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I use R for a poisson regression (GLM) to test for a difference in the number of years of schooling based on race (6 categories) and religion (3 categories), see structure of dataset and output below. Since the original poisson model suffers from rather significant overdispersion I conducted the same analysis but with a quasipoisson distribution.

Structure of dataset:

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GLM Quasipoisson regression:

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ANOVA - Test predictors relative to the full model:

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  1. The output of the quasipoisson regressions shows one statistically significant interaction. I read the following "use F tests with an empirical scale parameter instead of chi-squared (The R Book, Crawley M. J. 2013. p. 570)" as yet another suggested technique to deal with overdispersion. Hence, I conduct an ANOVA with an F-test for the model which then shows that the interaction term is not statistically significant. What is the correct way to interpret this result? Should I, based on the ANOVA state that the interaction is not statistically significant and then instead go on trying a model without the interaction term?
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Tests of significance are not used to determine whether or not variables are meant to be included in the model or not. If the goal is prediction or variable selection, you probably shouldn't be using quasilikelihood at all. That said, it looks like you've tested for the interaction of religion with race using the method that Crawley has advocated. The reason is that you need an estimate of the covariates and the dispersion parameter under the "reduced" model.

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    $\begingroup$ Ok, thanks for your answer. So the correct interpretation is that the interaction term is not statistically significant although one interaction is significant in the GLM output? $\endgroup$
    – JanC
    May 26, 2021 at 18:08
  • $\begingroup$ @JanC Not quite. According to Crawley, the significance test in the Pr(>|t|) is exactly the chi-square test that he advocates against. The "reduced" model requires a re-estimate of the dispersion parameter to give you the $F$ test. I think you can fit that full model as ~Race * Religion and the reduced model as ~Race*Religion - RaceMulato:ReligionEvangelic. Then the two models can be supplied to the aov command. $\endgroup$
    – AdamO
    May 27, 2021 at 15:46

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