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:
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.
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?
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
(relational, networked?)
Cognitive: items, tests, traits, person, social group,
neighborhood
(region, hemisphere—spatial!)
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?
Students in Classrooms
802 Students in 93 classrooms in 23 schools. Passage comprehension Wscores on Woodcock Johnson Language Proficiency Battery-Revised.
Multilevel Regression:
Random Intercept Model
Level 1 (i students)
Level 2 (j classrooms)
Yij = b0j+ eij
b0j = g00+ u0j
random residual for level 1
fixed intercept for level 2
(grand intercept)
random residual for level 2
(deviation from grand intercept)
By substitution, we get the full equation:
fixed random random
Yij = g00+ u0j + eij
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
Level 1 (i students)
Level 2 (j classrooms)
Yij = b0j+ eij
b0j = g00+ u0j
Yij
g00 u0j
random residual for level 1
fixed intercept for level 2
(grand intercept)
random residual for level 2
(deviation from grand intercept)
eij
Multilevel Regression:
SEM Diagram
1
u0j
Level 2 (j classrooms)
Level 1 (i students)
g00
Yij
fixed intercept for level 2
(grand intercept)
random residual for level 2
(deviation from grand intercept)
eij
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
HLM-style notation
SEM notation
1
a
Level 2 (j classrooms)
g00
Grand intercept
Variance of classroom deviations
y
t00
u0j
Variance of student deviations
Level 1 (i students)
q
Yij
eij
s2
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
1
a
Level 2 (j classrooms)
Grand intercept =
444.0
Variance of classroom deviations
y
89.8 (SD = 9.5)
u0j
Variance of student deviations
Level 1 (i students)
q
410.0 (SD = 20.2)
Yij
eij
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)
Student Classroom
Classroom Regressions
SEM
Outcome
1
1
Y11
2
1
Y21
3
2
Y32
4
2
Y42
5
3
Y53
6
3
Y63
1
Yi1 = h1 + ei1
a
hj
Yi2 = h2 + ei2
Yi3 = h3 + ei3
where h ~ N(a,y) e ~ N(0,q)
Yij
eij
y
q
Multilevel Regression = Multilevel SEM
Data Set (Excel, SPSS)
Student Classroom
Classroom Regressions
Classroom SEMs
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
e11
Y11
e21
Y21
e32
Y32
e42
Y42
e53
Y53
e63
Y63
where h ~ N(a,y) e ~ N(0,q)
h1
h2
h3
Multilevel Regression = Multilevel SEM
Classroom Regressions
Student Classroom
Classroom SEMs
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
e11
Y11
e21
Y21
e32
Y32
e42
Y42
e53
Y53
e63
Y63
where h ~ N(a,y) e ~ N(0,q)
h1
h2
h3
Classroom SEM: Expanded version
Classroom
1
Classroom
2
Classroom
3
e11
y
q
e21
q
Y11
h1
Y21
q
e42
q
e53
q
e63
q
Y32
y a
a
h2
y a
e32
Y42
Y53
Y63
h3
1
Classroom SEM: Expanded version
Classroom
1
Classroom
2
Classroom
3
e11
q
e21
q
e32
Y11
h1
Y21
y a
a
h2
y a
Y32
q
e42 Y42
q
e53 Y53
𝑌
1
q
1
e63𝑌𝑌 Y63
0
=
q
𝑌
0
11
21
32
42
𝑌53
𝑌63
y
0
0
1
1
0 0
0 0
h3
0
0
0
0
1
1
𝜂1
𝜂2
𝜂3
𝑒11
𝑒21
𝑒
+ 𝑒32
42
𝑒53
𝑒63
1
Classroom SEM: Expanded
version
y
Classroom
1
Classroom
2
Classroom
3
Matrix
Equation
for
outcomes
e11
q
e21
q
Y11
1
Y21
1
q
e42
q
e53
q
e63
q
Y32
1
Y42
1
Y53
1
Y63
1
e32
𝑌11
1 0
𝑌21
1 0
𝑌32
0 1
=
𝑌42
0 1
0 0
𝑌53
0 0
𝑌63
h1
y a
a
h2
y a
1
h3
(implicit) cross-level
linking matrix
𝑒11
0
𝑒21
0 𝜂1
𝑒32
0 𝜂
2 + 𝑒
42
0 𝜂
3
𝑒53
1
𝑒63
1
Classroom SEM: Concise version
Student Model
variance
of student
residuals
q
Classroom Model
Cross-level
link
l
eij
Yij
y
hj
variance between
classrooms
a
1
Latent mean
(across classrooms)
Classroom deviation
student residual
Model matrices
q
l
y
a
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
Data Set (Excel, SPSS)
Student Classroom
Classroom Regressions
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
Yi1 = h11 + Xi1h21 + ei1
Yi2 = h12 + Xi2h22 + ei2
Yi3 = h13 + Xi3h23 + ei3
Adding a Predictor
SEM
a1
y11
1
Classroom Regressions
Classroom Model
a2
y21
h1j
h2j
y22
Xij
Yij
q
Yi1 = h11 + Xi1h21 + ei1
Yi2 = h12 + Xi2h22 + ei2
Yi3 = h13 + Xi3h23 + ei3
eij
Student Model
Adding a Predictor
SEM
a1
y11
1
Model Matrices
Classroom Model
a2
y21
h1j
h2j
y22
𝛼 2,2
𝛼1
= 𝛼
2
Ψ2,2
𝜓11 𝜓12
=
𝜓21 𝜓22
Λ2,1 = 1 𝑋𝑖𝑗
Θ1,1 = 𝜃11
Xij
Observed Variable Matrices
Yij
q
eij
Student Model
1
𝑌11
1
𝑌21
𝑌32
0
=
𝑌42
0
𝑌53
0
𝑌63
0
𝑋11
𝑋21
0 1
0 1
0 0
0 0
0 0
0 0
𝑋32
𝑋42
0 1
0 1
0
0
0
0
0
0
0
0
𝑋53
𝑋63
𝜂11
𝑒11
𝜂21
𝑒21
𝜂12
𝑒32
+
𝜂22
𝑒42
𝜂13
𝑒53
𝜂23
𝑒63
Adding a Predictor
SEM
1
443.4
37.0
h1j
Classroom Regressions
Classroom Model
.85
-.34
(-.27)
h2j
.04
Xij
Yij
eij
234.6
Student Model
Not Just a Predictor: Two Outcomes
SEM: Random Slope
a1
y11
1
SEM: Bivariate Random Intercepts
Classroom Model
a2
y21
h1j
y22
1
y11
Classroom Model
a1 a2
y21
h1j
h2j
Yij
Yij
Xij
eij
e1ij
e2ij
h2j
y22
Xij
q
Student Model
q11
q21
q22
Student Model