What is the nature of the normality assumption in models for longitudinal data?

Question

I'm working on a longitudinal dataset to which I've been fitting non-linear mixed effects model in R. Regarding normality, I have a few questions:

1. Can I assume that a longitudinal data is normally distributed?
2. Do tests such as the Shapiro test apply to longitudinal data?
3. Is normality a strict requirement for mixed effects models?

Answer 1

1. There is nothing in the longitudinal nature of longitudinal data that necessitates it is normal. For example, there are longitudinal Bernoulli data (e.g., whether a child is out sick for every day of the school year). Even if data were continuous, there is nothing about that which necessitates the data be normal.

2. The Shapiro-Wilk test assumes the data tested are iid (i.e., independent). That assumption would be violated for longitudinal data.

3. I'm not sure what it means for normality to be a "strict requirement" of a model. However, linear mixed effects models typically assume normality in two senses:

   • First, they assume the residuals are normally distributed (really the errors, but that's more nuanced than you probably need to worry about). This isn't really different from non-mixed linear models. As with simpler models, if the deviations from normality are smaller, and/or you have more data, the violation of this assumption is less of a big deal.
   • Second, they tend to assume the random effects are normally distributed (although I believe it is possible to relax this and use the $t$ distribution for a more robust option). Again, the idea isn't that your sample is normally distributed, but that the population from which the random effects were drawn was normal. And again, violations of this assumption are less bad if you have more units and/or the deviations were minor.