hlm models. general analysis strategy baseline model - no predictors model 1- level 1 predictors...
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HLM Models
General Analysis Strategy
• Baseline Model - No Predictors
• Model 1- Level 1 Predictors
• Model 2 – Level 2 Predictors of Group Mean
• Model 3 – Variance of Level 1 Predictors
• Model 4 – Predictors of Level 1 Slopes
(cross-level interactions)
Model Evaluation• Variance Partitioning
– Do scores vary at individual vs. group level?– ICC, reliability
• Tests of Fixed Effects– Is each of the significantly different from 0?– T-test
• Tests of Variance Components– Is there significant variance in the parameter across groups?– Chi-square test
• Explained Variance– How well do predictors in each equation account for the outcome
variable?– R2
Testing Fixed Effects
is distributed approximately as t withdf = # groups - # level 2 predictors - 1
• Confidence Interval
• Wald Test
)ˆ(ˆ SEtCI crit
)ˆ(
ˆ
SEt
Test on Variance Component
• Chi-Square Test
df = # groups - # level 2 predictors - 1
)(
22
j
j
VAR
u
Baseline Model
– Combined model
– Analyses:• Compute ICC
• Chi-Square Test on unconditional level-2 variance
ijojij ry
joj u000
ijjij ruy 000
Model 1: Level 1 Explanatory Variables with Fixed Slopes
• Analyses– Wald tests on average slope (10)– Individual-Level R2
ijijjojij rXy 1
joj u000
101 j
Interpretation of level-1 intercept and level-2 variance (00)
• Uncentered– predicted Y when X=0
• Group Centered– group mean on Y
• Grand Centered– group mean adjusted for level-1 predictors
• Any other value (L)– predicted Y when X=L
Model 2: Level 2 Explanatory Variables
• Combined Equation
• Analysis– Wald tests on slopes for level 2 predictors (01)– Group-Level R2
– Chi-square test on residual level 2 variance
ijijjojij rXy 1
jjoj uW 00100 1 101 j
ijjijjij ruXWy 0100100 1
Level 2 Explanatory Variables: Contextual or Incremental Effects
• Does the environmental context matter?
• Does the group’s level on variable X influence behavior, beyond the influence of X as an individual difference?
• Include the same variable as a predictor at level 1 and level 2.
Contextual Effect
• Group Centered 01 = total group-level
relationship– combination of
individual and group-level effects
• Grand Centered 01 = unique group-
level relationship– e.g., contextual effect
ijijjojij rXy 1
jjoj uX 00100
101 j
Recommendations for Centering Level-1 Predictors
• Separate Models at different levels– Group Center
• Incremental or Contextual Effects– Grand Center
• Cross-Level Interactions– Group Center
• If you grand center, cross level interaction will be confounded with the level-2 interaction
Model 3: Random Slopes
• Combined Model
• Analyses 2 test on random slope (11)
– Can remove u’s if not significant
ijijjojij rXy 1
jjoj uW 00100 1
jj u1101
ijjijjijjij ruXuXWy 01100100 1
Model 4: Cross-Level Interactions (predictors of slopes)
• Combined Model
• Analyses– Wald test on cross-level interaction (11)– R2 for prediction of slope 2 test on random slope (11)
ijijjojij rXy 1 jjoj uW 00100 2
jjj uW 111101 2
ijjijjijjijjij ruXuXWXWy 0111100100 22
Model 4: Level-2 Centering
• W2 Uncentered 10 = level-1 slope of
X for a group with W2=0
• W2 Grand Centered 10 = average level-1
slope of X
11 not affected by centering of W2
ijijjojij rXy 1
jjoj uW 00100 2
jjj uW 111101 2
HLM Analysis Options
• Multiple Parameter Test
• Deviance Test
• Testing Homogeneity of Level 1 Variance
• Modeling Heterogeneous Level 1 Variance
• Create Residual File