Joop HoxUtrecht University

j.hox@uu.nlhttp://www.joophox.net

Introduction Multilevel AnalysisIntroduction Multilevel Analysis

Rens van de Schoota.g.j.vandeschoot@uu.nl / rensvandeschoot.wordpress.com

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Multilevel Regression ModelMultilevel Regression Model

Known in literature under a variety of names Hierarchical linear model (HLM) Random coefficient model Variance component model Multilevel model Contextual analysis Mixed Linear Model

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Hierarchical Data StructureHierarchical Data Structure

Three level data structure Groups at different levels may have different sizes Response (outcome) variable at lowest level Explanatory variables at all levels

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Examples? Examples?

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Traditional Approaches

Disaggregate all variables to the lowest level Do standard analyses (anova, multiple regression)

Aggregate all variables to the highest level Do standard analyses (anova, multiple regression)

Ancova with groups as factor

Some improvements: explanatory variables as deviations from their group mean have both

deviation score and disaggregated group mean as predictor (separates individual and group effects)

Why not? What is wrong with this?

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Problems With Standard Analysisof Hierarchical Data

Multiple Regression assumes independent observations independent error terms equal variances of errors for all observations (assumption of homoscedastic errors) normal distribution for errors

With hierarchical data observations are not independent errors are not independent different observations may have errors with different variances

(heteroscedastic errors)

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Problems With Standard Analysis Problems With Standard Analysis of Hierarchical Dataof Hierarchical Data

Observations in the same group are generally not independent they tend to be more similar than observations from different

groups selection, shared history, contextual group effects

The degree of similarity is indicated by the intraclass correlation rho:

Standard statistical tests are not at all robust against violation of the independence assumption

That is why we need special multilevel techniques!

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Sample size?Sample size?

Hox, J., van de Schoot. R., & Matthijsse, S. (2012). How few countries will do? Comparative survey analysis from a Bayesian perspective. Survey Research Methods, Vol.6, No.2, pp. 87-93

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Research questions I/IIIResearch questions I/III

Questions with respect to variables at the lowest level Intelligence (IQ) as predictor of school achievement (SA)

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Research questions II/IIIResearch questions II/III

Questions with respect to the influence of variables at a higher level on the dependent variable on the lowest level

Mean intelligence of a class (MIQ) as predictor of school achievement (SA); (control for individual IQ)

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Research questions III/IIIResearch questions III/III

Questions with respect to the interaction of variables on different levels (moderation effect)

The relation between intelligence and school achievement is not the same in all classes

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Graphical Picture of SimpleGraphical Picture of SimpleTwo-level Regression ModelTwo-level Regression Model

Outcome variable on pupil level Explanatory variables at both levels: individual & group Residual error at individual level Plus residual error at school level

school size

pupil sex grade

error

error

School levelSchool level

Pupil levelPupil level

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Regression analysisRegression analysis

In ordinary regression, with one explanatory variable X:

Yi= 0+ 1Xi+ ei

0 intercept,

1 regression slope,

ei residual error term

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Regression analysisRegression analysis

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Building the Multilevel Regression Building the Multilevel Regression Model: Random intercept modelModel: Random intercept model

In multilevel regression, at the lowest level: Yij= 0j+ 1jXij+ eij

0j intercept,

1j regression slope,

eij residual error term

subscript i for individuals, j for groups each group has its own intercept coefficient 0j

and its own slope coefficient 1j

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Building the Multilevel Regression Building the Multilevel Regression Model: Intercept only modelModel: Intercept only model

In multilevel regression, at the lowest level: Yij= 0j+ eij

Random intercept model: 0j= 00+ u0j

00 is the intercept of 0j u0j is the residual error term in the equation for 0j

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Building the Multilevel Regression Building the Multilevel Regression Model: Random intercept modelModel: Random intercept model

In multilevel regression, at the lowest level: Yij= 0j+ 1jXij+ eij

Random intercept model: 0j= 00+ u0j

00 is the intercept of 0j u0j is the residual error term in the equation for 0j

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Building the Multilevel Regression Building the Multilevel Regression Model: Random intercept modelModel: Random intercept model

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Building the Multilevel Regression Building the Multilevel Regression Model: Intercept only modelModel: Intercept only model

Yij= 0j+ 1jXij+ eij

Random intercept model: 0j= 00+ u0j

00 is the intercept of 0j u0j is the residual error term in the equation for 0j

Random slope model: 1j= 10+ u1j

10 is the intercept of ß1j u1j is the residual error term in the equation for 1j

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Difference with the usual Difference with the usual regression model:regression model:

Each class has a different intercept coefficient b0j and a different slope coefficient b1j

Since the intercept and the slope coefficients vary across the classes: random coefficients

=> Random intercept model & random slope model

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Building the Multilevel Regression Building the Multilevel Regression Model: Random slope modelModel: Random slope model

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BuildingBuilding the Multilevel Regression the Multilevel Regression Model: the Second (Group) LevelModel: the Second (Group) Level

Next step: explain the variation of the regression coefficients b0j

and b1j by introducing explanatory variables at the class level

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Building the Multilevel Regression Building the Multilevel Regression Model: the Second (Group) LevelModel: the Second (Group) Level

At the lowest (individual) level we have Yij= 0j+ 1jXij+ eij

0j= 00+ 01Zj+ u0j 00 and 01 are the intercept and slope to predict 0j from Zj

u0j is the residual error term in the equation for 0j

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Building the Multilevel Regression Building the Multilevel Regression Model: Cross level interactionModel: Cross level interaction

At the lowest (individual) level we have Yij= 0j+ 1jXij+ eij

0j= 00+ 01Zj+ u0j 00 and 01 are the intercept and slope to predict 0j from Zj

u0j is the residual error term in the equation for 0j

1j= 10+ 11Zj+ u1j 10 and 11 are the intercept and slope to predict ß1j from Zj

u1j is the residual error term in the equation for 1j

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Building the Multilevel Regression Building the Multilevel Regression Model: Single Equation VersionModel: Single Equation Version

At the lowest (individual) level we have Yij= 0j+ 1jXij+ eijand at the second (group) level 0j= 00+ 01Zj+ u0j 1j= 10+ 11Zj+ u1j

Combining (substitution and rearranging terms) gives Yij= 00+ 10Xij+ 01Zj+ 11ZjXij+ u1jXij+ u0j+ eij

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Building the Multilevel Regression Building the Multilevel Regression Model: Single Equation VersionModel: Single Equation Version

Yij= [00+ 10Xij+ 01Zj+ 11ZjXij] + [u1jXij+ u0j+ eij]

This equation has two distinct parts [00+ 10Xij+ 01Zj+ 11ZjXij] contains all the fixed coefficients,

it is called the fixed part of the model

[u1jXij+ u0j+ eij] contains all the random error terms, it is called the random part of the model

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Building the Multilevel Regression Building the Multilevel Regression Model: InterpretationModel: Interpretation

Yij = [00+ 10Xij+ 01Zj+ 11ZjXij] + [u1jXij+ u0j+ eij]

Several error variances e

2 variance of the lowest level errors eij

2u0 variance of the highest level errors u0j

2u1 variance of the highest level errors u1j

u01 covariance of u0j and u1j

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Full Multilevel Regression ModelFull Multilevel Regression Model

Explanatory variables at all levels

Higher level variables predict variation of lowest level intercept and slopes

Predicting the intercept implies a direct effect

Predicting slopes implies cross-level interactions

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Model ExplorationModel Exploration

1 Intercept-only model calculate intraclass correlation

2 Fixed model, 1st level predictor variables test individual slopes for significance

3 Model intercept by 2nd level predictor variables test for significance, how much intercept variance

explained?

4 Random coefficient model test if any 1st level slope has a significant variance

component (this is best done one-by-one)

5 Model random slopes by higher level variables: cross level interactions test for significance, how much slope variance is explained?

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Example: Popularity in SchoolsExample: Popularity in Schools

Outcome: popularity rating 100 classes, 2000 pupils Explanatory variables

Pupil level: sex (0=boy, 1=girl) Class level: teacher experience (in years)

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Graphical Picture of SimpleGraphical Picture of SimpleTwo-level Regression ModelTwo-level Regression Model

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Popularity Example:Popularity Example:Intercept-only ModelIntercept-only Model

Popularityij = 00+ u0j+ eij

Estimates (st. err.) 00 = 5.31 (.10) (This is just the overall average popularity)

e2 = 0.64 (.02)

2u0 = 0.88 (.13)

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Popularity Example:Popularity Example:Fixed ModelFixed Model

Popularityij = 00 + 10sexij + u0j + eij

Estimates (st. err.) 00 = 4.89 (.10),

10 = 0.84 (.03)

e2 = 0.46 (.02)

2u0 = 0.85 (.12)

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Popularity Example:Popularity Example:Fixed Model + Higher Level VariableFixed Model + Higher Level Variable

Popularityij = 00 + 10 sexij + 01 t.exp.j + u0j + eij

Estimates (st. err.) 00 = 3.56 (.17),

10 = 0.84 (.03),

01 = 0.09 (.01)

e2 = 0.46 (.02)

2u0 = 0.48 (.07)

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Popularity Example:Popularity Example:Random Coefficient ModelRandom Coefficient Model

Popularityij =

00 + 10 sexij + 01 t.exp.j + u0j + u1j sexij + eij

Estimates (st. err.) 00 = 3.34 (.16), 10 = 0.84 (.06), 01 = 0.11 (.01)

e2 = 0.39 (.01)

2u0 = 0.41 (.06)

u01 = 0.02 (.04) (covariance between intercept and slope)

2u1 = 0.27 (.05)

Slope variation for sex

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Popularity Example:Popularity Example:Random Coefficient Model + InteractionRandom Coefficient Model + Interaction

Popularityij = 00 + 10 sexij + 01 t.exp.j

+ 11 sexij t.exp.j + u0j + u1j sexij + eij

Estimates (st. err.) 00 = 3.31 (.16), 10 = 1.33 (.13), 01 = 0.11 (.01),

11 = -0.03 (.01)

e2 = 0.39 (.01)

2u0 = 0.40 (.06)

u01 = 0.02 (.04)

2u1 = 0.22 (.04)

Smaller, but still significant slope variation for sex

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5-day course Multilevel Analyses in Mplus 21-25 jan. 2013 http://www.uu.nl/faculty/socialsciences/NL/organisatie/graduateschool/promoveren/onderwijs%20voor

%20promovendi/courseoffering/Pages/Multilevel-Analyses-using-Mplus.aspx

The 9th International  Multilevel Conference is on March 27-28 (2013). http://multilevel.fss.uu.nl/

Prior to the conference (26th of March) a one-day course is taught by prof. Stef van Buuren on Mutiple Imputation of Multilevel missing data in MICE.

5th Mplus users meeting will be organized, 25th of March http://mplus.fss.uu.nl