for the estimation of proportions inm groups

A Bayesian Model II approach to the estimation of proportions inm groups (discussed by Novick, Lewis, and Jackson) is extended to obtain posterior marginal distributions for the proportions. It is anticipated that these will be useful in applications (such as Individually Prescribed Instruction) where decisions are to be made separately for each proportion, rather than jointly for the set of proportions. In addition, the approach is extended to allow greater use of prior information than previously and the specification of this prior information is discussed.

Educational Testing: Inferences In Relevant Subpopulations

essarily probabilistic in character. The essence of the scientific game is to make the description precise. The concept of population has received much attention in statistical literature in the works of Fisher (1956), Meehl and Rosen (1955), De Finetti (1974), and Lindley and Novick (1981). In this paper I attempt to show that the idea of relevant population is useful in considering a wide variety of otherwise unrelated problems in educational research. Much of the material in this paper is not new (see Novick, 1981a; Novick 1981c, Novick, 1982; Lindley & Novick, 1981). However, the emphasis and range of applications is distinctive. In places, some details are omitted with reference to one or more of the above papers.

Theory and Practice for the use of Cut-Scores for Personnel Decisions

Cut-scores are commonly used in industrial personnel selection, academic selection, minimum competence certification testing, and professional licensing, using simple and multiple-person/multiple-job category decision paradigms. Previous approaches have proposed cut-score solutions in a variety of applications using threshold, normal ogive, linear and discrete utility functions. This paper considers these results by investigating conditions on the posterior, likelihood and utility functions required for setting a cut-score in a Bayesian decision approach. Generalizing and extending results of Lehmann, Karlin, Ferguson and others, it is shown that cut-scores are appropriate in a wide range of applications, but they are less than universally appropriate. Following this, a general paradigm and computational algorithm for cut-score solutions is developed under the assumption that the conditions for a cut-score have been satisfied.