I wanted to ask for clarification on statistical approaches for a classic 2x2 pre-post design where I measure an outcome in a pre-intervention and post-intervention phase in two distinct groups.

Typically, what we want to investigate in these cases is the interaction, which tells us if the changes over time in a specific outcome differ between the two groups. To do this, we have three approaches available: Repeated Measures ANOVA (RM ANOVA), ANCOVA, and Linear Mixed Models (LMM).

RM ANOVA: It allows us to study the interaction, as well as the main effects of the between-subject factor (group) and the within-subject factor (time). So, besides telling us if there's a different change over time for the two groups, RM ANOVA also informs us if there's a general change over time (disregarding groups) and if there's a general difference between the two groups (disregarding time). However, the issue with RM ANOVA is that it doesn't allow us to covary the score of the outcome in the pre-intervention phase. This means that any significant pre-intervention difference between the two groups could influence the results and make their interpretation much less robust. The solution in this case would be to verify (with an independent samples t-test) that there are no pre-intervention differences between the two groups in the outcome.

ANCOVA with the pre-intervention outcome score as a covariate: It allows us to see if there are differences between the two groups post-intervention. It doesn't tell us if there's a different change over time between the two groups, if there's a general change over time, or if there's a general difference between the two groups. It doesn't evaluate the interaction, the main effect of time, or the main effect of group, unlike RM ANOVA. The advantage of ANCOVA is that it enables us to study the differences between the two groups ONLY in the post-intervention phase, accounting for pre-intervention differences in the outcome. This means that this model works well even in cases of large outcome differences between the two groups in the pre-intervention phase.

Therefore, RM ANOVA and ANCOVA provide quite different information. If there are differences in the outcome in the pre-intervention phase, it's advisable to use ANCOVA (less informative, as it considers fewer effects); otherwise, it's better to use RM ANOVA (more informative, as it considers more effects). Right?

The solution to all of this is LMM because, in addition to considering all the effects of RM ANOVA (interaction + the two main effects), it also models any differences in the outcome pre-intervention. So, is it always better to use an LMM, right? Is everything correct, or am I missing something?