8. Three and Four Level Models EPSY 8268 Three-Level Models We can conceptualize three level models. Consider students within classrooms within schools: children within classroom in school i njk j k, = 1, 2, …, classrooms within school j Jk k, = 1, 2, …, schools. k K = 1, 2, …, The unconditional model is: Student-level model Variance within level = + where ~ N(0, 2) 2 /(2 + + Yijk eijk eijk π0jk π β) Classroom-level model = + where ~ N(0, + + r0jk r0jk π0jk β00k π) π /(2 π β) School-level model = + where /(2 + + u00k u00k β00k γ000 ~ N(0, β) β π β) Reliability of = + 2/njk) and reliability of π /(π Generalized models: = + + + + + with student characteristics [a] Yijk a1ijk a2ijk aPijk eijk P π0jk π1jk π2jk … πPjk where ~ epjkl MN(0, σ2). = + with predictors [X] in the + 1 level-2 equations Qp P πpjk βp0k p is the associated level-1 predictor q is the level-2 predictor where ~ Tπ) rpjk MN(0, = + with predictors in the + 1 level-3 equations Spq Qp βpqk γpq0 s is the level-3 predictor where ~ Tβ). upqk MN(0, Four-Level Models
8. Three and Four Level Models EPSY 8268 Three-Level Models We can conceptualize three level models. Consider students within classrooms within schools: i = 1, 2, …, njk children within classroom j in school k, j = 1, 2, …, Jk classrooms within school k, k = 1, 2, …, K schools. The unconditional model is: Student-level model Variance within level Yijk = π0jk + eijk where eijk ~ N(0, 2) 2 /(2 + π + β) Classroom-level model π0jk = β00k + r0jk where r0jk ~ N(0, π) π /(2 + π + β) School-level model β00k = γ000 + u00k where u00k ~ N(0, β) β /(2 + π + β) Reliability of = π /(π + 2/njk) and reliability of Generalized models: Yijk = π0jk + π1jk a1ijk + π2jk a2ijk + … + πPjk aPijk + eijk with P student characteristics [a] where epjkl ~ MN(0, σ2). πpjk = βp0k + with Qp predictors [X] in the P + 1 level-2 equations p is the associated level-1 predictor q is the level-2 predictor where rpjk ~ MN(0, Tπ) βpqk = γpq0 + with Spq predictors in the Qp + 1 level-3 equations s is the level-3 predictor where upqk ~ MN(0, Tβ). Four-Level Models
8.Three-level-models.docx