Suppose I conduct an experiment to test the hypothesis that treatment A suppresses protein X. I am fortunate in that I work on kidneys, which come in pairs, however I am unfortunate in that I don't have much idea about how quickly or slowly any suppressive effect might appear. I therefore apply treatment A to six kidneys selected at random from six pairs (i.e. one kidney per pair gets the treatment); I apply a control treatment to the contralateral kidney in each pair. I measure protein X at 2 hours, 4 hours, and 6 hours following initiation of treatment. My hypothesis is not specific with respect to time of effect (I accept the design limitation here).
In my data I observe a group (treatment) effect, a time effect, and a 'pair' effect (concordance between kidneys within a pair). There appears to be a small within-pair (i.e. between group) difference in expression at the first time point, a bigger difference at the second, and an obvious difference by the third.
It strikes me that this data is both paired, and repeated-measures. I am not aware of a canonical statistical test for this situation. I could use a repeated-measures ANOVA and lose the increase in power due to pairing, or multiple paired T-tests and lose the time effect (and have to deal with multiple comparisons).
Alternatively I could fit a mixed model, with 'pair' as a random effect, and time and group as fixed effects. I'm not an expert, but this seems reasonable - it reflects the structure of the data/ experiment, I have 36 observations, that's three parameters to fit (is it? Or is it four? Does the intercept count if I've used effectively a random intercept like this?). I could then extract a p-value using lme4 and lmerTest, e.g. lmer(concentration~timepoint + group + (1|pair)), and see if there is a group effect.
Is this a correct treatment? I've read around this a bit and loosely understand various arguments re the meaning of significance with respect to mixed models, and about e.g. why lme4 doesn't report p-values by default. Nevertheless this is a fairly classic experimental design and my gut feeling is that testing significance in this manner is reasonable. I would be grateful for any expert advice. I would be particularly grateful for any relevance references.
