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The Relationship Between Fisher's Exact Test and Pearson's Chi-Square Test: A Bayesian Perspective

Published online by Cambridge University Press:  01 January 2025

Gregory Camilli
Gregory Camilli*
Affiliation:
Rutgers, The State University of New Jersey
*
Requests for reprints should be sent to Gregory Camilli, Rutgers University, 10 Seminary Place, New Brunswick, NJ 08903.
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Abstract

It is demonstrated in this paper that two major tests for 2 × 2 talbes are highly related from a Bayesian perspective. Although it is well-known that Fisher's exact and Pearson's chi-square tests are asymptotically equivalent, the present analysis shows that a formal similarity also exists in small samples. The key assumption that leads to the resemblance is the presence of a continuous parameter measuring association. In particular, it is shown that Pearson's probability can be obtained by integrating a two-moment approximation to the posterior distribution of the log-odds ratio. Furthermore, Pearson's chi-square test gave an excellent approximation to the actual Bayes probability in all 2×2 tables examined, except for those with extremely disproportionate marginal frequencies.

Keywords

2 × 2 tablesfour-fold tablesFisher's exact testchi-square testhypergeometric distributiondouble binomial distributionbinary datacategorical dataBayes theorem
Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 2 , June 1995 , pp. 305 - 312
Copyright
Copyright © 1995 The Psychometric Society

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