# Addressing Non-Normality of Random Effects in Linear Mixed Model

I have a paradigm where participants complete a reaction time task with three different stimulus 'Conditions'. I generate two contrasts for my mixed model. I produced the plots below to assess the normality of random effects. The random effect that is assessed here is (Condition|participant).

1. Can someone explain what is meant by 'random effect quantiles' and 'standard normal quantiles' (the axes labels)?

2. How is non-normality corrected? Transformations? Robust regression? Is this likely caused by participant outliers?

• It's hard to know what exactly these are showing without your code; are they the residuals from a model? . I'm also not sure why there are error bars. And they don't even look that non-normal. Dec 5, 2023 at 12:53

1. The quantiles on the x-axis represent what is often called 'theoretical quantiles' in QQ-plots; the distribution of quantiles in a standard normal distribution, in this case. The quantiles on the y-axis represent the observed quantiles, in this case the quantiles of the distribution of the predicted random effects.

Each dot represents a level of the random effect. If the predicted values of random effects follow a normal distribution, they would lie on (or close to) the straight line. Deviations indicate:

• Points below the straight line indicate the predicted values for those levels of the random effects are more concentrated than what would be expected for a normal distribution. I.e., some levels of the random effect have very similar or identical predicted values.

• Points above the straight line indicate the predicted values of the random effect are more spread out than what would be expected for a normal distribution.

In your left plot: The three left-most points indicate that for three levels of the random effect (probably three subjects?), the predicted value of the random intercept is close to -0.4. For a perfectly normal distribution, these values should be somewhat lower (more extreme / further from the mean). The two right-most points indicate that for two levels of the random effect, the predicted value of the random intercept is close to 0.75, while for a perfectly normal distribution, these value should be somewhat lower (less extreme / closer to the mean).

2. I have no clear answer, I hope others will chime in. Some suggestions:

• I would avoid transforming the response variable, because this will affect all components (both fixed and random) of your model.
• These plots indicate departures from the assumed distribution. With departures, point estimates (e.g., estimated fixed effects) can often be trusted, but variances are affected, and thereby statistical tests (p-values) may become less trustworthy. Note that most of the random-effects predictions lie on the straight line, mimicking a normal distribution. The plots show a few outliers, for which some of the CIs also do not cover the straight line. I am not sure how worried one should be about this.
• I would plot the predicted values of random effects (e.g., create a histogram), and perhaps try to identify which levels (subjects?) create these departures from normality. I am no big fan of throwing out outlying observations, but as a sensitivity analysis you could e.g., consider repeating the analyses without these subjects, to see if this affects conclusions.