Ron N. Forthofer

Guidelines for Analysis

In this chapter we describe the situations in which the WLS approach for categorical data can be used. We also define the type of functions involved and offer some guidance on their use. Lastly, we discuss different approaches to data analysis and model building and make some general recommendations.

Rank Correlation Methods

Rank correlation coefficients are statistical indices that measure the degree of association between two variables having ordered categories. Some well-known rank correlation coefficients are those proposed by Goodman and Kruskal (1954, 1959), Kendall (1955), and Somers (1962). Rank correlation methods share several common features. They are based on counts and are defined such that a coefficient of zero means “no association” between the variables and a value of +1.0 or -1.0 means “perfect agreement” or “perfect inverse agreement,” respectively.

Rank Choice Analysis

We encounter a new type of multiresponse situation in this chapter. People are asked to rank several choices, and the ranks they assign represent the multiple responses. The problem is to test whether there is a clear preference ordering. If the number of choices is large, the corresponding contingency table will have millions of cells. A direct analysis of such a large contingency table is impossible, but the method of analysis discussed in this chapter makes the table unnecessary.

Contingency Table Analysis: The WLS Approach

This chapter introduces the weighted least squares (WLS) method of categorical data analysis, also known as the GSK approach, developed by Grizzle, Starmer, and Koch. We begin with a definition and several examples of contingency tables and then discuss different functions of the contingency table frequencies that we may form. Next we present the variance-covariance matrix of a function and then show its use in applying the linear model to categorical data. Note that the development of the linear model to categorical data follows closely the presentation in Appendix B. The next sections define additive and multiplicative association; this discussion is followed by a demonstration that the hypothesis of no multiplicative association is equivalent to the hypothesis of independence of rows and columns in two-way tables. We begin with a discussion of the contingency table.

One Response and Two Factor Variables

This chapter details a step-by-step application of the WLS procedure on a small problem posing few complexities. The steps presented here will serve as a guide for the more complicated analyses presented in later chapters and thus should be mastered before proceeding further.

Multidimensional Contingency Tables

This book introduces workers in public health and public affairs to a general analytic methodology proposed by Grizzle, Starmer, and Koch (1969). This methodology is based on applications of the general linear model (the basis for both regression analysis and analysis of variance for continuous data) to categorical data. By categorical data we mean information measured on nominal or ordinal scales or grouped continuous data. Appendix B presents the general linear model in the analysis of variance situation; this appendix serves as a guide for the use of the general linear model with categorical data.

Log-Linear Models

The three preceding chapters have all used models in which the response variables were probabilities (Chapters 4 and 5) or a linear combination of probabilities (Chapter 6). In this chapter we consider a model in which the response function involves the natural logarithm of the response variable. The particular form of the logarithmic function that we will use is the logit. The logit function is defined as the natural logarithm of π divided by 1 - π:

Multiple Response Functions

\log it(\pi ) = \ln \left( {\frac{\pi } {{1 - \pi }}} \right)