A Conceptual Introduction to Multilevel Models as Structural Equations

1. A Conceptual Introduction to Multilevel Models as Structural Equations Lee Branum-Martin Georgia State University Language & Literacy Initiative A Workshop for the Society for the Scientific Study of Reading July 9, 2013 Hong Kong, China The analyses and software for this workshop were supported by the Institute of Education Sciences, U.S. Department of Education, through grants R305A10272 (Lee Branum-Martin, PI) and R305D090024 (Paras D. Mehta, PI) to University of Houston. The initial data collection was jointly funded by NICHD (HD39521) and IES (R305U010001) to UH (David J. Francis, PI). The opinions expressed are those of the author and do not represent views of these funding agencies.

2. Important concepts for students interested in high-quality education research Psychometrics/test theory is the basis for educational measurement. • Item Response Theory • Confirmatory Factor Analysis, Structural Equation Modeling • Direct tests of theory Multilevel models for nested data. • Longitudinal models (observations nested within persons) • Complex clustering (regular instruction + tutoring) • Mixed effects, random effects, and multilevel models can be fit in a number of different software packages.

3. Overall Goals for Today • Get an introductory understanding of how theory and models get represented in three crucial dialects of social science research: • Diagrams (accurate and complete) • Equations • a. Scalar equations for variables • b. Matrix equations for variables • c. Matrix representations of covariances • Code in different software • Apply these translations for simple multilevel models in some example software: Mplus, lme4, and xxm. • Get some experience with R.

4. Today’s Workshop • What is a multilevel model? • Conceptual basis: what is clustering? • Graphical approach: histograms, boxplots • Equations, data structure, diagram • Adding a predictor • Conceptual basis: what is a predictor? • Graphical approach: scatterplot • Equations, data structure, diagram • Extensions: bivariate to SEM?

5. Background Branum-Martin, L. (2013). Multilevel modeling: Practical examples to illustrate a special case of SEM. In Y. Petscher, C. Schatschneider & D. L. Compton (Eds.), Applied quantitative analysis in the social sciences (pp. 95-124). New York: Routledge. Singer, J. D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 24(4), 323-355. Mehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations models. Psychological Methods, 10(3), 259–284. West, B. T., Welch, K. B., & Gałecki, A. T. (2007). Linear mixed models : a practical guide using statistical software. Boca Raton: Chapman & Hall.

6. Nested Data: They’re everywhere Developmental: items, trials, days, persons Clinical: interview topics, sessions (days, weeks, months), persons, sites Cognitive: items, tests, traits, person, social group, neighborhood Neuropsychology: time (ms), electrode, person Education: items, tests, years, students, classrooms, schools (relational, networked?) (region, hemisphere—spatial!) If treatment is at one level, what does variability mean at lower and higher levels?

7. Students in Classrooms 802 Students in 93 classrooms in 23 schools. Passage comprehension W-scores on Woodcock Johnson Language Proficiency Battery-Revised.

8. Multilevel Regression: Random Intercept Model Yij = b0j+ eij b0j = g00+ u0j Level 1 (i students) random residual for level 1 fixed intercept for level 2 (grand intercept) Level 2 (j classrooms) random residual for level 2 (deviation from grand intercept) By substitution, we get the full equation: Yij= g00+ u0j + eij fixed random random proc mixed covtestdata = mydata; classclassroom; modely = / solution; random intercept / subject = classroom; run; Singer, J. D. (1998). "Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models." Journal of Educational and Behavioral Statistics 24(4): 323-355.

9. Multilevel Regression: Random Intercept Model Yij= b0j+ eij b0j = g00+ u0j Level 1 (i students) random residual for level 1 fixed intercept for level 2 (grand intercept) Level 2 (j classrooms) random residual for level 2 (deviation from grand intercept) Yij g00 • eij u0j

10. Multilevel Regression: SEM Diagram fixed intercept for level 2 (grand intercept) 1 g00 random residual for level 2 (deviation from grand intercept) Level 2 (j classrooms) Level 1 (i students) Yij random residual for level 1 • eij u0j Mehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations models. Psychological Methods, 10(3), 259–284.

11. Multilevel Regression: Variance components HLM-style notation SEM notation 1 Grand intercept g00 a Variance of classroom deviations t00 y Level 2 (j classrooms) Variance of student deviations s2 q Level 1 (i students) Yij • eij u0j Mehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations models. Psychological Methods, 10(3), 259–284.

12. Multilevel Regression: Results SEM notation Grand intercept = 444.0 1 a Variance of classroom deviations 89.8 (SD = 9.5) y Level 2 (j classrooms) Variance of student deviations 410.0 (SD = 20.2) q Level 1 (i students) Yij • eij u0j Intraclass correlation =

13. Model Results Classroom SD = 9.5 g00= 444.0 Student SD = 20.2

14. How Does a Multilevel Model Work? Data Set (Excel, SPSS) Classroom Regressions SEM a • Yi1 = h1 + ei1 1 y • Yi2 = h2 + ei2 Yij q • Yi3 = h3 + ei3 • eij hj • where h~ N(a,y) e ~ N(0,q)

15. Multilevel Regression = Multilevel SEM Data Set (Excel, SPSS) Classroom Regressions Classroom SEMs Y11 • Yi1 = h1 + ei1 Y21 Y32 • Yi2 = h2 + ei2 Y42 Y53 • Yi3 = h3 + ei3 Y63 • e21 • e32 • e42 • e53 • e11 • e63 h1 h2 h3 • where h~ N(a,y) e ~ N(0,q)

16. Multilevel Regression = Multilevel SEM Classroom Regressions Classroom SEMs Y11 • Yi1 = h1 + ei1 Y21 Y32 • Yi2 = h2 + ei2 Y42 Y53 • Yi3 = h3 + ei3 Y63 • e11 • e21 • e32 • e42 • e53 • e63 h1 h2 h3 • where h~ N(a,y) e ~ N(0,q)

17. Classroom SEM: Expanded version y Y11 q 1 Classroom 1 Y21 a q y Y32 a q Classroom 2 Y42 a q y Y53 q Classroom 3 Y63 q • e53 • e42 • e32 • e21 • e11 • e63 h3 h2 h1

18. Classroom SEM: Expanded version y Y11 q 1 Classroom 1 Y21 a q y Y32 a q Classroom 2 Y42 a q y Y53 q Classroom 3 Y63 q • e53 • e42 • e32 • e21 • e11 • e63 h3 h2 h1

19. Classroom SEM: Expanded version 1 1 y Y11 1 q 1 Classroom 1 Y21 1 a q y Y32 1 a q Classroom 2 Y42 1 a (implicit) cross-level linking matrix q y Y53 q Classroom 3 Y63 Matrix Equation for outcomes q • e42 • e53 • e21 • e11 • e63 • e32 h3 h2 h1

20. Classroom SEM: Concise version Student Model Classroom Model 1 Classroom deviation variance between classrooms Cross-level link variance of student residuals y l Yij a q Latent mean (across classrooms) student residual • eij hj q l y a Model matrices

21. Passage Comprehension Predicted by Word Attack 802 Students in 93 classrooms in 23 schools. W-scores on Woodcock Johnson Language Proficiency Battery-Revised.

22. Classroom Predictions of PC by WA 802 Students in 93 classrooms in 23 schools. W-scores on Woodcock Johnson Language Proficiency Battery-Revised.

23. Adding a Predictor Data Set (Excel, SPSS) Classroom Regressions • Yi1 = h11 + Xi1h21 + ei1 • Yi2 = h12 + Xi2h22 + ei2 • Yi3 = h13 + Xi3h23 + ei3

24. Adding a Predictor SEM Classroom Regressions Classroom Model a2 a1 y21 • Yi1 = h11 + Xi1h21 + ei1 1 y11 y22 • Yi2 = h12 + Xi2h22 + ei2 Xij Yij • Yi3 = h13 + Xi3h23 + ei3 • eij h1j h2j q Student Model

25. Adding a Predictor SEM Model Matrices Classroom Model a2 a1 y21 1 y11 y22 Xij Observed Variable Matrices Yij • eij h2j h1j q Student Model

26. Adding a Predictor SEM Classroom Regressions Classroom Model .85 443.4 -.34 1 37.0 .04 (-.27) Xij Yij • eij h2j h1j 234.6 Student Model

27. Not Just a Predictor: Two Outcomes SEM: Random Slope SEM: Bivariate Random Intercepts Classroom Model Classroom Model a2 a1 a1 a2 y21 y21 1 1 y11 y22 y22 y11 Xij Xij Yij Yij • eij • e2ij • e1ij h1j h2j h2j h1j q11 q22 q q21 Student Model Student Model