Paradoxes in Multidimensional Contingency Tables: What Do They Mean for CFA?

When Gustav Lienert set out his case for Configural Frequency Analysis (CFA) in the late 1960s and early 1970s, his starting point was a specific observation: in data with three or more dimensions, higher-order interactions can be present even when you fail to find any correlation between individual pairs of variables. That gap between pairwise correlation and genuine joint structure is what CFA was built to address.

Lienert followed this up in 1971 with Hierarchical CFA (HCFA), a procedure aimed at isolating the subset of variables that hangs together most tightly. The method works by collapsing the original contingency table step by step, following a hierarchy, until only two-dimensional tables remain. CFA is then run on every table produced along the way.

There is a catch that statisticians have recognized for well over a century. Collapsing a contingency table can produce results that look contradictory, the phenomenon most people now attach to Simpson's name (Simpson, 1951). A pattern visible in a combined table can reverse or vanish once the data are split, and the reverse can happen too.

A second trap comes from Lord (1967). It turns up when you compare groups that start from different baseline values, and the comparison can mislead depending on how those starting points are handled. This paper looks at a contingency-table form of that same problem rather than the continuous-measurement version Lord originally described.

The question running through all of this is what these two paradoxes imply once you try to read the output of a CFA. If collapsing tables can distort the picture, and baseline differences between groups can do the same, then the interpretation of configural results deserves a careful second look.

Joachim Krauth, Department of Experimental Psychology, University of Düsseldorf.