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