Measurement and Structural Models
1
Measurement Models and Structural Models
Structural Equation Models: Models that combine confirmatory factor analyses with
regression analyses.
They consist of two parts . . .
Measurement model. The first part is called the measurement model. It is the part that
specifies how the independent and dependent latent variables in the model are to be
measured – their indicators, and the relationships between the latent variables and the indicators.
The measurement model is usually a confirmatory factor analysis model with 1 or more latent
variable(s) and multiple indicators for each latent variable.
The measurement model specifies how each construct represented by a latent variable is
indicated by observed variables. It connects the latent variables in the model to the real world.
A measurement model is both an end and a means to an end depending on the researcher.
It’s an end if it specifies a way to measure a construct that has not been specified previously.
Some researchers, e.g., Biderman, are mainly interested in doing this.
For example, we’re working on a collection of models that specify a way to measure faking and
also the expression of participant affect. (A reviewer once remarked at the incongruity of
measuring both with the same type of model.) Although not new with our work, these issues
have not been pursued as far as we have taken them. The example that follows illustrates the
measurement models with which we’ve been working. By demonstrating that the measurement
model we created fits the data well, we have furthered our understanding of how persons respond
to personality questionnaires.
Structural model. Often, though, a measurement model is a means to an end. It is a way of
measuring a construct uncontaminated by measurement error in order to allow the testing of
relationships of that construct to other constructs. Those relationships, when incorporated into
the SEM, are called the structural model of the SEM. Many researchers are mainly interested
in the relationships. For them, the measurement model is simply a means to better assess the
constructs involved in those relationships.
Structural models are typically regression analyses involving the latent variables within a SEM.
The regression analyses can be between latent variables and other latent variables or between
latent variables and observed variables. They are called structural regressions by some
persons.
Measurement and Structural Models
2
Example 1A: A measurement model for Big Five personality characteristics and a faking
characteristic.
Below is an example of a measurement model from Wrensen & Biderman (2005). Participants responded
to a Big Five questionnaire twice – once under instructions to respond honestly, again under instructions
to “fake good”. The model specifies how the Big Five latent variables and a faking latent variable are to
be measured. Mike – Discuss it in detail, including the two types of correlated residuals.
Measurement Model
Chi-square = 564.595
df = 350
p = .000
CFI = .938
GFI = .838
RMSEA = .058
.77
.88
.82
.87
E
.27
.22
.24
.18
HETL1.67
e1
HETL2.77
e2
HETL3.40
e3
FETL1.16
e4
.55
FETL2 .57
.63
.42
e5
FETL3
e6
.17
.19
.17
.26
.51
A
.04
.93
.28
.46
.18
e7
HATL2.08
e8
HATL3.21
e9
FATL1.08
.12
.20
.00
HATL1.86
FATL2
.02
FATL3
C
-.08
.29
.12
.27
FA represents
.33
variance in the F
testlets that is
a) common across all
5 dimensions and
b) not attributable to
any of the Big Five
characteristics.
It’s a Faking Ability
factor.
S
.76
.34
.07
.57
.22
.10
.84
.72
.80
.77
.31
.26
.33
.75
.65
.66
.27
.19
FA
.27
.60
.63
.10
e11
e12
HCTL2.53
e14
HCTL3.66
e15
FCTL3
.74
.28
-.05
.35
e10
e13
FCTL2.62
.89
O
.40
.11
HCTL1.63
FCTL1.67
.09
.78
.77
.88
.79
.73
.76
.17
.17
.69
.17
.12
.03
e16
.07
e17
.39e18
.71
HSTL1
e19
.79
HSTL2.55
e20
HSTL3.63
e21
FSTL1.64
.09 e22
FSTL2.53
FSTL3
.08
e23
.15
.38e24
.71
.79
HOTL1.62
e25
.57
HOTL2.33
e26
HOTL3.51
e27
FOTL3
.66
.15
.84
FOTL1.43
FOTL2.43
These
covariances are
estimated
separately
because they’re
assumed to
reflect
similarities in
responses due
to similarities in
item wording.
.71
.16
.09 e28
-.03
e29
.22 e30
.21
These
covariances
are estimated
separately
because
they’re
assumed to
reflect slight
differences in
faking
specific to
the Big Five
dimensions.
We included
them to
improve
goodness-offit values.
We’ve since
quit
estimating
them,
although the
issue needs to
be revisited.
Measurement and Structural Models
3
Example 1B. Adding a Structural model of faking ability to the above measurement model.
The following schematic path diagram shows the measurement model above along with a
structural model relating faking ability to Cognitive Ability, Emotional Intelligence, Integrity,
Social Desirability, Self Monitoring, and the Order within the research procedure that the faking
instructions occurred (1st vs. 2nd). The structural model is in red. In this case, the structural
model is between exogenous observed variables and the endogenous Faking Ability latent
variable.
Mean of 3 standardized loadings: .88, .82, .87 (see above).
.85
E
H-E
.18
Mean of 3
residual
correlations.
.25
.55
.26
C
.51
F-E
.55
A
H-A
.39
.80
.14
.37
S
.07
F-A
H-C
.30
.83
.43
.71
O
.74
CA
.19*
EI
.16*
F-C
H-S
.19
.30
.37
.18*
.56
.21
F-S
FA
INT
H-O
-.24*
SD
-.02
.49
F-O
.37
.06
SM
-.13
Order
CA: Cognitive ability; EI: Emotional Intelligence; INT: Integrity; SD: Social Desirability; SM: Self
Monitoring; Order: Order of presentation of H and F conditions. Notice that all of the regression
relationships are partial, since we performed a multiple regression. The CA->FA relationship has been
replicated a couple of times – Smarter people are better able to fake when told to do so.
Measurement and Structural Models
4
Example 2. From Biderman, Nguyen, Mullins, & Luna (2008).
2A. Here is the measurement model. It’s similar to the above measurement model, with the
exception that it’s from only one administration of the Big Five. The model here was applied to
data obtained under instructions to respond honestly. So the latent variable that represents
variance not attributable to the Big Five is called M, for Method Bias. (We now know that it’s
more than method bias, but the letter ,M, has stuck.)
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
M
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
O1
O2
O3
O4
O5
O6
O7
O8
O9
O10
E
A
C
S
O
The model applied here is
called a bifactor model. It’s
called that because each observed
variable has two (hence “bi”) latent
variables influencing it. The first is
the trait latent variable. For
example, each Extraversion item is
influenced by the trait of
Extraversion.
But each item is also influence
by a 2nd latent variable – called M
in this model. Having each
observed variable influenced by
two latent variables is what causes
a model to be called a bifactor
model.
Bifactor models are not well
known in the CFA or SEM
literature, although they’re gaining
in popularity.
See Reise, S. P. (2012). The
rediscovery of bifactor
measurement models. Multivariate
Behavioral Research, 47, 667-696.
Measurement and Structural Models
5
2B. The Structural Model. Predicting supervisor ratings of performance.
In this diagram, I have made only the regression links of the structural model red. All of the Big
Five latent variables and M are part of both the measurement model and the structural model.
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
O1
O2
O3
O4
O5
O6
O7
O8
O9
O10
M
.174c
E
-.011
A
-.104
-.102
Cus
ts
P
Sal
es
Col
ls
C
-.142a
S
-.134a
We found M was the best predictor of Supervisor Ratings. We were puzzled about that until
recently when we discovered that under instructions to respond honestly, M appears to represent
O
the respondent’s affective state. Thus, persons
with positive affective states (happy people)
had higher ratings than those with negative affective states.