I'm running a Mixed effects model ANOVA with two fixed factors (condition, repetition) and one random factor (subject). Subsequently, a Tukey multiple comparisons test is performed.
Now I'd like to plot the means and standard errors (SEMs) of the single conditions in a single error bar plot, and report the p values between the conditions.
The problem: while in the Tukey test, I got significant differences and non-overlapping SEMs between certain means, for my plotted real/observed data the SEM bars overlap.
This is now counterintuitive, since commonly you would assume that in the case of overlapping, the means are not significantly different.
My question is:
is the difference between estimated marginal means and observed means due to having a random factor in my model, or what is the reason for the discrepancy?
how would you report the data? Would you still plot observed data with the p values and state that the p values are derived from the estimated model? Or would you plot estimated means and standard errors?
EDIT: I'm adding the multiple comparisons result for a sample case as well as the observed means and standard error plot in case this helps.