Studying Measurement Invariance Using
Confirmatory Factor Analysis
Measurement Invariance
Suppose we have a scale that is intended to measure a specified trait. The
scale is made up of multiple items (or subscales, or multi-item parcels). In
factor analytic terms, the items serve as indicators of the trait or factor in a
common factor model.
Suppose furthermore that we make use of this scale in samples from two or
more distinct populations (e.g., genders, cultural groups, racial groups, age
groups, occupational groups, etc.). We may wish to analyze resulting
measures in a variety of ways. For instance, we may wish to test the
difference between group means, or to analyze the relationship of the scale
to other variables in each group.
For any such use of scale scores, there is a critical assumption that the scale
is measuring the same trait in all of the groups. If that assumption holds,
then comparisons and analyses of those scores are acceptable and yield
meaningful interpretations. But if that assumption is not true, then such
comparisons and analyses do not yield meaningful results. This is the
general issue of measurement invariance (MI).
Issues of MI are also often relevant in longitudinal research. When a scale is
administered over repeated occasions to the same sample of people, the
question of MI involves the issue of whether the scale is measuring the same
construct at different occasions. This type of MI will be addressed later; for
now, we examine MI over multiple groups.
Studying MI Using Confirmatory Factor Analysis
A central principle of MI is that measures across groups are considered to be
on the same scale if relationships between the indicators and the trait are the
same across groups. This statement can be translated into factor analytic
terms: Given multiple items that make up a scale, if the loadings for those
items on the single underlying factor are the same across groups, then
measurement invariance is supported. When framed in these factor analytic
terms, this property is called factorial invariance, and represents one
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approach to the study of MI. As will be seen, invariance of factor loadings
is only one aspect of MI. The various aspects of MI can be investigated
using confirmatory factor analysis models. These models can be
supplemented with a model for structured means so as to allow for the study
of group differences in means on latent variables.
Review of Multi-Sample CFA Model with Structured Means
For a factor analysis model with nonzero means of MVs and LVs, we first
specify the following data model:
B œ 7 B  LB 0  $
where 7B represents a vector of intercept terms for the B variables, LB is the
factor loading matrix, 0 is the vector of scores on the LVs, and $ is the
vector of error of measurement terms corresponding to the B variables. We
also define , as the vector of means on the LVs.
From this data model, we can derive the following covariance structure:
OBB œ LB QLwB  K$
which expresses the population variances and covariances of the MVs as a
function of the parameters in LB , Q , and K$ .
We can also derive the following mean structure:
.B œ 7B  LB ,
This model expresses the population means of the B variables as a function
of the parameters in 7B , LB and ,.
The full model for covariances and means thus has 5 parameter matrices:
LB , Q , K$ , 7B , and ,.
A model is specified by designating fixed, free, and constrained parameters
in these 5 matrices. This model can be fit to data (sample covariance matrix,
– ) by obtaining estimates of the parameters such
W , and sample mean vector B
3
s
that the resulting implied population covariance matrix and mean vector (O
and .
s) are as similar as possible to their sample counterparts.
The model can also be generalized to the case of multiple samples. The
covariance structure is generalized to
Ð1Ñ
Ð1Ñ
wÐ1Ñ
Ð1Ñ
OBB œ LB QÐ1Ñ LB  K$
and the mean structure is generalized to
Ð1Ñ
Ð1Ñ
.B œ 7 Ð1Ñ B  LB ,Ð1Ñ
where 1 represents the 1th of K populations. The model is specified in terms
of the parameter matrices for each group, possibly including equality
constraints on selected parameters across groups. The model is fit to the
multiple samples simultaneously.
Model Specification for Investigating MI
The notion of MI usually is raised with reference to a single scale and the
question of whether it measures the same trait in different groups. This
question can be studied using the multi-sample CFA model with structured
means. In the simple MI case, there would be only one factor, and the
indicators of that factor would be the scale items (or subscales, parcels, etc.)
In the following developments, it should be kept in mind that all of these
methods are equally applicable to the case where there is more than one
factor and the question is whether the same LVs (plural) are being measured
across groups.
The multi-sample CFA model with structured means can be used to
investigate MI and to test for group differences in LV means. This is
achieved by testing a sequence of models, beginning with an unconstrained
model and progressively introducing equality constraints on parameters. It
is possible to define an a priori sequence of models to achieve this objective.
To see this, consider the case where the indicators do measure the same LV
across groups. What does this case imply with regard to invariance in model
parameters across groups? First, it implies that the loadings in LB will be
invariant, since those loadings represent the relationships of the indicators to
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the underlying trait. Does this level of invariance imply that any other
parameters must be invariant across groups. The answer to this question is
"no." To understand this, consider the data model:
B œ 7 B  LB 0  $
If this model holds for every individual in each population, then the factor
loadings in LB are the same for every individual. In this case, we have
simple MI and LB will be invariant across groups. But 7B need not be
invariant, since those intercepts do not involve the property of MI.
Considering the covariance structure given by
Ð1Ñ
Ð1Ñ
wÐ1Ñ
Ð1Ñ
OBB œ LB QÐ1Ñ LB  K$
it would not be necessary for Q or for K to be invariant across groups.
These matrices contain variances and covariances of exogenous LVs and of
errors of measurement, respectively. Even if we have MI, meaning we are
measuring the same LV in each group, these variances and covariances
could be different from group to group. Thus, a simple form of MI requires
factor loadings to be invariant across groups, but allows all other parameters
to vary from group to group. Without constraints on factor loadings, we do
not achieve this level of invariance. With additional constraints, we achieve
stronger forms of invariance. We can specify a sequence of models
accordingly, to represent different degrees of MI, beginning with the least
constrained and progressing to the most constrained model. The sequence of
models is defined so as to provide substantively useful information.
At each step in the sequence we are interested in (a) the fit of that model,
and (b) the degree of decrement in fit compared to the previous model. At
any step, if the model fits well and the decrement in fit is not significant,
then the constraints imposed at that step are deemed plausible in the
population. However, if at any step the decrement in fit caused by the
constraints introduced at that step is significant, then those constraints are
deemed implausible.
The material below is adapted from several sources, including Reise,
Widaman, and Pugh (1993), Meredith (1993), and Widaman and Reise
(1997), and Vandenberg and Lance (2000).
5
Invariance of Population Covariance Matrices
It is useful as a first step to test the null hypothesis that the population
covariance matrices are equal. Failure to reject this model implies that
equality of population covariance matrices is plausible, which in turn implies
that equality of parameters of the covariance structure model (factor
loadings, unique variances, and factor variances and covariances) is
plausible. In such a case, there is no need to carry out further tests of
invariance of these parameters. On the other hand, if invariance of
population covariance matrices is rejected, then one can begin a series of
tests to determine whether some, but not all, of the model parameters may be
invariant.
Configural Invariance (also called pattern invariance)
Configural invariance is defined as the same pattern of fixed and free factor
loadings (and other parameters) across groups, but no equality constraints.
This model implies that similar, but not identical, LVs are present in the 1
groups. This model is represented by the following set of equations:
Ð1Ñ
Ð1Ñ
wÐ1Ñ
Covariance Structure:
OBB œ LB QÐ1Ñ LB  KÐ1Ñ
Mean Structure:
.B œ 7 Ð1Ñ B  LB ,Ð1Ñ
Ð1Ñ
Ð1Ñ
According to this structure, group differences in variances, covariances, and
means, may be attributable to group differences in elements of all 5
parameter matrices.
In practice it is not possible to fit this model to data with all elements of 7 Ð1Ñ
and ,Ð1Ñ free in each group. These parameters would not be all identified.
The usual way around this problem in assessing configural invariance is to
omit the mean structure from the model and test the model using only the
covariance structure. There is no real loss in taking this approach, since
mean differences on LVs across groups would not be of interest without
invariance in factor loadings. In such a specification, one would need to set
Ð1Ñ
a scale for the LVs in the usual way. Beyond that, the matrices LB , Q Ð1Ñ ,
and KÐ1Ñ would be specified so as to have the same pattern of fixed and free
parameters across groups, but no equality constraints would be imposed.
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The model of configural invariance serves as a useful baseline model to
which we can compare more restrictive models.
Weak Factorial Invariance
Under weak factorial invariance, the factor loadings are constrained to be
equal across groups, but no other equality constraints are imposed. This
model implies that the same LVs are being measured across groups. The
model can be represented as follows:
Ð1Ñ
Covariance Structure:
OBB œ LB QÐ1Ñ LBw  KÐ1Ñ
Mean Structure:
.B œ 7 Ð1Ñ B  LB ,Ð1Ñ
Ð1Ñ
Note that the superscript 1 has been dropped from L , indicating that L does
not vary across groups. Again, the parameters in 7 Ð1Ñ and ,Ð1Ñ can not all be
estimated without further constraints. As a result, the weak factorial
invariance model is typically evaluated using only the covariance structure.
Considering that covariance structure, it can be seen that under weak
factorial invariance, differences in covariances among MVs (off-diagonal
Ð1Ñ
elements in OBB ) are attributable to differences among covariances of LVs
(off-diagonal elements in QÐ1Ñ ). That is, differences across groups in
relationships among MVs are attributable to differences across groups in
relationships among LVs.
Strong Factorial Invariance
Ð1Ñ
Strong factorial invariance is tested by specifying factor loadings in LB to
be invariant across groups, and also for intercepts in 7 Ð1Ñ B to be invariant
across groups. This condition is also called "scalar invariance." The
resulting model is then as follows:
Ð1Ñ
Covariance Structure:
OBB œ LB QÐ1Ñ LBw  KÐ1Ñ
Mean Structure:
.B œ 7 B  LB ,Ð1Ñ
Ð1Ñ
7
As with weak factorial invariance, this specification implies that the
measurement of the LVs is the same across groups. Furthermore, the
invariance in the intercepts in the mean structure allows for us to evaluate
mean differences in LVs across groups. The form of the mean structure
implies that any differences in means on the MVs are attributable to
differences in means on the LVs.
Although it is not possible to estimate factor means in ,Ð1Ñ in each group, the
differences among those means will be identified. Thus, to assess mean
differences, we must simply fix the vector ,Ð1Ñ in one group, and allow
elements of ,Ð1Ñ to be free in the remaining groups. This is usually achieved
by setting elements of ,Ð1Ñ to zero in one group. Resulting parameter
estimates for , in the remaining groups then reflect differences between
factor means in that group and the reference group.
Thus, under strong factorial invariance, group differences in covariances
among MVs and in means of MVs are attributable to group differences in
covariances and means on LVs.
Strict Factorial Invariance
Strict factorial invariance extends the previous model by invoking the
additional constraint that unique variances in KÐ1Ñ are invariant across
groups. The resulting model is as follows:
Ð1Ñ
Covariance Structure:
OBB œ LB QÐ1Ñ LBw  K
Mean Structure:
.B œ 7 B  LB ,Ð1Ñ
Ð1Ñ
The key difference between strict and strong factorial invariance involves
how the variances of the MVs (diagonal elements of O Ð1Ñ ) are accounted
for. Under strong factorial invariance, group differences in those variances
could be attributable to group differences in variances of LVs (in diagonals
of QÐ1Ñ ) as well as to group differences in error variances (in diagonals of
KÐ1Ñ ). Under strict factorial invariance, group differences in variances of
MVs are attributable only to group differences in variances of LVs, since
error variances are invariant across groups.
8
Strict factorial invariance is a highly constrained model and may often not
hold in practice. In fact, there is reason to expect that it would not hold,
even if strong factorial invariance does hold. Even if all populations come
from a common parent population with given error variances, it would be
expected that error variances would vary from one subpopulation to another.
Additional Models
Two additional sets of constraints could be of interest:
1) Invariant variances and covariances of LVs.
In this model, strict factorial invariance would be extended by constraining
Q to be invariant across groups. A test of the difference between this model
and the strict model would indicate whether such a constraint were plausible.
If so, the implication is that the entire covariance structure is invariant,
meaning that the population covariance matrix is invariant across groups and
has the same parametric structure.
2) Invariant means of LVs.
This model is an extension of strict factorial invariance and is specified by
constraining , to be invariant across groups. A test of the difference
between this model and the strict model would indicate whether this
constraint is plausible. If so, the implication is that the entire mean structure
is invariant, meaning that population means of the MVs and LVs are
invariant across groups. However, if the difference between this model and
the strict model is significant, then that is evidence that the population means
on the LVs are significantly different across groups.
What To Do When Factorial Invariance Is Not Supported
If an investigator carries out a study of factorial invariance and finds no
support for this property, what does he or she do?
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If configural variance is not supported, this is a serious problem. In that
case, the evidence argues against even similar factor patterns across groups.
One must either accept the fact that different LVs are being measured in
different groups, or attempt to identify and rectify the reason for the lack of
configural invariance. One approach to addressing the problem would be to
back up and reconsider the selection of indicators. One can examine results
and carry out more detailed analyses to determine whether configural
invariance would hold for a subset or different set of indicators.
If configural invariance is supported, but weak factorial invariance is not
supported, one can again either accept and attempt to assess the implications
of that fact or try to address it. One way to address it is to identify and
delete problematic indicators, so that weak factorial invariance would be
supported using a subset of the indicators. This approach is problematic,
however, because it can alter the nature of a standardized scale and is also
susceptible to capitalization on chance characteristics of the observed
samples. Another approach is to make use of the notion of "partial
measurement invariance," wherein equality constraints are imposed on some
but not all of the factor loadings (see Byrne, Shavelson, and Muthen, 1989).
If partial measurement variance is supported, however, it is highly debatable
whether subsequent comparisons of LV means across groups are
meaningful. In addition, a finding of partial measurement invariance could
be highly dependent on chance characteristics of the observed samples.
Example
The study of factorial invariance is illustrated using an example from Reise,
Widaman, and Pugh (1993). This example involves a 5-item scale of
negative affect administered to samples of American and Chinese students.
The items are adjectives: nervous, worried, jittery, tense, distressed.
Subjects responded to each item on a 5-point Likert scale. Sample sizes
were N=540 American students and N=598 Chinese students. Sample data
are provided in Table 1 of the Reise et al. paper. (Note: There are two
errors in this table: First, the entries listed as correlations are actually
covariances. Second, the entry for the relationship between worried and
tense for the Minnesota sample should be .77 rather than .74.)
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Following is the data file rwp1.dat for fitting a covariance structure (no
mean structure):
NERV WORRD JITTRY TENSE DSTRSD
.865
.29 1.232
.30 .25 1.020
.39 .31 .20 1.082
.40 .51 .42 .52 1.254
NERV WORRD JITTRY TENSE DSTRSD
1.254
.79 1.488
.66 .55 1.188
.73 .77 .63 1.416
.66 .84 .57 .92 1.563
Following are command files for fitting various models of measurement
invariance.
Command file for test of invariant population covariance matrices:
LISREL MODEL FOR INVARIANT SIGMA -- NANJING SAMPLE
DA NG=2 NI=5 NO=598
LA FI=RWP1.DAT
CM FI=RWP1.DAT
MO NX=5 NK=5 LX=ID TD=ZE PH=SY,FR
OU
LISREL MODEL FOR INVARIANT SIGMA -- MINNESOTA SAMPLE
DA NI=5 NO=540
LA FI=RWP1.DAT
CM FI=RWP1.DAT
MO PH=IN
OU
Command file rwpm1.spl for configural invariance:
LISREL MODEL FOR CONFIGURAL INVARIANCE -- NANJING SAMPLE
DA NG=2 NI=5 NO=598
LA FI=RWP1.DAT
CM FI=RWP1.DAT
MO NX=5 NK=1 LX=FR
LK
NEGAFF
11
FI LX 1
VA 1 LX 1
OU XM
LISREL MODEL FOR CONFIGURAL INVARIANCE -- MINNESOTA SAMPLE
DA NI=5 NO=540
LA FI=RWP1.DAT
CM FI=RWP1.DAT
MO
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
Command file rwpm2.spl for weak factorial invariance:
LISREL MODEL FOR WEAK FACTORIAL INVARIANCE -- NANJING SAMPLE
DA NG=2 NI=5 NO=598
LA FI=RWP1.DAT
CM FI=RWP1.DAT
MO NX=5 NK=1 LX=FR
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
LISREL MODEL FOR WEAK FACTORIAL INVARIANCE -- MINNESOTA
SAMPLE
DA NI=5 NO=540
LA FI=RWP1.DAT
CM FI=RWP1.DAT
MO LX=IN
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
Data file rwp2.dat for fitting both covariance and mean structures:
NERV WORRD JITTRY TENSE DSTRSD
.865
.29 1.232
12
.30 .25 1.020
.39 .31 .20 1.082
.40 .51 .42 .52 1.254
1.89 2.09 1.60 2.15 1.93
NERV WORRD JITTRY TENSE DSTRSD
1.254
.79 1.488
.66 .55 1.188
.73 .77 .63 1.416
.66 .84 .57 .92 1.563
2.17 2.52 2.01 2.35 2.29
Command file rwpm3.spl for strong factorial invariance:
LISREL MODEL FOR STRONG FACTORIAL INVARIANCE -- NANJING SAMPLE
DA NG=2 NI=5 NO=598
LA FI=RWP2.DAT
CM FI=RWP2.DAT
ME FI=RWP2.DAT
MO NX=5 NK=1 LX=FR TX=FR KA=FI
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
LISREL MODEL FOR STRONG FACTORIAL INVARIANCE -- MINNESOTA
SAMPLE
DA NI=5 NO=540
LA FI=RWP2.DAT
CM FI=RWP2.DAT
ME FI=RWP2.DAT
MO LX=IN TX=IN KA=FR
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
Command file rwpm4.spl for strict factorial invariance:
LISREL MODEL FOR STRICT FACTORIAL INVARIANCE -- NANJING SAMPLE
DA NG=2 NI=5 NO=598
LA FI=RWP2.DAT
CM FI=RWP2.DAT
13
ME FI=RWP2.DAT
MO NX=5 NK=1 LX=FR TX=FR KA=FI
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
LISREL MODEL FOR STRICT FACTORIAL INVARIANCE -- MINNESOTA
SAMPLE
DA NI=5 NO=540
LA FI=RWP2.DAT
CM FI=RWP2.DAT
ME FI=RWP2.DAT
MO LX=IN TX=IN KA=FR TD=IN
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
Command file rwpm5.spl for strict factorial invariance as well as invariant
factor variance:
LISREL MODEL FOR STRICT FACTORIAL INVARIANCE -- NANJING SAMPLE
FACTOR VARIANCE INVARIANT
DA NG=2 NI=5 NO=598
LA FI=RWP2.DAT
CM FI=RWP2.DAT
ME FI=RWP2.DAT
MO NX=5 NK=1 LX=FR TX=FR KA=FI
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
LISREL MODEL FOR STRICT FACTORIAL INVARIANCE -- MINNESOTA
SAMPLE
FACTOR VARIANCE INVARIANT
DA NI=5 NO=540
LA FI=RWP2.DAT
CM FI=RWP2.DAT
ME FI=RWP2.DAT
MO LX=IN TX=IN KA=FR TD=IN PH=IN
LK
14
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
Command file rwpm6.spl for strict factorial invariance as well as invariant
factor mean:
LISREL MODEL FOR STRICT FACTORIAL INVARIANCE -- NANJING SAMPLE
FACTOR MEAN INVARIANT
DA NG=2 NI=5 NO=598
LA FI=RWP2.DAT
CM FI=RWP2.DAT
ME FI=RWP2.DAT
MO NX=5 NK=1 LX=FR TX=FR KA=FI
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
LISREL MODEL FOR STRICT FACTORIAL INVARIANCE -- MINNESOTA
SAMPLE
FACTOR MEAN INVARIANT
DA NI=5 NO=540
LA FI=RWP2.DAT
CM FI=RWP2.DAT
ME FI=RWP2.DAT
MO LX=IN TX=IN KA=FR TD=IN KA=IN
LK
NEGAFF
FI LX 1
VA 1 LX 1
OU XM
Summary of Results
Invariant population covariance matrices:
;# œ )(Þ#), .0 œ "&ß rmsea = .091
15
This model fits very poorly, indicating that population covariance matrices
are almost certainly not invariant, which implies that at least some of the
covariance structure parameters are different between groups. We then
proceed to investigate hypotheses about different types of measurement
invariance.
Configural Invariance:
;# œ 75.30, .0 œ "!, rmsea=.11, nnfi=.91
For purposes of example, consider configural invariance to be plausible and
proceed to test weak factorial invariance:
;# œ 90.1!, .0 œ "4, rmsea=.095, nnfi=.93
Fit of this model is marginal. Comparison to configural invariance model
yields ;# œ 14.80, .0 œ 4, :  .01. But nnfi is better than for previous
model.
Proceed to test of strong factorial invariance:
;# œ 111.21, .0 œ "8, rmsea=.093, nnfi=.93
Under this model we can compare groups with respect to factor variances
s " œ .33 (.03) for the Chinese sample and .64
and means. For variances, 9
(.06) for the American sample, indicating higher variance on the negative
affect LV for the American sample. For means, ,
s œ .00 for the Chinese
sample (fixed), and .31 (.05) for the American sample, indicating a higher
mean on the negative affect LV for the American sample.
Proceed to test of strict factorial invariance:
;# œ 122.51, .0 œ 23, rmsea=.086, nnfi=.94
Fit of this model is marginal. The decrement in fit from the previous model
is tested by ;# œ 11.30, .0 œ 5, :  .05, indicating statistical rejection of
this constraint that error variances in the MVs are invariant across groups.
For the sake of the example, test two further models. In the present case,
these were tested by introducing additional constraints to the model of strict
16
invariance, but they could have also been specified by introducing these
constraints to the model of strong invariance.
Strict invariance, plus invariant factor variances:
;# œ 158.92, .0 œ 24, rmsea=.098, nnfi=.93
This constraint causes a significant decrease in fit (;# œ 36.41, .0 œ 1,
:  .001), meaning that this constraint is not plausible in the population.
This finding is not surprising given our earlier observation of factor different
variances in the two groups.
Strict invariance, plus invariant factor means:
;# œ 164.48, .0 œ 24, rmsea=.100, nnfi=.92
This constraint also causes a significant decrease in fit (;# œ 41.97, .0 œ 1,
:  .001), meaning that this constraint is not plausible in the population.
This finding is also not surprising given our earlier observation of different
factor means in the two groups.
17
Measurement Invariance over Time
Suppose that a multi-item scale intended to measure a particular construct is
administered to the same sample of individuals on repeated occasions. The
investigator may be interested in assessing aspects of change on the
construct, or in relationships of the construct to other variables at various
occasions. Obtaining valid information about such questions requires that
the same construct is being measured at each occasion. This is MI defined
across occasions rather than across different groups of people.
To investigate MI over time, one must conceive of modeling a population
covariance matrix that is of order pT x pT, where p is the number of
measured variables (items) and T is the number of occasions. This
covariance matrix D can be seen as being organized into blocks of order p x
p. Diagonal blocks D"" , D## , . . . DX X represent within-occasion covariance
matrices, and off-diagonal blocks D#" , D$" , etc., represent between-occasion
covariance matrices. Questions of MI refer to the factor structure of the
diagonal blocks, but the model represents the structure of the full matrix.
MI models can be defined and tested for this context much as in the multigroup context. Analyses, however, are single-sample analyses, with
constraints applied across occasions rather than across groups.
A typical sequence of models in this context might proceed through tests of
configural, weak, strong, and strict invariance. The question of invariance of
factor means is often of central interest.
18
References
Byrne, B. M., Shavelson, R. J., & Muthen, B. (19889). Testing for the
equivalence of factor covariance and mean structures: The issue of partial
measurement invariance. Psychological Bulletin, 105, 456-466.
Meredith, W. (1993). Measurement invariance, factor analysis, and
factorial invariance. Pyschometrika, 58, 525-543.
Reise, S. P., Widaman, K. F., & Pugh, R. H. (1993). Confirmatory
factor analysis and item response theory: Two approaches for exploring
measurement invariance. Psychological Bulletin, 114, 552-566.
Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of
the measurement invariance literature: Suggestions, practices, and
recommendations for organizational research. Organizational Research
Methods, 3, 4-70.
Widaman, K. F., & Reise, S. P. (1997). Exploring the measurement
invariance of psychological instruments: Applications in the substance use
domain. In K. J. Bryant, M. Windle, & S. G. West (Eds.) The science of
prevention: Methodological advances from alcohol and substance abuse
research, pp. 281-324. Washington, DC: APA.