A Conceptual Introduction to Multilevel Models as Structural Equations

Lee Branum-MartinGeorgia State University

Language & Literacy Initiative

A Workshop for theSociety for the Scientific Study of Reading

July 9, 2013Hong 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 TodayGet 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 variablesb. Matrix equations for variablesc. Matrix representations of covariances

3. Code in different softwareApply these translations for simple multilevel models in some example software: Mplus, lme4, and xxm.Get some experience with R.

Today’s Workshop

1. What is a multilevel model? a. Conceptual basis: what is clustering?b. Graphical approach: histograms, boxplotsc. Equations, data structure, diagram

2. Adding a predictora. Conceptual basis: what is a predictor?b. Graphical approach: scatterplotc. Equations, data structure, diagram

3. Extensions: bivariate to SEM?

BackgroundBranum-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.

If treatment is at one level, what does variability mean at lower and higher levels?

Developmental: items, trials, days, personsClinical: interview topics, sessions (days, weeks, months), persons, sitesCognitive: items, tests, traits, person, social group, neighborhoodNeuropsychology: time (ms), electrode, personEducation: items, tests, years, students, classrooms, schools

Nested Data: They’re everywhere

(region, hemisphere—spatial!)

(relational, networked?)

Students in Classrooms802 Students in 93 classrooms in 23 schools. Passage comprehension W-scores on Woodcock Johnson Language Proficiency Battery-Revised.

By substitution, we get the full equation:

Yij = g00+ u0j + eij

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)

fixed random 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

Level 1 (i students)

Level 2 (j classrooms)

Yij

g00

u0j

eij

1

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.

Multilevel Regression: Variance components

Variance of student deviations

Variance of classroom deviations

Level 1 (i students)

Level 2 (j classrooms)

Yij

g00

u0j

eij

1

t00

s2

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 notationSEM notation

a

y

q

Multilevel Regression: Results

Level 1 (i students)

Level 2 (j classrooms)

Yij

u0j

eij

1

SEM notation

a

y

q

Variance of student deviations

410.0 (SD = 20.2)

Variance of classroom deviations

89.8 (SD = 9.5)

Grand intercept = 444.0

Intraclass correlation =

Model Results

g00= 444.0

Classroom SD = 9.5

Student SD = 20.2

How Does a Multilevel Model Work?Data Set (Excel, SPSS) Classroom Regressions

Yij

hj

eij

1

SEM

ay

q

Student Classroom Outcome

1 1 Y11

2 1 Y21

3 2 Y32

4 2 Y42

5 3 Y53

6 3 Y63

Yi1 = h1 + ei1

Yi2 = h2 + ei2

Yi3 = h3 + ei3

where h ~ N( ,a y) e ~ N(0,q)

Multilevel Regression = Multilevel SEM

Student Classroom Outcome

1 1 Y11

2 1 Y21

3 2 Y32

4 2 Y42

5 3 Y53

6 3 Y63

Data Set (Excel, SPSS) Classroom Regressions

Y11 h1

e11

Classroom SEMs

Yi1 = h1 + ei1

Yi2 = h2 + ei2

Yi3 = h3 + ei3

where h ~ N( ,a y) e ~ N(0,q)

Y21e21

Y32 h2

e32

Y42e42

Y53 h3

e53

Y63e63

Multilevel Regression = Multilevel SEM

Student Classroom Outcome

1 1 Y11

2 1 Y21

3 2 Y32

4 2 Y42

5 3 Y53

6 3 Y63

Classroom Regressions Classroom SEMs

Yi1 = h1 + ei1

Yi2 = h2 + ei2

Yi3 = h3 + ei3

where h ~ N( ,a y) e ~ N(0,q)

Y11 h1

e11

Y21e21

Y32 h2

e32

Y42e42

Y53 h3

e53

Y63e63

Classroom SEM: Expanded version

Y11 h1

e11

Y21e21

Y32 h2

e32

Y42e42

Y53 h3

e53

Y63e63

1

a

y

q

y

y

a

a

q

qq

qqClassroom

1

Classroom 2

Classroom 3

Classroom SEM: Expanded version

Y11 h1

e11

Y21e21

Y32 h2

e32

Y42e42

Y53 h3

e53

Y63e63

1

a

y

q

y

y

a

a

q

qq

qqClassroom

1

Classroom 2

Classroom 3 [

𝑌 11

𝑌 21

𝑌 32

𝑌 42

𝑌 53

𝑌 63

]=[110000

001100

000011] [𝜂1𝜂2𝜂3]+[

𝑒11𝑒21𝑒32𝑒42𝑒53𝑒63

]

Classroom SEM: Expanded versionY11 h1

e11

Y21e21

Y32 h2

e32

Y42e42

Y53 h3

e53

Y63e63

1

a

y

q

y

y

a

a

q

qq

qqClassroom

1

Classroom 2

Classroom 3

[𝑌 11

𝑌 21

𝑌 32

𝑌 42

𝑌 53

𝑌 63

]=[110000

001100

000011] [𝜂1𝜂2𝜂3]+[

𝑒11𝑒21𝑒32𝑒42𝑒53𝑒63

]Matrix Equation

for outcomes

(implicit) cross-level linking matrix

1

1

1

1

1

1

Classroom SEM: Concise version

Yijhj

eij 1

y

aq

Classroom deviation

Latent mean (across classrooms)

student residual

variance of student residuals

variance between classrooms

Student Model Classroom Model

lCross-level

link

qModel matrices y al

Passage Comprehension Predicted by Word Attack802 Students in 93 classrooms in 23 schools. W-scores on Woodcock

Johnson Language Proficiency Battery-Revised.

Classroom Predictions of PC by WA802 Students in 93 classrooms in 23 schools. W-scores on Woodcock

Johnson Language Proficiency Battery-Revised.

Adding a Predictor

Student Classroom Outcome Predictor

1 1 Y11 X11

2 1 Y21 X21

3 2 Y32 X32

4 2 Y42 X42

5 3 Y53 X53

6 3 Y63 X63

Data Set (Excel, SPSS) Classroom Regressions

Yi1 = h11 + Xi1h21 + ei1

Yi2 = h12 + Xi2h22 + ei2

Yi3 = h13 + Xi3h23 + ei3

Adding a PredictorClassroom Regressions

Yi1 = h11 + Xi1h21 + ei1

Yi2 = h12 + Xi2h22 + ei2

Yi3 = h13 + Xi3h23 + ei3Yij

h1j

eij

1

SEM

a1

y11

q

h2j

Xij

y22

y21

a2

Student Model

Classroom Model

Adding a PredictorModel Matrices

Yij

h1j

eij

1

SEM

a1

y11

q

h2j

Xij

y22

y21

a2

Student Model

Classroom Model

[𝑌 11

𝑌 21

𝑌 32

𝑌 42

𝑌 53

𝑌 63

]=[1 𝑋 11 0 0 0 01 𝑋 21 0 0 0 00 0 1 𝑋 32 0 00 0 1 𝑋 42 0 00 0 0 0 1 𝑋 53

0 0 0 0 1 𝑋 63

] [𝜂11𝜂21𝜂12𝜂22𝜂13𝜂 23

]+[𝑒11𝑒21𝑒32𝑒42𝑒53𝑒63

]Observed Variable Matrices

𝛼2,2=[𝛼1𝛼2]Ψ 2,2=[𝜓 11

𝜓21

𝜓 12

𝜓 22]

Λ2,1=[1 𝑋𝑖𝑗 ]Θ1,1= [𝜃11 ]

Adding a PredictorClassroom Regressions

Yij

h1j

eij

1

SEM

443.4

37.0

234.6

h2j

Xij

.04-.34

.85

Student Model

Classroom Model

(-.27)

Not Just a Predictor: Two Outcomes

Yij

h1j

eij

1

SEM: Random Slope

a1

y11

q

h2j

Xij

y22

y21

a2

Student Model

Classroom Model

Yij

h1j

e1ij

1

SEM: Bivariate Random Intercepts

a1

y11

q11

h2j

y22

y21

a2

Student Model

Classroom Model

Xij

e2ij

q22q21