I have a linear mixed effects (LME) model with a single nominal-valued fixed effect with 4 levels, and two nested random effects (measurements from cells at each level of the fixed effect, with multiple cells within each subject) w/ random slopes for the fixed effect. If I use a likelihood ratio test to compare the model with and without the fixed effect, I get a non-significant p-value, which would make me conclude that the fixed effect isn't significant. However, the confidence intervals for 2 of the four levels (so really 2/3, because the first is captured by the intercept) are quite negative, not close to including 0.
So, including the variable doesn't improve the model, but the expected values for two of the levels are clearly significant. Should I trust the likelihood ratio test, and ignore the confidence intervals of the coefficients? Report both, with an explanation of the disparity (and any suggestions on what that explanation might be - perhaps too much variability between subjects)? Or am I approaching this wrong?