A multilevel model is defined as y = Xβ + Zη + ǫ

Thus there are 3 different kinds of residuals:  

1. Marginal residuals: y − Xβ (= Zη + ǫ)
2. Conditional residuals: y − Xβ − Zη (= ǫ)
3. Random effects: y − Xβ − ǫ (= Zη)

Marginal residuals:

• Should be mean 0, but may show grouping structure
• May not be homoskedastic!
• Good for checking fixed effects, just like linear regr.  

Conditional residuals:

• Should be mean zero with no grouping structure
• Should be homoskedastic!
• Good for checking normality of ǫ, outliers  

Random effects:

• Should be mean zero with no grouping structure
• May not be be homoskedastic!
• Good for checking normality of , outliers

In R (if results is an mer object), the command residuals(results) gives you the conditional residuals.

# checking the normality of conditional residuals:
qqnorm(resid(results), main="Q-Q plot for conditional residuals")

# checking the normality of the random effects (here random intercept):
qqnorm(ranef(resuls)$Name_of_group_variable$`(Intercept)`, main="Q-Q plot for the random intercept")

The answer is partly copied from the following PowerPoint slide deck: http://www.stat.cmu.edu/~brian/463/week07/14%20-%20lmer%20model%20selection%20and%20residuals.pdf

A multilevel model is defined as $y = Xβ + Zη + ǫ$

Thus there are 3 different kinds of residuals:

1. Marginal residuals: $y − Xβ\ (= Zη + ǫ)$
2. Conditional residuals: $y − Xβ − Zη\ (= ǫ)$
3. Random effects: $y − Xβ − ǫ\ (= Zη)$

Marginal residuals:

• Should be mean 0, but may show grouping structure
• May not be homoskedastic!
• Good for checking fixed effects, just like linear regression.

Conditional residuals:

• Should be mean zero with no grouping structure
• Should be homoskedastic!
• Good for checking normality of ǫ, outliers

Random effects:

• Should be mean zero with no grouping structure
• May not be be homoskedastic!
• Good for checking normality of , outliers

In R (if results is an mer object), the command residuals(results) gives you the conditional residuals.

# checking the normality of conditional residuals:
qqnorm(resid(results), main="Q-Q plot for conditional residuals")

# checking the normality of the random effects (here random intercept):
qqnorm(ranef(resuls)$Name_of_group_variable$`(Intercept)`, 
       main="Q-Q plot for the random intercept")

The answer is partly copied from the following PowerPoint slide deck pdf.