Hierarchical Linear Model

Advance from Random Intercept Model (simple case of hierarchical linear model) by introducing random slopes. RIM handle group difference with respect to average value of independent variable.

Y_ij = β_0j + β_1j X_ij + R_ij

• β_0j is group intercept
  • let’s further split it become Y₀₀ + U_0j
• β_1j is group slope (group coefficient)
  • let’s further split it become Y₁₀ + U_1j

become…

Y_ij = γ₀₀ + U_0j + γ₁₀ x_ij + U_1j x_ij + R_ij , note this extra part is missing from Random Intercept Model. It is the “random” part for slope aka level 2 residual.

• explain about the interaction between group and X
• group are characterize by 2 random effect (intercept is B0 and slope is B1)

Reorganize into symbol γ₀₀ + γ₁₀ x_ij can consider as fixed part of the model, U_0j + U_1j x_ij + R_ij consider as random part aka residuals.

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Heteroscedasticity

The spread / variability of residual (error) in a regression model.

Similar to MMDA module, we want no cone shape in the “Constant variance of the error terms” Test.

Another example would be,

• High social economic status (SES) student produce similar result regardless what school they attend.
• Low SES student make huge different in performance, leading bigger variance in the outcome of which school they attend.
• School then add a components of variance for low SES student.