Subject Variation Modelled by Random Effects I'm wondering how variation between subjects is accounted for in data, specifically when one or a few subjects show a significant effect in the opposite direction to other subjects.
For example, if subjects are tested on 2 days and some subjects perform better on day 1 but others perform better on day 2.
How is this unsystematic variation accounted for in a linear mixed model with random effects for the subject? Is there an outcome measure for model fit for each subject separately (as an indication of outliers)?

I found the answer to this; I'm sure there will be more resources for those interested in the maths underlying linear mixed models, but this describes the overall principle.
This problem relates to the idea of homogeneity of regression slopes. With the standard linear model (fixed effects only e.g., ANCOVA), the relationship between the outcome and covariates must be the same across groups. This assumption is not necessary with linear mixed models, as the variance across groups (for e.g., participants) is included in the model.
Similarly, linear mixed models can be useful in cases of Simpson's paradox (a statistical illusion where the association between two variables appears in different groups, but disappears or reverses when this grouping structure is not taken into account; see  Jaeger, Graff, Croft, & Pontillo, 2011).
