Random Intercept Model

aka Ordinary Least Square (OLS) model

Some brief introduction, before start.

Fixed Effect (Taken only 1 group, level 1)

• Explain about individual
• Simple Model / Multiple Regression Model / Classical Model
• Y_ij = B₀ + B₁X_ij + R_ij , handle individual effect
• Y_ij = B₀ + B₁X_ij + B₂Z_j + R_ij , introduce group effect
• Y_ij = B₀ + B₁X_ij + B₂Z_j + B₃Z_jX_ij + R_ij , introduce interaction effect
• aka OLS model which just take nesting structure account at the minimum level, using Z_j

Exception, when all group sample size n_j are equal to one. The researcher need to have no qualied about using the model because nesting structure are not take place in this situation. The focus entirely on the regression coefficient, rather than how the variability of Y is fascinate within the group and between the group.

Random Effect (Taken multiple group, multi level)

• Within Effect (Individual)
• Between Effect (Group)
• Each level corresponds to a population. Eg. for a study of students in schools, there is a population of schools and a population of students.
• Mean (x) in one group maybe different to another group. X itself have a positive between group variance (interaction effect).
• Y_ij = B_0j + B_1jX_ij + B₂Z_j + R_ij , introduce group coefficient effect
• Y_ij = B_0j + B_1jX_ij + R_ij , removed group effect from group coefficient effect
• Y_ij = γ₀₀ + U_0j + γ₁₀ x_ij + R_ij , replace B_0j with γ₀₀+U_0j as
  • Y₀₀ is average of intercept,
  • U_0j is the main effect of the group aka level 2 residual,
  • Y₁₀ is average coefficient for X
• aka OLS model that do take the nesting structure into account; having larger number of regression parameter

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Empty Model aka ANOVA

because it contains not a single explanatory variable. Only contain random group and random variation within group.

Y_ij = γ₀₀ + U_0j + R_ij

• Y₀₀ is the sum of general mean
• U_0j is a random effect at group level
• R_ij is random effect at the individual level