Presentation on theme: "A Conceptual Introduction to Multilevel Models as Structural Equations"— Presentation transcript:
1A Conceptual Introduction to Multilevel Models as Structural Equations Lee Branum-MartinGeorgia State UniversityLanguage & Literacy InitiativeA Workshop for theSociety for the Scientific Study of ReadingJuly 9, 2013Hong Kong, ChinaThe 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.
2Important concepts for students interested in high-quality education research Psychometrics/test theory is the basis for educational measurement.Item Response TheoryConfirmatory Factor Analysis, Structural Equation ModelingDirect tests of theoryMultilevel 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.
3Overall 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)Equationsa. Scalar equations for variablesb. Matrix equations for variablesc. Matrix representations of covariancesCode in different softwareApply these translations for simple multilevel models in some example software: Mplus, lme4, and xxm.Get some experience with R.
4Today’s Workshop What is a multilevel model? Adding a predictor Conceptual basis: what is clustering?Graphical approach: histograms, boxplotsEquations, data structure, diagramAdding a predictorConceptual basis: what is a predictor?Graphical approach: scatterplotExtensions: bivariate to SEM?
5BackgroundBranum-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.
6Nested 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, schoolsIf treatment is at one level, what does variability mean at lower and higher levels?(relational, networked?)(region, hemisphere—spatial!)
7Students in Classrooms 802 Students in 93 classrooms in 23 schools. Passage comprehension W-scores on Woodcock Johnson Language Proficiency Battery-Revised.
8Multilevel Regression: Random Intercept Model Yij = b0j+ eijb0j = g00+ u0jrandom residual for level 1Level 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 + eijfixedrandomproc 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):
9Multilevel Regression: Random Intercept Model Yij = b0j+ eijb0j = g00+ u0jrandom residual for level 1random residual for level 2 (deviation from grand intercept)fixed intercept for level 2 (grand intercept)Level 1 (i students)Level 2 (j classrooms)Yijg00u0jeij
10Multilevel Regression: SEM Diagram 1fixed intercept for level 2 (grand intercept)g00u0jrandom residual for level 2 (deviation from grand intercept)Level 2 (j classrooms)eijLevel 1 (i students)Yijrandom residual for level 1Mehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations models. Psychological Methods, 10(3), 259–284.
11Multilevel Regression: Variance components SEM notationayqHLM-style notation1Grand interceptg00Variance of classroom deviationst00u0jLevel 2 (j classrooms)Variance of student deviationss2eijLevel 1 (i students)YijMehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations models. Psychological Methods, 10(3), 259–284.
12Multilevel Regression: Results SEM notationayq1Grand intercept =444.0Variance of classroom deviations89.8 (SD = 9.5)u0jLevel 2 (j classrooms)Variance of student deviations(SD = 20.2)eijLevel 1 (i students)YijIntraclass correlation = 𝑣(𝑏𝑒𝑡𝑤𝑒𝑒𝑛) 𝑣(𝑡𝑜𝑡𝑎𝑙) = =.18
14How Does a Multilevel Model Work? Data Set (Excel, SPSS)Classroom RegressionsSEMStudentClassroomOutcome1Y112Y213Y324Y425Y536Y631aYi1 = h1 + ei1yhjYi2 = h2 + ei2YijqYi3 = h3 + ei3eijwhere h ~ N(a,y) e ~ N(0,q)
15Multilevel Regression = Multilevel SEM Data Set (Excel, SPSS)Classroom RegressionsClassroom SEMsStudentClassroomOutcome1Y112Y213Y324Y425Y536Y63e11Y11Yi1 = h1 + ei1h1e21Y21e32Y32h2Yi2 = h2 + ei2e42Y42e53Y53h3Yi3 = h3 + ei3e63Y63where h ~ N(a,y) e ~ N(0,q)
16Multilevel Regression = Multilevel SEM Classroom RegressionsClassroom SEMsStudentClassroomOutcome1Y112Y213Y324Y425Y536Y63Y11h1e11Y21e21Y32h2e32Y42e42Y53h3e53Y63e63Yi1 = h1 + ei1Yi2 = h2 + ei2Yi3 = h3 + ei3where h ~ N(a,y) e ~ N(0,q)
17Classroom SEM: Expanded version yY11h1e11Y21e21Y32h2e32Y42e42Y53h3e53Y63e63Classroom 1qqaya1qClassroom 2aqyClassroom 3qq
26Adding a Predictor h1j h2j Xij Yij eij .85 443.4 -.34 37.0 .04 (-.27) SEMClassroom RegressionsClassroom Model1.85443.4-.3437.0.04(-.27)h1jh2jXijYijeij234.6Student Model
27Not Just a Predictor: Two Outcomes SEM: Random SlopeYijh1je1ij1SEM: Bivariate Random Interceptsa1y11q11h2jy22y21a2Student ModelClassroom ModelXije2ijq22q21Classroom Model1a2a1y21y11y22h1jh2jXijYijeijqStudent Model
28Bivariate Random Intercept Model fixedY1ij = g100+ u10j + e1ijY2ij = g200+ u20j + e2ijOutcome 1 (Spanish)Classroom random effects are correlatedStudent random effects are correlatedOutcome 2 (English)Mehta, P. D. and M. C. Neale (2005). "People are variables too: Multilevel structural equations models." Psychological Methods 10(3): 259–284.