Residual Analysis and Goodness of Fit

Residual analysis, when used to assess goodness of fit in a linear mixed effects model, is usually done with the same methods as in linear or nonlinear models. However, two type of residuals arise with a mixed effects model. If interest centers on a particular individual then conditional residuals (sc ) are of interest 

If however, interest centers on a individual averaged across all possible random effects, then the marginal residuals (sm ) are of interest 

If there are no random effects in the model, then the two residuals are equal. Residual analysis can be done on either the raw conditional or marginal residuals. However, as with the linear model, residual analysis in this manner is complicated by the fact that the variance of the residuals is unknown. A data point with a small residual may be of more importance because it has a smaller standard error than an observation with a larger residual but correspondingly larger standard error. 

raw residuals are often standardized by some quantity, usually its standard deviation, to account for the unequal variance of the residuals. If the estimate of the standard deviation is based on the observation in question the method is called internal studentization, but if the observation is not included in the estimate of the standard deviation, then the method is called external studentization. Studentized residuals can be estimated 

Scaled residuals have zero mean and are approximately uncorrelated. Once the residuals are scaled or standardized then it is appropriate to compare residuals directly for influence and outliers. 

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