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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 R305D (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.

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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.

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Overall Goals for Today Get an introductory understanding of how theory and models get represented in three crucial dialects of social science research: 1.Diagrams (accurate and complete) 2.Equations a. Scalar equations for variables b. Matrix equations for variables c. Matrix representations of covariances 3.Code in different software Apply these translations for simple multilevel models in some example software: Mplus, lme4, and xxm. Get some experience with R.

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Today’s Workshop 1.What is a multilevel model? a.Conceptual basis: what is clustering? b.Graphical approach: histograms, boxplots c.Equations, data structure, diagram 2.Adding a predictor a.Conceptual basis: what is a predictor? b.Graphical approach: scatterplot c.Equations, data structure, diagram 3.Extensions: bivariate to SEM?

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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 ). 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), 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.

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If treatment is at one level, what does variability mean at lower and higher levels? 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 Nested Data: They’re everywhere (region, hemisphere—spatial!) (relational, networked?)

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

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By substitution, we get the full equation: Y ij = 00 + u 0j + e ij Multilevel Regression: Random Intercept Model Y ij = 0j + e ij 0j = 00 + u 0j 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) fixedrandom 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):

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Multilevel Regression: Random Intercept Model Y ij = 0j + e ij 0j = 00 + u 0j 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) Y ij 00 u 0j e ij

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Multilevel Regression: SEM Diagram Level 1 ( i students) Level 2 ( j classrooms) Y ij 00 u 0j e ij random residual for level 1 random residual for level 2 (deviation from grand intercept) fixed intercept for level 2 (grand intercept) Mehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations models. Psychological Methods, 10(3), 259–284.

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Multilevel Regression: Variance components Variance of student deviations Variance of classroom deviations Level 1 ( i students) Level 2 ( j classrooms) Y ij 00 u 0j e ij 00 Grand intercept Mehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations models. Psychological Methods, 10(3), 259–284. HLM-style notation SEM notation

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Multilevel Regression: Results Level 1 ( i students) Level 2 ( j classrooms) Y ij u 0j e ij SEM notation Variance of student deviations (SD = 20.2) Variance of classroom deviations 89.8 (SD = 9.5) Grand intercept = 444.0

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Model Results = Classroom SD = 9.5 Student SD = 20.2

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How Does a Multilevel Model Work? Data Set (Excel, SPSS)Classroom Regressions Y ij jj e ij SEM StudentClassroomOutcome 11Y 11 21Y 21 32Y 32 42Y 42 53Y 53 63Y 63 Y i1 = 1 + e i1 Y i2 = 2 + e i2 Y i3 = 3 + e i3 where ~ N( ) e ~ N(0, )

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Multilevel Regression = Multilevel SEM StudentClassroomOutcome 11Y 11 21Y 21 32Y 32 42Y 42 53Y 53 63Y 63 Data Set (Excel, SPSS)Classroom Regressions Y 11 11 e 11 Classroom SEMs Y i1 = 1 + e i1 Y i2 = 2 + e i2 Y i3 = 3 + e i3 where ~ N( ) e ~ N(0, ) Y 21 e 21 Y 32 22 e 32 Y 42 e 42 Y 53 33 e 53 Y 63 e 63

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Multilevel Regression = Multilevel SEM StudentClassroomOutcome 11Y 11 21Y 21 32Y 32 42Y 42 53Y 53 63Y 63 Classroom RegressionsClassroom SEMs Y i1 = 1 + e i1 Y i2 = 2 + e i2 Y i3 = 3 + e i3 where ~ N( ) e ~ N(0, ) Y 11 11 e 11 Y 21 e 21 Y 32 22 e 32 Y 42 e 42 Y 53 33 e 53 Y 63 e 63

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Classroom SEM: Expanded version Y 11 11 e 11 Y 21 e 21 Y 32 22 e 32 Y 42 e 42 Y 53 33 e 53 Y 63 e 63 Classroom 1 Classroom 2 Classroom 3

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Classroom SEM: Expanded version Y 11 11 e 11 Y 21 e 21 Y 32 22 e 32 Y 42 e 42 Y 53 33 e 53 Y 63 e 63 Classroom 1 Classroom 2 Classroom 3

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Classroom SEM: Expanded version Y 11 11 e 11 Y 21 e 21 Y 32 22 e 32 Y 42 e 42 Y 53 33 e 53 Y 63 e 63 Classroom 1 Classroom 2 Classroom 3 Matrix Equation for outcomes (implicit) cross-level linking matrix

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Classroom SEM: Concise version Y ij jj e ij Classroom deviation Latent mean (across classrooms) student residual variance of student residuals variance between classrooms Student ModelClassroom Model Cross-level link Model matrices

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

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

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Adding a Predictor StudentClassroomOutcomePredictor 11Y 11 X 11 21Y 21 X 21 32Y 32 X 32 42Y 42 X 42 53Y 53 X 53 63Y 63 X 63 Data Set (Excel, SPSS)Classroom Regressions Y i1 = 11 + X i1 21 + e i1 Y i2 = 12 + X i2 22 + e i2 Y i3 = 13 + X i3 23 + e i3

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Adding a Predictor Classroom Regressions Y i1 = 11 + X i1 21 + e i1 Y i2 = 12 + X i2 22 + e i2 Y i3 = 13 + X i3 23 + e i3 Y ij 1j e ij SEM 2j X ij Student Model Classroom Model

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Adding a Predictor Model Matrices Y ij 1j e ij SEM 2j X ij Student Model Classroom Model Observed Variable Matrices

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Adding a Predictor Classroom Regressions Y ij 1j e ij SEM 2j X ij Student Model Classroom Model (-.27)

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Not Just a Predictor: Two Outcomes Y ij 1j e ij SEM: Random Slope 2j X ij Student Model Classroom Model Y ij 1j e 1ij SEM: Bivariate Random Intercepts 2j Student Model Classroom Model X ij e 2ij

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Bivariate Random Intercept Model Y 1ij = u 10j + e 1ij Y 2ij = u 20j + e 2ij Outcome 1 (Spanish) 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. fixed Classroom random effects are correlated Student random effects are correlated

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