I have run a sequence of nested multi-level models using lmer, and looked at both the summary output giving all the coefficients, etc, as well as the Anova output which does a chi-squared test on each of the factors.
In the first model without the interaction, the factor medium is significant, looking at both the chi-squared test output and at the coefficients from the summary output.
In the second model when the interaction is added, the first-level factor medium is no longer significant in the chi-squared test output yielded by the Anova function. (It isn't significant in the summary output where we get the coefficient information, but I believe I understand how to interpret that output - it just means that the medium is not significant for the reference group I believe?)
But how do I interpret the non-significant outcome of the chi-squared test for the first-level factor medium in the presence of the added interaction?
If it helps, here is a simplified version of my model:
And here is the Anova (chi-squared tests) output for Model 1 (without the interaction) and Model 2 (with the interaction):
Chisq Df Pr(>Chisq) medium 42.03818323 1 0.0000 gender 6.082823479 1 0.0137
Chisq Df Pr(>Chisq) medium 2.23141795 1 0.1352 gender 12.95254908 1 0.0003 medium:gender 6.920056392 1 0.0085
I'd be happy to read a reference on this, but I just can't find anything in any of my books on how to interpret the chi-squared tests in the presence of an interaction - they all seem to focus entirely on interpreting the regression coefficients only. So if anyone could help to explain the correct interpretation, or refer me to a good reference that specifically talks about how to interpret the chi-squared tests of main effects in the presence of interactions, I'd be grateful!