Similar to Tyler's question about the t-test, is Pearson's chi-squared in the GLM?


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Note that a test is not a model (though it uses a model). A GLM is a model, not a test (though it can have associated tests).

"Pearson chi-square" refers to multiple things -- tests for a specified set of multinomial proportions (chi-square "goodness of fit") as well as tests for independence or homogeneity in two-way or multiway tables (among other uses of the term).

There are some count data GLMs (e.g. Poisson, multinomial) for which you can construct tests that correspond to some of the usual chi-squared tests, but they're not Pearson chi-square.

They're based on an asymptotic approximation (Wilks, 1938) to the distribution of $-2\log\Lambda$ in likelihood ratio tests, and they correspond to things like the G-test.

These likelihood ratio tests and the corresponding Pearson chi-squared tests are both in the Cressie-Read family of power-divergence statistics, and are asymptotically equivalent --- in practice, if your sample sizes are large, they tend to give very similar results.

[The earliest discussion of the G test appears to be by Wilks in 1935, and I think the name G test comes from Woolf (1957).]


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