1. 1 Hierarchical Linear Modeling David A. Hofmann Kenan-Flagler Business School University of North Carolina at Chapel Hill Academy of Management August, 2007

2. 2 Overview of Session 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. 3 Why Multilevel Methods

4. 4 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 Methods

5. 5 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 Methods

6. 6 Meso Paradigm –Micro OB –Macro OB –Call for shifting focus: Contextual variables into Micro theories Behavioral variables into Macro theories Longitudinal Paradigm –Intraindividual change –Interindividual differences in individual change Why Multilevel Methods

7. 7 What is HLM

8. 8 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. 9 What is HLM 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. 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. 11 What is HLM Level 1: Regression lines estimated separately for each unit Level 2: Variance in Intercepts predicted by between unit variables Variance in slopes predicted by between unit variables Y ij X ij

12. 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:Y ij = ß 0j + ß 1j X ij + r ij Level 2:ß 0j =  00 +  01 (Group j ) + U 0j ß 1j =  10 +  11 (Group j ) + U 1j { j subscript indicates parameters that vary across groups}

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

14. 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. 15 What is HLM 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 Helping Mood High Proximity Low Proximity

16. 16 What is HLM For those who like equations … Here are the equations for this model Level 1: Helping ij = ß 0j + ß 1j (Mood ij ) + r ij Level 2: ß 0j =  00 +  01 (Proximity j ) + U 0j ß 1j =  10 +  11 (Proximity j ) + U 1j ß 0j

17. 17 How is HLM Different

18. 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 …

19. 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 IDGrpHelpMoodProx 11555 21345 31435 41455 522110 6233 7233 8224

20. 20 How is HLM Different Then you would run OLS regression Regress helping onto mood and proximity Equation: Help = b 0 + b 1 (Mood) + b 2 (Prox) + b 3 (Mood*Prox) + e ij Independence of e ij 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. 21 How is HLM Different OLS regression equation (main effect): Helping ij = b 0 + b 1 (Mood) + b 2 (Prox.) + e ij The HLM equivalent model (ß 1j is fixed across groups): Level 1:Helping ij = ß 0j + ß 1j (Mood) + r ij Level 2:ß 0j =  00 +  01 (Prox.) + U 0j ß 1j =  10

22. 22 How is HLM Different Form single equation from HLM models –Simple algebra –Take the level-1 formula and replace ß 0j and ß 1j with the level-2 equations for these variables Help = [  00 +  01 (Prox.) + U 0j ] + [  10 ] (Mood) + r ij =  00 +  10 (Mood) +  01 (Prox.) + U 0j + r ij =  00 + ß 1j (Mood) +  01 (Prox.) + [U 0j + r ij ] OLS = b 0 + b 1 (Mood) + b 2 (Prox.) + e ij Only difference –Instead of e ij you have [U 0j + r ij ] –No violation of independence, because different components are estimated instead of confounded

23. 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. 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

25. 25 Estimating Models in HLM

26. 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:Helping ij = ß 0j + ß 1j (Mood) + r ij Level 2:ß 0j =  00 +  01 (Prox.) + U 0j ß 1j =  10

27. 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. 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. 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. 30 Estimating Models in HLM 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 Helping Mood High Proximity Low Proximity

31. 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 = b 1 (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 Variable1s 51 41 51 31 21 41 51 Mean = 4.0

32. 32 Estimating Models in HLM One-way ANOVA - no Level-1 or Level-2 predictors (null) Level 1:Helping ij = ß 0j + r ij Level 2: ß 0j =  00 + U 0j where: ß 0j = mean helping for group j  00 = grand mean helping Var ( r ij ) =  2 = within group variance in helping Var ( U 0j ) =    between group variance in helping Var (Helping ij ) = Var ( U 0j + r ij ) =   +  2 ICC =   / (   +  2 )

33. 33 Estimating Models in HLM Random coefficient regression model –Add mood to Level-1 model ( no Level-2 predictors) Level 1:Helping ij = ß 0j + ß 1j (Mood ij ) + r ij Level 2:ß 0j =  00 + U 0j ß 1j =  10 + U 1j where:  00 = mean (pooled) intercepts (t-test)  10 = mean (pooled) slopes (t-test; Hypothesis 1) Var ( r ij ) =  2 = Level-1 residual variance (R 2, Hyp. 1) Var ( U 0j ) =    variance in intercepts (related Hyp. 2) Var (U 1j ) = variance in slopes (related Hyp. 3) R 2 = [σ 2 owa - σ 2 rrm] / σ 2 owa] R 2 = [(σ 2 owa +   owa) – (σ 2 rrm +   rrm)] / [σ 2 owa +   owa)]

34. 34 Estimating Models in HLM Intercepts-as-outcomes - model Level-2 intercept (Hyp. 2) –Add Proximity to intercept model Level 1:Helping ij = ß 0j + ß 1j (Mood ij ) + r ij Level 2:ß 0j =  00 +  01 (Proximity j ) + U 0j ß 1j =  10 + U 1j where:  00 = Level-2 intercept (t-test)  01 = Level-2 slope (t-test; Hypothesis 2)  10 = mean (pooled) slopes (t-test; Hypothesis 1) Var ( r ij ) = Level-1 residual variance Var ( U 0j ) =   = residual inter. var (R 2 - Hyp. 2) Var (U 1j ) = variance in slopes (related Hyp. 3) R 2 = [   rrm -   intercept ] / [   rrm ] R 2 = [(σ 2 rrm +   rrm) – (σ 2 inter +   inter)] / [σ 2 rrm +   rrm)

35. 35 Slopes-as-outcomes - model Level-2 slope (Hyp. 3) –Add Proximity to slope model Level 1:Helping ij = ß 0j + ß 1j (Mood ij ) + r ij Level 2: ß 0j =  00 +  01 (Proximity j ) + U 0j ß 1j =  10 +  11 (Proximity j ) + U 1j 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 ( r ij ) = Level-1 residual variance Var ( U 0j ) = residual intercepts variance Var (U 1j ) = residual slope var (R 2 - Hyp. 3) Estimating Models in HLM

36. 36 Other Issues

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

38. 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 r ij 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. 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 More recent paper (ORM) –30/30 still somewhat holds –Factor in cost of sample level-1 and level-2 –Additional formulas for computing power

40. 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. 41 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. 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. 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. 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. 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. 46 Centering Decisions An illustration from Hofmann & Gavin (1998): –15 Groups / 10 Observations per group –Individual variables: A, B, C, D –Between Group Variable: G j G = f (A j, B j ) Thus, if between group variance in A & B (i.e., A j & B j ) is accounted for, G j should not significantly predict the outcome –Run the model: Grand Mean Group Mean Group + mean at level-2

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

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

49. 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. 50 Centering Decisions Centering decisions are also important when investigating cross-level interactions Consider the following model: Level 1:Y ij = ß 0j + ß 1j (X grand ) + r ij Level 2:ß 0j =  00 + U 0j ß 1j =  10 The ß 1j 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

51. 51 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 (Hofmann & Gavin, 1999, J. of Management) –Created two data sets Cross-level interaction, no between-group interaction Between-group interaction, no cross-level interaction

52. 52

53. 53 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

54. 54 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

55. 55 Do You Really Need HLM? Alternatives for Estimating Hierarchical Models

56. 56 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)

57. 57 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 )

58. 58 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;

59. 59

60. 60 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/

61. 61 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.htmhttp://www.ssicentral.com/hlm/hlmref.htm