Is there a minimum number of observations needed to justify the use of random slopes in a logistic regression model? I have a mixed-effects logistic regression model with 584 observations, which models the outcome of a variable linguistic phenomenon. There are three predictor variables and I I used random intercepts to take into account non-independence in the data (i.e., many observations come from the same word). A reviewer asks why I did not also use random slopes. When I was designing the model, I didn't do this because I didn't think I had enough data to justify random slopes. But I realize that this was just my intuition and that I don't know how much data I would need exactly to warrant random slopes. Are there guidelines on how many observations one needs to justify random slopes? If this is discussed in the literature anywhere, I'd be grateful for references.
 A: For a mixed-effects model, 584 first level observations is fine. The problem is how many second level units do you have. It is often considered that you should have at least 50 second level groups. Check this and this papers for more information.
The sample size requirements are not different for random intercepts and random slopes. The correct substantive answer you should give to your reviewer is that you did not specify random slopes because you did not have any theoretical reason to believe that the slopes vary across groups. But apparently this is not the case. You may check if other similar studies used random slopes. If there is not any that has done so, this is itself an important substantive reason for assuming that the slopes do not vary across groups.
However, maybe your data is different and you have the chance do identify something new. If the reviewer asked you why you did not use random slopes, he is probably suggesting you to try to do so. Therefore, you should try it, and check model fit statistics (-2LL, AIC, BIC). If the model fit is not improved by the random slope you have also a statistical argument to provide to the reviwer and for keeping your original model. If the model is significantly improved with random slopes, then you could use them.
