# Zero-inflated mixed models and negative trend in residuals

I am trying to fit a mixed model in lme4 using the lmer.

the model form is:

model<- lmer(entropy ~ X1 * X2 + (1| video) + (1| subject), REML = FALSE


entropy is a continuous variable but normalised now between 0 and 1. X1 is a centred and scaled continuous predictor, and X2 is a categorical predictor with 4 levels. I should note that entropy was computed from a variable that had a too many zeros, and so, during the normalisation of the variable I had to add a very small number number to avoid undefined logarithms. As a result, the distribution of entropy is somewhat bimodal:

I am aware that the distribution of the outcome does not matter for the assumptions of linear models (whereas the residuals distribution do). However, the residual plot for this model shows a decreasing trend which I am worried about.

I am not quite sure what to make of this pattern, or how to improve this model. I initially fitted this using lmer, tried log-transforming and it didn't help. when I extract and compute the actual correlation between the fitted and residuals it is actually close to zero, which is puzzling considering this trend.

the qq-norm plot doesn't look terrible qqnorm(resid(model)

Histograms of the residuals look pretty normal, but I am aware histograms are not appropriate to check for the random distribution of residuals assumption of linear models.

I tried including other predictors that are of interest as well as running a few glmer (e.g. binonimal) models but I can't seem to resolve this.

Any suggestions on how to remedy this? How worried should I be about the first residual plot above?

Thanks, any suggestions are very much appreciated!

• So entropy was log-transformed? What did its distribution look like prior to transformation? Perhaps you could use a generalized mixed model to model the variable in its original scale? – Erik Ruzek Apr 21 at 15:17
• Hi Henrik, thanks for the response, it wasn't quite log-transformed, it was normalised by diving entropy by the log of maximum entropy. This normalisation is deemed to be necessary in my field, so I don't think I can run the models on the non normalised values – Helio Cuve Apr 21 at 16:47
• Also even the non normalised entropy scores have a relatively similar distribution to those, so I would still have the same problem – Helio Cuve Apr 21 at 16:55
• I see. As mentioned in the linked thread by Ben Bolker, you could use robust estimation to deal with violations to the usual regression assumptions. See stats.stackexchange.com/questions/457680/… for some additional resources. – Erik Ruzek Apr 21 at 17:22