Diagnostic Checks for Multilevel Models

for all j 6= l. The lengths of the vectors yj , β, and δj , respectively, are nj , r, and s. Like in all regression-type models, the explanatory variables X and Z are regarded as fixed variables, which can also be expressed by saying that the distributions of the random variables ǫ and δ are conditional on X and Z. The random variables ǫ and δ are also called the vectors of residuals at levels 1 and 2, respectively. The variables δ are also called random slopes. Level-two units are also called clusters. The standard and most frequently used specification of the covariance matrices is that level-one residuals are i.i.d., i.e.,