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# Hierarchical Linear Modeling HLM by liaoqinmei

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```									     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
(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
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
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
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
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
parameter
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
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
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
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
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
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
(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

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