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Hierarchical Linear Modeling and Related Methods David A. Hofmann Department of Management Michigan State University Ex

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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
- Work groups in departments
- Departments in organizations
- Organizations in environments
- Consequently, we have constructs that describe:
- Individuals
- Work groups
- Departments
- Organizations
- Environments

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

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

- 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

- 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

- 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

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

- 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 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 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

HLM: A Simple Example

- Individual variables
- Helping behavior (DV)
- Individual Mood (IV)
- Group variable
- Proximity of group members

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 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

- 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 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 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 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

- 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

- 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

- 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 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 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 Decisions

- Grand Mean Centering

Centering Decisions

- Group Mean Centering

Centering Decisions

- Group Mean Centering with A, B, C, D Means in Level-2 Model

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 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

- 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

- 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

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

- 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)

- 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 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 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 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/jqq / (qq + vqqj )

The Reported “Reliability”qq

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

ß^qj

ß^qj

ß^qj

ß^qj

ß^qj

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 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).

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 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 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/

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 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 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 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.

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