View
214
Download
0
Category
Tags:
Preview:
Citation preview
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
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
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
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
Recommended