I will present a set of routines to conduct a one-factor confirmatory factor analysis in Stata. The use of Mata in programming will be highlighted. Corrections for non-normality, as common in the structural equation modeling literature, will be demonstrated. Indications for further development into multifactor models and, eventually, structural equation models, will be given.

Stas Kolenikov

English

Research Papers in Economics

cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
cfa1: Conﬁrmatory Factor Analysis
with a Single Factor
Stas Kolenikov
Department of Statistics
University of Missouri-Columbia
NASUG, Boston, MA, July 24, 2006cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Outline
1 Factor analysis
2 Implementation
3 Demonstration
4 Extensionscfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Factor analysis
If one has p variables y1,...,yp, are there q < p factors
explaining most of the variability in y’s?
• Exploratory factor analysis: ﬁnd (simple) covariance
structure in the data; a standard multivariate technique
— see [MV] factor
• Conﬁrmatory factor analysis: upon having formulated a
theoretical model, see if it ﬁts the data; estimate the
parameters and assess goodness of ﬁt. Simplest of
structural equation models (SEM)
• Principal components analysis is neither of the above,
but closer to EFA in spiritcfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Factor analysis
If one has p variables y1,...,yp, are there q < p factors
explaining most of the variability in y’s?
• Exploratory factor analysis: ﬁnd (simple) covariance
structure in the data; a standard multivariate technique
— see [MV] factor
• Conﬁrmatory factor analysis: upon having formulated a
theoretical model, see if it ﬁts the data; estimate the
parameters and assess goodness of ﬁt. Simplest of
structural equation models (SEM)
• Principal components analysis is neither of the above,
but closer to EFA in spiritcfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Factor analysis
If one has p variables y1,...,yp, are there q < p factors
explaining most of the variability in y’s?
• Exploratory factor analysis: ﬁnd (simple) covariance
structure in the data; a standard multivariate technique
— see [MV] factor
• Conﬁrmatory factor analysis: upon having formulated a
theoretical model, see if it ﬁts the data; estimate the
parameters and assess goodness of ﬁt. Simplest of
structural equation models (SEM)
• Principal components analysis is neither of the above,
but closer to EFA in spiritcfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Factor analysis
If one has p variables y1,...,yp, are there q < p factors
explaining most of the variability in y’s?
• Exploratory factor analysis: ﬁnd (simple) covariance
structure in the data; a standard multivariate technique
— see [MV] factor
• Conﬁrmatory factor analysis: upon having formulated a
theoretical model, see if it ﬁts the data; estimate the
parameters and assess goodness of ﬁt. Simplest of
structural equation models (SEM)
• Principal components analysis is neither of the above,
but closer to EFA in spiritcfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Path diagrams
A typical CFA model looks like...
y1 y2 y3 y4
ξ1
δ1 δ2 δ3 δ4
1
1
0.80 .8
1.2
0.4 0.5 0.6 1.5
.
1cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Equation notation
...which is the same as saying
ξ1 is unobserved, V[ξ1] = 1 = φ
y1 = λ1ξ1 + δ1, V[δ1] = 0.4 = θ1,λ1 = 1,
y2 = λ2ξ1 + δ2, V[δ2] = 0.5 = θ2,λ2 = 0.8,
y3 = λ3ξ1 + δ3, V[δ3] = 0.6 = θ3,λ3 = 0.8,
y4 = λ4ξ1 + δ3, V[δ4] = 1.5 = θ4,λ4 = 1.2cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Likelihood formulation
y = Λξ1 + δ,
Cov[y] = ΛφΛ0 + Θ = Σ(θ),
lnL = −
np
2
ln(2π) −
n
2
ln|Σ| −
n − 1
2
trSΣ−1(θ)
where S is the sample covariance matrixcfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Likelihood formulation
y = Λξ1 + δ,
Cov[y] = ΛφΛ0 + Θ = Σ(θ),
lnL = −
np
2
ln(2π) −
n
2
ln|Σ| −
n − 1
2
trSΣ−1(θ)
where S is the sample covariance matrixcfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Likelihood formulation
y = Λξ1 + δ,
Cov[y] = ΛφΛ0 + Θ = Σ(θ),
lnL = −
np
2
ln(2π) −
n
2
ln|Σ| −
n − 1
2
trSΣ−1(θ)
where S is the sample covariance matrixcfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Identiﬁcation conditions
Not all parameters are necessarily estimable...
• Number of parameters ≤ p(p + 1)/2
• Eξ1 is not identiﬁed, assumed zero
• Only λkφ1/2, or ratios λk/λj, are identiﬁed
• Set φ = 1
• Set one of λk = 1cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Identiﬁcation conditions
Not all parameters are necessarily estimable...
• Number of parameters ≤ p(p + 1)/2
• Eξ1 is not identiﬁed, assumed zero
• Only λkφ1/2, or ratios λk/λj, are identiﬁed
• Set φ = 1
• Set one of λk = 1cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Identiﬁcation conditions
Not all parameters are necessarily estimable...
• Number of parameters ≤ p(p + 1)/2
• Eξ1 is not identiﬁed, assumed zero
• Only λkφ1/2, or ratios λk/λj, are identiﬁed
• Set φ = 1
• Set one of λk = 1cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Stata implementation
• Stata’s ml model lf structure
• Identiﬁcation: by the ﬁrst indicator, or by φ = 1;
implemented as constraints supported by ml
• Improper solutions workarounds: what if ˆ θk ≤ 0?
• Goodness of ﬁt tests
• Corrections for multivariate kurtosis traditional for SEM
literature (Satorra-Bentler standard errors and χ2)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Stata implementation
• Stata’s ml model lf structure
• Identiﬁcation: by the ﬁrst indicator, or by φ = 1;
implemented as constraints supported by ml
• Improper solutions workarounds: what if ˆ θk ≤ 0?
• Goodness of ﬁt tests
• Corrections for multivariate kurtosis traditional for SEM
literature (Satorra-Bentler standard errors and χ2)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Stata implementation
• Stata’s ml model lf structure
• Identiﬁcation: by the ﬁrst indicator, or by φ = 1;
implemented as constraints supported by ml
• Improper solutions workarounds: what if ˆ θk ≤ 0?
• Goodness of ﬁt tests
• Corrections for multivariate kurtosis traditional for SEM
literature (Satorra-Bentler standard errors and χ2)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Stata implementation
• Stata’s ml model lf structure
• Identiﬁcation: by the ﬁrst indicator, or by φ = 1;
implemented as constraints supported by ml
• Improper solutions workarounds: what if ˆ θk ≤ 0?
• Goodness of ﬁt tests
• Corrections for multivariate kurtosis traditional for SEM
literature (Satorra-Bentler standard errors and χ2)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Stata implementation
• Stata’s ml model lf structure
• Identiﬁcation: by the ﬁrst indicator, or by φ = 1;
implemented as constraints supported by ml
• Improper solutions workarounds: what if ˆ θk ≤ 0?
• Goodness of ﬁt tests
• Corrections for multivariate kurtosis traditional for SEM
literature (Satorra-Bentler standard errors and χ2)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Mata usage
• Normal likelihood, observation by observation; a lot of
st view’s and Cholesky decompositions. Earlier versions
used mkmat... that was a disaster!
• Satorra-Bentler corrections:
[ acov(b θ) = (n − 1)−1 ˆ ∆0Vn ˆ ∆
−1 ˆ ∆0VnΓnVn ˆ ∆
 ˆ ∆0Vn ˆ ∆
−1
ˆ ∆ =
∂σ
∂θ


b θ
,Vn = 1/2D0(A−1
n ⊗ A−1
n )D,
An
p
−→ Σ,vecΣ = D vechΣ
Γn =
1
n − 1
X
i
(bi − ¯ b)(bi − ¯ b)0
bi = (yi − ¯ y)(yi − ¯ y)0
• Mata functions for D (vec, invvech), ˆ ∆ (analytic
derivatives, but with lots of matrix operations), b (st view)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Mata usage
• Normal likelihood, observation by observation; a lot of
st view’s and Cholesky decompositions. Earlier versions
used mkmat... that was a disaster!
• Satorra-Bentler corrections:
[ acov(b θ) = (n − 1)−1 ˆ ∆0Vn ˆ ∆
−1 ˆ ∆0VnΓnVn ˆ ∆
 ˆ ∆0Vn ˆ ∆
−1
ˆ ∆ =
∂σ
∂θ


b θ
,Vn = 1/2D0(A−1
n ⊗ A−1
n )D,
An
p
−→ Σ,vecΣ = D vechΣ
Γn =
1
n − 1
X
i
(bi − ¯ b)(bi − ¯ b)0
bi = (yi − ¯ y)(yi − ¯ y)0
• Mata functions for D (vec, invvech), ˆ ∆ (analytic
derivatives, but with lots of matrix operations), b (st view)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Mata usage
• Normal likelihood, observation by observation; a lot of
st view’s and Cholesky decompositions. Earlier versions
used mkmat... that was a disaster!
• Satorra-Bentler corrections:
[ acov(b θ) = (n − 1)−1 ˆ ∆0Vn ˆ ∆
−1 ˆ ∆0VnΓnVn ˆ ∆
 ˆ ∆0Vn ˆ ∆
−1
ˆ ∆ =
∂σ
∂θ


b θ
,Vn = 1/2D0(A−1
n ⊗ A−1
n )D,
An
p
−→ Σ,vecΣ = D vechΣ
Γn =
1
n − 1
X
i
(bi − ¯ b)(bi − ¯ b)0
bi = (yi − ¯ y)(yi − ¯ y)0
• Mata functions for D (vec, invvech), ˆ ∆ (analytic
derivatives, but with lots of matrix operations), b (st view)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Syntax
cfa1 varlist [if ...] [in ...]
[[pweight=weight]],
unitvar free posvar
constraint(numlist)
cluster(varname) svy
robust vce(oim|opg|robust|sbentler)
from(starting values) level(#)
ml optionscfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Demonstration
Do something!cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Broader CFA models?
y1 y2 y3
ξ1
y4 y5 y6
ξ2
δ1 δ2 δ3 δ4 δ5 δ6
φ11 φ22 φ12
1
λ2
λ3 1
λ5
λ6
θ1 θ2 θ3 θ4 θ5 θ6
.
1
cfa (xi1: y1 y2 y3) (xi2: y4 y5 y6)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
General structural equation
models?
x1 x2 x3
ξ1
δ1 δ2 δ3
η1
ζ1
y1  1θ4
y2  2θ5
y3  3θ6
z1 φ11
φ22 φ12
1 λ2
λ3
β11
β12
θ1 θ2 θ3 σ1
θ4
.
1
sem (xi1: x1 x2 x3) (eta1 = .xi1 z1: y1 y2 y3)
Parsing structural equation models is generally a nightmare...cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
GLLAMM
The minimal syntax for a CFA example for gllamm package
g long id = n
reshape long y, i(id) j(k)
g byte one = 1
tab k, gen(d)
eq main: d1 d2 d3 d4
gllamm y d*, eq(main) i(id) nocons s(main)cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Mplus
The minimal syntax for Mplus SEM package (text ﬁle input
and output)
Title: Simple CFA example
Data: File = cfa-example.txt;
Type = individual;
Variable:
Names = y1 y2 y3 y4;
Analysis:
Type = general;
Model:
xi by y1 y2 y3 y4;cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
SAS PROC CALIS
The minimal syntax for SAS PROC CALIS
proc calis data = cfa-example;
lineqs
y1 = f1 + e1,
y2 = l2 f1 + e2,
y3 = l3 f1 + e3,
y4 = l4 f1 + e4;
std
e1 = theta1,
e2 = theta2,
e3 = theta3,
e4 = theta4,
f1 = phi;
run;
Rigid variable names (F for factors/latent variables, E and D
for errors and disturbances).cfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
LISREL
TI PRELIS processing of CFA example
DA NI = 4 NO = 200 MI = -999
LA
y1 y2 y3 y4
RA = cfa-example.txt
OU RA = cfa-example.psf
TI Estimation of CFA example
DA MA = CM NI = 4 NO = 200
RA = cfa-example.psf
LK xi
MO NK = 1 NX = 4
LX = FU, FR
TD = DI, FR
FI LX(1,1) VA 1 LX(1,1)
PD
OU ME = ML EFcfa1: Simple
CFA Models
Stas
Kolenikov
U of Missouri
Factor
analysis
Implementation
Demonstration
Extensions
Download info
net from ///
http://www.missouri.edu/˜kolenikovs/stata
net install cfa1
findit cfa1

Confirmatory factor analysis in Stata

http://repec.org/nasug2006/NASUG-Kolenikov-cfa1.pdf

Confirmatory factor analysis in Stata

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Research Papers in Economics