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.

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.

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.

Today’s Workshop What is a multilevel model? Adding a predictor 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 Extensions: bivariate to SEM?

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.

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 If treatment is at one level, what does variability mean at lower and higher levels? (relational, networked?) (region, hemisphere—spatial!)

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

Multilevel Regression: Random Intercept Model Yij = b0j+ eij b0j = g00+ u0j random residual for level 1 Level 1 (i students) 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 proc mixed covtest data = mydata; class classroom; model y = / 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.

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

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

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

Multilevel Regression: Results SEM notation a y q 1 Grand intercept = 444.0 Variance of classroom deviations 89.8 (SD = 9.5) u0j Level 2 (j classrooms) Variance of student deviations 410.0 (SD = 20.2) eij Level 1 (i students) Yij Intraclass correlation = 𝑣(𝑏𝑒𝑡𝑤𝑒𝑒𝑛) 𝑣(𝑡𝑜𝑡𝑎𝑙) = 89.8 89.8+410 =.18

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

How Does a Multilevel Model Work? Data Set (Excel, SPSS) Classroom Regressions SEM Student Classroom Outcome 1 Y11 2 Y21 3 Y32 4 Y42 5 Y53 6 Y63 1 a Yi1 = h1 + ei1 y hj Yi2 = h2 + ei2 Yij q Yi3 = h3 + ei3 eij where h ~ N(a,y) e ~ N(0,q)

Multilevel Regression = Multilevel SEM Data Set (Excel, SPSS) Classroom Regressions Classroom SEMs Student Classroom Outcome 1 Y11 2 Y21 3 Y32 4 Y42 5 Y53 6 Y63 e11 Y11 Yi1 = h1 + ei1 h1 e21 Y21 e32 Y32 h2 Yi2 = h2 + ei2 e42 Y42 e53 Y53 h3 Yi3 = h3 + ei3 e63 Y63 where h ~ N(a,y) e ~ N(0,q)

Multilevel Regression = Multilevel SEM Classroom Regressions Classroom SEMs Student Classroom Outcome 1 Y11 2 Y21 3 Y32 4 Y42 5 Y53 6 Y63 Y11 h1 e11 Y21 e21 Y32 h2 e32 Y42 e42 Y53 h3 e53 Y63 e63 Yi1 = h1 + ei1 Yi2 = h2 + ei2 Yi3 = h3 + ei3 where h ~ N(a,y) e ~ N(0,q)

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

Classroom SEM: Expanded version Y11 h1 e11 Y21 e21 Y32 h2 e32 Y42 e42 Y53 h3 e53 Y63 e63 1 a y q Classroom 1 Classroom 2 Classroom 3 𝑌 11 𝑌 21 𝑌 32 𝑌 42 𝑌 53 𝑌 63 = 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 𝜂 1 𝜂 2 𝜂 3 + 𝑒 11 𝑒 21 𝑒 32 𝑒 42 𝑒 53 𝑒 63

Classroom SEM: Expanded version Y11 h1 e11 Y21 e21 Y32 h2 e32 Y42 e42 Y53 h3 e53 Y63 e63 1 a y q Classroom 1 Classroom 2 Classroom 3 1 (implicit) cross-level linking matrix 𝑌 11 𝑌 21 𝑌 32 𝑌 42 𝑌 53 𝑌 63 = 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 𝜂 1 𝜂 2 𝜂 3 + 𝑒 11 𝑒 21 𝑒 32 𝑒 42 𝑒 53 𝑒 63 Matrix Equation for outcomes

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

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

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

Adding a Predictor Yi1 = h11 + Xi1h21 + ei1 Yi2 = h12 + Xi2h22 + ei2 Data Set (Excel, SPSS) Classroom Regressions Student Classroom Outcome Predictor 1 Y11 X11 2 Y21 X21 3 Y32 X32 4 Y42 X42 5 Y53 X53 6 Y63 X63 Yi1 = h11 + Xi1h21 + ei1 Yi2 = h12 + Xi2h22 + ei2 Yi3 = h13 + Xi3h23 + ei3

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

Observed Variable Matrices Adding a Predictor SEM Model Matrices Classroom Model 1 𝛼 2,2 = 𝛼 1 𝛼 2 a2 a1 y21 Ψ 2,2 = 𝜓 11 𝜓 21 𝜓 12 𝜓 22 y11 y22 h1j h2j Xij Λ 2,1 = 1 𝑋 𝑖𝑗 Θ 1,1 = 𝜃 11 Observed Variable Matrices Yij 𝑌 11 𝑌 21 𝑌 32 𝑌 42 𝑌 53 𝑌 63 = 1 𝑋 11 0 0 0 0 1 𝑋 21 0 0 0 0 0 0 1 𝑋 32 0 0 0 0 1 𝑋 42 0 0 0 0 0 0 1 𝑋 53 0 0 0 0 1 𝑋 63 𝜂 11 𝜂 21 𝜂 12 𝜂 22 𝜂 13 𝜂 23 + 𝑒 11 𝑒 21 𝑒 32 𝑒 42 𝑒 53 𝑒 63 eij q Student Model

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

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

Bivariate Random Intercept Model fixed Y1ij = g100+ u10j + e1ij Y2ij = g200+ u20j + e2ij Outcome 1 (Spanish) Classroom random effects are correlated Student random effects are correlated Outcome 2 (English) Mehta, P. D. and M. C. Neale (2005). "People are variables too: Multilevel structural equations models." Psychological Methods 10(3): 259–284.