1 hierarchical linear modeling and related methods david a. hofmann department of management...

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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

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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 StructuresHierarchical Data Structures

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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

Hierarchical Data StructuresHierarchical Data Structures

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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

Click to edit Master title styleClick to edit Master title styleTheoretical ParadigmsTheoretical Paradigms

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Some Substantive QuestionsSome 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.

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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

Statistical & Methodological OptionsStatistical & Methodological Options

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HLM OverviewHLM 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

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Regression lines estimated separately for each unit

Level 1

Level 2

Var. Intercepts = Modeled with between unit variables

Var. Slopes = Modeled with between unit variables

Yij

Xij

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Some Substantive Questions: Applications of HLMSome 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

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Some Preliminary definitions:

– Random coefficients/effects» Coefficients/effects that are assumed to vary across

units– Common Random coefficients/effects

Within unit interceptsWithin unit slopesLevel 2 residual

– Fixed effects» Effects that do not vary across units

– Common Fixed effectsLevel 2 interceptLevel 2 slope

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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)***

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A set of example hypotheses:

Answering them using HLM

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HLM: A Simple ExampleHLM: A Simple Example

Individual variables

– Helping behavior (DV)

– Individual Mood (IV)

Group variable

– Proximity of group members

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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

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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)

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HLM: Hypothesis TestingHLM: 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 )

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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)

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Intercepts-as-outcomes - model Level-2 intercept (Hyp. 2)– Add Proximity to intercept model


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)

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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:





Var ( rij ) = Level-1 residual variance

Var ( U0j ) = residual intercepts variance

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


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Statistical AssumptionsStatistical 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).

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Statistical PowerStatistical 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

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Centering Decisions:

Scaling of Level-1 Predictors

(It’s important and confusing)

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Centering DecisionsCentering 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

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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.]

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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 +

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Grand Mean Centering

Variables included in:

Level 1 Model Level 2 Model Parameter Est. Gj

Null Gj .445**

Aij Gj .209**

Aij Bij Gj .064

Aij Bij Cij Gj .007

Aij Bij Cij Dij Gj -.028

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Group Mean Centering



Null Gj .445**

Aij - Aj Gj .445**

Aij - Aj Bij - Bj Gj .445**

Aij - Aj Bij - Bj Cij - Cj Gj .445**

Aij - Aj Bij - Bj Cij - Cj Dij - Dj Gj .445**

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



Null Gj .445**

Aij - Aj Gj Aj .300*

Aij - Aj Bij - Bj Gj Aj Bj .132

Aij - Aj Bij - Bj Cij - Cj Gj Aj Bj Cj .119

Aij - Aj Bij - Bj Cij - Cj Dij - Dj Gj Aj Bj Cj Dj .099

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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

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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

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Model Cross-level Interaction Between-group interactionParameter t p Parameter t p

00

-1.86 -.27 ns 2.10 .63 ns

013.10 .91 ns 10.16 6.23 <.01

10

-.71 -.73 ns .01 .01 ns

Level-1: Yij = 0j + 1j(Xij - X..) + eij Level-2:

0j = 00 + 01G1j + u0j

1j = 10 + 11G1j + u1j

113.51 7.24 <.01 .72 3.80 <.01

00

-.22 -.03 ns .10 .07 ns

01.19 .10 ns .07 .19 ns

02

1.59 .44 ns .80 1.16 ns

03-.19 -.22 ns 3.77 22.50 <.01

10

-.52 -.49 ns -.20 -.51 ns

Level-1: Yij = 0j + 1j (Xij - Xj) + rij

Level-2: 0j = 00 + 01 Xj + 02 Gj + 03 ( XjGj ) + u

0j1j = 10 + 11Gj + u

1j

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4.52 8.43 <.01 .19 .95 ns

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Centering Decision: Theoretical ParadigmsCentering 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

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Centering Decisions: Theoretical ParadigmsCentering 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

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Hierarchical Linear Models:

Let’s take a look at the software

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HLM versus OLS regression

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HLM versus OLSHLM 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 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.

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HLM Estimation:

A brief overview

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HLM EstimationHLM 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

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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 )

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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.

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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


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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 )

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The Reported “Reliability”The Reported “Reliability”

ß^qj

ß^qj

ß^qj

ß^qj

ß^qj

ß^qj

ß^qj

ß^qj

ß^qj

ß^qjqq

vqqj

vqqj

vqqj

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)

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)

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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.

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HLM Estimation (random level-1 coefficients)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).

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Do You Really Need HLM?

Alternatives for Estimating Hierarchical Models

Part I: Cross Level Models

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SAS: Proc MixedSAS: 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)

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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)

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SAS: Proc MixedSAS: Proc MixedModel

00

01

10

11

2 00

11

One-way ANOVA HLM

L1: Helpingij = 0j + rij

L2: 0j = 00 + U0j Proc Mixed

31.39

31.39

--

--

--

--

--

--

31.76

31.76

99.82

99.81

--

--

Random coefficient regression

L1: Helpingij = 0j + 1j (Moodij) + rij HLM

L2: 0j = 00 + U0j


31.42

31.43

--

--

3.01

3.01

--

--

5.61

5.61

45.63

45.64

.13

.13

Intercepts-as-outcomes


L2: 0j = 00 + 01 (Proximityj ) + U0j


24.92

24.91

1.24

1.24

3.01

3.01

--

--

5.61

5.61

41.68

41.68

.13

.13

Slopes-as-outcomes


L2: 0j = 00 + 01 (Proximityj ) + U0j

L2: 1j = 10 + 11 (Proximityj ) + U1j Proc Mixed

25.14

25.14

1.19

1.19

2.06

2.06

.18

.18

5.61

5.61

42.95

42.94

.02

.02

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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/

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Do You Really Need HLM?

Alternatives for Estimating Hierarchical Models

Part II: Longitudinal Models

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Latent Growth Curve ModelsLatent 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

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Intercept Slope

Y1 Y5Y2 Y3 Y4

Int. Slope

1 0

1 1

1 2

1 3

So what is this doing?

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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

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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

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Intercept Slope

Y1 Y5Y2 Y3 Y4

Individual Predictor

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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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