Good evening to everyone!

The limma package (and the related papers) defines the "random effect" in a mixed effect model as a "variable" (e.g. "patient_ID"). However, the lme4 package define the random effect as a coefficient of the model (intercept or slope) free to change based on a certain blocking variable. Then... what is a "random effect"? A whole variable or a coefficient?

In particular, in limma, it's the intercept or the slope (or both) that is free to change according to the block of the mixed effet model?

I got it wrong or there is some other information that I miss?

Thank you very much for your help

Leandro

There are many ways to describe random effect models, but they are all equivalent. In some fields of statistics, they are described as variance component models.

lme4 allows very general mixed models whereas limma allows only one additive random effect, similar to a classical "randomized block" design. In the case of one additive random effect, the limma and lme4 models are essentially equivalent. The differences are (i) that limma allows negative as well as positive intra-block correlations, whereas in lme4 negative correlations are set to zero and (ii) limma applies a constraint to the estimated intra-block correlations across genes in order to share information between genes and hence increase precision.