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Hierarchical Linear Modeling and Related Methods David A. Hofmann Department of Management Michigan State University Expanded Tutorial SIOP Annual Meeting April 16, 2000. Hierarchical Data Structures. Hierarchical nature of organizational data Individuals nested in work groups

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slide1
Hierarchical Linear Modeling

and Related Methods

David A. Hofmann

Department of Management

Michigan State University

Expanded Tutorial

SIOP Annual Meeting

April 16, 2000

hierarchical data structures
Hierarchical Data Structures
  • Hierarchical nature of organizational data
    • Individuals nested in work groups
    • Work groups in departments
    • Departments in organizations
    • Organizations in environments
  • Consequently, we have constructs that describe:
    • Individuals
    • Work groups
    • Departments
    • Organizations
    • Environments
hierarchical data structures3
Hierarchical Data Structures
  • Hierarchical nature of longitudinal data
    • Time series nested within individuals
    • Individuals
    • Individuals nested in groups
  • Consequently, we have constructs that describe:
    • Individuals over time
    • Individuals
    • Work groups
theoretical paradigms
Theoretical Paradigms

Click to edit Master title style

  • Meso Paradigm (House et al., 1995; Tosi, 1992):
    • Micro OB
    • Macro OB
    • Call for shifting focus:
      • Contextual variables into Micro theories
      • Behavioral variables into Macro theories
  • Longitudinal Paradigm (Nesselroade, 1991):
    • Intraindividual change
    • Interindividual differences in individual change
some substantive questions
Some Substantive Questions
  • Kidwell et al., (1997), Journal of Management
    • Dependent: Organizational citizenship behavior
    • Individual: Job satisfaction and organizational commit.
    • Group: Work group cohesion
  • Deadrick et al., (1997), Journal of Management
    • Dependent: Employee performance
    • Within individual: Performance over time (24 weeks)
    • Between individual: Cognitive & psychomotor ability
  • Question

Given variables at different levels of analysis,

how do we go about investigating them.

statistical methodological options
Statistical & Methodological Options
  • Aggregate level
    • Discard potentially meaningful variance
    • Ecological fallacies, aggregation bias, etc.
  • Individual level
    • Violation of independence assumption
    • Complex error term not dealt with
    • Higher units tested based on # of lower units
  • Hierarchical linear models
    • Models variance at multiple levels
    • Addresses independence issues
    • Straightforward conceptualization of multilevel data
hlm overview
HLM Overview
  • Two-stage approach to multilevel modeling
    • Level 1: within unit relationships for each unit
    • Level 2: models variance in level-1 parameters (intercepts & slopes) with between unit variables

Level 1: Yij = ß0j + ß1j Xij + rij

Level 2: ß0j = 00 + 01 (Groupj ) + U0j

ß1j = 10 + 11 (Groupj ) + U1j

slide8
Yij

Regression lines estimated separately for each unit

Xij

Level 1

Level 2

Var. Intercepts = Modeled with between unit variables

Var. Slopes = Modeled with between unit variables

some substantive questions applications of hlm
Some Substantive Questions: Applications of HLM
  • Kidwell et al., (1997), Journal of Management
    • Individual level job satisfaction and organizational commitment positively related to OCB
    • Cohesion will be positively related to OCB exhibited by employees beyond that accounted for by satisfaction and commitment
    • The relationships between commitment/satisfaction and OCB will be stronger in more cohesive groups
  • Deadrick et al., (1997), Journal of Management
    • Are there inter-individual differences in performance over time
    • Do individual differences in ability account for these inter-individual differences
hlm overview10
HLM Overview

Some Preliminary definitions:

  • Random coefficients/effects
    • Coefficients/effects that are assumed to vary across units
      • Common Random coefficients/effects

Within unit intercepts

Within unit slopes

Level 2 residual

  • Fixed effects
    • Effects that do not vary across units
      • Common Fixed effects

Level 2 intercept

Level 2 slope

hlm overview11
HLM Overview
  • Estimates provided:
    • Level 1 parameters (intercepts, slopes)
    • Level-2 parameters (intercepts, slopes)**
    • Variance of Level-1 residuals
    • Variance of Level-2 residuals***
    • Covariance of Level-2 residuals
  • Statistical tests:
    • t-test for parameter estimates (Level-2, fixed effects)**
    • Chi-Square for variance components (Level-2, random effects)***
a set of example hypotheses answering them using hlm
A set of example hypotheses:

Answering them using HLM

hlm a simple example
HLM: A Simple Example
  • Individual variables
    • Helping behavior (DV)
    • Individual Mood (IV)
  • Group variable
    • Proximity of group members
hlm a simple example14
HLM: A Simple Example
  • Hypotheses

1. Mood is positively related to helping

2. Proximity is positively related to helping after controlling for mood

      • On average, individuals who work in closer proximity are more likely to help; a group level main effect for proximity after controlling for mood

3. Proximity moderates mood-helping relationship

      • The relationship between mood and helping behavior is stronger in situations where group members are in closer proximity to one another
hlm a simple example15
HLM: A Simple Example
  • Necessary conditions
    • Systematic within and between group variance in helping behavior
    • Mean level-1 slopes significantly different from zero (Hypothesis 1)
    • Significant variance in level-1 intercepts (Hypothesis 2)
    • Significant variance in level-1 slopes (Hypothesis 3)
    • Variance in intercepts significantly related to Proximity (Hypothesis 2)
    • Variance in slopes significantly related to Proximity (Hypothesis 3)
hlm hypothesis testing
HLM: Hypothesis Testing
  • One-way ANOVA - no Level-1 or Level-2 predictors (null)

Level 1: Helpingij = ß0j + rij

Level 2: ß0j = 00 + U0j

  • where:

ß0j = mean helping for group j

00 = grand mean helping

Var ( rij ) = 2 = within group variance in helping

Var ( U0j ) = between group variance in helping

Var (Helping ij ) = Var ( U0j + rij ) =  + 2

ICC =  / ( + 2 )

hlm hypothesis testing17
HLM: Hypothesis Testing
  • Random coefficient regression model
    • Add mood to Level-1 model ( no Level-2 predictors)

Level 1: Helpingij = ß0j + ß1j (Mood) + rij

Level 2: ß0j = 00 + U0j

ß1j = 10 + U1j

  • where:

00 = mean (pooled) intercepts (t-test)

10 = mean (pooled) slopes (t-test; Hypothesis 1)

Var ( rij ) = Level-1 residual variance (R 2, Hyp. 1)

Var ( U0j ) = variance in intercepts (related Hyp. 2)

Var (U1j ) = variance in slopes (related Hyp. 3)

hlm hypothesis testing18
HLM: Hypothesis Testing
  • Intercepts-as-outcomes - model Level-2 intercept (Hyp. 2)
    • Add Proximity to intercept model

Level 1: Helpingij = ß0j + ß1j (Mood) + rij

Level 2: ß0j = 00 + 01 (Proximity) + U0j

ß1j = 10 + U1j

  • where:

00 = Level-2 intercept (t-test)

01 = Level-2 slope (t-test; Hypothesis 2)

10 = mean (pooled) slopes (t-test; Hypothesis 1)

Var ( rij ) = Level-1 residual variance

Var ( U0j ) = residual inter. var (R2 - Hyp. 2)

Var (U1j ) = variance in slopes (related Hyp. 3)

hlm hypothesis testing19
HLM: Hypothesis Testing
  • Slopes-as-outcomes - model Level-2 slope (Hyp. 3)
    • Add Proximity to slope model

Level 1: Helpingij = ß0j + ß1j (Mood) + rij

Level 2: ß0j = 00 + 01 (Proximityj) + U0j

ß1j = 10 + 11 (Proximityj ) + U1j

  • where:

00 = Level-2 intercept (t-test)

01 = Level-2 slope (t-test; Hypothesis 2)

10 = Level-2 intercept (t-test)

11 = Level-2 slope (t-test; Hypothesis 3)

Var ( rij ) = Level-1 residual variance

Var ( U0j ) = residual intercepts variance

Var (U1j ) = residual slope var (R2 - Hyp. 3)

statistical assumptions
Statistical Assumptions
  • Linear models
  • Level-1 predictors are independent of the level-1 residuals
  • Level-2 random elements are multivariate normal, each with mean zero, and variance qq and covariance qq’
  • Level-2 predictors are independent of the level-2 residuals
  • Level-1 and level-2 errors are independent.
  • Each rij is independent and normally distributed with a mean of zero and variance 2 for every level-1 unit i within each level-2 unit j (i.e., constant variance in level-1 residuals across units).
statistical power
Statistical Power
  • Kreft (1996) summarized several studies
    • .90 power to detect cross-level interactions 30 groups of 30
    • Trade-off
      • Large number of groups, fewer individuals within
      • Small number of groups, more individuals per group
  • My experience
    • Cross-level main effects, pretty robust
    • Cross-level interactions more difficult
    • Related to within unit standard errors and between group variance
centering decisions scaling of level 1 predictors it s important and confusing
Centering Decisions:

Scaling of Level-1 Predictors

(It’s important and confusing)

centering decisions
Centering Decisions
  • Level-1 parameters are used as outcome variables at level-2
  • Thus, one needs to understand the meaning of these parameters
  • Intercept term: expected value of Y when X is zero
  • Slope term: expected increase in Y for a unit increase in X
  • Raw metric form: X equals zero might not be meaningful
centering decisions24
Centering Decisions
  • 3 Options
    • Raw metric
    • Grand mean
    • Group mean
  • Kreft et al. (1995): raw metric and grand mean equivalent, group mean non-equivalent
  • Raw metric/Grand mean centering
    • intercept var = adjusted between group variance in Y
  • Group mean centering
    • intercept var = between group variance in Y

[Kreft, I.G.G., de Leeuw, J., & Aiken, L.S. (1995). The effect of different forms of centering in Hierarchical Linear Models. Multivariate Behavioral Research, 30, 1-21.]

centering decisions25
Centering Decisions
  • An illustration:
    • 15 Groups / 10 Observations per
    • Within Group Variance: f (A, B, C, D)
    • Between Group Variable: Gj
      • G = f (Aj, Bj )
      • Thus, if between group variance in A & B (i.e., Aj & Bj ) is accounted for, Gj should not significantly predict the outcome
    • Run the model:
      • Grand Mean
      • Group Mean
      • Group +
centering decisions26
Centering Decisions
  • Grand Mean Centering
centering decisions27
Centering Decisions
  • Group Mean Centering
centering decisions28
Centering Decisions
  • Group Mean Centering with A, B, C, D Means in Level-2 Model
centering decisions29
Centering Decisions
  • Centering decisions are also important when investigating cross-level interactions
  • Consider the following model:

Level 1: Yij = ß0j + ß1j (Xgrand) + rij

Level 2: ß0j = 00 + U0j

ß1j = 10

  • Bryk & Raudenbush (1992) point out that 10 does not provide an unbiased estimate of the pooled within group slope
    • It actually represents a mixture of both the within and between group slope
    • Thus, you might not get an accurate picture of cross-level interactions
centering decisions30
Centering Decisions
  • Bryk & Raudenbush make the distinction between cross-level interactions and between-group interactions
    • Cross-level: Group level predictor of level-1 slopes
    • Group-level: Two group level predictors interacting to predict the level-2 intercept
  • Only group-mean centering enables the investigation of both types of interaction
  • Illustration
    • Created two data sets
      • Cross-level interaction, no between-group interaction
      • Between-group interaction, no cross-level interaction
centering decision theoretical paradigms
Centering Decision: Theoretical Paradigms
  • Incremental
    • group adds incremental prediction over and above individual variables
    • grand mean centering
    • group mean centering with means added in level-2 intercept model
  • Mediational
    • individual perceptions mediate relationship between contextual factors and individual outcomes
    • grand mean centering
    • group mean centering with means added in level-2 intercept model
centering decisions theoretical paradigms
Centering Decisions: Theoretical Paradigms
  • Moderational
    • group level variable moderates level-1 relationship
    • group mean centering provides clean estimate of within group slope
    • separates between group from cross-level interaction
    • Practical: If running grand mean centered, check final model group mean centered
  • Separate
    • group mean centering produces separate within and between group structural models
hierarchical linear models let s take a look at the software
Hierarchical Linear Models:

Let’s take a look at the software

hlm versus ols
HLM versus OLS
  • Investigate the following model using OLS:

Helpingij = ß0 + ß1 (Mood) + ß2 (Prox.) + rij

  • The HLM equivalent model (ß1j is fixed across groups):

Level 1: Helpingij = ß0j + ß1j (Mood) + rij

Level 2: ß0j = 00 + 01(Prox.) + U0j

ß1j = 10

  • Form single equation from two HLM equations:

Help = [00 + 01(Prox.) + U0j ] + [10 ] (Mood) + rij

= 00 + 10 (Mood) + 01(Prox.) + U0j + rij

= 00 + ß1j(Mood) + 01(Prox.) + [U0j + rij]

Independence Assump.

hlm estimation a brief overview
HLM Estimation:

A brief overview

hlm estimation
HLM Estimation
  • Types of effects estimated
    • Level-2 fixed effects / Level-1 random effects
    • Variance / covariance components
      • Estimated using maximum likelihood using EM algorithm
  • Purposes of HLM model
    • Inferences about level-2 effects
    • Estimating level-1 relationships for particular unit
    • Each purpose requires efficient estimates
      • level-2 effects => efficient estimates of level-2 regression coefficients
      • particular level-1 units => most efficient estimate of level-1 regression coefficients
hlm estimation fixed effects
HLM Estimation (fixed effects)
  • General level-1 model (matrix):

Yj = Xjßj + rj rj ~ N(0, 2 I )

  • The OLS estimator of ßj is given by:

ß^j = (Xj’Xj)-1 Xj’Yj

  • The dispersion, or variance in ß^j is given by:

Var(ß^j ) = Vj = 2 (Xj’Xj)-1

  • which means:

ß^j = ßj + ej ej ~ N (0, Vj )

hlm estimation fixed effects40
HLM Estimation (fixed effects)
  • General model at level-2:

ßj = Wj  + uj uj ~ N( 0, T )

  • Substituting the equations yields a single combined model:

ß^j = Wj  + uj + ej

  • where the dispersion of ß^j given Wj is

Var (ß^j ) = Var (uj + ej ) = T + Vj =  j

  • which equals parameter dispersion + error dispersion.
hlm estimation fixed effects41
HLM Estimation (fixed effects)
  • The Generalized Least Squares (GLS) estimator for is:

^ = (  Wj’  j-1 Wj )-1 Wj’  j-1ß^j

  • which is a standard OLS regression estimate except each group’s data are weighted by its precision matrix ( j-1).
  • The dispersion of ^ follows:

Var (^ ) = ( Wj’  j-1Wj )-1

the reported reliability
The Reported “Reliability”
  • The diagonal elements of T (e.g., qq ) and Vj (e.g., vqqj ) can be used to form a “reliability” index for each OLS level-1 coefficients:

reliability (ß^qj ) = qq / (qq + vqqj )

  • Because sampling variance (vqqj ) of ß^j will be different among the j units, each level-2 unit has a unique reliability index. The overall reliability can be summarized by computing the average reliability across j units:

reliability (ß^q ) = 1/jqq / (qq + vqqj )

the reported reliability43
The Reported “Reliability”

ß^qj

vqqj

ß^qj

vqqj

ß^qj

vqqj

Between group variance in

parameters is considered

systematic whereas the

variance around each estimate

is considered error. Thus, the

reliability equals the ratio of:

True variance / (True + error)

ß^qj

vqqj

ß^qj

qq

ß^qj

ß^qj

ß^qj

ß^qj

ß^qj

hlm estimation random level 1 coefficients
HLM Estimation (random level-1 coefficients)
  • Purpose: To obtain most efficient estimates of parameters for a particular level-1 unit.
    • Two estimates are available
      • the OLS estimate, ß^j
      • the predicted value from level-2, ß^^j = Wj ^

where ^ is the GLS estimate described previously

    • Obviously, if two estimates are available, the best estimate is likely to be some combination of these two estimates.
hlm estimation random level 1 coefficients45
HLM Estimation (random level-1 coefficients)
  • A composite level-1 estimate:

ß*j = j ß^j + ( I - j ) Wj ^

  • where j = T ( T + Vj )-1
  • which is the ratio of the parameter dispersion of ßj relative to the dispersion of ß^j (i.e., the ratio of “true” parameter variance over “observed” parameter variance).
  • Thus, the composite level-1 estimate is a weighted combination of the level-1 and level-2 estimate where each estimate is weighted proportional to its reliability. This is the most efficient estimate of the level-1 coefficient for any given unit (lowest mean square error; Raudenbush, 1988).
do you really need hlm alternatives for estimating hierarchical models part i cross level models
Do You Really Need HLM?

Alternatives for Estimating Hierarchical Models

Part I: Cross Level Models

sas proc mixed
SAS: Proc Mixed
  • SAS Proc Mixed will estimate these models
  • Key components of Proc Mixed command language
    • Proc mixed
      • Class
        • Group identifier
      • Model
        • Regression equation including both individual, group, and interactions (if applicable)
      • Random
        • Specification of random effects (those allowed to vary across groups)
sas proc mixed48
SAS: Proc Mixed
  • Key components of Proc Mixed command language
    • Some options you might want to select
      • Class: noitprint (suppresses interation history)
      • Model:
        • solution (prints solution for random effects)
        • ddfm=bw (specifies the “between/within” method for computing denominator degrees of freedom for tests of fixed effects)
      • Random:
        • sub= id (how level-1 units are divided into level-2 units)
        • type=un (specifies unstructured variance-covariance matrix of intercepts and slopes; i.e., allows parameters to be determined by data)
sas proc mixed50
SAS: Proc Mixed
  • Key references
    • Singer, J. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 23, 323-355.
  • Available on her homepage
    • http://hugse1.harvard.edu/~faculty/singer/
do you really need hlm alternatives for estimating hierarchical models part ii longitudinal models
Do You Really Need HLM?

Alternatives for Estimating Hierarchical Models

Part II: Longitudinal Models

latent growth curve models
Latent Growth Curve Models
  • Structural equation programs can be used to model
    • Interindividual differences in intraindividual change
    • Predictors of these change patterns
  • How does it work
    • Analyze covariance matrix of interrelationships among repeated measures of outcome
    • “Flip” the logic of factor analysis
      • Typically program estimates factor loadings and factors are interpreted in relation to factor loadings
      • In these models, you fix all of the factor loadings and interpret variance in factors in accordance to factor loadings that you specify
latent growth curve models53
Latent Growth Curve Models

Intercept

Slope

Int. Slope

1 0

1 1

1 2

1 3

Y1

Y2

Y3

Y4

Y5

So what is this doing?

latent growth curve models54
Latent Growth Curve Models
  • Estimating a set of regression equations

Yt =  Intercept +   Slope + t

  • which translates into

Y1

Y2

Y3

Y4

Y5

1

1

1

1

1

0

1

2

3

4

Intercept

Slope

=

+

latent growth curve models55
Latent Growth Curve Models
  • Variance in factors
    • Individual differences in intercepts and slopes
    • Why
      • In factors analysis, variance in factors equals variability across persons on latent construct
      • Could create a factor score for each individual; variability in factor scores conceptually represents variance in factor across persons
      • Same applies here
        • Fixing factor loadings defines factors as intercept and linear trend
        • Variance in factors represents variance across persons in intercepts and slopes
latent growth curve models56
Latent Growth Curve Models

Individual

Predictor

Intercept

Slope

Y1

Y2

Y3

Y4

Y5

latent growth curve models57
Latent Growth Curve Models
  • Key references
    • McArdle, J.J., & Epstein, D. (1987). Latent growth curves within developmental structural equation models. Child Development, 58, 110-133.
    • Muthen, B.O. (1991). Analysis of longitudinal data using latent variable models with varying parameters. In L.M. Collins, & J.L. Horn, (Eds.), Best Methods for the Analysis of Change (pp. 37-54).
    • Ployhart, R.E., & Hakel, M.D. (1998). The substantive nature of performance variability: Predicting interindividual differences in intraindividual change. Personnel Psychology, 51, 859-901.
    • Chan, D. (1998). The conceptualization and analysis of change over time: An integrative approach to incorporating longitudinal mean and covariance structures analysis (LMACS) and multiple indicator latent growth modeling (MLGM). Organizational Research Methods, 1, 421-483.
questions about today or about your own research
Questions:

About Today or

About Your Own Research

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