The combined model, = Yi* AXiβ + εi And in the model of observed data, = + Yi Mi AXiβ Mi εi The number of parameters is + + 1)/2, where is the number of fixed effects and f T(T f T is the number of intended observations per person. This is the most complex of the models to be compared. The remaining models impose constraints, and thus can be compared using a likelihood ratio test. Homogenous Level-1 Variance Level 1 in matrix notation, = + where ~ (0, and = This indicates Yi* Aπi N εi Σ) Σ σ2 IT. that the variance is the same for all time points. t Level 2 includes covariates, and because of the level-1 homogenous variance, degrees of freedom are available to estimate randomly varying intercepts and slopes: In matrix notation, = + , provided is large enough, intercepts/slopes can be Xi ri T πi β randomly varying. The combined model is = + + which is = + Yi* A Xi Airi ei A Xi β β εi = Var (Airi + = + IT = ei) AτA Var(εi) σ2 Δ The number of parameters is + + 1)/2 + 1, where is the dimension of In this case, f r(r r τ. must be less than r T. Heterogeneous (varying) Level-1 Variance The level-1 model is the same as the homogenous level-1 variance model, where Var(ei) = = where is diagonal with elements , the variance associated with diag{}, Σ Σ occasion where = 1, This indicates that the variance varies over time. t, t T. …, The number of parameters is + + 1)/2 + where can be no larger than 1. When f r(r T, r T – = 1, the results duplicate those from the unrestricted model. r T – Log-Linear Model for Level-1 Variance The model above has varying level-1 variance, which assumes a unique level-1 variance for every occasion. A more parsimonious model would specify a functional association between occasion (time or age) and the variance. In this case, = and also diag{}, Σ
The combined model, Yi* = AXiβ + εi And in the model of observed data, Yi = Mi AXiβ + Mi εi The number of parameters is f + T(T + 1)/2, where f is the number of fixed effects and T is the number of intended observations per person. This is the most complex of the models to be compared. The remaining models impose constraints, and thus can be compared using a likelihood ratio test. Homogenous Level-1 Variance Level 1 in matrix notation, Yi* = Aπi + εi where ~ N (0, Σ) and Σ = σ2 IT. This indicates that the variance is the same for all t time points. Level 2 includes covariates, and because of the level-1 homogenous variance, degrees of freedom are available to estimate randomly varying intercepts and slopes: In matrix notation, πi = Xi β + ri , provided T is large enough, intercepts/slopes can be randomly varying. The combined model is Yi* = A Xi β + Airi + ei which is = A Xi β + εi Var(εi) = Var (Airi + ei) = AτA + σ2 IT = Δ The number of parameters is f + r(r + 1)/2 + 1, where r is the dimension of τ. In this case, r must be less than T. Heterogeneous (varying) Level-1 Variance The level-1 model is the same as the homogenous level-1 variance model, where Var(ei) = Σ = diag{}, where Σ is diagonal with elements , the variance associated with occasion t, where t = 1, …, T. This indicates that the variance varies over time. The number of parameters is f + r(r + 1)/2 + T, where r can be no larger than T – 1. When r = T – 1, the results duplicate those from the unrestricted model. Log-Linear Model for Level-1 Variance The model above has varying level-1 variance, which assumes a unique level-1 variance for every occasion. A more parsimonious model would specify a functional association between occasion (time or age) and the variance. In this case, Σ = diag{}, and also
HMLM.v2.docx