u1j ~ N(0, 00) and ~ N(0, 11) with covariance 01. u0j u1j Generalization of the Model , (q = 0,1,…,Sq) qs Wsj uqj = Var(uqj) qq = Cov(uqj, uqj) qq Note that each coefficient can be modeled at level-2 as one of three general forms: 1. a fixed level-1 coefficient; e.g., = qj q0, 2. a non-randomly varying level-1 coefficient, e.g., 3. a randomly varying level-1 coefficient, e.g., = + or a level-1 coefficient qj q0 uqj with both non-random and random sources of variation, . Simpler Submodels from and Regression The simplest possible HLM is equivalent to a one-way ANOVA with random effects. = 0j + , where ~ N(0, 2). Yij rij rij 0j = 00 + , where ~ N(0, 00). u0j u0j = 00 + + Yij u0j rij Var(Yij) = Var(u0j + = 00 + 2 rij) The intraclass correlation coefficient is . This model is an and is an important tool for comparison with conditioned unconditional model models with explanatory variables at either or both levels.
u1j u0j ~ N(0, 00) and u1j ~ N(0, 11) with covariance 01. Generalization of the Model , qs (q = 0,1,…,Sq) Wsj uqj Var(uqj) = qq Cov(uqj, uqj) = qq Note that each coefficient can be modeled at level-2 as one of three general forms: 1. a fixed level-1 coefficient; e.g., qj = q0, 2. a non-randomly varying level-1 coefficient, e.g., 3. a randomly varying level-1 coefficient, e.g., qj = q0 + uqj or a level-1 coefficient with both non-random and random sources of variation, . Simpler Submodels from and Regression The simplest possible HLM is equivalent to a one-way ANOVA with random effects. Yij = 0j + rij , where rij ~ N(0, 2). 0j = 00 + u0j , where u0j ~ N(0, 00). Yij = 00 + u0j + rij Var(Yij) = Var(u0j + rij) = 00 + 2 The intraclass correlation coefficient is . This model is an unconditional model and is an important tool for comparison with conditioned models with explanatory variables at either or both levels.
2.Two-level-models-handouts.docx