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Hierarchical Linear Modeling HLM by liaoqinmei


									     Questions From Yesterday
• Equation 2: r-to-z transform
   – Equation is correct
   – Comparable to other p-value estimates (z = r sqrt[n])
• ANOVA will not be able to detect a group effect
  that has alternating + and – ICC
   – Effect defined in terms of between and within group
     variability rather than being represented individually
• SPSS Advanced Models can be ordered at the VU
  Bookstore for $51
  Hierarchical Linear Modeling
• Theoretical introduction
  –   Introduction to HLM
  –   HLM equations
  –   HLM interpretation of your data sets
  –   Building an HLM model
• Demonstration of HLM software
• Personal experience with HLM tutorial
      General Information and
• HLM can be used on data with many levels
  but we will only consider 2-level models
• The lowest level of analysis is Level 1 (L1),
  the second lowest is Level 2 (L2), and so on
• In group research, Level 1 corresponds to
  the individual level and Level 2 corresponds
  to the group level
• Your DV has to be at the lowest level
  When Should You Use HLM?
• If you have mixed variables
• If you have different number of
  observations per group
• If you think a regression relationship varies
  by group
• Any time your data has multiple levels
        What Does HLM Do?
• Fits a regression equation at the individual
• Lets parameters of the regression equation
  vary by group membership
• Uses group-level variables to explain
  variation in the individual-level parameters
• Allows you to test for main effects and
  interactions within and between levels
The Level 1 Regression Equation
• Predicts the value of your DV from the values of
  your L1 IVs (example uses 2)
• Equation has the general form of
   Yij = B0j + B1j * X1ij + B2j * X2ij + rij
• “i” refers to the person number and “j” refers to
  the group number
• Since the coefficients B0, B1, and B2 change from
  group to group they have variability that we can
  try to explain
          Level 2 Equations
• Predict the value of the L1 parameters using
  values of your L2 IVs (example uses 1)
• Sample equations:
  B0j = G00 + G01 * W1j + u0j
  B1j = G10 + G11 * W1j + u1j
  B2j = G20 + G21 * W1j + u2j
• You will have a separate equation for each
            Combined Model
• We can substitute the L2 equations into the L1
  equation to see the combined model
  Yij = G00 + G01 * W1j + u0j
  + (G10 + G11 * W1j + u1j) X1ij
  + (G20 + G21 * W1j + u2j) X2ij + rij
• Cannot estimate this using normal regression
• HLM estimates the random factors from the model
  with MLE and the fixed factors with LSE
• L1 regression equation:
   Yij = B0j + B1j * X1ij + B2j * X2ij + rij
• B0j tells us the value of Yij when X1ij = 0
  and X2ij = 0
• Interpretation of B0j therefore depends on
  the scale of X1ij and X2ij
• “Centering” refers to subtracting a value
  from an X to make the 0 point meaningful
       Centering (continued)
• If you center the Xs on their group mean
  (GPM) then B0 represents the group mean
  on Yij
• If you center the Xs on the grand mean
  (GRM) then B0 represents the group mean
  on Yij adjusted for the group’s average
  value on the Xs
• You can also center an X on a meaningful
  fixed value
        Estimating the Model
• After you specify the L1 and L2 parameters
  you need to estimate your parameters
• We can examine the within and between
  group variability of L1 parameters to
  estimate the reliability of the analysis
• We examine estimates of L2 parameters to
  test theoretical factors
    Interpreting Level 2 Intercept
• L2 intercept equation
    B0j = G00 + G01 * W1j + u0j
•   G00 is the average intercept across groups
•   If Xs are GPM centered, G01 is the relationship
    between W1 and the group mean (main effect of
•   If Xs are GRM centered, G01 is the relationship
    between W1 and the adjusted group mean
•   u0 is the unaccounted group effect
     Interpreting Level 2 Slope
• L2 slope equation
   B1j = G10 + G11 * W1j + u1j
• G10 is the average slope (main effect of X)
• G11 is relationship between W1 and the
  slope (interaction between X and W)
• u1 is the unaccounted group effect
      Building a HLM Model
• Start by fitting a random coefficient model
  – All L1 variables included
  – L2 equations only have intercept and error
• Examine the L2 output for each parameter
  – If there is no random effect then parameter does
    not vary by group
  – If there is no random effect and no intercept
    then the parameter is not needed in the model
       Building a HLM Model
• Build the full intercepts- and slopes-as-
  outcomes model
  – Use L2 predictor variables to explain variability
    in parameters with group effects
  – Remove L2 predictors from equations where
    they are unable to explain a significant amount
    of variability

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