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I have data where each participant is giving binary responses (let's say a/b) to 9 questions each in two different categories.

I ran a 2x2 chi-square that has in the columns, {a, b} and in the rows {category1, category2}.

Is this okay given that each participant is providing multiple answers (presumably each participant will contribute to the counts in all four of the cells)?

I am thinking, for instance, if I was running a regression, I would need to include a participant random effect term because the data points are not independent. Is there a similar problem with my chi-square?

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    $\begingroup$ Yes, Pearson's chi-squared test also assumes independence. The regression with a random effect would be a better approach. $\endgroup$
    – TPM
    Apr 17, 2018 at 19:15

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You are correct--this calls for a random effect model. A straightforward implementation would be a multilevel logistic regression, predicting a/b from category 1/2 while accounting for the clustering at the participant level. This is an instance of a generalized linear mixed model.

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  • $\begingroup$ What we’re expecting is an interaction where we see higher counts in the category a/response 1 cell, higher counts in the category b/response 0 cells, and lower counts in the other two cells. Is a “response ~ category + (1|participant)” model best equipped to capture this type of crossover effect? $\endgroup$
    – Ashish
    Apr 18, 2018 at 20:22
  • $\begingroup$ Yes. A 2x2 table only has one degree of freedom. It always has the pattern you're describing when one cell changes but the marginals do not. $\endgroup$ Apr 19, 2018 at 11:15
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Yes, there is a problem with your $\chi^2$. A simple approach and pragmatic approach would be to sum up each individual's responses and perform a linear regression regressing the sum score on predictor variables of interest. The substantive findings from this should not be too different from those from the GLMM.

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