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Multi-level modelling (Random intercept MLM (If introducing macro level…

- - - - % reduction from null model
        
        level 2
      - level 1:use residual
- - - - β0 + βj = β0j
  - - - u0j varies depending on group membership
      - γ00 is fixed
    - - – u0j assumed normally distributed with mean of 0 and variance 
        2
        
        group
      - – εij assumed normally distributed with mean 0 and variance σ
        2
        
        indivudal
      - total variance = σ
        2 + 
        2
- - - - Yij = β0j + β1j x ij + εij
    - - β0j = γ00 + γ01 zj + u0j
      - β1j = γ10 + u1j
    - - γ00 + γ10 x ij + γ01 zj + γ11 zj
        x ij + u0j + u1j x ij + εij
- - - - problematic for micro level or micro-macro propositions
    - - Shift of meaning from ind. scores to means
      - neglect of original data structure.
      - precentv examinations of cross-level relationships
      - ecological fallacy: association at macro level might not translate to association at micro-level.
      - aggregation bias
        
        inflated statistical effects when means are used to replace individual scores.
  - - - Apply a grouping variable to all individuals.
      - problems
        
        multiplication of units
        
        risk of type 1 error increased
        
        observations within a macro level could be related
- - - - grand mean centering
      - group mean centering
- - - - 2.These relationships used as outcome variables for variables at level 2. (different linear relationships for different groups)