Confirmatory Factor Analysis

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Confirmatory Factor Analysis
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In principal components analysis and factor
analysis, the goals of the estimation procedure
determine the solution (e.g., variance-maximizing
linear combinations or simple structure). In
confirmatory factor analysis, the theory or
expectations of the researcher determines the
solution. The researcher anticipates rather than
discovers a structure to the data.
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The quality of those expectations are then
judged by how well they match the original data
(e.g., the variance-covariance matrix for the
observed variables). The expectations imply
certain things about the variance-covariance
matrix, which can be tested for goodness of fit.
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In confirmatory factor analysis, expectations are
articulated very clearly. Those expectations then
dictate the elements of certain matrices that
reflect the relations between latent variables
(i.e., common factors) and observed variables,
the variances and covariances for latent
variables, and the variances and covariances for
specific factors (i.e., errors).
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Some elements of these matrices are dictated to
have certain values (e.g., 0) and others are free
to be estimated. Once the matrices have been
estimated, they can be used to reconstruct the
variance-covariance matrix for the original
variables. The similarity of this reconstructed
matrix to the actual variance-covariance matrix
indicates the quality of the solution and the
original guiding expectations.
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Expectations can be represented in a measurement
model that indicates how latent variables are
related to observed variables and to other latent
variables.
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x1
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X3
X2
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A measurement model implies that the observed
variables are weighted linear combinations of the
latent variables.
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These linear combinations resemble those from
exploratory factor analysis. A key difference is
that each of the common factors does not
contribute to all observed variables. The
underlying theory has dictated that some weights
are 0. If that assumption is wrong, then the
ability to reconstruct the variances and
covariances among the observed variables will
suffer.
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The assumptions about the weights are represented
in the L matrix, with some values fixed to be 0
and others free to be estimated. These weights
represent the relations between the observed
variables and the latent variables. They contain
the same kind of information as the structure and
pattern matrices in exploratory factor analysis.
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The assumptions about the weights may resemble
simple structure from exploratory factor
analysis, but that is not required. The
particular weights that are proposed should
reflect the underlying theory being tested.
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The guiding theoretical model also will make
assumptions about the number and relationships
among the latent variables. That defines the F
matrixthe variance-covariance matrix for the
latent variables.
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The model assumes that each person has a score on
each latent variable and a score on an error
latent variable for each measure.
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Because of expectations about the weights, the
linear combinations can be simplified.
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The model also makes assumptions about the
variances and covariances of the error latent
variables. These are contained in the Qd matrix.
When the errors are assumed to be uncorrelated,
the off-diagonal elements of this matrix will be
zero. This is an often made, but often
unrealistic assumption. Confirmatory factor
models allow correlated errors to be explicitly
modeled.
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This model implies uncorrelated errors.
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The covariance (and variances) among the observed
variables can be estimated from the latent
variable parameters. If the model-implied
parameters are correct, then the reproduced
variances and covariances (S) should be close to
the observed variances and covariances (S).
17
More generally
Reminder X is a linear combination of latent
variables and so its variance can be obtainable
from the variance-covariance matrix for the
latent variables along with the weights for
creating the linear combination.
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The estimation procedure most commonly used in
confirmatory factor analysis is maximum
likelihood estimation. This approach seeks the
parameter estimates that maximize the probability
of the data that were actually obtained. The
approach is best understood by comparing it to
the more familiar ordinary least squares approach
to estimation.
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In ordinary least squares, we seek a parameter
estimate that minimizes an error function. The
sample mean is a good example. There is no other
location for central tendency (e.g., mode,
median) that does a better job of minimizing the
following
Provided we find this rule satisfactory, then the
mean provides a good way of capturing the most
typical score in a distribution.
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Similarly, the regression coefficients in
multiple regression are derived so that they
minimize
Provided this rule is satisfactory, then the
regression weights are optimal for prediction.
21
As an alternative estimation procedure, maximum
likelihood finds the parameter estimates that
maximize the probability of the data. The maximum
likelihood estimate for the sample mean and
variance finds the values that maximize the
following
22
Note the explicit assumption that the data are
normally distributed. If that assumption is in
error, then the normal probability density
function will not provide an optimal solution to
the problem.
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Provided the data are normally distributed, the
maximum likelihood estimates for m and s make the
obtained data more likely than any other
parameter estimates. The estimation process also
produces standard errors, making hypothesis tests
possible as well. But, the validity of these
hypothesis tests rests on the validity of the
normality assumption.
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The approach can be extended to multivariate data
as well. We could seek the maximum likelihood
estimates for a bivariate normal distribution
25
Three bivariate normal distributions varying only
in the value of r. The validity of estimates of r
rely on the validity of the assumption of
bivariate normality.
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In the context of confirmatory factor analysis,
maximum likelihood estimates represent model
parameters that make the obtained data most
likely, within the constraints imposed by the
model.
27
All of the model assumptions are contained in the
reproduced variance-covariance matrix, S. The
probability density function, assuming
multivariate normality, is
28
The likelihood function that is maximized is thus
29
It is more convenient to work with the log of the
likelihood function, and, the function can be
simplified
30
The last part of this formula tends toward an
identity matrix as the reproduced
variance-covariance matrix approaches the actual
variance-covariance matrix. The trace of that
matrix product will be larger as it approaches an
identity matrix. A chi-square test allows a test
of the badness of fit of the reproduced and
obtained covariance matrices.
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The quality of the original model and its ability
to reproduce the actual variance-covariance
matrix is more easily gauged by the
goodness-of-fit index (GFI). The numerator of the
ratio tends toward zero as the reproduced
variance-covariance matrix approaches the actual
variance-covariance matrix. This index is similar
to R2 in multiple regression.
32
The estimation procedure can capitalize on
chance, so the adjusted goodness-of-fit index
(AGFI) was created to account for that. It is
similar to the adjusted R2 in multiple regression.
33
The mental abilities data set can be used to
compare the exploratory and confirmatory
approaches to factor analysis.
Hypothetical data (N 500) were created for
individuals completing a 12-section test of
mental abilities. All variables are in standard
form.
34
The scree test clearly shows the presence of
three factors
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On average, the three factors extracted can
accounted for about half of the variance in the
individual subtests.
36
Factor analysis accounts for less variance than
principal components and rotation shifts the
variance accounted for by the factors.
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The initial extraction . . .
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Oblique rotation . . .
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The correlations among the factors . . .
40
The confirmatory approach begins with an explicit
model that implies the elements of the key
matrices.
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Parameter estimates.
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Tests of significance for parameter estimates t
values.
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CHI-SQUARE WITH 51 DEGREES OF FREEDOM 55.50 (P
0.31) ESTIMATED NON-CENTRALITY PARAMETER (NCP)
4.50 90 PERCENT CONFIDENCE INTERVAL FOR NCP
(0.0 26.77) MINIMUM FIT FUNCTION VALUE
0.11 POPULATION DISCREPANCY FUNCTION VALUE (F0)
0.0090 90 PERCENT CONFIDENCE INTERVAL FOR F0
(0.0 0.054) ROOT MEAN SQUARE ERROR OF
APPROXIMATION (RMSEA) 0.013 90 PERCENT
CONFIDENCE INTERVAL FOR RMSEA (0.0
0.032) P-VALUE FOR TEST OF CLOSE FIT (RMSEA lt
0.05) 1.00 EXPECTED CROSS-VALIDATION INDEX
(ECVI) 0.22 90 PERCENT CONFIDENCE INTERVAL FOR
ECVI (0.21 0.26) ECVI FOR SATURATED MODEL
0.31 ECVI FOR INDEPENDENCE MODEL
3.98 CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66
DEGREES OF FREEDOM 1962.12 INDEPENDENCE AIC
1986.12 MODEL AIC 109.50 SATURATED AIC
156.00 INDEPENDENCE CAIC 2048.69 MODEL CAIC
250.29 SATURATED CAIC 562.74 ROOT MEAN SQUARE
RESIDUAL (RMR) 0.028 STANDARDIZED RMR
0.028 GOODNESS OF FIT INDEX (GFI) 0.98 ADJUSTED
GOODNESS OF FIT INDEX (AGFI) 0.97 PARSIMONY
GOODNESS OF FIT INDEX (PGFI) 0.64 NORMED FIT
INDEX (NFI) 0.97 NON-NORMED FIT INDEX (NNFI)
1.00 PARSIMONY NORMED FIT INDEX (PNFI)
0.75 COMPARATIVE FIT INDEX (CFI)
1.00 INCREMENTAL FIT INDEX (IFI) 1.00 RELATIVE
FIT INDEX (RFI) 0.96 CRITICAL N (CN) 696.82
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CHI-SQUARE WITH 51 DEGREES OF FREEDOM 55.50 (P
0.31) (This test models the variances and
covariances as implied by the parameter
expectations, df 78-12-12-3) CHI-SQUARE FOR
INDEPENDENCE MODEL WITH 66 DEGREES OF FREEDOM
1962.12 (This test only models the variances of
the variables and assumes all covariances are 0,
df 78-12) GOODNESS OF FIT INDEX (GFI)
0.98 ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.97
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CORRELATION MATRIX TO BE ANALYZED
V1 V2 V3
V4 M1 M2
-------- -------- -------- --------
-------- -------- V1 1.00
V2 0.52 1.00 V3 0.52
0.48 1.00 V4 0.54 0.54
0.49 1.00 M1 0.16
0.22 0.19 0.23 1.00 M2
0.22 0.28 0.23 0.23
0.48 1.00 M3 0.19 0.21
0.13 0.17 0.47 0.46
M4 0.22 0.23 0.23 0.17
0.48 0.49 R1 0.23
0.25 0.29 0.23 0.14 0.22
R2 0.22 0.17 0.21
0.17 0.17 0.23 R3 0.28
0.22 0.26 0.22 0.18
0.23 R4 0.27 0.25 0.24
0.26 0.21 0.23
CORRELATION MATRIX TO BE ANALYZED
M3 M4 R1 R2
R3 R4 --------
-------- -------- -------- --------
-------- M3 1.00 M4
0.50 1.00 R1 0.15 0.28
1.00 R2 0.11 0.19
0.47 1.00 R3 0.17 0.25
0.50 0.51 1.00 R4
0.15 0.23 0.51 0.52 0.52
1.00
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FITTED COVARIANCE MATRIX
V1 V2 V3 V4 M1
M2 -------- --------
-------- -------- -------- --------
V1 1.00 V2 0.53 1.00
V3 0.51 0.49 1.00 V4
0.54 0.52 0.50 1.00
M1 0.21 0.20 0.20 0.21
1.00 M2 0.21 0.21
0.20 0.21 0.48 1.00 M3
0.21 0.20 0.19 0.20
0.46 0.47 M4 0.22 0.21
0.21 0.22 0.49 0.50
R1 0.24 0.23 0.22 0.23
0.19 0.20 R2 0.23
0.23 0.22 0.23 0.19 0.19
R3 0.25 0.24 0.23
0.24 0.20 0.20 R4 0.25
0.24 0.23 0.25 0.20
0.21 FITTED COVARIANCE MATRIX
M3 M4 R1 R2
R3 R4 -------- --------
-------- -------- -------- --------
M3 1.00 M4 0.48 1.00
R1 0.19 0.20 1.00 R2
0.19 0.20 0.48 1.00
R3 0.20 0.21 0.50 0.50
1.00 R4 0.20 0.21
0.51 0.51 0.53 1.00
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FITTED RESIDUALS V1
V2 V3 V4 M1
M2 -------- -------- --------
-------- -------- -------- V1
0.00 V2 -0.01 0.00 V3
0.01 -0.01 0.00 V4
0.00 0.02 -0.01 0.00 M1
-0.05 0.01 -0.01 0.02
0.00 M2 0.01 0.07 0.03
0.02 0.00 0.00 M3
-0.02 0.01 -0.06 -0.04 0.01
-0.01 M4 0.00 0.01
0.02 -0.04 -0.01 -0.01 R1
-0.01 0.02 0.07 -0.01
-0.05 0.02 R2 -0.01 -0.05
-0.01 -0.06 -0.03 0.04
R3 0.03 -0.02 0.03 -0.03
-0.02 0.03 R4 0.02
0.00 0.01 0.02 0.01 0.03
FITTED RESIDUALS M3
M4 R1 R2 R3
R4 -------- -------- --------
-------- -------- -------- M3
0.00 M4 0.01 0.00 R1
-0.04 0.08 0.00 R2
-0.08 -0.01 -0.01 0.00 R3
-0.02 0.04 0.00 0.01
0.00 R4 -0.05 0.01 0.00
0.01 -0.01 0.00
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STANDARDIZED RESIDUALS
V1 V2 V3 V4 M1
M2 -------- --------
-------- -------- -------- --------
V1 0.00 V2 -0.87 0.00
V3 0.77 -0.64 0.00 V4
0.17 1.20 -0.66 0.00
M1 -1.43 0.45 -0.21 0.63
0.00 M2 0.16 2.27
0.79 0.66 0.19 0.00 M3
-0.51 0.37 -1.84 -1.12
0.81 -0.40 M4 0.05 0.38
0.63 -1.40 -0.49 -1.08
R1 -0.30 0.59 2.23 -0.24
-1.44 0.68 R2 -0.35
-1.68 -0.27 -1.83 -0.76
1.11 R3 1.02 -0.62 0.92
-0.92 -0.59 0.81 R4
0.60 0.12 0.21 0.57 0.36
0.83 STANDARDIZED RESIDUALS
M3 M4 R1 R2
R3 R4 --------
-------- -------- -------- --------
-------- M3 0.00 M4
1.00 0.00 R1 -1.10 2.45
0.00 R2 -2.38 -0.34
-0.32 0.00 R3 -0.70 1.15
-0.03 0.74 0.00 R4
-1.44 0.45 -0.21 0.74 -0.91
0.00
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!Confirmatory Factor Analysis Model, Mental
Abilities
QPLOT OF STANDARDIZED RESIDUALS
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3.5
STANDARDIZED RESIDUALS
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The chosen model fits the data quite well. How
would other models do? A complete confirmatory
analysis would not only test the preferred model
but also examine alternative models to assess how
easily they could account for the data. To the
extent that reasonable alternatives exist, the
preferred model must be considered with more
caution.
53
Other models can easily be tested.
Verbal
Math
Analytic
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CHI-SQUARE WITH 54 DEGREES OF FREEDOM 214.02 (P
0.0) CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66
DEGREES OF FREEDOM 1962.12 GOODNESS OF FIT
INDEX (GFI) 0.93 ADJUSTED GOODNESS OF FIT
INDEX (AGFI) 0.90
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F1
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CHI-SQUARE WITH 54 DEGREES OF FREEDOM 770.57 (P
0.0) CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66
DEGREES OF FREEDOM 1962.12 GOODNESS OF FIT
INDEX (GFI) 0.73 ADJUSTED GOODNESS OF FIT
INDEX (AGFI) 0.61
63
F1
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CHI-SQUARE WITH 53 DEGREES OF FREEDOM 488.89 (P
0.0) CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66
DEGREES OF FREEDOM 1962.12 GOODNESS OF FIT
INDEX (GFI) 0.84 ADJUSTED GOODNESS OF FIT
INDEX (AGFI) 0.76
68
Verbal
Math
Analytic
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d4
d9
d10
d11
d12
d6
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Oops!
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CHI-SQUARE WITH 51 DEGREES OF FREEDOM 742.32 (P
0.0) CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66
DEGREES OF FREEDOM 1962.12 GOODNESS OF FIT
INDEX (GFI) 0.75 ADJUSTED GOODNESS OF FIT
INDEX (AGFI) 0.61
73
Verbal
Math
Analytic
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d4
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CHI-SQUARE WITH 54 DEGREES OF FREEDOM 1296.47
(P 0.0) CHI-SQUARE FOR INDEPENDENCE MODEL WITH
66 DEGREES OF FREEDOM 1962.12 GOODNESS OF FIT
INDEX (GFI) 0.70 ADJUSTED GOODNESS OF FIT
INDEX (AGFI) 0.56
78
Just because one model fits the data well, that
does not mean that other models cant fit the
data well too. It is important to test
alternative models to provide some sense of how
well the preferred model performs against
competitors.
79
One distinct advantage of confirmatory factor
analysis over exploratory approaches is that
goodness-of-fit indices can determine more
precisely how well the solution fits the
data. The principal components analysis of the
Ofir and Simonson (2001) Need for Cognition data
(N 201) suggested a single component solution.
80
1. I would prefer complex to simple
problems. 2. I like to have the
responsibility of handling a situation that
involves a lot of thinking. 3.
Thinking is not my idea of fun. 4. I
would rather do something that requires little
thought than something that is sure to challenge
my thinking abilities. 5. I try to
anticipate and avoid situations where there is
likely a chance I will have to think in depth
about something. 6. I find joy in
deliberating hard and for long hours.
7. I only think as hard as I have to.
8. I prefer to think about small, daily projects
to long-term ones. 9. I like tasks
that require little thought once Ive learned
them. 10. The idea of relying on thought
to make my way to the top appeals to me.
11. I really enjoy a task that involves coming
up with new solutions to problems. 12.
Learning new ways to think doesnt excite me very
much. 13. I prefer my life to filled
with puzzles that I must solve. 14. The
notion of thinking abstractly appeals to me.
15. I would prefer a task that is
intellectual, difficult, and important to one
that is somewhat important but does not require
much thought. 16. I feel relief rather
than satisfaction after completing a task that
required a lot of attention. 17. Its
enough for me that something gets the job done I
dont care how or why it works.. 18. I
usually end up deliberating about issues even
though they do not affect me personally.
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Each items is rated using the following scale 1
very characteristic of me 2 somewhat
characteristic of me 3 neutral 4 somewhat
uncharacteristic of me 5 very uncharacteristic
of me
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One component is the best solution here, but is
it a good solution?
Only 32 of the variance is accounted for by the
component.
84
One-factor confirmatory factor analysis
Parameter estimates.
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One factor confirmatory factor analysis t-values
for parameter estimates. All parameter estimates
are significantly different from 0.
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The fit statistics suggest that the model does
not provide an especially good description of the
variances and covariances among the Need for
Cognition items

• CHI-SQUARE WITH 135 DEGREES OF FREEDOM 422.61
  (P 0.0)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 153
  DEGREES OF FREEDOM 1535.33
• GOODNESS OF FIT INDEX (GFI) 0.81
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.76
  

89
Does a two-factor model fit better, with separate
latent variables for items scored in positive
versus negative directions?
90
Better, but still not great

• CHI-SQUARE WITH 134 DEGREES OF FREEDOM 361.07
  (P 0.0)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 153
  DEGREES OF FREEDOM 1535.33
• GOODNESS OF FIT INDEX (GFI) 0.85
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.81

91
The next step? Probably scale refinement. The
confirmatory factor analysis may only tell us
that the presumed model is not adequate to
explain the pattern of variances and covariances
in the data. It wont necessarily identify a
specific problem that needs to be addressed.
92

• Additional important issues in confirmatory
  factor analysis
• Nested models
• Model identification
• Correlation versus covariance matrices
• Other goodness-of-fit indices
• Model modification (exploratory confirmatory
  factor analysis)

93
Careful use of confirmatory factor analysis
should test competing models and attempt to
examine nested models that place more
restrictions on the original formulation in an
attempt to determine the most parsimonious
account of the data.
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A multitrait-multimethod matrix problem.
95
The standard multitrait-multimethod matrix is
used to determine the reliability, convergent
validity, and discriminant validity for measures
of multiple traits collecting using different
methods.
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Parameter estimates for the hypothesized model.
98
An excellent fit

• CHI-SQUARE WITH 39 DEGREES OF FREEDOM 46.69 (P
  0.19)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.98
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.97

99
An alternative model might argue that there is no
method variance and so the method latent
variables can safely be dropped.
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Is there method variance? Restricting the model
to just traits tests whether eliminating method
variance matters.
102
Clearly there is method variance because
eliminating this part of the original model
reduces the goodness of fit.

• CHI-SQUARE WITH 51 DEGREES OF FREEDOM 695.28 (P
  0.0)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.71
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.56

103
Another alternative model could argue that all of
the covariance among items is due to common
methods of measurementa pure method artifact
model.
Paper
IRT
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Is trait variance important? Restricting the
model to just methods determines if eliminating
traits reduces the goodness of fit.
106
A poor fit. Clearly trait variance is necessary.

• CHI-SQUARE WITH 54 DEGREES OF FREEDOM 1184.25
  (P 0.0)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.65
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.49

107
Perhaps there are elements of the methods that
are commonboth might use the same question
content, require the same response type (e.g.,
True vs. False), etc. This can be modeled by
allowing the method latent variables to be
correlated.
Paper
IRT
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Can the model be improved by allowing the methods
factors to be correlated?
110
No, that does not improve matters. Still a poor
fit

• CHI-SQUARE WITH 53 DEGREES OF FREEDOM 1094.18
  (P 0.0)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.66
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.50

111
Are the separate traits really just a single
latent variable?
If so, then there is no evidence for discriminant
validity.
Trait
Paper
IRT
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Can the basic methods-only model be improved by
adding a single trait latent variable?
114
Including trait information, but assuming that
only a single trait latent variable exists does
not fit the data adequately

• CHI-SQUARE WITH 42 DEGREES OF FREEDOM 601.40 (P
  0.0)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.80
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.62

115
Previously, adding a correlation between methods
latent variables improved fit.
Does it make the fit adequate now?
Trait
Paper
IRT
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Does it help to let the methods latent variables
correlate?
118
Still not an acceptable fit

• CHI-SQUARE WITH 41 DEGREES OF FREEDOM 513.62 (P
  0.0)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.82
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.65

119
Trait
A pure single trait model could also be proposed.
120
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121
Does a one-factor trait model provide an adequate
fit?
122
Definitely not

• CHI-SQUARE WITH 54 DEGREES OF FREEDOM 1334.96
  (P 0.0)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.65
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.50

123
Of the several models tested, the original
multitrait multimethod model provided an
excellent fit and apparently much better fit than
the alternatives. Is a simpler version of that
model possible?
124
For example, can all weights for a latent
variable be set equal?
125
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126
Can all weights for a particular latent variable
be set equal?
127
Yes, that is nearly as good as the original model
and provides a more parsimonious account of the
data

• CHI-SQUARE WITH 58 DEGREES OF FREEDOM 65.11 (P
  0.24)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.98
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.97

128
Can the model be further constrained so that ALL
trait loadings are the same, and, ALL methods
loadings are the same?
129
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130
Can all trait loadings be set equal and all
methods loadings be set equal?
131
Still an excellent fit and more parsimony

• CHI-SQUARE WITH 61 DEGREES OF FREEDOM 66.72 (P
  0.29)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.98
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.97

132
Can the latent trait correlations be set equal?
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134
Can the latent trait correlations be set equal?
135
Still not much reduction in fit

• CHI-SQUARE WITH 63 DEGREES OF FREEDOM 67.01 (P
  0.34)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.98
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.97

136
Can the error variances (not shown) be set equal
as well?
137
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138
Can the error variances be set equal?
139
A very simple model, with excellent fit

• CHI-SQUARE WITH 74 DEGREES OF FREEDOM 77.09 (P
  0.38)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.98
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.97

140
When a parameter in one model is fixed or
constrained in a second model, the models are
said to be nested. The difference between their
respective chi-square values is itself a
chi-square, with degrees of freedom equal to the
difference in the two models degrees of freedom.
The new chi-square tests whether the two models
have significantly different fit to the data.
141
The Identification Problem
A model must be identified before valid and
unique parameter estimates can be obtained.
Identification means that there is sufficient
information in the data and model to provide
unique estimation of the free parameters.
Identification is obtained by restricting the
model in some way. Example There are an infinite
number of solutions to the following simple
model X Y 15. A unique solution can only
be found by restricting the model in some way,
for example, by requiring that X 5. Then the
unique solution for Y is 10.
142
Similarly, and with greater complexity,
confirmatory factor models must have a sufficient
number of restrictions before the parameters can
be estimated uniquely. Models are restricted by
either forcing some parameters to values of 0 or
constraining some parameters to be equal to other
parameters. One important condition for
identification is called the order condition.
This means that the number of parameters
estimated in the model is less than or equal to
the number of distinct values in the
variance-covariance matrix (S). Another important
requirement is that latent variables have a
specified scale.
143
Scaling For a model to be identified, the latent
variables must be given a scale. This can be done
by either standardizing the latent variables
(setting the diagonals of the F matrix to 1.00)
or by setting one l for each latent variable to
1.00 to give that latent variable the same scale
as the observed variable. Unfortunately, this
can produce different results for the tests of
individual parameters.
144
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145
Parameter estimates when latent variable scales
are set by standardizing the F matrix.
146
The t-values for testing the parameter values for
significance.
147
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148
Parameter estimates when latent variable scales
are set by fixing elements of the L matrix to
1.00.
149
The t values for testing the parameter values for
significance.
150
Depending on the method of scaling, the t values
for parameter estimates can vary.
Standardized F matrix
Scaling by l
151
The method of scaling has no effect on the
goodness-of-fit. Both approaches will produce the
same overall goodness of fit

• CHI-SQUARE WITH 39 DEGREES OF FREEDOM 46.69 (P
  0.19)
• CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES
  OF FREEDOM 2346.60
• GOODNESS OF FIT INDEX (GFI) 0.98
• ADJUSTED GOODNESS OF FIT INDEX (AGFI) 0.97

152
Because the method of scaling has no effect on
the overall goodness-of-fit, a test of any given
parameter can be obtained by taking the
difference between the goodness of fit chi-square
with that parameter free and the goodness of fit
chi-square when that parameter is fixed. The
difference is a test (c2, 1 df) of whether the
two models are the same, which is the same as
asking if the parameter is 0.
153
Fixing f21 to 0 in the l scaled model produces an
overall goodness of fit c2 of 52.63 (df 40).
The same goodness of fit is obtained in the model
that uses a standardized F matrix with f21 set to
0. The model that has f21 free produces an
overall goodness of fit c2 of 46.69 (df
39). The difference is a c2 of 5.94 (df 1),
significant at p lt .05.
154
Analyzing Correlation versus Covariance Matrices
Confirmatory factor analysis models are based on
the decomposition of covariance matrices, not
correlation matrices. The solutions hold,
strictly speaking, for the analysis of covariance
matrices. To the extent that the solution depends
on the scale of the variables, analyses based on
covariance matrices and correlation matrices can
differ.
155
Other Goodness-of-Fit Indices
In addition to the chi-square goodness of fit
test and the fit indices (GFI, AGFI), another
common fit statistic is based on the difference
between the original covariance matrix and the
covariance matrix implied by the model.
Root-Mean-Squared Residual (RMR)
Small values are desirable, indicating a good
reproduction of the original covariance matrix.
156
One other commonly reported fit index is the
Root-Mean-Square Error of Approximation (RMSEA),
a chi-square goodness of fit statistic, adjusted
for degrees of freedom and sample size
157
Some Common Rules of Thumb for Model Fit
158
Model Modification(exploratory confirmatory
factor analysis)
When a presumed model does not fit, changes to
the model can be tested for goodness-of-fit,
but the exercise is no longer confirmatory and
may capitalize on chance and other biases.
Cross-validation is essential in such cases. Some
software provides guidance in the modification
process by identifying the fixed parameters that
could be set free to improve the model fit.
159
In LISREL, the modification indices are the
changes in the goodness-of-fit c2 that would
result from setting that parameter free.
160
In LISREL, the modification indices are the
changes in the goodness-of-fit c2 that would
result from setting that parameter free.
161
The parameter estimates if set free are also
provided. The approach assumes that only a single
parameter is set free at a time.
162
Reliability and Attenuation Confirmatory factor
analysis has a close relation to classical
measurement theory. When the latent variables are
standardized, the l are the correlations between
each variable and the true score. The square of
these l are the individual item reliabilities. If
these are averaged, they can be used in the
Spearman-Brown formula to provide an estimate of
standardized coefficient alpha
163
An alternative estimate of reliability is
164
For the Need for Cognition data, the one factor
confirmatory factor analysis parameter estimates
yield reliability estimates of a .90 r2c
.91 A good reminder that reliability is a faulty
indicator of dimensionality.
165
fVM
Math
Verbal
Analytic
l1V
l5M
X5
X1
Inherent in the measurement model is the
classical measurement theory notion of
attenuation. The correlation between X1 and X5 is
estimated by l1VfVMl5M.
166
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167
When used wisely, confirmatory factor analysis is
a powerful tool that can test well-specified
models and compare competing models. Like any
statistical procedure, however, it can be biased
when it is used in a more exploratory manner (as
in model modification). In those cases, careful
cross-validation is necessary to insure the
validity of the best fitting model.