Introduction

Logit models are a class of models used to explore the relationship of a dichotomous dependent variable to one or more independent variables. In these models, the logit, or log-odds (i.e., the natural logarithm of the odds), that the dependent variable has a specific given value is analyzed as a linear function of the independent variables. Logit models are analogous to ordinary regression models in which the expected value of a continuous dependent variable is expressed as a linear combination/function of one or more independent variables, in much the same way as hierarchical log-linear models have been earlier shown to be analogs of the analysis of variance (ANOVA) class of models. Generally utilized algorithms for the estimation of logit models exploit such analogy.

As social scientists we are often concerned with the problem of explaining and predicting behavior. Often, the dependent variable that describes behavior is a continuous variable. In that case we can employ standard parametric inferential procedures (like multiple regression analysis), which allow inferences about “average” population behavior given a random sample of data from a population of individuals.

In most observational coding systems, however, the dependent variable is not continuous, but instead is a set of alternatives that are discrete or “quantal.” Efforts to analyze behavior observed as discrete outcomes or events thus involve a class of models with discrete, or qualitative, dependent variables. Such models are generally referred to in the social science literature as “quantal choice models” or “quantal response analysis.”

Model Building and Selection of Categorical Data Modelsy

The selection of a log-linear model for describing a given data set becomes increasingly complicated as the number of variables increases, because of the rapid increase in possible associations and interactions. It is normally too cumbersome and demanding a task to fit all possible models once the number of dimensions exceeds three, and we thus need rules of thumb for selecting a specific model over others. As in fitting regression models, we have to balance two contradictory goals or objectives. On the one hand, we would prefer a model complex enough to afford a reasonably good fit of the data. On the other hand, we want a model that is relatively simple to interpret and that is parsimonious – one that smoothes rather than overfits the data.

Even though the partitioning and ordering properties of the likelihood ratio X2 statistic are very useful in assessing the relative worthiness of several (nested) candidate log-linear models, the process of finding the most appropriate – “best” fitting and parsimonious –(hierarchical) log-linear model is by no means an easy task. In general, we are able to fit all possible models for a three-way table; however, the number of hierarchical models to be fitted increases dramatically as the number of dimensions grows. Thus, for anything larger than a three-way table, we usually have to consider model selection strategies to limit the number to be evaluated for any given problem.

The Problem

This chapter outlines a few approaches to the basic problem of auto-contingency. What is the problem? Briefly, in sequential analysis we want to determine if we can predict the occurrence of code B at time t + 1 by knowing that code A occurred at time t. Usually we have a dyad we are observing, and the B codes come from observing one person's behavior while the A codes come from observing the other person's behavior. The standard we employ to evaluate the prediction is the amount of improvement over the unconditional probability of, p(B). However, there is another standard that makes a lot more sense when we try to establish a causal link between A and B. That standard follows from the fact that knowing that B occurred at time t would lead, in many cases, to gain in prediction over and above the unconditional, p(B). Hence, the test for sequential connection could be made more stringent by comparing p(B| A) not with p(B) but with the residual from p(B) obtained from predicting B's occurrence by own B's past behavior. This could be viewed simply as an adjustment to p(B) in the numerator of our usual z-scores for sequential connection. The nature of this adjustment is the subject of this chapter. Currently, we have little experience with such solutions, and thus our review is entirely theoretical. As we gain experience, we will be able to say more about the relative merits of the alternative reviewed here.

This book describes models of category counts from the perspective of the social, environmental and behavioural scientist. The literature on the analysis of category counts, also variously referred to as contingency table, categorical or cross-tabulated data analysis, has developed apace since the heady era of previous decades, and the aim has been to capture the flavour of recent developments. Naturally, in a diverse subject area of growing importance and application, it is necessary to be selective, and the book draws in particular on a fairly coherent body of literature judged to be relevant to the non-statistician or applied statistician primarily interested in data analysis, without, it is hoped, disparaging the contribution made by writers on more-general or theoretical topics.

Since the focus of the book is application, particular attention is given to the analysis of real data sets. Chapter 1 contains an introduction to basic concepts and notation in the context of the two-way table and in Chapter 2 the notion of the multiway table is developed. Chapter 3 investigates the fitting of alternative models to the multiway table, with consideration given to the important topic of multiple-hypothesis tests. Chapter 4 contains a discussion of issues related to the sample design, and outlines some straightforward methods of handling data obtained by means other than simple random sampling. Chapter 5 features methods of analysing large, unwieldy tables and tables where quantitative values or scores are assigned to variable levels, and Chapter 6 integrates various themes pursued in earlier chapters in the special context of observations over time.

Introduction

This book describes models of category counts where each count is the observed frequency with which a unique combination of the levels of categorical variables occurs in a sample. This observed frequency provides an estimate of the probability of the variable-level combination in the population. The variablelevel combinations are jointly displayed as a contingency table. Consider, for example, the simple two-way table formed by observing two categorical (as opposed to continuous) variables. Table 1.2 is a typical example with rows representing the categories of the variable residential area (A) and columns representing the categories of the variable preferred political party (B). For convenience, both A and B are treated as dichotomies, that is, each has two levels. More-complex data sets involving polytomies and more than two variables (and hence multiway tables) are investigated at length later; we remain with the two-way table in this first chapter to introduce some basic notation and concepts.

It is important at the outset to make explicit any assumptions that are implied by a methodology and, therefore, upon which the results of any analysis are conditional. The mere act of drawing up a contingency table represents a conscious structuring of a multifarious reality prior to analysis in which the variation present is subdivided according to distinct categorical variables. Although this may enable one to obtain plausible models, it is, it seems, an imposed world-view that may not be universally shared.

With dichotomous variables an n-variable analysis means 2n cells in the ensuing <>in-way table, whereas the inclusion of polytomies among the set of n variables produces more than 2n cells and less-compact marginal tables. A consequence is that even conceptually simple models may contain a surfeit of parameters corresponding to the enlarged marginal tables, thus making data interpretation somewhat more difficult. Also, the extra table cells may spread the sample too thinly to warrant the usual distributional assumptions about Y2 and X2.

This chapter is largely concerned with these and related problems arising from large unwieldy tables. It contains a description of some recent progress towards parsimonious modelling which involves the resolution of large sets of interaction parameters and takes account of the ordinal or higher-level values often attached to the levels of polytomies. Alternative strategies that are considered include the deletion of cells that are aberrant from the perspective of a simple model adequately fitting the majority of table cells, and category amalgamation and variable elimination. These latter ploys do have consequences which are discussed subsequently, though they remind us that table design is not an immutable phenomenon and that tables can be modified, not simply as a last resort to overcome the sparseness of observations and superfluity of parameters that often accompanies the analysis of polytomies, but as a means of illustrating and clarifying the meaning of interactions. The chapter is loosely organised as two halves. The first half mainly examines how simplification can be achieved via the models fitted to complex tables. The second half examines more-closely the role played by modifications to the table itself.

This chapter describes methods of analysing table counts obtained by categorising the same individual on at least two occasions. A sample comprising a number of such individuals yields what are normally referred to as panel data, and these are a common source of information on mass changes in opinion, status or behaviour. As a result of attrition of the panel and the difficulties associated with high-dimensional tables, panels are more often two-wave (i.e. two-occasion) than multiwave, and univariate rather than multivariate, and although these more-complex data are not ignored, we naturally commence with the simple one-variable two-wave panel.

A variety of models are introduced, initially in the context of this two-way table, to deal with features commonly associated with counts from temporal observations. Three common phenomena are the preponderance of identical responses at different times (loyalty or inertia), the dependence of transitions between categories on intercategory distances (distance), and the directional balance of intercategory transitions (symmetry). Loyalty, distance and symmetry effects are often in evidence together and in association with other traits in two-way tables, and are the fundamental components of the various models discussed in the text. They are also important elements of models devised for multiwave and multivariate panels described in the second half of the chapter.

As well as internal coherence, an attempt is made to provide wider integration by drawing together themes introduced in previous chapters. These include screening, STP, incomplete table analysis and uniform association. The topic of many repeated observations in close sequence producing dependent observations (see Chapter 4) is also briefly reconsidered.

Sampling models

Up to now we have been largely concerned with identifying structural models and have paid little heed to the fact that one particular sampling model is implicitly assumed. In fact, a number of sampling models could have been adopted, as outlined by Bishop et al (1975, p. 435). This means that we have assumed that the observed data have been collected in one particular way, namely by choosing a sample of a predetermined size at random from a population. Such a scheme will produce an observed frequency in any cell of a multidimensional contingency table that reflects the underlying unknown probability that a randomly chosen member of the population will occupy the cell.

There are in fact three basic interrelated sampling models appropriate to a variety of data collection schemes. One such model is the Poisson distribution, which is applicable if we can assume that the frequency in each of N cells of the multidimensional contingency table is the outcome of N independent Poisson processes. This sampling scheme requires no a priori fixing of the total sample size and, therefore, would be appropriate if the phenomena of interest were observed for a fixed time period, at the end of which the frequencies in each cell, each the outcome of a Poisson process, could be totalled to arrive at a previously unspecified aggregate. Such a scheme would be approximately appropriate if we were conducting a traffic survey to investigate the relationship between vehicle type and whether or not the speed limit was broken, by observing a stretch of road for a given time period.

This appendix contains selected ‘listings’ of GENSTAT directives that reproduce results given in the text. A number of alternative computer programs are available, some of which are mentioned on appropriate pages, and thus access to the GENSTAT system is not a prerequisite for practical application of the models under discussion. However, GENSTAT (which is a close relative of the GLIM system) is a highly acclaimed package which can be used generally throughout the book and which has been found particularly useful for fitting some of the more intricate models of Chapters 5 and 6.

In fact, GENSTAT is more a programming language than a conventional statistics package, and this gives it immense flexibility. The large number of directives available to perform a wide variety of functions in areas such as matrix algebra, time series analysis, optimisation and generalised linear modelling, calls for some guidance, in the form of the listings below, as to those appropriate to models of category counts. From these the analyst will be able to work out how to fit a range of models additional to those specifically discussed.

The compact nature of the programming language enables some otherwise tedious and lengthy procedures to be quickly implemented. This is readily evident from the listings below, though these can be developed and improved in elegance once some familiarity with the system has been gained, for the aim at this juncture is simply to provide the reader unfamiliar with GENSTAT with an entry point into practical modelling with the minimum of fuss.