How can I compare one group to the mean of all other groups in a longitudinal mixed-effects regression?

Question

I have a longitudinal mixed-effects regression comparing change in depression between two timepoints across 12 groups. I'd like to know if the control group is significantly less effective in reducing depression than the average of the other 11 groups. The model to compare each group to each other looks like this:

m1_phq9 <- lmer(phq9_score ~ time * group + (1|pid),data=df)

Would this solution work?

df<- df %>% mutate(passive_vs_all=ifelse(group=='passive','trout','active') %>% factor %>% fct_rev())

m2_phq9_all_vs_passive <- lmer(phq9_score ~ time * passive_vs_all + (1|group/pid),data=mega_alltimes)

I'd appreciate any insights on setting up the contrast for this kind of model in R.

Answer 1

I think your method will probably work, but it is most streamlined to use a downstream package such as emmeans to compute contrasts (see the relevant vignette via vignette("comparisons", package = "emmeans"), particularly section 4); I think the following code will probably work, assuming your control group is the first level of the factor:

em <- emmeans(m1_phq9, specs = ~group)
contrast(em, list(ctrl.vs.all = c(-1, rep(1/11, 11)))

i.e. you want to compute the estimated value/CIs/p-value etc. of

$$ -\mu_1 + \frac{\mu_2 + \dots + \mu_{12}}{11} = \bar\mu_{2:12} - \mu_1 $$

You'll get a warning about "Results may be misleading due to involvement in interactions". This is a fundamental interpretational problem — among-group comparisons will depend on the baseline used, and if the interaction is too large it may just not make sense to do a simple comparison across groups ... However, while you shouldn't proceed unthinkingly, you can mitigate this problem (comparisons across groups depend on the baseline time used for the comparison) by making sure that (1) you're using sum-to-zero contrasts (e.g. set options(contrasts = c("contr.sum", "contr.poly")) before fitting the model) and (2) that all continuous predictors have their zero reference level in a sensible place (e.g., mean-center them). See e.g. Schielzeth 2010 Methods in Ecology and Evolution ...