HLM in 60 minutes

1
HLM in 60 minutes

• Jim Hinterlong, Ph.D.
• FSU College of Social Work
• February 22, 2006

2
Acknowledgments

• Shameless borrowing from ISR course
• Raudenbush, S. W., Bryk, A. S. (2002).
  Hierarchical linear models (2nd Ed.). Sage.
• www.wtgrantfoundation.org
• Optimal design software (free download)

3
Agenda

• General Overview
• HLM with longitudinal data
• HLM with clustered, cross-sectional data
• Tips
• Summary

4
Multilevel data

• Hierarchical data (HD) structures are common.
• Nested (repeated) observations
• Nested units
• Dual-nested structures (units and obs.)

5
Repeated Observations

• Individual change over time
• Within person change
• Change in trajectories across individuals
• Between-person differences in change
• Growth Curve Modeling

6
Nested Units (Organizations)

• Shared effects on clustered units
• Upper level units naturally occurring

7
Problems faced with HD

• Aggregation bias
• Disaggregation bias (neglect nesting)
• Correlated (misestimated) error (ICC)
• Heterogeneity of regression
• Ecological fallacy (inferential errors)
• Unbalanced designs
• HLM can address these issues

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HLM a.k.a.

• Multilevel linear models (Socio)
• Mixed-effects or Random Effects (Bio)
• Random Coefficient Reg. (Econ)
• Covariance Components Models (Stats)

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Advantages of HLM

• Relaxed assumptions vis-à-vis OLS
• Flexibility in research design
• Incomplete observations
• Unequal data collection lags
• Identifies temporal patterns
• Includes time-varying predictors
• Allows interactions with time

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HLM

• Fits a regression line at individual level one
  for each level 1 unit
• Allows parameter estimates to vary by group
  membership
• Allows testing of main effects and interactions
  within and between levels

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HLM Assumes

• Time is linearly associated with Y
• Residuals are normal
• Equality of variances
• Independence of uppermost level units
• Variability among units at each level

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HLM with longitudinal data
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Repeated Observations

• Can changes in opposite naming over time be
  attributed to differences in IQ?
• Level-1
• DV Score (opposites named)
• IV Week-1 (to aid intercept interpretation)
• Level-2
• IV IQ (grand-mean centered)

• Centering
• Affects the interpretation of the intercept.
• Can be group mean, grand mean, or around a
  specific value.

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Level-1 Model Examined
r
Betas are latent variables model-based estimates
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Growth Record w/Trend, single case
300
250
Opposites naming Score
b0j
200
b1j
150
100
50
0
0
2
3
4
5
1
Week
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Level-2 Model
Mean intercept across Level-2 units
Intercept
Slope
Mean slope across Level-2 units
Level-2 Fixed Effect
Level-2 Random Effect
Combined Model
Yij g00 g01Xj g10tij g11Xjtij uoj
u1jtij rij
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Modeling Change

• Level-1 (within-person model)
• Level-2 (between-person model)

r
r
i ith person j jth group

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Model Equations
MIXED MODEL
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HLM Output
Summary of the model specified (in equation
format) -----------------------------------------
---------- Level-1 Model Y B0 B1(WEEK)
R Level-2 Model B0 G00 G01(IQ) U0 B1
G10 G11(IQ) U1
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First Step Is HLM needed?

• Random Coefficient model for Level-1
• All Level-1 variables
• Level-2 only intercept and residual
  (unconditional)
• Check Level-2 output
• If there is a random effect, parameter varies by
  group
• If no random effect or intercept at L2, the
  parameter is not needed in model

YES! (Although not shown here)
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Level-1 OLS regressions -----------------------
Level-2 Unit INTRCPT1 WEEK slope
--------------------------------------------------
---------------------------- 1
196.70 34.20 2 218.00
28.50 3 151.40
47.90 4 206.40 21.90
5 199.90 9.90
6 165.60 29.60 7
159.60 32.60 8
92.90 37.90 9 138.70
22.70 10 219.80
26.80 The average OLS level-1 coefficient
for INTRCPT1 164.37429 The average OLS
level-1 coefficient for WEEK
26.96000 Sigma_squared 159.47714
Standard Error of Sigma_squared 26.95656
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Tau INTRCPT1,B0 1159.38147 -165.31460
WEEK,B1 -165.31460 99.29807
Standard Errors of Tau INTRCPT1,B0
304.41620 78.28048 WEEK,B1
78.28048 31.82128 Tau (as correlations)
INTRCPT1,B0 1.000 -0.487 WEEK,B1 -0.487
1.000 ----------------------------------------
------------ Random level-1 Reliability
coefficient estimate --------------------------
-------------------------- INTRCPT1, B0
0.912 WEEK, B1
0.757 -------------------------------------
---------------
Correlation between Initial status and growth rate
Reliability of distinguishing between
individuals Reliability of latent variables B0
and B1
l0j
l1j
As Nj increases, reliability increases
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Final estimation of fixed effects ---------------
--------------------------------------------------
--------------------------------------------------
------
Standard Approx. Fixed Effect
Coefficient Error T-ratio
d.f. P-value ------------------------------
--------------------------------------------------
----------------------------------------- For
INTRCPT1, B0 INTRCPT2, G00
164.374 6.026170 27.277 33 0.000
IQ, G01 -0.114
0.415881 -0.273 33 0.787 For
WEEK slope, B1 INTRCPT2, G10
26.9600 1.936075 13.925 33 0.000
IQ, G11 0.4329
0.121468 3.564 33 0.001
--------------------------------------------------
--------------------------------------------------
---------------------
HLM also produces a model with robust standard
errors Compare these with the non-robust. Sig.
differences indicate assumption checking is
needed.
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Exploratory Level-2 Residuals
IQ is not related to initial status (B0j)
Empirical Bayes estimates (corrected OLS
residuals borrowing strength from complete
cases)
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Exploratory Level-2 Residuals
IQ is related to slope (Bij)
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Final estimation of variance components
--------------------------------------------------
--------------------------------------------------
--------------------------- Random Effect
Standard Variance df Chi-square
P-value Deviation
Component ---------------------------------------
--------------------------------------------------
----------------------------------- INTRCPT1,
U0 34.04969 1159.38147 33
398.49456 0.000 WEEK slope, U1
9.96484 99.29807 33 143.96334
0.000 level-1, R 12.62843
159.47714 -------------------------------------
--------------------------------------------------
--------------------------------------
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Individual Growth Curves
First 10 respondents
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Effect of IQ on Score Trajectory
300
IQ 9.5 (75th tile) IQ - 9.5 (25th tile)
250
Opposites naming Score
200
150
100
50
0
0
2
3
4
5
1
Week
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Examining Level-1 Residuals
Detecting Outliers
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Level-1 Residuals
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Level-1 Residuals
Testing for normality
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Unconditional Model of Growth in Opposites-naming
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Linear model of Growth in Opposite-naming
(Effects of IQ)
Didnt have time To compute
Model comparison c2 8.46, df 2, p 0.014
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• HLM with
• Clustered, Cross-sectional Data

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Nested Unit (Organization)Analysis options

• One-Way ANOVA
• Significant unconditional model reveals
  clustering effect
• Means-as-Outcome
• Contextual Effects
• Random Coefficients
• Random Intercept Slopes as Outcomes

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Other HLM Options

• HLM3 three-level models
• HGLM binary, count, nonlinear
• HMLM multivariate allows latent variable
  analysis
• Cross-classed Models observations fit under 2
  upper-level clusters

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Tips

• Create separate dataset for each level
• Test descriptive stats in HLM against those
  produced in SPSS
• ID variable is ID of higher order unit
• Upper-most level cannot contain time-varying
  covariates
• Use Full Maximum Likelihood
• Produces gs
• Allows likelihood ratio testing of nested models
  (use deviance, which is -2LL)

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Advantages of HLM

• Models hierarchical data structure
• Adjusts level-1 estimates for measurement error
  (only DV not IV)
• Permits/Uses heterogeneity of regression
• Allows categorical/continuous predictors
• Produces correct standard errors

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Advantages of HLM (cont.)

• Can model discontinuous outcomes
• Allows non-linear models
• Handles missing data at Level-1

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Software 3 general options
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End