Properties of a Model for Ordered Categorical Dependent Variables.

Abstract

The analysis of ordered categorical dependent variables is a particularly acute problem in social research, medicine, and biology; no method of analysis is generally accepted as correct. In this dissertation, we extend the mathematical basis for a class of models suitable to ordered variables and examine their application to specific problems. The models are natural extensions of probit and logit analysis. Let Y be a categorical r and om variable, and the probability that Y = k, given independent variables, x, be Pr(Y = k(VBAR)x). We examine models of the form F('-1)(Pr(Y (LESSTHEQ) k(VBAR)x)) = g(x,b), where F is a two-parameter distribution function, and g(x,b) is either linear or nonlinear in b. We extend the mathematical basis for linear functions, g(x,b) = x'b, to include r and om as well as fixed variables, which permits us to properly apply the models to additional problem. We show that the family of linear functions, g(x,b) = x'b, forms a complete set, that is, any model for a categorical dependent variable has an equivalent linear model of this form. We define saturated models and lack of fit statistics, which test the fit of two-parameter distribution functions, F. In two applications, the models yield substantive conclusions different from those previously reported. A linear model is proposed as the first stage in two-stage estimation problems. This method can reduce the bias of parameter estimates from second stage regressions whose independent variables are r and om ordered categories. We propose specific applications to economic and social survey problems, such as analyses of personal income. We examine graphical methods based on the proposed models. Graphical methods using empirical transformations can aid both in choosing a model, and in evaluating sources of lack of fit. However, they usually require large sample sizes. We describe two analysis techniques to evaluate the assumptions of the proposed models. One technique is suitable for fixed independent variables, and the other for r and om variables. These approximate techniques permit simple evaluation of a proposed linear model, and aid in the diagnosis of observed lack of fit. Both techniques can be used with sparse data, and may be viewed as alternatives to graphical methods, in such cases. Residual analysis is a conceptually complex problem for ordered categorical variables. The analysis of residuals, both graphically and statistically, is useful both in evaluating a particular model, and comparing alternative models. While simple displays of residuals are useful, other methods are difficult to extend to ordered categorical variables. We describe a gradient projection algorithm for estimating model parameters, and compare alternative numerical techniques on several examples. A simple starting estimate was found to speed convergence for the examples considered. Iteration with Fisher's Information matrix converged faster than iteration with Gauss's matrix. Modification of the matrix diagonal (Marquardt's method, similar to ridge regression) almost always speeded convergence with Gauss's matrix but never with Fisher's. The linear and nonlinear models substantially reduce problems of estimability, and therefore convergence, due to sparse and patterned data in contingency tables. This permits us to design and effectively analyze experiments and surveys which yield sparse data.