hierarchical linear modelshierarchical data structures many social and natural phenomena have a...
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Hierarchical Linear Models
Joseph Stevens, Ph.D., University of Oregon(541) 346-2445, [email protected]
© Stevens, 2007
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Overview and resourcesOverviewWeb site and links: www.uoregon.edu/~stevensj/HLMSoftware:
HLMMLwinNMplusSASR and S-PlusWinBugs
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Workshop Overview
Preparing dataTwo Level modelsTesting nested hierarchies of modelsEstimationInterpreting resultsThree level modelsLongitudinal modelsPower in multilevel models
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Hierarchical Data Structures
Many social and natural phenomena have a nested or clustered organization:
Children within classrooms within schoolsPatients in a medical study grouped within doctors within different clinics Children within families within communitiesEmployees within departments within business locations
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Grouping and membership in particular units and clusters are important
Grouping and membership in particular units and clusters are important
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Hierarchical Data Structures
More examples of nested or clustered organization:Children within peer groups within neighborhoodsRespondents within interviewers or ratersEffect sizes within studies within methods (meta-analysis)Multistage samplingTime of measurement within persons within organizations
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Simpson’s Paradox:
Quiz 1 Quiz 2 Total
Gina 60.0% 10.0% 55.5%
Sam 90.0% 30.0% 35.5%
Clustering Is Important
Well known paradox in which performance of individual groups is reversed when the groups are combined
Quiz 1 Quiz 2 Total
Gina 60 / 100 1 / 10 61 / 110
Sam 9 / 10 30 / 100 39 / 110
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Simpson’s Paradox: Other Examples2006 US School study:
• In past research, private schools achieve higher than public schools
• Study was expected to provide additional support to the idea that private and charter schools perform better
• USED study (using multilevel modeling):
• Unanalyzed math and reading higher for private schools
• After taking demographic grouping into account, there was littledifference between public and private and differences were almost equally split in favor of each school type
2006 US School study:
• In past research, private schools achieve higher than public schools
• Study was expected to provide additional support to the idea that private and charter schools perform better
• USED study (using multilevel modeling):
• Unanalyzed math and reading higher for private schools
• After taking demographic grouping into account, there was littledifference between public and private and differences were almost equally split in favor of each school type
1975 Berkeley sex bias case:
• UCB sued for bias by women applying to grad school
• Admissions figures showed men more likely to be admitted
• When analyzed by individual department, turned out that no individual department showed a bias; • Women applied to low admission rate departments
• Men applied more to high admission rate departments
1975 Berkeley sex bias case:
• UCB sued for bias by women applying to grad school
• Admissions figures showed men more likely to be admitted
• When analyzed by individual department, turned out that no individual department showed a bias; • Women applied to low admission rate departments
• Men applied more to high admission rate departments
“When the Oakies left Oklahoma and moved to California, it raised the IQ of both states.”
– Will Rogers
“When the Oakies left Oklahoma and moved to California, it raised the IQ of both states.”
– Will Rogers
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Participant (i) Cluster (j) Outcome (Y) Predictor (X)
1 1 5 1
2 1 7 3
3 2 4 1
4 2 6 4
5 3 3 3
6 3 5 5
7 4 2 4
8 4 4 6
9 5 1 5
10 5 3 7
Hypothetical Data Example from Snijders & Bosker (1999), n = 2, j=5, nj = N = 10
Hypothetical Data Example from Snijders & Bosker (1999), n = 2, j=5, nj = N = 10
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Model Summary
.333a .111 .000 1.826Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), Xa.
Coefficientsa
5.333 1.453 3.671 .006-.333 .333 -.333 -1.000 .347
(Constant)X
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: Ya.
Y = 5.333 -.333(X) + r
Interpretation: There’s a negative relationship between the predictor X and the outcome Y, a one unit increase in X results in .333 lower Y
All 10 cases analyzed without taking cluster membership into account:All 10 cases analyzed without taking cluster membership into account:
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X
876543210
Y
8
7
6
5
4
3
2
1
0
Y = 5.333 -.333(X) + rY = 5.333 -.333(X) + r
β0
β1
This is an example of a disaggregated analysis
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Participant (i)
Cluster (j) Outcome (Y)
Predictor (X)
1 1 5 1
2 1 7 3
3 2 4 2
4 2 6 4
5 3 3 3
6 3 5 5
7 4 2 4
8 4 4 6
9 5 1 5
10 5 3 7
Cluster (j) Outcome (Y) Predictor (X)
1 6 2
2 5 3
3 4 4
4 3 5
5 2 6
Another alternative is to analyze data at the aggregated group levelAnother alternative is to analyze data at the aggregated group level
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Y = 8.000 -1.000(X) + r
Interpretation: There’s a negative relationship between the predictor X and the outcome Y, a one unit increase in X results in 1.0 lower Y
The clusters are analyzed without taking individuals into account:The clusters are analyzed without taking individuals into account:
Coefficientsa
8.000 .000 . .-1.000 .000 -1.000 . .
(Constant)MEANX
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: MEANYa.
Model Summary
1.000a 1.000 1.000 .000Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), MEANXa.
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This is an example of a disaggregated analysis
Y = 8.000 -1.000(X) + r
MEAN X
76543210
MEA
N Y
8
7
6
5
4
3
2
1
β1
β0
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X
876543210
Y
8
7
6
5
4
3
2
1
0
A third possibility is to analyze each cluster separately, looking at the regression relationship within each group
ijr+−+= )XX(00.1YY jijjij
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Y = 8.000 -1.000(X) + r
Multilevel regression takes both levels into account:Multilevel regression takes both levels into account:
ijr+−+−= )XX(00.1)X(00.100.8Y jijjij
ijr+−+= )XX(00.1YY jijjij
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X
876543210
Y
8
7
6
5
4
3
2
1
0
Taking the multilevel structure of the data into account: Taking the multilevel structure of the data into account:
Within group regressions Between Groups Regression
Total Regression
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Why Is Multilevel Analysis Needed?
Nesting creates dependencies in the dataDependencies violate the assumptions of traditional statistical models (“independence of error”, “homogeneity of regression slopes”)Dependencies result in inaccurate statistical estimates
Important to understand variation at different levels
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Decisions About Multilevel Analysis
Properly modeling multilevel structure often matters (and sometimes a lot)Partitioning variance at different levels is useful
tau and sigma (σ 2Y = τ + σ 2)
policy & practice implications
Correct coefficients and unbiased standard errorsCross-level interactionUnderstanding and modeling site or cluster variability
“Randomization by cluster accompanied by analysis appropriate to randomization by individual is an exercise in self-deception and should be discouraged” (Cornfield, 1978, pp.101-2)
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Preparing Data for HLM Analysis
Use of SPSS as a precursor to HLM assumedHLM requires a different data file for each level in the HLM analysisPrepare data first in SPSS
Clean and screen dataTreat missing dataID variables needed to link levelsSort cases on ID
Then import files into HLM to create an “.mdm” file
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Creating an MDM file
Example:Go to “Examples” folder and then “Appendix A” in the HLM directoryOpen “HSB1.sav” and “HSB2.sav”
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Start HLM and choose “Make new MDM file” from menu; followed
by “Stat package input”
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For a two level HLM model stay with the default “HLM2”
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Click “Browse”, identify the level 1 data file and open it
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Click on “Choose Variables”
Check off the ID linking variable and all variables to
be included in the MDM file
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Provide a name for the MDM file (use .mdm as the suffix)Click on “Save mdmt file” and
supply a name for the syntax command file
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Click on “Make MDM”
Results will briefly appear
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Click on “Check Stats” to see, save, or print results
The click “Done”
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You should see a screen like this. You can now begin to specify your HLM model
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Two-Level HLM Models
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The Single-Level, Fixed Effects Regression Model
Yi = β0+ β1X1i + β2X2i +…+ βkXki + ri
The parameters βkj are considered fixed One for all and all for oneSame values for all i and j; the single level model
The ri ’s are random: ri ~ N(0, σ) and independent
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The Multilevel Model
Takes groups into account and explicitly models group effectsHow to conceptualize and model group level variation?How do groups vary on the model parameters?Fixed versus random effects
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Fixed vs. Random Effects
Fixed Effects represent discrete, purposefully selected or existing values of a variable or factor
Fixed effects exert constant impact on DVRandom variability only occurs as a within subjects effect (level 1)Can only generalize to particular values used
Random Effects represent more continuous or randomly sampled values of a variable or factor
Random effects exert variable impact on DVVariability occurs at level 1 and level 2Can study and model variabilityCan generalize to population of values
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Fixed vs. Random Effects?
Use fixed effects ifThe groups are regarded as unique entitiesIf group values are determined by researcher through design or manipulationSmall j (< 10); improves power
Use random effects ifGroups regarded as a sample from a larger populationResearcher wishes to test effects of group level variablesResearcher wishes to understand group level differencesSmall j (< 10); improves estimation
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Fixed Intercepts Model
The simplest HLM model is equivalent to a one-way ANOVA with fixed effects:
Yij = γ00 + rij
This model simply estimates the grand mean (γ00) and deviations from the grand mean (rij)Presented here simply to demonstrate control of fixed and random effects on all parameters
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Note the equation has no “u” residual term; this creates a fixed effect model
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ANOVA Model (random intercepts)
A simple HLM model with randomly varying interceptsEquivalent to a one-way ANOVA with random effects:
Yij = β0j + rijβ0j = γ00 + u0j
Yij = γ00 + u0j + rij
Note the addition of u0j allows different intercepts for each j unit, a random effects model
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ANOVA ModelIn addition to providing parameter estimates, the ANOVA model provides information about the presence of level 2 variance (the ICC) and whether there are significant differences between level 2 unitsThis model also called the Unconditional Model (because it is not “conditioned” by any predictors) and the “empty” modelOften used as a baseline model for comparison to more complex models
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Variables in HLM Models
Outcome variables Predictors
Control variablesExplanatory variables
Variables at higher levelsAggregated variables (Is n sufficient for representation?)Contextual variables
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Conditional Models: ANCOVA
Adding a predictor to the ANOVA model results in an ANCOVA model with random intercepts:
Note that the effect of X is constrained to be the same fixed effect for every j unit (homogeneity of regression slopes)
Yij = β0j + β1(X1) + rij
β0j = γ00 + u0j
β1 = γ10
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Conditional Models: Random Coefficients
An additional parameter results in random variation of the slopes:
Both intercepts and slopes now vary from group to group
Yij = β0j + β1(X1) + rij
β0j = γ00 + u0j
β1j = γ10 + u1j
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Standardized coefficients
Standardized coefficient at level 1:β0j (SDX / SDY)
Standardized coefficient at level 2:γ00 (SDX / SDY)
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Modeling variation at Level 2:Intercepts as Outcomes
Yij = β0j + β1jX1ij + rij
β0j = γ00 + γ0jWj + u0j
β1j = γ10 + u1j
Predictors (W’s) at level 2 are used to model variation in intercepts between the j units
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Modeling Variation at Level 2: Slopes as Outcomes
Yij = β0j + β1jX1ij + rij
β0j = γ00 + γ0jWj + u0j
β1j = γ10 + γ1jWj + u1j
Do slopes vary from one j unit to another?W’s can be used to predict variation in slopes as well
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Variance Components Analysis
VCA allows estimation of the size of random variance components
Important issue when unbalanced designs are usedIterative procedures must be used (usually ML estimation)
Allows significance testing of whether there is variation in the components across units
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Estimating Variance Components: Unconditional Model
= τ0 + σ2
Var(Yij) = Var(u0j) + Var(rij)
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Final estimation of variance components:---------------------------------------------------------------------------------------------------Random Effect Standard Variance df Chi-square P-value
Deviation Component---------------------------------------------------------------------------------------------------INTRCPT1, U0 14.38267 206.86106 14 457.32201 0.000level-1, R 32.58453 1061.75172---------------------------------------------------------------------------------------------------
Statistics for current covariance components model--------------------------------------------------Deviance = 21940.853702Number of estimated parameters = 2
HLM Output
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Variance explained
R2 at level 1 = 1 – (σ2
cond + τcond) / (σ2uncond + τuncond)
R2 at level 2 = 1 – [(σ2
cond / nh) + τcond] / [(σ2uncond / nh) + τuncond]
Where nh = the harmonic mean of n for the level 2 units (k / [1/n1 + 1/n2 +…1/nk])
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Comparing models
Deviance testsUnder “Other Settings” on HLM tool bar, choose “hypothesis testing”Enter deviance and number of parameters from baseline model
Variance explainedExamine reduction in unconditional model variance as predictors added, a simpler level 2 formula:
R2 = (τ baseline – τ conditional) / τ baseline
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Deviance Test Results
Statistics for current covariance components model------------------------------------------------------------Deviance = 21615.283709Number of estimated parameters = 2
Variance-Covariance components test------------------------------------------------------------Chi-square statistic = 325.56999Number of degrees of freedom = 0P-value = >.500
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Testing a Nested Sequence of HLM Models1. Test unconditional model2. Add level 1 predictors
Determine if there is variation across groups If not, fix parameterDecide whether to drop nonsignificant predictorsTest deviance, compute R2 if so desired
3. Add level 2 predictorsEvaluate for significanceTest deviance, compute R2 if so desired
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Example
Use the HSB MDM file previously created to practice running HLM models:
UnconditionalLevel 1 predictor fixed, then randomLevel 2 predictor
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Statistical Estimation in HLM Models
Estimation MethodsFMLRMLEmpirical Bayes estimation
Parameter estimation Coefficients and standard errorsVariance Components
Parameter reliabilityCenteringResidual files
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Estimation Methods: Maximum Likelihood Estimation (MLE) Methods
MLE estimates model parameters by estimating a set of population parameters that maximize a likelihood functionThe likelihood function provides the probabilities of observing the sample data given particular parameter estimatesMLE methods produce parameters that maximize the probability of finding the observed sample data
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Estimation Methods
Full: Simultaneously estimate the fixed effects and the variance components.
Goodness of fit statistics apply to the entire model
(both fixed and random effects)
Check on software default
Full: Simultaneously estimate the fixed effects and the variance components.
Goodness of fit statistics apply to the entire model
(both fixed and random effects)
Check on software default
Restricted: Sequentially estimates the fixed effects and then the variance components
Goodness of fit statistics (deviance tests) apply only to the random effects
RML only tests hypotheses about the VCs (and the models being compared must have identical fixed effects)
Restricted: Sequentially estimates the fixed effects and then the variance components
Goodness of fit statistics (deviance tests) apply only to the random effects
RML only tests hypotheses about the VCs (and the models being compared must have identical fixed effects)
RML – Restricted Maximum Likelihood, only the variance components are included in the likelihood function
RML – Restricted Maximum Likelihood, only the variance components are included in the likelihood functionFML – Full Maximum Likelihood, both the regression coefficients and the variance components are included in the likelihood function
FML – Full Maximum Likelihood, both the regression coefficients and the variance components are included in the likelihood function
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Estimation Methods
RML expected to lead to better estimates especially when j is smallFML has two advantages:
Computationally easierWith FML, overall chi-square tests both regression coefficients and variance components, with RML only variance components are testedTherefore if fixed portion of two models differ, must use FML for nested deviance tests
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Computational Algorithms
Several algorithms exist for existing HLM models:Expectation-Maximization (EM)Fisher scoringIterative Generalized Least Squares (IGLS)Restricted IGLS (RIGLS)
All are iterative search and evaluation procedures
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Model Estimation
Iterative estimation methods usually begin with a set of start valuesStart values are tentative values for the parameters in the model
Program begins with starting values (usually based on OLS regression at level 1)Resulting parameter estimates are used as initial values for estimating the HLM model
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Model EstimationStart values are used to solve model equations on first iterationThis solution is used to compute initial model fitNext iteration involves search for better parameter valuesNew values evaluated for fit, then a new set of parameter values triedWhen additional changes produce no appreciable improvement, iteration process terminates (convergence)Note that convergence and model fit are very different issues
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IntermissionIntermission
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Centering
No centering (common practice in single level regression)Centering around the group mean ( )Centering around the grand mean (M )A known population meanA specific meaningful time point
jX
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Centering: The Original Metric
Sensible when 0 is a meaningful point on the original scale of the predictor
For example, amount of training ranging from 0 to 14 daysDosage of a drug where 0 represents placebo or no treatment
Not sensible or interpretable in many other contexts, i.e. SAT scores (which range from 200 to 800)
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0 10 20 30 40 50
)0(0 == ijijj XYEβ
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Centering Around the Grand Mean
Predictors at level 1 (X’s) are expressed as deviations from the grand mean (M): (Xij – M)Intercept now expresses the expected outcome value (Y) for someone whose value on predictor X is the same as the grand mean on that predictorCentering is computationally more efficientIntercept represents the group mean adjusted for the grand mean - M
Variance of β0j = τ00, the variance among the level-2 unit means adjusted for the grand mean
jX
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0 10 20 30 40 50
) ( 000 γβ == ijijj XYE
Centering Around the Grand Mean
β01
β02
β03
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Centering Around the Group Mean
Individual scores are interpreted relative to their group mean The individual deviation scores are orthogonal to the group meansIntercept represents the unadjusted mean achievement for the groupUnbiased estimates of within-group effects May be necessary in random coefficient models if level 1 predictor affects the outcome at both level 1 and level 2Can control for unmeasured between group differencesBut can mask between group effects; interpretation is more complex
)( XXij −
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Centering Around the Group Mean
Level 1 results are relative to group membership
Intercept becomes the unadjusted mean for group j
Should include level 2 mean of level 1 variables to fully disentangle individual and compositional effects
Variance β0j is now the variance among the level 2 unit means
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0 10 20 30 40 50
)25 ( then 25, X If 11011 === ii XYEβ
Centering Around the Group Mean
)20 ( then 20, X If 22022 === ii XYEβ)18 ( then 18, X If 33033 === ii XYEβ
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Parameter estimation
Coefficients and standard errors estimated through maximum likelihood procedures (usually)
The ratio of the parameter to its standard error produces a Wald test evaluated through comparison to the normal distribution (z)In HLM software, a more conservative approach is used:
t-tests are used for significance testing t-tests more accurate for fixed effects, small n, and nonnormal distributions)
Standard errorsVariance components
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Parameter reliability
Analogous to score reliability: ratio of true score variance to total variance (true score + error)In HLM, ratio of true parameter variance to total variabilityFor example, in terms of intercepts, parameter reliability, λ, is:
)//()(/)( 2200
2000 jjjj nYVarVar σττβλ +==
Total variance of the sample means (observed)
True variance of the sample means (estimated)
Variance of error of the sample meansTrue variance of the
sample means (estimated)
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ICC ( ρI )
nj .05 .10 .20
5 .21 .36 .56
10 .34 .53 .71
20 .51 .69 .83
30 .61 .77 .88
50 .72 .85 .93
100 .84 .92 .96
)1(1
Ij
Ijj n
nρ
ρλ
−+=
Parameter reliability
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Parameter reliability
)1(1
Ij
Ijj n
nρ
ρλ
−+=
n per cluster
100503020105
Para
met
er R
elia
bilit
y1.0
.8
.6
.4
.2
0.0
ICC=.05
ICC=.10
ICC=.20
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Predicting Group Effects
It is often of interest to estimate the random group effects (β0j, β1j)This is accomplished using Empirical Bayes (EB) estimationThe basic idea of EB estimation is to predict group values using two kinds of information:
Group j dataPopulation data obtained from the estimation of the regression model
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Empirical Bayes
If information from only group j is used to estimate then we have the OLS estimate:
If information from only the population is used to estimate then the group is estimated from the grand mean:
j
N
j
j YNn
Y 1
..00 ∑=
==γ
jj Y=0β
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Empirical Bayes
A third possibility is to combine group level and population informationThe optimal combination is an average weighted the parameter reliability:
The results in the “posterior means” or EB estimates
0000 )1( γλβλβ jjjjEB −+=
The larger the reliability, the greater the weight of the group mean
The smaller the reliability, the greater the weight of the grand mean
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Bayesian EstimationUse of prior and posterior information improves estimation (depending on purpose)Estimates “shrink” toward the grand mean as shown in formulaAmount of shrinkage depends on the “badness” of the unit estimate
Low reliability results in greater shrinkage (if λ = 1, there is no shrinkage; if λ = 0, shrinkage is complete, γ00)Small n-size within a j unit results in greater shrinkage, “borrowing”from larger units
0000 )1( γλβλβ jjjjEB −+=
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OLS Intercept
201816141210864
EB In
terc
ept
20
18
16
14
12
10
8
6
4 Rsq = 0.9730
Reliability=.733
OLS Slope
86420-2-4
EB S
lope
4
3
2
1
0 Rsq = 0.3035
Reliability=.073
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High School
Achi
evem
ent
22
17
12
7
2
EB
OLS
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HLM Residual FilesImportant outcome information from an HLM analysis can be saved for each level of the analysis in a “residual file”
Residual files contain parameter estimates and other variables from the analysisResidual files can be save in statistical package format (SPSS, SAS, etc.)
Residual files can be used for diagnostic evaluation of statistical model assumptionsResidual files can be used to estimate and further describe or analyze effects among the units at each level
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Example: Creating Residual Files
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Three level modelsLevel-1 (p students)
Yijk = π0jk + π1jk(apijk)+ eijkLevel-2 (j classrooms)
π0jk = βp0k + βp1k(Xqjk) + rp0k
π1jk = βp1kj + βp1k(Xqjk) + rp1kLevel-3 (k schools)
βp0k = γpq0 + γpqs(Wsk) + upqk
βp1k = γpq1 + γpqs(Wsk) + upqk
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Partitioning variance in the three level modelProportion of variance within classrooms (individual
student differences) = σ2 / (σ2 + τπ + τβ )
Proportion of variance between classrooms within schools = τπ / (σ2 + τπ + τβ )
Proportion of variance between schools = τβ / (σ2 + τπ + τβ )
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Three level example
Example:Go to “Examples” folder and then “Chapter 4” in the HLM directoryOpen “EG1.sav”, “EG2.sav”, and “EG3.sav”
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Longitudinal modelsLevel 1 defined as repeated measurement occasionsLevels 2 and 3 defined as higher levels in the nested structureFor example, longitudinal analysis of student achievement
Level 1 = achievement scores at times 1 – tLevel 2 = student characteristicsLevel 3 = school characteristics
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Longitudinal models
Two important advantages of the MLM approach to repeated measures:
Times of measurement can vary from one person to anotherData do not need to be complete on all measurement occasions
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Longitudinal models
Level-1Ytij = π0ij + π1ij(time)+ etij
Level-2π0ij = β00j + β01j(Xij) + r0ij
π1ij = β10j + β11j(Xij) + r1ijLevel-3
β00j = γ000 + γ001(W1j) + u00j
β10j = γ100 + γ101(W1j) + u10j
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An id variable is needed to link the level 1 file to level 2
An id variable is also needed to link the level 2 file to level 3
A variable is also needed to indicate the time of occurrence of each measurement occasion
Each row represents a measurement occasion
The set of rows represent the repeated measures for one
participant
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Fitting a growth trajectory
0
2
4
6
8
10
12
14
16
9 10 11
Age
Ach
ieve
men
t
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104
600
620
640
660
680
700
720
740
760
GRADE 6 GRADE 7 GRADE 8
grade level
mat
hem
atic
s ac
hiev
emen
t
Linear growth for individual students
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630640650660670680690700710720730
GRADE6 GRADE7 GRADE8
Grade Level
Mea
n M
athe
mat
ics
Ach
ieve
men
t
Average linear growth by school
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Curvilinear Longitudinal modelsLevel-1
Ytij = π0ij + π1ij(time)+ π2ij(time2)+ etijLevel-2
π0ij = β00j + β01j(Xij) + r0ijπ1ij = β10j + β11j(Xij) + r1ijπ2ij = β20j + β21j(Xij) + r2ij
Level-3β00j = γ000 + γ001(W1j) + u00j
β10j = γ100 + γ101(W1j) + u10jβ20j = γ200 + γ201(W2j) + u20j
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Grade
9876
EB E
stim
ated
Ter
raN
ova
Scor
e800
700
600
500
Curvilinear growth for individual students
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Testing a Nested Sequence of HLM Longitudinal Models1. Test unconditional model2. Test Level 1 growth model3. After establishing the level 1 growth model, use it as the
baseline for succeeding model comparisons 4. Add level 2 predictors
Determine if there is variation across groups If not, fix parameterDecide whether to drop nonsignificant predictorsTest deviance, compute R2 if so desired
5. Add level 3 predictorsEvaluate for significanceTest deviance, compute R2 if so desired
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Regression Discontinuity and Interrupted Time Series Designs: Change in Intercept
ijijiijiiij TreatmentTimeY επππ +++= 210
ijijiiij TimeY εππ ++= 10
ijijiiiij TimeY επππ +++= 120 )(When Treatment = 1:
When Treatment = 0: Treatment is coded 0 or 1
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Regression Discontinuity and Interrupted Time Series Designs
Treatment effect on level: )( 20 ii ππ +
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Regression Discontinuity and Interrupted Time Series Designs: Change in Slope
ijijiijiiij imeTreatmentTTimeY επππ +++= 310
ijijiiij TimeY εππ ++= 10
When Treatment = 1:
When Treatment = 0:Treatment time expressed as 0’s
before treatment and time intervals post-treatment (i.e., 0, 0, 0, 1, 2, 3
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Regression Discontinuity and Interrupted Time Series Designs
Treatment effect on slope: )( 3iπ+
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Change in Intercept and Slope
ijijiiijiiij imeTreatmentTTreatmentTimeY εππππ ++++= 3210
ijijiiij TimeY εππ ++= 10
When Treatment = 0:
Effect of treatment on intercept
Effect of treatment on slope
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Regression Discontinuity and Interrupted Time Series Designs
Effect of treatment on intercept
Effect of treatment on slope
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Analyzing Randomized Trials (RCT) with HLM
When random assignment is accomplished at the participant level, treatment group is dummy coded and included in the participant level data fileWhen random assignment is accomplished at the cluster level, treatment group is dummy coded and included in the cluster level data file
Treatment can be used to predict intercepts or slopes as outcomesAnother strength of this approach is the ability to empirically model treatment variation across clusters (i.e., replication)
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Power in HLM Models
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Using the Optimal Design Software
The Optimal Design Software can also be used to estimate power in a variety of situationsThe particular strength of this software is its application to multilevel situations involving cluster randomization or multisite designsAvailable at:
http://sitemaker.umich.edu/group-based/optimal_design_software
Optimal Design
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Factors Affecting Power in CRCTSample Size
Number of participants per cluster (N)Number of clusters (J)
Effect SizeAlpha levelUnexplained/residual varianceDesign Effects
Intraclass correlation (ICC)Between vs. within cluster varianceTreatment variability across clustersRepeated measuresBlocking and matching
Statistical control
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Effect of Unexplained Variance on Power
Terminology: “error” versus unexplained or residualResidual variance reduces power
Anything that decreases residual variance, increases power (e.g., more homogeneous participants, additional explanatory variables, etc.)
Unreliability of measurement contributes to residual varianceTreatment infidelity contributes to residual varianceConsider factors that may contribute to residual between cluster variance
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Effect of Design Features on Statistical Power
Multicollinearity (and restriction of range)
Statistical model misspecificationLinearity, curvilinearity,…Omission of relevant variablesInclusion of irrelevant variables
)1( 212
21
212y
b 2.1y rxs
s−Σ
=
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The number of clusters has a stronger influence on power than the cluster size as ICC departs from 0
JnSE )/)1((4)ˆ( 01
ρργ −+=
The standard error of the main effect of treatment is:
As ρ increases, the effect of n decreasesIf clusters are variable (ρ is large), more power is gained by increasing the number of clusters sampled than by increasing n
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Power in Studies with a Small Number of Clusters
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Fixed vs. Random EffectsFixed Effects Model
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Effect of Effect Size Variability ( )
2δσ
nJn)/22 +(
=στγ
Variance of the treatment effect across clusters
2δσ
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Randomization as a Control TacticWhat does randomization accomplish?
Controls bias in assignment of treatment (works OK with small J)Turns confounding factors into randomly related effects (equivalence vs. randomness; does not work well with small J)
Applying an underpowered, small CRCT may not be sufficient to achieve rigor
Consider other design approaches (e.g., interrupted time series,regression discontinuity designs)Aggressively apply other tactics for experimental or statistical control
Not all designs are created equalNo one design is best (e.g., randomized trials)
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Improving Power Through Planned Design
Evaluate the validity of inferences for the planned designDesign to address most important potential study weaknessesRealistic appraisal of study purpose, context, and odds of successImportance of fostering better understanding of the factors influencing powerPlanning that tailors design to study context and setting
Strategies for cluster recruitmentPrevention of missing dataPlanning for use of realistic designs and use of other strategies like blocking, matching, and use of covariates
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Design For Statistical Power
Stronger treatments!Treatment fidelityBlocking and matchingRepeated measuresFocused tests (df = 1)Intraclass correlationStatistical control, use of covariatesRestriction of range (IV and DV)Measurement reliability and validity (IV and DV)
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Jaccard, J., Turrisi, R., & Wan, C. K. (1990). Interaction effects in multiple regression. Thousand Oaks: Sage.
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Pedhazur, E. J. (1997). Multiple regression in behavioral research (3rd Ed.). Orlando, FL: Harcourt Brace & Company.
Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials, Psychological Methods, 2(2), 173-185.
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ed.). Newbury Park: Sage. Raudenbush, S.W., Bryk, A.S., Cheong, Y.F., & Congdon, R.T. (2004). HLM 6: Hierarchical linear and
nonlinear modeling. Chicago, IL: Scientific Software International.Rumberger, R.W., & Palardy, G. J. (2004). Multilevel models for school effectiveness research. In D.
Kaplan (Ed.), The Sage handbook of quantitative methodology for the social sciences. Thousand Oaks,CA: Sage Publications.
Snijders, T. A. B., & Bosker, R. J. (1999). Multilevel analysis: An introduction to basic and advanced multilevel modeling. London: Sage.
Seltzer, M. (2004). The use of hierarchical models in analyzing data from experiments and quasi-experiments conducted in field settings. In D. Kaplan (Ed.), The Sage handbook of quantitativemethodology for the social sciences. Thousand Oaks, CA: Sage Publications.
Spybrook, J., Raudenbush, S., & Liu, X.-f. (2006). Optimal design for longitudinal and multilevel research: Documentation for the Optimal Design Software. New York: William T. Grant Foundation.
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