I am running multilevel regressions with random slopes and binary outcome (type=twolevel random). I am mainly interested in the extent to which the relationship x->y varies across clusters.
Due to my limited statistics knowledge I have difficulties interpreting the random slope coefficient. When I estimate the same model with Stata (gllamm) or R (lme4) the output provides odds ratios for the "fixed part" in addition to across-cluster variances. Is the mean slope on logit scale or is it a linear regression coefficient? How can I interpret it?
The mean slope is on a logit scale so you can consider odds ratios. Note that interpreting this model requires some study - see for example the reference that we give when teaching:
Larsen & Merlo (2005). Appropriate assessment of neighborhood effects on individual health: Integrating random and fixed effects in multilevel logistic regression. American Journal of Epidemiology, 161, 81-88.