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The article comments on conditional independence and log linear models for multi-dimensional contingency tables. Frequency data in the form described by a contingency table occur in many different fields of research, as indicated by the abundant literature available. The approach to their analysis varies from the relatively ad hoc application of the Chi-square tests for various relationships of independence to the specification of a model for which parameter estimates of effects, as well as tests, may be used. A number of features may be listed for the log linear model which make it convenient, at least if no theoretically more acceptable model is available. The log linear model is the simplest form of model which automatically fills the boundary conditions that all predicted proportions for all cells of the table lie in the interval. It is the only model for which the parameters can be estimated in a situation of inverse sampling, i.e., when random samples are taken from each sub-population corresponding to a value of the dependent variable. In addition, since it is directly analogous to the linear models of normal theory, the statistical theory of inference-making is relatively simple, primarily because sufficient statistics exist for all the parameters.

The sociologist who has accumulated data on a subject is soon faced with the problem of interpreting these data. He then must reorganize, combine, eventually reduce them. For this he uses mathematical models. If his variables are real numbers, he can choose among a variety of standard statistical models which most frequently involve linear structures, as these are the most easily worked out. The simplest probabilistic model involving attributes is the multinomial model, where the basic assumption is that the probability that an event belongs to a certain class depends only on this class. Statisticians have long been interested in multinomial models and have provided with a certain number of results. One approach has been to assimilate the linear model for proportions with an analysis-of-variance model, which also involves attributes. The theory of tests concerning multinomial models is largely based on tests of the chi-square type. The aim of this article is to discuss a statistical treatment for the estimations of the parameters, regions of confidence and tests of hypotheses.