Is it a must to include a random slope in a mixed model? I am learning about fitting mixed models and I find when it is justified to include or exclude a random slope rather confusing.
Some tutorials suggest that although the maximal random structure should be specified at the outset, the random slope should be kept only if it contributes to extra explanatory power, i.e. is significant and hence mathematically justified to stay in the model.
Other papers suggest that the random slope should be kept regardless of its significance when a cross-level interaction effect is included in the model; but does this assertion hold when no cross-level interaction but only a main effect is being investigated? Under what circumstances should I decide to keep or discard the random slope component from my model?
 A: There is considerable disagreement on this topic.
I like to keep it simple. If you have a priori reasons to believe that the fixed effect in question should vary by subject (or whatever the grouping variable is) then you should fit random slopes. Obviously, this is provided that the data supports such a model. Often a model containing random slopes will have a singular fit, either because the correlation between the slopes and intercepts is estimated close to, or at, +/- 1, or because the variance in the random slopes is estimated close to, or at, 0. In the former case, a model fitted without such correlation can be fitted, but in the latter case the only solution is usually to remove the random slopes.
If the model converges without warnings, then I would retain the random slopes. I would not do any statistical test to decide whether to retain them, because I had a priori reasons to include them in the first place and just because they weren't significant in this sample is not a reason to exclude them if I believe they should be present in the population.
Having said that, I can't really argue with the approach of testing with a likelihood ratio, and removing the slopes if they don't offer an improved fit, on the basis of model parsimony. It's just my preference to retain them.
I am not familiar with the argument to retain random slopes if a cross-level interaction is fitted in the fixed part of the model. Does this refer to random slopes for the interaction only ?
