# Multiple Chi-Squared Tests

I have cross classified data in a 2 x 2 x 6 table. Let's call the dimensions response, A and B. I fit a logistic regression to the data with the model response ~ A * B. An analysis of deviance of that model says that both terms and their interaction are significant.

However, looking at the proportions of the data, it looks like only 2 or so levels of B are responsible for these significant effects. I would like to test to see which levels are the culprits. Right now, my approach is to perform 6 chi-squared tests on 2 x 2 tables of response ~ A, and then to adjust the p-values from those tests for multiple comparisons (using the Holm adjustment).

My question is whether there is a better approach to this problem. Is there a more principled modeling approach, or multiple chi-squared test comparison approach?

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 I once asked the same question on the R mailing list, and didn't get a response. I'd suggest you to change your title since your question is regarding "post hoc analysis of chi square - to detect the cause of the significance" (a shorter titles then the one I proposed would be better :) ) – Tal Galili Aug 2 '10 at 20:33 Just look at the betas for your culprits.... And use a poisson, log-linear model. You then get the same thing as what the chi-square test gives you, but you get all of the different tests at once. – probabilityislogic Jan 11 '12 at 20:46

You should look into "partitioning chi-squared". This is similar in logic to performing post-hoc tests in ANOVA. It will allow you to determine whether your significant overall test is primarily attributable to differences in particular categories or groups of categories.

A quick google turned up this presentation, which at the end discusses methods for partitioning chi-squared.

http://www.ed.uiuc.edu/courses/EdPsy490AT/lectures/2way_chi-ha-online.pdf

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Interesting. Did you ever come across an R implementation of this ? – Tal Galili Aug 4 '10 at 3:16
No, not directly. However, R will give you everything you need to do this--such as: the observed counts, the expected values, and the residuals for each cell. x <- matrix(c(12, 5, 7, 7), ncol = 2) chisq.test(x)$expected chisq.test(x)$observed chisq.test(x)\$residuals – Brett Magill Aug 4 '10 at 15:55
I'll give you the tick, since this should be useful for my research life. However, this approach is applicable to an i x j matrix. However, my question involves an i x j x k matrix, – JoFrhwld Aug 4 '10 at 17:02
Chi-square partitioning is extensible to multi-way contingency tables. Here's the article that Agresti cites in his book, in fact... H. O. Lancaster (1951) "Complex Contingency Tables Treated by the Partition of χ2" Journal of the Royal Statistical Society. Series B (Methodological), Vol. 13, No. 2 – Brett Magill Aug 4 '10 at 18:13

The unprincipled approach is to discard the disproportionate data, refit the model and see if logit/conditional odds ratios for response and A are very different (controlling for B). This might tell you if there's cause for concern. Pooling the levels of B is another approach. On more principled lines, If you're worried about relative proportions inducing Simpson's paradox, then you can look into the conditional and marginal odds ratios for response/A and see if they reverse.

For avoiding multiple comparisons in particular, the only thing that occurs to me is to use a hierarchical model which accounts for random effects across levels.

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I don't know exactly what your goals are, or why they are what they are. But rather than hypothesis testing, I usually recommend focusing attention on predictions and confidence intervals.

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