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Hierarchical Linear Modeling David A. Hofmann Kenan-Flagler Business School

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Hierarchical Linear Modeling

David A. Hofmann

Kenan-Flagler Business School

University of North Carolina at Chapel Hill

CARMA Webcast

- 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

- 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 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

- 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

- 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

- 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

- 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

- Step 1
- Mathematically, not really a two step process, but this helps in understanding what is going on

- 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

- 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}

- Think about a simple example
- Individual variables
- Helping behavior (DV)
- Individual Mood (IV)

- Group variable
- Proximity of group members

- Individual variables

- 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

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

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

- 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

- 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

- “Traditional” regression analysis for our example
- Compute average proximity score for the group
- “Assign” this score down to everyone in the group

- 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

- 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

- 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

- 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

- 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 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

- Coefficients/effects that are assumed to vary across units
- 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

- Level 2 intercept, Level 2 slope

- Effects that do not vary across units

- 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)***

- 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

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

- 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)

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

- 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

- What do you get if you regress a variable onto a vector of 1’s and nothing else; equation:
- How much variance can 1’s account for
- Zero
- All variance forced to residual

- 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 )

- 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

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

- 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

- Add Proximity to intercept model
- 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)

- 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

- Add Proximity to slope model
- 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

- Assumptions
- Statistical power
- Centering level-1 predictors
- Additional resources

- 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

- 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 …

- Select your level-1 predictor(s): 1
- 1 for (job satisfaction)
- 2 for (pay satisfaction)

- 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

- 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

- 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.]

- intercept var = between group variance in Y

- 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

- Does NOT estimate incremental models

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

- 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

- Grand Mean Centering
- What do you see happening here … what can we conclude?

- Group Mean Centering
- What do you see happening here … what can we conclude?

- Group Mean Centering with A, B, C, D Means in Level-2 Model
- What do you see happening here … what can we conclude?

- 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

- 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

- 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

- 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

- 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)

- Class

- 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)

- Some options you might want to select

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;

- 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/

- 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