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A linear mixed effects model allows you to specify random subject-level effects, or a (non-zero) covariance structure for within-subject residuals, or both. I've read a number of separate explanations of subject-level random effects and residual covariance structures, but I don't understand how these relate to each other or whether it's ever appropriate to use both in the same model.

For example, if you specify a random intercept by subject and also a compound symmetry covariance structure, those both predict that the observations for a given participant will tend to be higher or lower than the relevant fixed-effects mean. Are they completely redundant with each other? Is it just an artifact of the software interface that I'm even able to include both?

On the other hand, you could specify a subject-specific intercept and an autoregressive covariance pattern, which don't entirely overlap. Does this mean that an autoregressive pattern will be used to model each data point's deviations from the value predicted by the fixed and random effects?

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