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If I have a GLM with a number of explanatory variables, where one is a categorical variable with levels "no treatment" "treatment A" "treatment B"

Assuming that it is reasonable to combine the 2 treatment groups together experimentally, what statistical assumptions do I need to check?

I wish to include to include the predictor variable treatment with levels 0 and 1 in the model.

For ANOVA I would check that it was reasonable to assume similar in groups variance for treatment A and treatment B

Does that assumption apply to GLM and linear regression as well? are there others to consider?

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To check whether you can join the levels, a simple likelihood ratio test will do (the model with joined levels is a submodel of the one where they are separate).

As usual, this test is conditional on the correctness of the larger model (though hardly anybody ever bothers to check that), so technically, you need to assert the assumptions of your splitlevels model first. Note that checking the similar variance between two levels is not enough! Since you have multiple explanatory variables, what you really should check it the feasibility of equal variances in each 'cell', i.e. in each combination of levels from all predictors. Once again, typically, it is hardly possible to check this.

Finally (if the LR-test gives you enough confidence to join the levels), you should check the correctness of your reduced model, i.e.: once again checking the equal variance in each new 'cell' (note that the previous check on the old set of cells does not imply that the variances will be equal in the 'joined' sets, although it is highly likely if the reduced model came up ok: the means were then rather close and the variances were equal in each set of two old cells that were joined).

In reality, once again, this kind of thing is really hard to check (lots of cells with typically small numbers of observations, depending on your field of research). But remember that the univariate checks (like you appeared to suggest) in no way validate the equality of variances per cell. They are just an unfortunately accepted practice of soothing the mind.

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