172021-09-23 08:20:16PeterWatson162021-09-23 08:18:48PeterWatson152021-09-23 08:10:19PeterWatson142016-02-26 11:07:52PeterWatson132014-09-04 14:37:37PeterWatson122014-09-04 08:51:07PeterWatson112014-09-04 08:50:51PeterWatson102014-09-04 08:33:01PeterWatson92014-09-04 08:30:52PeterWatson82014-09-04 08:30:34PeterWatson72014-09-04 08:27:59PeterWatson62013-03-08 10:17:13localhostconverted to 1.6 markup52012-12-06 12:57:47PeterWatson42012-12-06 12:57:27PeterWatson32012-12-06 12:57:00PeterWatson22012-12-06 12:56:27PeterWatson12012-12-06 12:55:42PeterWatson

$$\rho = \frac{\mbox{variance(between subjects)}}{\mbox{(variance(between subjects) + variance(within subjects))}}$$ In the wikipedia it states 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). The Snijders and Bosker effect size formed from pooling the intercept and residual variances may be evaluated using the below R syntax incorporating a function written by Adam Wagner. T statistics associated with fixed regression terms may also be used (see here.) which can be converted into Cohen's d and r (correlation) effect sizes for pairwise group and covariate predictors respectively. which gives R-squareds as output for each model effect (usually > 0) You can also compute Cohen's d effect sizes directly using the eff_size function which works in conjunction with the emmeans which one can output from a regression, in this case from the mixed model routine, lmer. An example of its use is below. Thanks to Andrea Kusec for this. m.bads2 time = 1: group emmean SE df lower.CL upper.CL 1 93.9 4.41 104 85.1 102.6 2 84.2 4.48 105 75.3 93.1 3 90.5 4.94 127 80.8 100.3 ]]> eff_size(m.bads2, sigma=sigma(modelbadsgroup2), edf = df.residual(modelbadsgroup2)) time = 1: contrast effect.size SE df lower.CL upper.CL 1 - 2 0.739 0.441 104 -0.1350 1.612 1 - 3 0.255 0.400 104 -0.5387 1.049 2 - 3 -0.484 0.415 105 -1.3068 0.340 ]]>Reference Snijders TAB and Bosker RJ (1999) Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling. Sage:London.