(Note: Parts where I've encountered problems are in bold to make them easily findable.)
I have data from a longitudinal study of twins. Several biomarkers are measured, usually 4 measurements each (twin 1 baseline, twin 2 baseline, twin 1 follow-up, twin 2 follow-up). This is an observational study, so there is no expected systematic change between baseline and follow-up, but people within a twin pair may change together due to shared environmental factors (eg. both twins could have higher follow-up measurements because they happened to have colds the day of their follow-up visit).
I want to look at correlation with self over time vs. correlation with co-twin. I know that ICCs are generally used here (since twin 1 and twin 2 are arbitrary assignments), and that if I just had non-longitudinal twin data, I could do a mixed model and calculate my ICCs as
s2_sibship / (s2_sibship + s2_residuals) giving me the percent variance explained by sibship. (Where s2_variablename is the model output sigma-squared for that variable.)
But since I have longitudinal data, I have two observations per person and two people per sibship. So if I fit a model with one random effect for sibship and one random effect for person I see three ways to calculate self-over-time ICCs, and two ways to calculate ICCs for sibship:
icc_self <- s2_person / (s2_person + s2_sibship + s2_residuals)
icc_self_adj <- s2_person / (s2_person + s2_residuals)
icc_self_nested <- (s2_person + s2_sibship) / (s2_person + s2_sibship + s2_residuals)
icc_pair <- s2_sibship / (s2_person + s2_sibship + s2_residuals)
icc_pair_adj <- s2_sibship / (s2_sibship + s2_residuals)
With _adj being an ICC adjusted for the other effect as discussed here and _nested treating person as being nested within sibship. I don't like _nested here, because it assumes self-over-time ICC will always be higher than sibship ICC (ignores shared environment, eg. the scenario where they both have a cold at follow-up).
To work around having to deal with the two random effects, I thought I maybe I could calculate 4 ICCs separately -- if I did self-over-time with just one member of each pair, I wouldn't need a random effect for sibship, and if I did sibship ICC and just one timepoint, I wouldn't need a random effect for person:
icc_self_twin1s: s2_person / (s2_person + s2_residuals)) using only twin 1s
icc_self_twin2s: Same as above but using only twin 2s
icc_pair_baseline: s2_sibship / (s2_sibship + s2_residuals)) using only baseline data
icc_pair_followup: Same as above but using only follow-up data
And then use meta-analysis to combine the twin1 and twin2 self ICCs, and to combine the baseline and followup pair ICCs. But I am using bootstrap confidence intervals for my ICCs, like described here, and when I look at my distribution of boot-strap estimates it's not normal, there's a long left tail. So I'm having trouble up with a reasonable standard error to use for meta-analysis.
I also thought of adding a random effect for visit (baseline or follow-up) nested within sibship to capture possible changes in shared environment over time, but I'm not sure how useful/accurate that would be given that I only have 2 people x 2 visits per sibship and already have two random effects (person and sibship). I also don't know where this would fit within my ICC calculations if I did that.
Here are examples of what I get for each ICC with four of my biomarkers:
A B C D icc_self 0.43 0.36 0.25 0.52 icc_self_adj 0.67 0.59 0.49 0.69 icc_self_nested 0.78 0.75 0.74 0.77 icc_self_twin1s 0.79 0.79 0.82 0.89 icc_self_twin2s 0.80 0.69 0.62 0.32 icc_pair 0.35 0.38 0.48 0.25 icc_pair_adjusted 0.62 0.60 0.65 0.52 icc_pair_baseline 0.32 0.57 0.59 0.35 icc_pair_followup 0.44 0.15 0.49 0.23
How should I approach this problem to get correlation with self over time and correlation between co-twins? And do you have any suggested citations that I could use to justify the recommended approach to reviewers?
(The three parts where I've encountered trouble with various approaches are in bold above.)