# How to calculate Cohen's d between multilevel model predictions at two points in the domain?

I would like to get a Cohen's d between model predictions at different points in the domain of a mixed effects model. I have model predictions and I have bootstrapped standard errors. I mention the SEs in case they are relevant to the answer we may get.

The predictor is continuous time (days since treatment onset). It is standard in the literature to have a categorical predictor for time, but in our study, assessments were not taken at a consistent time after treatment onset so we modeled time as a continuous predictor. However, we would like to be able to compare easily to other similar studies in the literature.

Is there a way I can calculate a Cohen's d between, for example, participant assessments at day 0 and day 14?

EDIT: Would it be correct to use the formula $$(\mu_{y | x = 0} - \mu_{y|x=14})/\sigma_{y|x}$$ where $$\sigma_{y|x}$$ is the standard deviation of the residuals and the $$\mu$$ terms are the predictions at their respective values of $$x$$? (I am assuming $$\sigma$$ is equal at both values of $$x$$ because that is an assumption of the linear model.)

• I'll probably come back with a more detailed response later, but this paper has a lot to say about effect sizes for linear mixed models, and the important point is that $d$ should take into account both the SD of the random effects (e.g. differences between subjects) and the SD of the residuals (noise), since the residuals in simple analysis like the t test capture both. – Eoin Sep 12 '20 at 20:35