Questions From Yesterday

1
Questions From Yesterday

• Equation 2 r-to-z transform
• Equation is correct
• Comparable to other p-value estimates (z r
  sqrtn)
• 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

2
Hierarchical Linear Modeling (HLM)

• 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

3
General Information and Terminology

• 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

4
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

5
What Does HLM Do?

• Fits a regression equation at the individual
  level
• 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

6
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

7
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
  parameter

8
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

9
Centering

• 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

10
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 groups average value on the Xs
• You can also center an X on a meaningful fixed
  value

11
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

12
Interpreting Level 2 Intercept Parameters

• 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 W1)
• If Xs are GRM centered, G01 is the relationship
  between W1 and the adjusted group mean
• u0 is the unaccounted group effect

13
Interpreting Level 2 Slope Parameters

• 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

14
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

15
Building a HLM Model (continued)

• 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