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I use R for a mixed-effect binomial regression (GLM) to test for a difference in whether people voted (coded as 0/1) in an election based on race (6 categories) and religion (3 categories), see structure of dataset and output below.

Structure of dataset:

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mixed-effect Binomial regression:

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ANOVA - Test predictors relative to the full model:

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Question: as shown in the output for the model there are some statistically significant interaction between race and religion. However, as I control the significance of the interaction term relative to the full model by performing an analysis of variance for the interaction term the overall effect of the interaction seems not to be statistically significant. What is the correct way to interpret these results? Can there be statistically significant interactions between races/religions although the ANOVA shoes there are the interaction term is not statistically significant?

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This is more or less the same problem as finding significant differences in a post hoc analysis, but not the omnibus test in a regular ANOVA.

The likelihood ratio test shows you that adding the interaction term does not significantly improve the model. What you should do with that information depends on your goal: If you want to compare combinations of race and religion rather than estimate their overall effect, then this is the right model. If you are trying to find a parsimonious model for voting, then you could leave out the interaction.

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I agree with the answer by Frans. This kind of thing can easily happen with interactions involving factors with several levels, or in a simple ANOVA with a factor with several levels. The overall effect of the variable may not be "significant" while contrasts between some of the levels might be. This is completely normal. One school of thought says that if the overall level of significance for the variable does not meet the threshold then we should not look at individual contrasts. I don't really subscribe to that approach. As mentioned by Frans, a lot depends on the use for the model. If you have reason to believe that there should be an interaction, then leave it in and do not be too concerned about p-values reaching whatever threshold you choose.

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