1. 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

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 group’s 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