I'm analysing some data from a pilot study using longitudinal linear mixed models. Because of conventions in psychology it's common to report effect-size in Cohen's $d$ for interpretability. I found this post which led me to the package EMAtools and the function lme.dscore().
To help clarify my question, it may help with a description of the data-structure and the type of effect size I want to describe. I have 20 patients followed for 10 weeks giving weekly ratings. I want to describe the within-group change effect size. Using the lme4 package I've modelled simulated data like this: lmer(y ~ weeks + (weeks | id), data = df), so random intercepts and random slopes and a fixed effect of time/weeks.
A visual description of simulated data may also help clarify:
Here are examples of the random effects (random intercept and slopes) for 20 simulated individuals across 10 weeks.
I want to describe the fixed effect in terms of $d$. The lme.dscore() function appears to use this formula:
$$d=\frac{2t}{\sqrt{df}}$$
Which I'm familiar with in the context of extracting Cohen's d from simple t-tests, but I'm unsure if it makes sense in this context?
To compare, a colleague of mine cited Feingold (2013) and used a formula that looked like this (if I interpreted her correctly):
$$d=\frac{b*duration}{SD_{pre}}$$
These two methods result in pretty different estimates that can diverge a lot. For example the lme.dscore() method is much more affected by the variance in the slopes, something the Feingold method barely gets affected by - as long as the average effect of the slopes $b$ is the same that method yields the same d-score. This seems weird to me if we think of effect size as a measure of probability an individual will improve. A situation with low variability between slopes seems different from a situation with high variability of slopes, in terms of how certain the effect is. (Or in other words shouldn't intra-individual correlation, or in a simpler setup pre-post correlation, affect the estimate?)
Would be thankful to learn whether there's a problem with either method of converting to $d$, or at least someone pointing me in the right direction. I'm currently learning towards Feingold method since I have a precedent in my colleague's work there.
