Contingency Table

Historical Development

Early work on CATEGORICAL DATA ANALYSIS in the beginning of the 20th century was primarily concerned with the analysis of contingency tables. The well-known debate between Karl Pearson and G. Udny Yule sparked interest in contingency table analysis. How would we analyze a 2 × 2 contingency table? Pearson was a firm believer in continuous bivariate distributions underlying the observed counts in the cross-classified table; Yule was of the opinion that variables containing discrete categories such as inoculation versus no inoculation would be best treated as discrete without assuming underlying distributions. Pearson's contingency coefficient was calculated based on the approximated underlying correlation of bivariate normal distributions collapsed into discrete crossclassifications, whereas Yule's Q was computed using a function of the ODDS RATIO of the observed discrete data directly.

Contributions throughout the 20th century can often be traced by the names of those inventing the statistics or tests for contingency table analysis. Pearson chisquare test, Yule's Q, Fisher's exact test, Kendall's tau, Cramér's V, Goodman and Kruskal's tau, and the Cochran-Mantel-Haenszel test (or the MantelHaenszel test) are just some examples. These statistics are commonly included in today's statistical software packages (e.g., proc freq in SAS generates most of the above).

Table 1 A Contingency Table of Parents' Socioeconomic Status (SES) and Mental Health Status (expected values in parentheses)
	Mental Health Status
Parent's SES	Well	Mild Symptom	Moderate Symptom	Impaired
A (high)	64 (48.5)	94 (95.0)	58 (57.1)	46 (61.4)
B	57 (45.3)	94 (88.8)	54 (53.4)	40 (57.4)
C	57 (53.1)	105 (104.1)	65 (62.6)	60 (67.3)
D	72 (71.0)	141 (139.3)	77 (83.7)	94 (90.0)
E	36 (49.0)	97 (96.1)	54 (57.8)	78 (62.1)
F (low)	21 (40.1)	71 (78.7)	54 (47.3)	71 (50.9)

Contingency table analysis can be viewed as the foundation for contemporary categorical data analysis. We may see the series of developments in log-linear modeling in the latter half of the 20th century as refinements in analyzing contingency tables. Similarly, the advancement in latent trait and latent class analysis is inseparable from the statistical foundation of contingency table analysis.

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