Background: I’m analyzing data with mixed-models (lmer in lme4) from an experiment that had RTs and Error Rates as dependent variables. This is a repeated-measures design with approximately 300 measurements for each of the 190 human subjects. The fixed-effects are 1 between-subjects experimental manipulation (dichotomous), 2 within-subjects experimental manipulations (both dichotomous), and 1 subject variable (continuous, centered). My uncorrelated random effects are the participants, and 2 stimulus characteristics. For the mixed-models, I’ve coded the experimental manipulations as a -.5/+.5 contrasts so that the parameters are estimates of the experimental effects and the intercept should be the grand mean.
The grand mean produced by the RT model (740 ms) does not match the mean I get if I average all of the individual trials (730 ms). Why does this happen?
A related question: the GLMM (binomial distribution, logistic link function) for error rates produces a parameter estimate with an associated Z-score that has an absolute value over 2, but when I look at the means (determined the same way as above) to examine this difference they are tiny and almost identical (0.01353835 vs. 0.01354846). What are the values that I can provide that support the reliable parameter estimate?
I have a feeling the discrepancy between my calculated means and the model estimates has something to do with the random factors (perhaps the grouping by subjects), but I’m not sure exactly what.
If I want to display descriptive statistics along with the table of mixed model estimates, how should these descriptive be determined? Any points to references, examples, etc. will be greatly appreciated.
If this is all just a brain fart on my part, please let me know that too.
Edit: It is probably also important to mention that the amount of trials and types of trials contributed are not the same for every person. The between-subject manipulation changes the proportions of the different trial types presented, and for RTs only correct trials were analyzed. There were, however, very few errors made.
Best Answer
With regards to your general observation that the grand mean predicted by the mixed effects model is not equal to the grand mean of the data, I believe that this is an expected behaviour and attributable to the fact that shrinkage occurs in mixed effects models. My understanding of shrinkage is that, taking your example of participants as random effects, the model's estimate of a given participant's effect is influenced by not only that participant's data but also the data from the other participants. The consequence of this is that each participant's effect becomes shrunk towards the grand mean in proportion to how deviant their raw mean is (as well as the relative variability of their data, I believe). The result is that the predictions from the model for even simple descriptives like the grand mean can vary from their raw counterparts.
For more on shrinkage and it's rationale, here's a link to section 7.3 "Shrinkage in mixed effects models" (bottom of page 275) from Baayen's "Analyzing Linguistic Data" text.