Multilevel Modeling: Introduction

1. Multilevel Modeling:Introduction Chongming Yang, Ph.D Social Science Research Institute Social Capital Group Meeting, Spring 2008

2. New Paradigm in Data Analysis “In the past twenty years we have witnessed a paradigm shift in the analysis of correlational data. Confirmatory factor analysis and structural equation modeling have replaced exploratory factor analysis and multiple regression as the standard methods. We are currently in the early stages of a paradigm shift in the analysis of experimental data. Multilevel modeling is replacing ANOVA. Certainly ANOVA will remain a basic tool in the social psychological research, but it can no longer be considered the only technique” Kenny, D.A. Kashy, D.A., & Bolger, N. (1998). Data analysis in psychology. In D.T. Gilbert, S.T. Fiske, & G. Lindzey (Eds.) The Handbook of Social Psychology, Vol. 1 (pp233-265). New York: McGraw-Hill.

3. Alternative Labels • Hierarchical Linear Model (HLM) • Random Coefficient Model • Variance Component Model • Multilevel Model • Contextual Analysis • Mixed Linear Model

4. Hierarchical Data Structure • Response (outcome) variable at lowest level • Grouping at higher levels • Explanatory (predictive) variables at all levels • Assuming sampling at all levels

5. Two Types • Persons nested within a group • Repeated measures nested within a person

6. Example of Multilevel Data

7. Properties of Hierarchical Data • Observations are interdependent, more similar within groups than from different groups due to shared history, contextual effects, etc. • Errors are not independent (longitudinal data)

8. Standard Modeling Assumptions • Independent observations • Independent errors • Equal variances of errors for all observations

9. Consequences of Ignoring Hierarchical Data Properties • Smaller standard errors for regression coefficients, thus • Spurious effects

10. Design-based Approach • Apply standard analysis with sampling weights to adjust standard errors, common in survey research

11. Design Effects of Two-level Data • Intraclass Correlation  = between-level variance/total variance • Design Effect n/[1+(n-1)] where n = average cluster size (=>2 warrants a multilevel analysis)

12. Another Look

13. Intercepts & Slopes for Each Class y 0 X

14. Class Level Summary

15. Modeling Intercepts & Slopes • 0 = g0 + u0 • 1 = g10 + u1 when variances of u0 andu1 are zero, there are no group differences in 0 and 1. Thus variances of u0 andu1 are very important parameters.

16. Model-based ApproachMultilevel Modeling • (Multiple Equations) Multilevel Model: yi= 0 + 1xi + ri 0 = g00 + u0 1 = g10 + u1 • (Single Equation) Mixed Model: yi= g00+u0+g10xi+u1xi+ri

17. Multilevel Modeling (with 2nd Level Predictors) • (Multiple Equations) Multilevel Model: yi= 0 + 1xi + ri 0 = g00 + g01zj + u0(main effects) 1 = g10 + g11zj + u1 (cross-level interaction) • (Single Equation) Mixed Model: yi= g00+g01zj+u0+g10xi+g11zjxi+u1xi+ri

18. Rearranged Single Equation • yi= [g00 + g10xi + g01zj + g11zjxi] (fixed effects) + [u1xi + u0 + ri] (random effects) • Parameters to be estimated: intercept: g00 slopes: g10, g01,g11 variances: r, u0, u1 covariances: among rs(in longitudinal data), u0 &u1, gs

19. Fixed or Random?

20. Cross-level Interaction • Appears in a single equation as product term, not in multiple equations • The effect of a lower level variable depends on upper level variables • Example: The effect of students’ aptitude on math achievement depends teachers’ competence

21. Estimation • Restricted Maximum Likelihood: Variance components are included in the likelihood function, regression coefficients are estimated in a second step (less biased against variance) • Full Maximum Likelihood: Both variance components and regression coefficients are included in the likelihood function (variances are slightly underestimated.

22. Deviance • -2 times log-likelihood Function, 2 distribution, can be used for model comparison, • The smaller, the better fit

23. Explore HLM Program • Create MDM • Specify and run a model • Interpret parameters in the output

24. Model Exploration Procedures • Start with an intercept-only model (Calculate intraclass correlation) • Add 1st level predictors for a fixed model (Test individual slopes) • Model intercept by 2nd level predictors (Test significance & amount of variance explained) • Random coefficient model (Test variance component of 1st level slopes one by one) • Model Random slopes predicted by higher level variables (Test significance and amount variance explained)

25. Longitudinal Data y 0 Time 1 2 3 4

26. Unconditional Growth Model • 1st level: Occasion y = p0 + p1t + r • 2nd level: Person p0 = g00 + u0 p1 =g10 + u1

27. Parameters to Interpret • Means of Intercept (g00) & Slope(g10) • Variances of Intercept (u0) & Slope (u1) • Covariance/Correlation of Intercept & Slope (u0 & u1)

28. Extended Model • Occasion level: Time-variant covariate x y = p0 + p1t + p3x + r • Person level: time-invariant covariate z p0 = g00 + g01zj + u0 p1 =g10 + g11zj + u1 p2 =g20 + g21zj + u2

29. Nonlinear Growth (by Recoding T Variable) • Linear: 0, 1, 2, 3… (0, 1, 2.5, 3.5…) • Quadratic: 0, 1, 4, 9… • Logarithmic: 0, 0.69, 1.10, 1.39… • Exponential: 0, 1.72, 6.39, 19.09

30. Explore HLM Program Chapter 4 Example • Create MDM • Specify and run a model • Interpret parameters in the output

31. Explore the SAS program • Identify levels of the variables in the data • Identify which variables could have main and/or interaction effects • Identify random coefficients and then their variances in the output

32. Minimum Sample Size • Cluster level: > 20 • Individual level: =>1

33. Obtain Standardized Coefficients Standardize continuous variables to obtain standardized coefficients

34. Further Topics • Categorical dependent variables • Multivariate dependent variables • Latent variables + mediating effects (multilevel structural equation modeling) • Power & Sample Size • ...

35. Further Resources • http://gseweb.harvard.edu/~faculty/singer/ • www.ats.ucla.edu/stat/sas/default.htm • SSRI consultants • …