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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. 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

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  1. Hierarchical Linear Modeling David A. Hofmann Kenan-Flagler Business School University of North Carolina at Chapel Hill CARMA Webcast

  2. 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

  3. Why Multilevel Methods

  4. 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

  5. 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

  6. 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

  7. What is HLM

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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}

  13. What is HLM • Think about a simple example • Individual variables • Helping behavior (DV) • Individual Mood (IV) • Group variable • Proximity of group members

  14. 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

  15. 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

  16. 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

  17. How is HLM Different

  18. 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 

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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 

  25. Estimating Models in HLM

  26. 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

  27. 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)***

  28. 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

  29. 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)

  30. 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

  31. 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

  32. 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 )

  33. 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)

  34. 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)

  35. 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)

  36. Other Issues

  37. Other Issues • Assumptions • Statistical power • Centering level-1 predictors • Additional resources

  38. 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).

  39. 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

  40. 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 …

  41. 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: ___

  42. 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

  43. 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

  44. 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.]

  45. 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

  46. 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

  47. Centering Decisions • Grand Mean Centering • What do you see happening here … what can we conclude?

  48. Centering Decisions • Group Mean Centering • What do you see happening here … what can we conclude?

  49. 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?

  50. 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

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