- a bayesian mixed model is just what I would have used here, great choice and a good sign for power
- the effect sizes are definitely very small. I would not overstate the story here.
That said, I also don't believe that facebook is the primary story here, because other findings all suggest tiny effect sizes (see the famous emotional contagion experiment) and because we don't have a clear, convincing theoretical model IMO. There are just so many intra-individual confounders that would need to be included, including some that have dynamic, self-reinforcing effects.
Still, great to have more robust work here.
[edit] PS: just to throw a methodological nitpick out: Some bayesians are going to have a heart attack while reading "the uncertainty cutoff of 97.5% for posterior probabilities".