Mixed models for longitudinal data

 Time is level-1.

https://link.springer.com/article/10.1007/s10869-017-9491-z

Mixed-effects models can be used with nested data, even if no group-level effects are expected or evident.

Failing to use mixed effects to model nested data can increase the risk of type I error (for group-level effects) and type II error (for lower-level effects).

Centering only applies to level-1 vars.

Group-mean centering of a level-1 variable fundamentally changes the interpretation of the level-2 parameter estimate for the analogue of the same variable. In group-mean centered models, level-2 parameter estimates represent overall group effects; in raw or grand-mean models, level-2 parameter estimates represent differences in slopes between individual-level and group-level relationships. When level-1 variables are group-mean centered, one can mistakenly conclude the level-2 group-mean analogue of the lower-level variable represents a test of an emergent effect when it actually represents a test of the total effect.

Group-mean centering changes the conceptual meaning of the level-1 construct such that the term now reflects relative position in a group rather than absolute values. “People high in X tend to be high on Y,” (raw or grand-mean centered) and “people who are higher than their group on X also tend to be higher on Y than their group” (group-mean centered X).

Raw data and grand-mean centering produce equivalent models, with a slight technical advantage going to grand-mean centering because it reduces the correlation between slopes and intercepts.

Tests of emergent effects at level-2 are most easily accomplished using raw (or grand-mean) variables whereas tests of whether the level-2 variable is related to the group average of the dependent variable is most easily accomplished by group-mean centering the level-1 variable.

In mixed-effects models, higher-level variables predict the group means of the lower-level outcome.

In mixed-effects models, emergent effects identifying different relationships between level-1 variables and their group mean analogs likely represent important changes in the meaning of constructs across levels.

Substantial ICC (2) values are not necessary for identifying emergent group-level effects (but they help). The ICC(2), like the ICC(1), can be estimated from an ANOVA.

It is helpful and informative to test and report whether level-1 slopes randomly vary among groups prior to conducting a test of a cross-level interaction, but the results of such tests should not prevent subsequently testing cross-level interactions.

When testing cross-level interactions with raw or grand-mean centered variables, it is important to use group-mean centering to verify model results.