Handbook of Multilevel Analysis, edited by Jan de Leeuw and Erik Meijer c 2008 Springer, New York 3 Diagnostic Checks for Multilevel Models Tom A. B. Snijders 1,2 and Johannes Berkhof 3 1 University of Oxford 2 University of Groningen 3 VU University Medical Center, Amsterdam 3.1 Specification of the Two-Level Model This chapter focuses on diagnostics for the two-level Hierarchical Linear Model (HLM). This model, as defined in chapter 1, is given by y j = X j β + Z j δ j + ǫ j , j =1,...,m, (3.1a) with  ǫ j δ j  ∼N  ∅ ∅  ,  Σ j (θ) ∅ ∅ Ω(ξ)  (3.1b) and (ǫ j , δ j ) ⊥ (ǫ ℓ , δ ℓ ) (3.1c) for all j = ℓ. The lengths of the vectors y j , β, and δ j , respectively, are n j , 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-2 units are also called clusters. The standard and most frequently used specification of the covariance matrices is that level-1 residuals are i.i.d., i.e., Σ j (θ)= σ 2 I nj , (3.1d) where I nj is the n j -dimensional identity matrix; and that either all elements of the level-2 covariance matrix Ω are free parameters (so one could identify Ω with ξ), or some of them are constrained to 0 and the others are free parameters.