= Effect size for multilevel models =

$$\rho = \frac{\mbox{variance_{between subjects}}}{\mbox{(variance{between subjects} + variance{within subjects}}}$$

I read in the wikipedia that this "design effect" is used with cluster observations. If we fit a mixed model with a '''random subject-specific intercept''', the clusters are the observations within a participant e. g. the 7 times the participant chose a fruit or a snack. The "design effect" is

D_{eff} = 1 + (m-1) $$\rho$$

where m is the number of observations in each cluster (e.g. number of repeated measures per subject).