I used SPSS (v. 25) conducted a pair of nested multtlevel models (aka linear mixed models) where the outcome criterion is itself also nested within participants. I followed Singer and Willett's (2003) recommendations to include time as a level-1 (within-participant) predictor.
The "base" comparison model included no predictors except a fixed-effect (between-participant, level-2 factor) intercept and the random-effect time I mentioned just above.
The model I compared against that base model included those same two factors (i.e., intercept and time) and only added a fixed-effect, dummy-coded gender term.
The -2 log likelihood (-2 LL) for the base model was 28316. The -2 LL for the new extended model that added in gender was 27341. The deviance for this difference is 28316 - 27341 = 975. With 1 df, this difference is highly significant (critical χ2 = 5.02).
However, when reporting the inferential test for the gender term, SPSS indicates that the F-score for this term is 3.05, which is not significant (with dfs of 1 & 903.8 and α = .05).
In other words, adding the gender term made for a significantly better-fitting model. However, the term that was added that made for this better fit is itself not significant.
I am at a loss how to interpret this in practical terms. It seems that gender should be taken into consideration when understanding the outcome variable, but how do I reconcile explaining that gender itself is not significant?