Academic journal article Canadian Journal of Experimental Psychology

Multiway Frequency Analysis for Experimental Psychologists

Academic journal article Canadian Journal of Experimental Psychology

Multiway Frequency Analysis for Experimental Psychologists

Article excerpt

Abstract Many research designs in experimental psychology generate data that are fundamentally discrete or categorical in nature, and produce multiway tables of frequencies. Despite an extensive and, more recently, accessible literature on the topic, multiway frequency analysis is rarely used in experimental psychology. A reason may be the form of exposition in the literature, with emphases and concerns far removed from those of the typical experimental psychologist. An approach to multiway frequency analysis for experimental psychologists is described that has the features we want: asymmetrical designs, factors assessed for their respective main and interactive effects in a manner analogous to ANOVA, and the ability to handle within-subject designs.

Since the seminal work of Goodman (1971, for example), and through the now classic texts of Bishop, Fienberg, and Holland (1975) and Haberman (1978, 1979), the last 30 years has seen a revolution in the analysis of multiway, categorical frequency data. The development and discussion of this revolution have occurred predominantly in the social science literature. And yet, despite the fact that many of our research designs produce data that are fundamentally in the form of multiway tables of frequencies, and the ready availability of accessible, book-length expositions on the topic (e.g., Agresti, 1990; Gilbert, 1993; Kennedy, 1983; Wickens, 1989), and a few, prominently placed if specialized, articles in experimental psychology journals (e.g., Olzak & Wickens, 1983; Wickens, 1993), the use of multiway frequency analysis within experimental psychology is rare.

There is probably any number of reasons for this state of affairs, but a few figure prominently. First, most textbooks on statistical methods for experimental psychologists have no mention of multiway frequency analysis: The theory and the computational techniques presented for handling frequency data are appropriate for contingency tables of at most two dimensions, and, often, further limited to 2 x 2 tables. Such two-dimensional tables typically are analyzed with the goodness-of-fit approach (re-characterized as tests of independence) developed early in the history of statistics by Karl Pearson (Pearson, 1900, 1904, 1911, 1916).1

Second, presumably because of their familiarity with complex ANOVA designs, experimental psychologists are most comfortable with experimental (i.e., asymmetrical, one dependent variable) designs and the assessment of the statistical significance of individual effects on the dependent variable. In contrast, even in the more accessible of expositions in the multiway frequency literature, the presentations routinely are concerned with correlational (i.e., bidirectional or symmetrical) designs with no variable serving as the dependent variable. These analyses typically proceed by systematically or otherwise investigating various combinations (the "model") of what are confusingly (for the experimental psychologist) referred to as "main" and "interaction" effects on the cell frequencies themselves. A log-linear model is accepted typically when it fits the cell frequencies with no significant residual using a minimum of higher-order combinations, although other acceptance criteria are sometimes invoked. Even for relatively small tables, there is an often bewildering complex of patterns of associations, and hence, models to be considered.2

Even assuming one such model were selected as fitting the data exceptionally well, it still would not answer the question of interest for the typical experimental psychologist: Is a given effect (i.e., the unique influence of one variable or combination thereof on the dependent variable) significant? Instead, confronted with cross-classified frequency data of more than two dimensions, many experimental psychologists reduce the data analysis to a series of classic tests of independence of the various two-way contingency tables (the dependent variable and, successively, each of the independent variables), successively collapsed over the remaining dimensions. …

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