Chapter 4 DIAGNOSTIC CHECKS FOR MULTILEVEL MODELS

and ( j , δj) ⊥ ( `, δ`) (4.1c) for all j 6= `. 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 and δ are conditional onX andZ. The random variables and δ are also called the vectors of residuals at levels 1 and 2, respectively. The variables δ are also called random slopes. The standard and most frequently used specification of the covariance matrices is that level-one residuals are i.i.d., i.e.,