Hierarchical Linear Modeling
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Hierarchical Linear Modeling David A. Hofmann Kenan-Flagler Business School University of North Carolina at Chapel Hill CARMA Webcast. Overview of Session. Structure of the Webcast Audience cannot ask questions Structured presentation around typical questions Nested questions throughout

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Hierarchical linear modeling david a hofmann kenan flagler business school

Hierarchical Linear Modeling

David A. Hofmann

Kenan-Flagler Business School

University of North Carolina at Chapel Hill

CARMA Webcast


Overview of session

Overview of Session

  • Structure of the Webcast

    • Audience cannot ask questions

    • Structured presentation around typical questions

    • Nested questions throughout

  • Overview

    • Why are multilevel methods critical for organizational research

    • What is HLM (conceptual level)

    • How does it differ from “traditional” methods of multilevel analysis

    • How do you do it (estimating models in HLM)

    • Critical decisions and other issues


Why multilevel methods

Why Multilevel Methods


Why multilevel methods1

Why Multilevel Methods

  • 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


Why multilevel methods2

Why Multilevel Methods

  • 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


Why multilevel methods3

Why Multilevel Methods

  • Meso Paradigm (House et al., 1995; Klein & Kozlowski, 2000; 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


What is hlm

What is HLM


What is hlm1

What is HLM

  • Hierarchical linear modeling

    • The name of a software package

    • Used as a description for broader class of models

      • Random coefficient models

      • Models designed for hierarchically nested data structures

  • Typical application

    • Hierarchically nested data structures

    • Outcome at lowest level

    • Independent variables at the lowest + higher levels


What is hlm2

What is HLM

  • I think I’ve got … but what might be some examples

    • Organizational climate predicting individual outcomes over and above individual factors

    • Organizational climate as a moderator of individual level processes

    • Individual characteristics (personality) predicting differences in change over time (e.g., learning)

    • Organizational structure moderating the relationship between individual characteristics

    • Industry characteristics moderating relationship between corporate strategy and performance


What is hlm3

What is HLM

  • Yes, but what IS it …

    • HLM models variance at two levels of analysis

    • At a conceptual level

      • Step 1

        • Estimates separate regression equations within units

        • This summarizes relationships within units (intercept and slopes)

      • Step 2

        • Uses these “summaries” of the within unit relationships as outcome variables regressing them on level-2 characteristics

    • Mathematically, not really a two step process, but this helps in understanding what is going on


What is hlm4

What is HLM

  • For those who like figures …

Yij

Level 1: Regression lines estimated separately for each unit

Xij

  • Level 2:

    • Variance in Intercepts predicted by between unit variables

    • Variance in slopes predicted by between unit variables


What is hlm5

What is HLM

  • For those who like equations …

  • 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

      { j subscript indicates parameters that vary across groups}


What is hlm6

What is HLM

  • Think about a simple example

    • Individual variables

      • Helping behavior (DV)

      • Individual Mood (IV)

    • Group variable

      • Proximity of group members


What is hlm7

What is HLM

  • 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


What is hlm8

What is HLM

Helping

High Proximity

Low Proximity

Mood

  • Overall, positive relationship between mood and helping (average regression line across all groups)

  • Overall, higher proximity groups have more helping that low proximity groups (average intercept green/solid vs. mean red/dotted line)

  • Average slope is steeper for high proximity vs. low proximity


What is hlm9

What is HLM

  • For those who like equations …

  • Here are the equations for this model

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

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

    ß1j = 10 + 11 (Proximityj) + U1j

ß0j

ß0j

=00 + 01 (Proximityj) + U0j

ß0j

ß0j

ß0j


How is hlm different

How is HLM Different


How is hlm different1

How is HLM Different

  • Ok, this all seems reasonable …

  • And clearly this seems different from “traditional” regression approaches

    • There are intercepts and slopes that vary across groups

    • There are level-1 and level-2 models

  • But, why is all this increased complexity necessary …

  • Good question … glad you asked 


How is hlm different2

How is HLM Different

  • “Traditional” regression analysis for our example

    • Compute average proximity score for the group

    • “Assign” this score down to everyone in the group


How is hlm different3

How is HLM Different

  • Then you would run OLS regression

  • Regress helping onto mood and proximity

  • Equation:

    Help = b0 + b1(Mood) + b2 (Prox) + b3 (Mood*Prox) + eij

  • Independence of eij component assumed

    • But, we know individuals are clustered into groups

    • Individuals within groups more similar than individuals in other groups

    • Violation of independence assumption


How is hlm different4

How is HLM Different

  • OLS regression equation (main effect):

    Helpingij = b0 + b1 (Mood) + b2 (Prox.) + eij

  • 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


How is hlm different5

How is HLM Different

  • Form single equation from HLM models

    • Simple algebra

    • Take the level-1 formula and replace ß0j and ß1j with the level-2 equations for these variables

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

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

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

      OLS = b0 + b1 (Mood) + b2 (Prox.) + eij

      Only difference

    • Instead of eij you have [U0j + rij]

    • No violation of independence, because different components are estimated instead of confounded


How is hlm different6

How is HLM Different

  • HLM

    • Models variance at multiple levels

    • Analytically variables remain at their theoretical level

    • Controls for (accounts for) complex error term implicit in nested data structures

    • Weighted least square estimates at Level-2


Estimating models in hlm

Estimating Models in HLM

  • Ok, I have a sense of what HLM is

  • I think I’m starting to understand that it is different that OLS regression … and more appropriate

  • So, how do I actually start modeling hypotheses in HLM

  • Good question, glad you asked 


Estimating models in hlm1

Estimating Models in HLM


Estimating models in hlm2

Estimating Models in HLM

Some Preliminary definitions:

  • Random coefficients/effects

    • Coefficients/effects that are assumed to vary across units

      • Within unit intercepts; within unit slopes; Level 2 residual

  • Fixed effects

    • Effects that do not vary across units

      • Level 2 intercept, Level 2 slope

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

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

        ß1j = 10


Estimating models in hlm3

Estimating Models in HLM

  • Estimates provided:

    • Level-2 parameters (intercepts, slopes)**

    • Variance of Level-2 residuals***

    • Level 1 parameters (intercepts, slopes)

    • Variance of Level-1 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)***


Estimating models in hlm4

Estimating Models in HLM

  • Hypotheses for our simple example

    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


Estimating models in hlm5

Estimating Models in HLM

  • 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)

    • Variance in intercepts significantly related to Proximity (Hypothesis 2)

    • Significant variance in level-1 slopes (Hypothesis 3)

    • Variance in slopes significantly related to Proximity (Hypothesis 3)


Estimating models in hlm6

Estimating Models in HLM

Helping

High Proximity

Low Proximity

Mood

  • Overall, positive relationship between mood and helping (average regression line across all groups)

  • Overall, higher proximity groups have more helping that low proximity groups (average intercept green/solid vs. mean red/dotted line)

  • Average slope is steeper for high proximity vs. low proximity


Estimating models in hlm7

Estimating Models in HLM

  • Pop quiz

    • What do you get if you regress a variable onto a vector of 1’s and nothing else; equation:

      variable = b1(1s) + e

    • b-weight associated with 1’s equals mean of our variable

    • This is what regression programs do to model the intercept

  • How much variance can 1’s account for

    • Zero

    • All variance forced to residual


Estimating models in hlm8

Estimating Models in HLM

  • 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 )


Estimating models in hlm9

Estimating Models in HLM

  • Random coefficient regression model

    • Add mood to Level-1 model ( no Level-2 predictors)

      Level 1:Helpingij = ß0j + ß1j (Moodij) + 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 ) = 2 = Level-1 residual variance (R 2, Hyp. 1)

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

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

R2 = [σ2 owa - σ2 rrm] / σ2 owa

R2 = [(σ2 owa +  owa) – (σ2 rrm +  rrm)] / [σ2 owa +  owa)


Estimating models in hlm10

Estimating Models in HLM

  • Intercepts-as-outcomes - model Level-2 intercept (Hyp. 2)

    • Add Proximity to intercept model

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

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

    R2 = [  rrm - intercept ] / [ rrm ]

    R2 = [(σ2 rrm +  rrm) – (σ2 inter +  inter)] / [σ2 rrm +  rrm)


Estimating models in hlm11

Estimating Models in HLM

  • Slopes-as-outcomes - model Level-2 slope (Hyp. 3)

    • Add Proximity to slope model

      Level 1:Helpingij = ß0j + ß1j (Moodij) + 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)


Other issues

Other Issues


Other issues1

Other Issues

  • Assumptions

  • Statistical power

  • Centering level-1 predictors

  • Additional resources


Other issues2

Other Issues

  • 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).


Other issues3

Other Issues

  • 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


Other issues4

Other Issues

  • Picture this scenario

    • Fall of 1990

    • Just got in the mail the DOS version of HLM

    • Grad student computer lab at Penn State

    • Finally, get some multilevel data entered in the software and are ready to go

    • Then, we are confronted with …


Other issues5

Other Issues

  • Select your level-1 predictor(s): 1

    • 1 for (job satisfaction)

    • 2 for (pay satisfaction)

  • How would you like to center your level-1 predictor Job Satisfaction?

  • 1 for Raw Score

  • 2 for Group Mean Centering

  • 3 for Grand Mean Centering

  • Please indicate your centering choice: ___


  • Other issues6

    Other Issues

    • HLM forces to you to make a choice in how to center your level-1 predictors

    • This is a critical decision

      • The wrong choice can result in you testing theoretical models that are inconsistent with your hypotheses

      • Incorrect centering choices can also result in spurious cross-level moderators

        • The results indicate a level-2 variable predicting a level-1 slope

        • But, this is not really what is going on


    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 decisions1

    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 decisions2

    Centering Decisions

    • Bottom line

      • Grand mean centering and/or raw metric estimate incremental models

        • Controls for variance in level-1 variables prior to assessing level-2 variables

      • Group mean centering

        • Does NOT estimate incremental models

          • Does not control for level-1 variance before assessing level-1 variables

          • Separately estimates with group regression and between group regression


    Centering decisions3

    Centering Decisions

    • An illustration from Hofmann & Gavin (1998):

      • 15 Groups / 10 Observations per group

      • Individual variables: 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 + mean at level-2


    Centering decisions4

    Centering Decisions

    • Grand Mean Centering

    • What do you see happening here … what can we conclude?


    Centering decisions5

    Centering Decisions

    • Group Mean Centering

    • What do you see happening here … what can we conclude?


    Centering decisions6

    Centering Decisions

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

    • What do you see happening here … what can we conclude?


    Centering decisions7

    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

    • The ß1j 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 decisions8

    Centering Decisions

    • Let’s draw a 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

    • Exploration of centering implications for multilevel research (Hofmann & Gavin, 1998)

      • Show how you can get spurious cross-level findings

      • Discuss centering choices in the context of different multilevel paradigms


    Centering decision

    Centering Decision

    • 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 decisions9

    Centering Decisions

    • 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


    Do you really need hlm alternatives for estimating hierarchical models

    Do You Really Need HLM?

    Alternatives for Estimating Hierarchical 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 mixed1

    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 mixed2

    SAS: Proc Mixed

    proc means;

    run;

    data; set;

    {grand mean center mood}

    moodgrd = mood-5.8388700;

    data; set;

    proc mixed noitprint;

    class id;

    model helping = / solution;

    random intercept / sub=id;

    proc mixed noitprint;

    class id;

    model helping = moodgrd/ solution ddfm=bw ;

    random intercept moodgrd/

    sub=id type=un;

    proc mixed noitprint;

    class id;

    model helping = moodgrd proxim / solution ddfm=bw ;

    random intercept moodgrd / sub=id type=un;

    proc mixed noitprint;

    class id;

    model helping = moodgrd proxim moodgrd*proxim / solution ddfm=bw ;

    random intercept moodgrd / sub=id type=un;

    run;


    Sas proc mixed3

    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/


    Resources

    Resources

    • http://www.unc.edu/~dhofmann/hlm.html

      • PowerPoint slides

      • Annotated output

      • Raw data + system files

      • Link to download student version of HLM

      • Follows chapter:

        • Hofmann, D.A., Griffin, M.A., & Gavin, M.B. (2000). The application of hierarchical linear modeling to management research. In K.J. Klein, & S.W.J. Kozlowski, (Eds.), Multilevel theory, research, and methods in organizations: Foundations, extensions, and new directions. Jossey-Bass, Inc. Publishers.

      • Proximity = Group Cohesion

    • Also: http://www.ssicentral.com/hlm/hlmref.htm


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