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Red de Revistas Científicas de América Latina y el Caribe, España y Portugal
539
In substantive research that focuses on multiple-group
comparisons, it is typically assumed that the assessment scale is
operating equivalently across the groups of interest. That is to say,
there is presumed equality of: (a) factorial structure (i.e., same
number of factors and pattern of item loadings onto these factors),
(b) perceived item content, (c) factor loadings (i.e., similar size of
item estimates), and (d) when comparison of latent factor means of
interest, the item intercepts (i.e., the item means). Development of
a method capable of testing for such multigroup equivalence (i.e.,
invariance) derives from the seminal con
f
rmatory factor analytic
(CFA) work of Jöreskog (1971), which is rooted in the analysis
of covariance structures.
In a critically important extension of
Jöreskog’s CFA work, Sörbom (1974) made possible tests for
the invariance of latent factor means (commonly referred to in
reverse as latent mean differences), based on the analysis of mean
and covariance structures (i.e., the moment matrix). Structural
equation modeling (SEM) is the premier analytic strategy capable
of testing these assumed measurement equivalencies, in addition
to testing for latent mean differences across groups.
A review of the early SEM literature reveals applications of
multigroup testing for measurement invariance to be blatantly
sparse in its
f
rst decade of existence. Indeed, it was not until
the mid-1980s and early 1990s that this methodological strategy
actually started to take hold, with most researchers focused on
construct validation issues related to construct dimensionality
ISSN 0214 - 9915 CODEN PSOTEG
Copyright © 2017 Psicothema
www.psicothema.com
The maximum likelihood alignment approach to testing for approximate
measurement invariance: A paradigmatic cross-cultural application
Barbara M. Byrne
1
and Fons J.R. van de Vijver
2
1
University of Ottawa and
2
University of Tilburg
Abstract
Resumen
Background:
The impracticality of using the con
f
rmatory factor analytic
(CFA) approach in testing measurement invariance across many groups
is now well known. A concertedeffort to addressing these encumbrances
over the last decade has resulted in a new generation of alternative
methodological procedures that allow for approximate, rather than exact
measurement invariance across groups. The purpose of this article is
twofold: (a) to describe and illustrate common dif
f
culties encountered when
tests for multigroup invariance are based on traditional CFA procedures
and the number of groups is large, and (b) to walk readers through the
maximum likelihood (ML) alignment approach in testing for approximate
measurement invariance.
Methods:
Data for this example application
derive from an earlier study of family functioning across 30 cultures that
include responses to the Family Values Scale for 5,482 university students
drawn from 27 of these30 countries. Analyses were based on the Mplus
7.4 program.
Results:
Whereas CFA tests for invariance revealed 108
misspeci
f
ed parameters that precluded tests for latent mean differences,
noninvariant results were well within the acceptable range for the
alignment approach thereby substantiating the trustworthiness of the latent
mean estimates and their comparison across groups.
Conclusion:
The
alignment approach in testing for approximate measurement invariance
provides an automated procedure that can overcome important limitations
of traditional CFA procedures in large-scale comparisons.
Keywords:
Alignment optimization, large-scale measurement invariance,
cross-cultural comparisons.
El enfoque de alineamiento de máxima verosimilitud para evaluar de
forma aproximada la invarianza de medida: una aplicación intercultural
paradigmática.
Antecedentes:
la
imposibilidad
de
utilizar
el
análisis
factorial con
f
rmatorio (AFC) para evaluar la invarianza de medida para
muchos grupos es bien conocida. El objetivo de este artículo es doble: (a)
describir e ilustrar las di
f
cultades que se encuentran cuando las pruebas
para evaluar la invarianción multigrupo se basan en los procedimientos
tradicionales de AFC y el número de grupos es grande, y (b) mostrar a los
lectores el método de alineamiento de máxima verosimilitud para evaluar
la invarianza de medida aproximada.
Método:
los datos provienen de un
estudio intercultural previo sobre funcionamiento familiar que incluye 30
culturas. Se aplicó la Escala de Valores Familiares a 5.482 estudiantes
universitarios de 27 de estos 30 países. Los análisis se realizaron con
el programa Mplus 7.4.
Resultados:
los métodos basados en el AFC
generaron 108 parámetros mal especi
f
cados, lo cual hace inviable la
comparación de las diferencias de medias latentes. Con el método de
alineamiento se obtuvieron resultados de invarianza dentro de un rango
acceptable, lo cual da solidez a las estimaciones de las medias latentes y su
comparación a través de los grupos.
Conclusion:
el método de alineamiento
para la evaluación de la invarianza de medida aproximada proporciona
un procedimiento automatizado que puede superar las importantes
limitaciones de los métodos tradicionales basados en el AFC.
Palabras clave:
método de alineamiento optimizado, invarianza de
medida, comparaciones interculturales.
Psicothema 2017, Vol. 29, No. 4, 539-551
doi: 10.7334/psicothema2017.178
Received: April 16, 2017 • Accepted: August 21, 2017
Corresponding author: Barbara M. Byrne
School of Psychology
University of Ottawa
Otawa, Ontario (Canada)
e-mail: bmbch@nottawa.ca
Barbara M. Byrne and Fons J.R. van de Vijver
540
equivalence (e.g., Byrne, 1989; Byrne & Shavelson, 1987; Marsh &
Hocevar, 1985) and/or assessment scale equivalence (e.g., Byrne,
1988, 1991; Drasgow & Kanfer, 1985) across groups. The next
twenty years, however, witnessed rapidly expanding application
of this methodological strategy as evidenced from a review of
scholarly journals that revealed the publication of 40 articles from
1980 to 1989, 210 articles from 1990 to 1999, and a remarkable
2,545 articles from 2000 to 2009 (Rutkowski & Svetina, 2014), all
of which were limited to within-country comparisons.
Despite this increase in tests for measurement invariance
per se, a recent study of the frequency of invariance tests in the
Journal of Cross-Cultural Psychology
, a journal that specializes
in
cross-cul
tural
comparisons
,
revealed
tha
t
only
17%
of
the
studies conducted such tests (Boer, Hanke, & He, in press). Thus,
even if invariance tests are becoming more commonly applied
within national boundaries, there is still a long way to go before
they become routinely applied in cross-cultural studies.
Furthermore, it is important to note that the same pattern of
growth has not been evident with respect to tests for latent mean
differences. Rather, reports in the literature of such research
have been scant. However, in a follow-up review of the literature
subsequent to the earlier work of Vandenberg and Lance (2000),
Schmitt and Kuljanin (2008) reported a substantial increase in
the frequency with which these tests for latent mean differences
have been conducted. One possible explanation for this short-
term increase could be linked to the publication of pedagogical
papers (e.g., Byrne & Stewart, 2006; Little, 1997), as well as book
chapters published during this time that focused exclusively on
this procedure (e.g., Byrne, 1998, 2001, 2006).
Clearly, the volume of literature addressing the testing of
measurement invariance surely re
f
ects on the current heightened
awareness of researchers regarding this critical preliminary step
in the conduct of multigroup mean comparisons. Nonetheless,
upon closer scrutiny of this literature, it becomes evident that
the lion’s share of these tests for invariance has been limited to
comparisons across two groups, with only a modicum of studies
testing for equivalence across at least three groups (see, e.g., Byrne
& Campbell, 1999; Munet-Vilaró,
Gregorich, & Folkman, 2002;
Robert, Lee, & Chan, 2006; Woehr, Arciniega, & Lim, 2007),
and become increasingly scant as the number of groups under
test increase. The ultimate question here then, is why this
F
nding
should be so?
In broad terms, the answer to this query has been shown
to lie in the restrictiveness of CFA procedures in testing for
measurement invariance. More speci
F
cally, it stems from the
requirement that (a) all non-target factor loadings in multifactor
models are constrained to zero across groups, (b) there are zero
error covariances among the indicator variables across groups,
and (c) when testing latent mean differences is of interest, the
indicator variable intercepts are equivalent. Indeed, it has recently
become customary to refer to this original CFA approach to testing
for measurement invariance as the “exact” approach (see, e.g.,
Zercher, Schmidt, Cieciuch, & Davidov, 2015). When this CFA
approach is used in testing for invariance across a large number
of groups, results typically yield poor model
F
t underscored by
numerous modi
F
cation indexes, thereby leading Asparouhov and
Muthén (2014) to note its impracticality for use in large-scale
studies. In addition, Byrne and van de Vijver (2010) detailed two
aspects of this CFA methodological procedure that contribute
importantly to the impracticality of its use: (a) establishment of
a group-appropriate structure of the con
F
gural model (Horn &
McArdle, 1992), and (b) limited functionality of all SEM software
in comparing only one group at a time with each of the other
groups. Both of these latter two procedures involve an abundance
of time and labor intensity that becomes progressively more
demanding as the number of groups increase (Details related to
these three issues follow later.)
Taken together, these three aspects of the CFA approach to tests
for multigroup invariance make it cumbersome and impractical
in large-scale assessment. As a result, these limitations have
remained a major impediment to advancing our substantive
knowledge of cross-group differences within the context of a wide
variety of disciplines and in the conduct of numerous important
large scale studies both nationally and cross-nationally. A few
examples of such studies are as follows: (a) construct validation
studies in which researchers wish to test for the equivalence of an
assessment scale, theoretical construct, or nomological network
across multiple national, international, or cross-cultural groups;
(b) large-scale cross-national and cross-cultural educational
surveys of academic achievement in various subject areas such as
the Programme for International Student Assessment (PISA) and
Trends in International Mathematics and Science Study (TIMSS;
see, e.g., Marsh, Abduljabbar, Parker, Morin, Abdelfattah, &
Nagengast, 2014); and/or (c) large-scale sociological studies
such as the European Social Survey (ESS; see, e.g., Davidov,
Cieciuch, Mueleman, Schmidt, Algesheimer, & Hausherr, 2015),
the European Value Study (EVS), and the World Value Survey
(WVS).
Frustrated and hampered by these limitations of the multigroup
CFA approach to tests for measurement invariance, the past 6 to
8 years has seen a growing number of researchers, particularly
those interested in cross-national comparisons, actively testing
out alternative methodological strategies capable of achieving the
same goals, albeit without the same limitations. This progression
of new testing strategies began with a procedure that allowed
for either the deletion of particular groups due to their failure to
meet the constraints of invariance (see, e.g., Davidov, 2008) or
for the deletion and replacement of particular items that failed
to demonstrate invariance (see, e.g., Thalmayer & Saucier, 2014).
This initial path to addressing the CFA limitations was soon
followed by the introduction of two methodological strategies that,
in contrast to the CFA approach, allowed for tests of
approximate
,
rather than
exact
measurement invariance: (a) exploratory
structural equation modeling (ESEM; Asparouhov & Muthén
,
2009; and (b) Bayesian structural equation modeling (BSEM;
Muthén & Asparouhov, 2012). Finally, Asparouhov and Muthén
(2014) recently introduced the new and unique technique of
alignment
in testing for measurement invariance when the number
of groups is large.
Our primary intent in this article is to walk readers through
an example application of the alignment strategy based on data
used in reanalysis of a previous study that tested for measurement
invariance and latent mean differences across 27 cultural groups
(see Byrne & van de Vijver, 2010). More speci
F
cally, our purpose
is twofold: (a) to describe and illustrate the common dif
F
culties
encountered when tests for multigroup invariance are based on
traditional CFA procedures and the number of groups is large;
and (b) to outline and illustrate the ML alignment approach to
multigroup tests for invariance based on the same 27-country
data.
The maximum likelihood alignment approach to testing for approximate measurement invariance: A paradigmatic cross-cultural application
541
Given that the CFA approach to measurement invariance is now
well known, we begin with only a brief overview of this traditional
multigroup testing strategy. Next, we elaborate on, and illustrate
the problems noted earlier in using the CFA approach to test for
invariance based on a previous attempt to acquire such information
for a large scale study comprising 27 countries (see Byrne &
van de Vijver, 2010). These problematic issues are followed by
a brief description of ESEM and BSEM, the two initially
introduced alternate approaches to multigroup CFA that focus
on approximate, rather than on exact measurement invariance,
and cite a few example applications of each. We then move on to
a description of the alignment approach to tests for invariance,
explain how it addresses the above-noted CFA limitations, and
outline the steps involved in testing for multigroup invariance and
latent mean differences based on ML estimation. Finally, based on
the same data used in the earlier Byrne and van de Vijver (2010)
study, we walk the reader through each of the steps comprising
use of the alignment approach in testing for approximate
measurement invariance and latent mean differences based on ML
estimation. The paper is written in a didactic mode that embraces
a nonmathematical, rather than a statistically-oriented approach
to the topic and is intended as a guide for researchers interested in
applying this methodology but who may be somewhat uncertain of
the testing strategy involved.
Traditional CFA approach to tests for multigroup invariance
Testing for multigroup invariance entails a hierarchical set of
steps that should always begin with determination of a well-
f
tting
baseline
model for each group separately. Once these baseline models
are established, their separate model speci
f
cations are combined
thereby representing a multigroup baseline model
. In technical
terms, this initial multigroup model is termed the
con
f
gural model
(Horn & McArdle, 1992) and is the
f
rst and least restrictive one
to be tested. With the con
f
gural model, only the extent to which
the same pattern (or con
f
guration) of
f
xed and freely estimated
parameters holds across groups is of interest and thus no equality
constraints are imposed. It is this multigroup model for which
sets of parameters are subsequently put to the test of equality in a
logically-ordered and increasingly restrictive fashion. In contrast
to the con
f
gural model, all remaining tests for measurement
equivalence involve the speci
f
cation
of
increas
ingly
restrictive
cross-group equality constraints for particular parameters.
Limitations of CFA approach with large-scale studies
In a study designed speci
f
cally to illustrate the extent to which
the CFA approach to testing for invariance can be problematic
when applied to large-scale and widely diverse cultural groups,
Byrne and van de Vijver (2010) were unable to structure a well-
f
tting con
f
gural model, despite a precise and systematic attempt to
identify sources of noninvariance and misspeci
f
cation. As a result,
they could not test for multigroup equivalence. Consistent with
Asparouhov and Muthén (2014) as well as Rutkowski and Svetina
(2014), Byrne and van de Vijver (2010) concluded this approach to
be completely impractical and attributed the dif
f
culties to stem
from at least three aspects of the CFA procedure as follows:
1. Given that assessment scales are often group-speci
f
c in
the way they operate, it has been customary to establish a
baseline model before testing for multigroup equivalence.
These models should exhibit the best-
f
tting, yet most
parsimonious model representing data for a particular group.
Although typically, these baseline models are the same for
each group, they need not be (see Bentler, 2005; Byrne et al.,
1989). For example, it may be that the best-
f
tting model for
one group includes an error covariance or a cross-loading,
but not so for other groups under study. Presented with such
f
ndings, Byrne et al. (1989) showed that by implementing
a condition of
partial measurement invariance
, multigroup
analyses can still continue given that the recommended
conditions for some are met. As noted earlier, these
f
nal
best-
f
tting baseline models are then combined to form the
multigroup model, commonly termed the con
f
gural model.
This technique however, only works well when the number
of groups is small (as illustrated later).
2.
Given the somewhat impossible task of determining baseline
models for a large number of groups, we then began with a
con
f
gural model for which the same hypothesized factorial
structure was speci
f
ed for all groups simultaneously.
However, there are numerous challenges associated with
such a multigroup model, some of which might relate to
translation issues, sample comparability (the study employed
convenience samples of students), and/or differential
applicability of item contents (e.g., the sample of countries
differed in the importance of the extended family). Not
surprisingly, we were unable to attain a well-
f
tting model.
Goodness-of-
f
t results revealed the robust CFI values to be
.837 and the RMSEA values to be .066 (based on the EQS
62 program [Bentler, 2005]). A major complicating factor
here arises from the CFA analysis itself in that all non-target
factor loadings are
f
xed to zero, with only the hypothesized
speci
f
ed loadings being freely estimated. In theory, these
restricted
zero
loadings
are
expected
to
hold
across
all
groups under test. In practice, however, this is typically not
the case, thereby leading to a poorly
f
tting model and a
substantial number of misspeci
f
ed parameters as indicated
by the modi
f
cation indexes which, in turn, may result in a
totally false model (Muthén & Asparouhov, 2014).
3. Common to all SEM programs is the process of testing for
the equality of constrained parameters by comparing two
groups at a time. For example, given four groups, the program
initially compares Group 1 with Group 2, then with Group
3 and then with Group 4. The researcher must subsequently
respecify the input
f
le such that on the next run, Group 2 is
compared with Group 3 and then, with Group 4. The
f
nal
respeci
f
cation and testing of the input
f
le compares Group
3 with Group 4. Thus, it is easy to see that conducting a
comparison of group pairs across 27 countries
is rendered
an exceedingly tedious, if not impossible task!
The ESEM and BSEM approaches to testing for approximate
measurement invariance
The ESEM approach.
Building upon the strengths of both
exploratory factor analyses (EFA) and the more traditional
CFA strategy, development of the ESEM approach represents
a combined synthesis of both methodologies that enables a less
restrictive testing for the equivalence of factorial structures.
More speci
f
cally, in contrast to CFA in which all non-target
Barbara M. Byrne and Fons J.R. van de Vijver
542
factor loadings (i.e., cross-loadings) and error covariances (i.e.,
residual covariances) are
f
xed to, and presumed to be zero, ESEM
allows these parameters to be freely estimated. These non-zero
loadings of items on non-target factors are a common feature in
personality and attitude measurement, where instruments with a
high dimensionality are used and it is dif
f
cult to specify items
that tap only into the target factor (e.g., Costa & McCrae, 1995),
which is exacerbated in cross-cultural work. Thus, although
the factor structure is similarly hypothesized across the ESEM
and CFA procedures, model speci
f
cation of both the factor
loadings and error covariances differ. Consistent with other SEM
procedures, ESEM provides access to all the usual parameters,
standard errors, and
f
t indexes, and also allows for rotation of the
original measurement model. ESEM is considered to be supported
by the data if the target loadings (i.e., factor loadings of items
designed speci
f
cally to measure the latent factors of interest) are
substantially higher than the non-target cross-loadings and the
model exhibits a satisfactory goodness-of-
f
t to the sample data
(Davidov, Meueleman, Cieciuch, Schmidt, & Billiet, 2014).
Asparouhov and Muthén (2009) contend that the primary
advantage of ESEM over other modeling practices is its seamless
incorporation of the EFA and SEM models, including the use of
f
t
statistics and invariance tests in the case of multigroup analyses.
They further note that in most applications involving multiple
factors, several steps are involved in the process of determining
hypothesized model structure.
First
, EFA is typically used to
discover and formulate the factor structure.
Second,
the researcher
uses an ad hoc procedure to mirror this EFA structure as an SEM
model having a CFA measurement speci
f
cation. However, as
Asparouhov and Muthén (2009) importantly note, not only does
the ESEM approach accomplish this task in a single step, but it can
avoid potential pitfalls pertinent to the EFA to CFA conversion.
For an extended list of advantages in using ESEM and/or a detailed
statistical explanation of the analytic process, readers are referred
to Asparouhov and Muthén (2009).
Although the ESEM approach can be used in testing for
approximate measurement invariance across few as well as many
groups, it is particularly valuable when the number of groups
under test is large and represents diverse cultural samples as
evidenced from our review of this relatively new, but rapidly
growing literature base. We further found the primary interest of
most reported ESEM studies to focus on issues of approximate
measurement invariance related to assessment scales; these
included personality scales (see, e.g., Bowden, Saklofske, van de
Vijver, Sudarshan, & Eysenck, 2016; Ion, Iliescu, Aldhafri, Rana,
Ratanadilok, Widyanti, & Nedelcea, 2017; Ispas, Iliescu, Ilie,
& Johnson, 2014; Marsh, Lüdtke, Muthén, Asparouhov, Morin,
Trautwein, & Nagengast, 2010) and attitude scales (see, e.g.,
Ozakinci, Boratav, & Mora, 2011).
The BSEM approach.
As with ESEM, the BSEM approach
focuses on the extent to which the measurement parameters are
approximately, rather than exactly invariant across groups. In
contrast to ESEM, BSEM is conducted solely within a Bayesian
framework. As such, all parameters are considered to be variables,
with their distribution described by a prior probability distribution
(Davidov, Cieciuch, Meuleman, Schmidt, Algesheimer, &
Hausherr, 2015; van de Schoot, Kluytmans, Tummers, Lugtig, Hox,
& Muthén, 2013). Referred to more commonly as “informative
priors”, these distributions are applicable to any constrained
parameter in an SEM model (Muthén & Asparouhov, 2012). Of
particular interest in BSEM, however, is the extent to which these
priors allow for slight differences between factor loadings and/or
intercepts across groups, thereby going beyond the requirement
of strict equivalence in the CFA approach. Indeed, van de Schoot
et al. (2013) contend that in testing for invariance, a researcher
can assume that differences between these two sets of parameters
are
approximately
equal. Thus, in allowing for some degree of
uncertainty, speci
f
cation of a small amount of variance (e.g., 0.01
or 0.05) around the difference in factor loadings and intercepts
could be considered reasonable (Zercher, Schmidt, Cieciuch,
& Davidov, 2015). But, to what extent can these differences be
considered “reasonable”? To date, there are no established rules or
recommendations regarding which variance values of the loadings
and intercepts may be considered small, medium or large, or the
extent to which factor loadings or intercepts maybe considered
suf
f
ciently diverse as to be interpreted in a different manner.
These limitations need to be acknowledged.
In addition to the prior distribution, which is crucial to BSEM,
the Bayesian approach to these analyses requires two additional
components: (a) the likelihood function of the data, which contains
all information pertinent to the parameters, and (b) the posterior
distribution which represents a synthesis of both the prior and
the likelihood function. The posterior distribution comprises
updated information through a balance of prior knowledge and
the observed data (van de Schoot et al., 2013). Over and above
the major interest in attaining approximate invariance pertinent
to factor loadings and intercepts, Muthén and Asparouhov (2012)
have outlined and illustrated how the use of informative priors can
also be used to study error covariances (i.e., residuals) among the
factor indicators.
In our review of the applied BSEM literature, we found most
studies to focus on testing for approximate invariance related
to an assessment scale (e.g., De Bondt & Van Petegem, 2015), a
subset of items from an assessment scale designed to measure
the same construct (Zercher, Schmidt, Cieciuch, & Davidov,
2015), and a major international survey instrument (Davidov
et al., 2015). In addition, both van de Schoot et al. (2013) and
Muthén and Asparouhov (2012) present example applications
of BSEM. Finally, for an exceptionally well-written article that
clearly explains both the concept of, and reason for approximate
measurement invariance, in addition to the appropriate application
of BSEM in the attainment of such invariance, we refer readers to
van de Schoot et al. (2013).
The alignment approach to testing for multigroup invariance and
latent mean differences
In broad terms, the overarching difference between the CFA
approach to tests for the multigroup invariance of an assessment
scale and that of the alignment approach lies with the absence
of speci
f
ed equality restrictions of both the factor loadings and
observed variable (i.e., item, indicator) intercepts across groups.
Consistent with both the ESEM and BSEM strategies, alignment
allows for a pattern of
approximate
measurement invariance in
the data (Asparouhov & Muthén, 2014).
In contrast to CFA, the
alignment method begins with a common con
f
gural model (i.e.,
no consideration of group baseline models) and then automates
the closeness of the factor loading estimates in the process of
establishing the most optimal measurement invariance pattern,
all of which substantially simpli
f
es the test for measurement
The maximum likelihood alignment approach to testing for approximate measurement invariance: A paradigmatic cross-cultural application
543
invariance. It does so by incorporating a simplicity function
similar to the rotation criteria used in EFA (Asparouhov &
Muthén, 2014). Cieciuch, Davidov, and Schmidt (in press) note that
one extremely valuable advantage of the alignment procedure in
testing for approximate measurement invariance and latent mean
differences is that the optimization process automatically takes
the non-invariance of all factor loading and intercept parameters
into account in the process of means estimation, thereby yielding
mean values that are more trustworthy than those calculated
without this strategy. Unquestionably, a major strength of the
alignment procedure is that it automates and greatly simpli
f
es
tests for invariance across a large number of groups. It can be
particularly advantageous when the groups represent countries
wherein noninvariance is expected to be large due to cultural and
country differences as “existing methods are simply not practical
for handling such complexity” (Muthén & Asparouhov, 2014, p.
10). Recent research has shown that the alignment approach to
testing measurement invariance is quite feasible, even when the
number of groups is large as 92 (see Munck, Barber, & Torney-
Purta, in press).
Although alignment can be based on either ML or Bayes
estimation, except for the example applications presented in
Asparouhov and Muthén (2014), we were able to locate only one
Bayesian application based on real data (as opposed to simulated
data; see De Bondt & Van Petegem, 2015). Based on simulated
data, only van de Schoot et al. (2013) appear to have tested
the use of alignment across groups based on both the ML and
Bayesian estimators. This dearth of applications clearly relates to
the newness of these methodological strategies (see Davidov et
al., 2014). Based on the recommendation of other methodologists,
as well as his own work in the
f
eld, van de Schoot has advised
that the Bayesian approach to alignment is in need of much more
simulation work in order to be more explicit about the exact priors
to use (R. van de Schoot, personal communication, November 3,
2016). In light of this recommendation, together with a virtual void
in the literature of alignment applications based on ML estimation
at this time, we considered it most constructive to focus on the ML
approach. Our intent is to illustrate and address the ML alignment
procedure by walking readers through a cross-cultural application
based on 27 countries.
Based on the assumption that the number of noninvariant
measurement parameters, as well as the extent of measurement
noninvariance can be held to a minimum, the alignment method
is capable of estimating the factor loadings, item intercepts, factor
means and factor variances. As such, alignment optimization
enables the estimation of trustworthy means despite the presence of
some measurement noninvariance. This process involves two steps
and ultimately leads to a modi
f
ed con
f
gural model that exhibits
the same model
f
t, albeit with substantially less noninvariance.
The purpose of Step 1 is to establish a base (or root) con
f
gural
model that represents the best-
f
tting model among all multigroup
factor analytic models having no cross-group constraints as the
factor loadings and indicator intercepts are freely estimated for
each group; the factor means and factor variances, on the other
hand, are
f
xed at 0.00 and 1.00, respectively. In Step 2, the factor
means and variances are freely estimated and this con
f
gural
model undergoes an optimization process such that for every
group factor mean and variance parameter, there are factor loading
and intercept parameters that yield the same likelihood estimation
as the con
f
gural model. The ultimate aim of this process, for
each group
, is to choose values of both the factor mean and
factor variance that minimize the total amount of measurement
noninvariance (i.e., it minimizes the total loss simplicity).
Asparouhov and Muthén (2014; p. 497) note that the point at which
this minimization process terminates will occur where “there
are few large noninvariant measurement parameters and many
approximately noninvariant parameters rather than many medium-
sized noninvariant measurement parameters”. They compare this
result with that of EFA rotation for which the aim is to identify
either large or small loadings, rather than midsized loadings (For a
more statistical description of these analyses, readers are referred
to Asparouhov & Muthén, 2014, pp. 496-497).
Once this minimization point has been reached, alignment
analyses then focus on a comparison of factor means and factor
variances across groups, albeit allowing for approximate invariance
in lieu of the more rigid measurement invariance required in the
CFA approach. These invariance results derive from use of a so-
called “post-estimation algorithm” capable of identifying for each
measurement parameter (i.e., factor loadings and item intercepts),
the largest invariant set of groups for which the parameter is not
statistically signi
f
cant from the average value for that parameter
across all groups included in the invariant set of groups. In contrast,
for each group not included in the invariant set of groups, the
same parameter is considered to be statistically different from the
average value. To prevent false positive noninvariance results, this
algorithm conducts multiple pairwise comparisons across groups
based on
p
values < .05. Once alignment estimation has been
completed, additional tests can identify measurement parameters
that are approximately invariant and those that are not. Details
related to these alignment analyses are now described.
Results of the alignment analyses derive from a series
of coordinated steps.
First
, identi
f
cation of a starting set of
invariant groups must be established. This procedure involves the
assessment of every factor loading and item intercept parameter
in the model. That is, given
P
parameters and
G
Groups, there
will be (not considering a few parameters that are constrained
for
f
xating scales of the latent variables)
P
×
G
factor loading
and
P
×
G
item intercept parameters. Ultimately then, for each
of these parameters, a set of groups is identi
f
ed for which the
parameter is noninvariant; this set of groups will be different for
every parameter (personal communication, B. Muthén, November,
24, 2015). Based on the conduct of pairwise tests for each pair of
groups, two groups are then connected if the
p
-value obtained by
this comparison is larger than .01 (Asparouhov & Muthén, 2014).
Second
, from these comparisons, the largest connected set for
this parameter is determined and then serves as the starting set of
groups.
Third
, the starting set is then modi
f
ed such that: (a) the
average parameter for the current invariance set is computed, and
(b) for each group in this set, a test of signi
f
cance is conducted to
compare the parameter value for each group with the average value
computed for the current invariance set. A new group is added to
the invariant set if the
p
-value is > .001; if the
p
-value is < .001,
the group is removed from the invariant set.
Finally
, this process
is repeated until the invariant set stabilizes. That is, no groups are
either added to, or removed from the invariant set.
Turning now to the example data and statistical analyses, we
begin with speci
f
cation and testing of the con
f
gural model within
the framework of CFA methodology, followed by application of
the ML alignment method in testing for multigroup invariance
and latent mean differences.
Barbara M. Byrne and Fons J.R. van de Vijver
544
Method
Sample
Data used in this example alignment application derive from
a large project designed to measure family functioning across 30
cultures (Georgas, Berry, van de Vijver, Kagitcibasi, & Poortinga,
2006). Our interest in the present study lies with responses to the
Family Values Scale (FVS; Georgas, 1999) for 5,482 university
students drawn from 27 of these 30 countries
(deletions due
to technical complexities); sample sizes ranged from
n
= 65
(Ukraine) to
n
= 450 (Pakistan). Selection of countries focused
on representation of the major geographical and cultural regions
of the world so as to maximize eco-cultural variation in known
family-related context variables such as economic factors and
religion (Georgas et al., 2006). Thus, countries were selected from
north, central, and south America; north, east, and south Europe;
north, central, and south Africa; the Middle East; west and east
Asia; and Oceania.
The FV Scale was administered in university classroom
settings and response data collected by the research team trained
in each country. All members of each team were indigenous to
their home country.
Instrumentation
The FV Scale is an 18-item measure having a 7-point Likert
scale that ranges from 1 (
strongly disagree
) to 7 (
strongly agree
).
Items were derived from an original 64-item pool and selected in
such a way that the expected factors (hierarchy and family/kin
relationships) would be well represented. Based on EFA
f
ndings
that revealed near-zero loadings for 4 items (see Byrne & van de
Vijver, 2010; van de Vijver, Mylonas, Pavlopoulos & Georgas,
2006) we included only 14 of the 18 items in our application (For
additional information related to the data, instrumentation, and/or
ethical approval see Georgas et al., 2006.)
Internal consistency coef
f
cients were computed by factor for
the total sample; Cronbach’s coef
f
cient alpha was .87 for the
Hierarchy Scale and .80 for the Relationships Scale. Country-wise
analyses showed a median alpha coef
f
cient of .78 (IQR = .10) for
the
f
rst scale and .74 (IQR = .11) for the second scale.
The hypothesized model
The CFA model of FV Scale structure is shown schematically
in Figure 1. This model hypothesized a priori that, for each
cultural group: (a) the FV Scale is most appropriately represented
by a 2-factor structure comprising the constructs of Family
Hierarchy and Family/Kin Relations, (b) each observed variable
(i.e., FV Scale item) has a nonzero loading on the factor it was
designed to measure, and zero loadings on the other factor, (c) the
two factors are correlated, and (d) measurement error terms are
uncorrelated.
Statistical analyses
All analyses were based on the Mplus 7.4 program (Muthén &
Muthén, 1998-2015).
Given evidence of non-normality of the data for some countries,
the robust MLR estimator was used for both the CFA and
alignment analyses. Although speci
f
c distributional assumptions
such as normality of item parameters is not required in the use
of alignment (Muthén & Asparouhov, 2014), we retained the
MLR estimator in the interest of consistency. Model goodness-
of-
f
t related to the CFA application was based on the following
robust indexes: the Comparative Fit Index (CFI; Bentler, 1990),
the Root Mean Square Error of Approximation (RMSEA; Steiger,
1990), together with its 90% con
f
dence interval. In addition, the
Standardized Root Mean Square Residual (SRMR) is reported.
Results
Con
f
rmatory factor analytic approach to test for measurement
invariance and latent mean differences
For consistency with the Alignment analyses conducted later
in this study, in addition to exemplifying the known dif
f
culties in
ITEM 1
ITEM 3
ITEM 4
ITEM 6
ITEM 15
ITEM 18
ITEM 2
ITEM 5
ITEM 4
ITEM 9
ITEM 10
ITEM 11
ITEM 12
ITEM 14
FAMILY ROLES
HIERARCHY
F1
FAMILY/KIN
RELATIONS
F2
Figure 1.
CFA model of family values scale structure
The maximum likelihood alignment approach to testing for approximate measurement invariance: A paradigmatic cross-cultural application
545
attempts to establish baseline models for multigroup data noted
earlier, we began with speci
f
cation of the postulated con
f
gural
model based on the 27-country database. Goodness-of-
f
t statistics
were as follows:
χ
2
(2053)
= 4202.223; CFI = 0.869; RMSEA = 0.072,
90% CI = 0.069,
0.075; SRMR = 0.80. As expected, based on our
previous study, results revealed 108 misspeci
f
ed parameters having
values > .10 (36 cross-loadings; 72 error covariances). Pakistan
exhibited the highest number of misspeci
f
ed parameters (6 cross-
loadings; 17 error covariances), while three countries yielded
no
ev
idence
o
f
m
isspec
i
f
cation (Bulgaria, France, Ukraine).
The breakdown of these modi
f
cation indices is summarized in
Table 1. Of critical importance is the fact that these misspeci
f
ed
parameters not only vary widely across the 27 countries, but in
addition, are minimally replicable across these groups.
These results make it is easy to see why testing for invariance
across a large number of groups is fraught with problems when
based on the CFA approach. We turn now to alignment factor
analysis in testing for invariance, which takes a much less rigid
approach to these analyses.
ML alignment approach to tests for measurement invariance and
latent mean differences
The alignment approach to these analyses began with the
con
f
gural model and consistent with the CFA method, was based
on robust ML estimation (MLR). However, in contrast to CFA, as
noted earlier (see Step 1; base model), the factor means and factor
variances for each group were
f
xed to 0.0 and 1.0, respectively,
and all factor loading and item intercept parameters were freely
estimated. The optimization process comprising the Step 2
analyses subsequently results in a modi
f
cation of the con
f
gural
model such that the amount of noninvariance has been minimized,
without compromising model
f
t.
There are two types of alignment optimization that can be
speci
f
ed –
free alignment
and
f
xed alignment
. Whereas free
alignment optimization estimates the factor mean of Group 1 as an
additional parameter,
f
xed alignment optimization assumes that
this parameter (for Group 1) is
f
xed to 0.0 and as such, serves as
the reference group. Based on a simulation study comparing these
two types of alignment optimization, Asparouhov and Muthén
(2014) found that in cases where the number of groups is greater
than 2 and given evidence of measurement noninvariance, the free
alignment parameters are more accurate than estimates based on
f
xed alignment optimization.
Based on these
f
ndings, and following Asparouhov and
Muthén’s (2014) example application, we initiated the alignment
optimization process for the con
f
gural model based on the free
alignment approach. As was the case for the Asparouhov and
Muthén, study, the Mplus output yielded the following warning
message: “
Standard error comparison indicates that the free
alignment model may be poorly identi
f
ed. Using the Fixed
Table 1
CFA con
f
gural model: Summary of modi
f
cation indices > 0.10 by country
a
Misspeci
f
ed parameters
Country
Factor cross-loadings
Error covariances
1. Greece
2. Germany
3. United Kingdom
4. Netherlands
5.
Cyprus
7.
Hong Kong
8.
Brazil
10. South Korea
11. Mexico
12. Nigeria
13. Canada
14. United States
15. Turkey
16. Indonesia
17. Japan
19. France
20. Spain
21. Algeria
22. Georgia
23. Ukraine
24. Saudi Arabia
25. Chile
26. Bulgaria
27. Pakistan
28. Ghana
29. Iran
30. India
3
3
1
0
0
3
1
0
2
3
1
1
1
1
3
0
0
0
1
0
2
3
0
6
0
0
1
1
3
2
1
1
9
5
5
2
5
1
2
1
2
1
0
1
1
0
0
7
1
0
17
1
1
2
a
Although three countries were excluded from the analyses, the originally designated
numbers assigned to the original 30 countries were retained, thus accounting for
the mismatch between the number of countries in the analyses and their assigned
identi
f
cation numbers
Table 2
Factor means by country based on free ML alignment analysis
Factor 1
Factor 2
Country (Group number)
Family roles
Hierarchy in family/
kin relations
Greece (1)
-1.428
-1.213
Germany (2)
-1.877
-1.834
United Kingdom (3)
-1.658
-1.204
Netherlands (4)
-2.342
-1.717
Cyprus (5)
-1.263
-0.558
Hong Kong (7)
-1.003
-1.357
Brazil (8)
-1.232
-0.772
South Korea (10)
-0.059
-1.285
Mexico (11)
-0.919
-0.945
Nigeria (12)
0.590
-0.151
Canada (13)
-1.915
-0.876
United States (14)
-1.376
-1.029
Turkey (15)
-1.604
-1.118
Indonesia (16)
0.416
-0.298
Japan (17)
-1.426
-2.331
France (19)
-1.546
-1.139
Spain (20)
-2.514
-1.272
Algeria (21)
0.660
0.033
Georgia (22)
0.427
0.535
Ukraine (23)
-1.203
-0.764
Saudi Arabia (24)
0.697
-0.223
Chile (25)
-1.007
-0.902
Bulgaria (26)
-0.898
-0.889
Pakistan (27)
0.610
-0.506
Ghana (28)
-0.466
-0.214
Iran (29)
0.003
-0.481
India (30)
0.552
-0.552
Barbara M. Byrne and Fons J.R. van de Vijver
546
alignment option may resolve this problem
”. In using the
f
xed
alignment approach, Asparouhov and Muthén (2014) suggest
that the country having the factor mean value closest to 0.0
be speci
f
ed as the reference group. All factor mean values by
country as reported in the free alignment analytic output are
shown in Table 2. A review of these values reveals Country 29
(Iran) to have a combination of Factor 1 and Factor 2 means
closest to 0.0. Thus, the con
f
gural model was respeci
f
ed as
a
f
xed alignment analysis with the two factor means for Iran
constrained to 0.0, and the factor means for the remaining 26
countries freely estimated.
Noninvariance results.
Evidence of noninvariance pertinent
to both the factor loadings and item intercepts by country is
reported in Table 3. There are many more noninvariant item
intercepts than there are noninvariant factor loadings, a pattern
that is certainly consistent with the usual results found in tests for
invariance (e.g., Crane, Belle, & Larson, 2004; Meiring, van de
Vijver, Rothmann, & Barrick, 2005).
In reviewing these results,
it is somewhat surprising to
f
nd 7 of the 14 items having factor
loadings that exhibit no signi
f
cant noninvariance across the 27
countries. Of particular import are two items for which both
the factor loadings and item intercepts are completely invariant
– FVS5 (“Parents should teach proper behavior”) and FVS14
(“Children should respect grandparents”). These two items would
appear to be especially useful in making comparisons across these
27 countries. Over and above these two items, there is one item
in Factor 1 (FVS15; Mother should accept father’s decisions) and
4 items in Factor 2 that were found to be invariant across the 27
countries (FVS 8: Children should take care of elderly parents;
FVS9: Children should help with chores; FVS10:
Problems should
be resolved within the family; FVS12: Children should honor
family’s reputation). Outside of these 9 invariant parameters (7
factor loadings; 2 item intercepts), all others showed some degree
of noninvariance ranging from 1 incidence for factor loadings and
from 1 to 12 for intercepts across the 27 countries. Taken together,
these results certainly illuminate the complexity involved in
attempts to the attain cross-group equivalence of both the factor
loadings and item intercepts related to psychological assessment
scales not only when the number of groups is large, but also when
the groups are of a cross-cultural nature.
Our noninvariant
f
ndings are well within the 25% cutpoint
proposed
by
Mu
thén
and
Asparouhov
(2014
)
in
prov
id
ing
a
reasonable rule of thumb for determining the trustworthiness of
latent mean estimates derived from alignment results. Given 14
items and 27 groups, our
f
nding of 7 noninvariant parameters
(of a total 378 parameters) reveals evidence of factor loading
Table 3
ML alignment: Approximate measurement invariance (noninvariance) of the Family Values Scale (FVS) over 27 countries
a
Factor Loadings
Factor 1
Item
Country
FVS1
FVS3
FVS4
FVS6
FVS15
FVS18
1 2 3 4 5 7 8 10 11 12 13 14 15
(16)
17 19 20 21 (22) 23 24 25 26 27
(28)
29 30
1 2 3 4 5 7
(8)
10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17
(19)
20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16
(17)
19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16
(17)
19 20 21 22 23 24 25 26 27 28 29 30
Factor 2
FVS2
FVS5
FVS8
FVS9
FVS10
FVS11
FVS12
FVS14
1 2 3 4 5 7 8 10 11 12 13 14 15
(16)
17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11
(12)
13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
Item Intercepts
FVS1
FVS2
FVS3
FVS4
FVS5
FVS6
FVS8
FVS9
FVS10
FVS11
FVS12
FVS14
FVS15
FVS18
1 2 3 4 5 7 8 10 11 12 13 14 15
(16)
17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15
(16)
17 19
(20)
21
(22)
23 24 25 26 27 28 29 30
1 2 3 4 5
(7) (8) (10)
11 12 13 14
(15)
16 17
(19)
20 21 22 23 24 25 26 27 28 29 30
1 2
(3) (4) (5)
7 8
(10)
11 12
(13) (14)
15 16 17 19 20 21 22 23 24
(25)
(26)
27 28 29
(30)
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 7 8 10 11 12 13 14 15
(16) (17)
19
(20)
21 22 23 24 25 26 27 28
(29)
30
(1)
2 3 4 5
(7) (8) (10)
11
(12)
13 14 15
(16)
17 19 20
(21) (22)
23
(24)
25 26
(27)
28 29
(30)
1 2 3 4 5 7 8 10
(11)
12 13 14
(15)
16 17 19 20 21
(22)
23
(24)
25 26 27 28
(29)
30
1 2 3 4 5
(7)
8 10 11 12
(13) (14)
15 16 17 19 20 21 22 23 24 25 26 27 28
(29)
30
1 2 3 4 5 7
(8)
10
(11) (12) (13) (14)
15
(16)
17 19 20
(21)
22 23
(24) (25)
26
(27) (28)
29
(30)
1 2 3 4 5 7 8 10 11
(12)
13 14 15
(16)
17 19 20 21
(22)
23
(24)
25 26 27 28 29
(30)
1 2 3 4 5 7 8 10 11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30
1 2 3 4 5
(7)
8
(10)
11 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29
(30)
1 2 3 4 5 7 8 10 11 12 13 14
(15)
16 17 19 20 21 22 23 24 25
(26)
27 28 29 30
a
Noninvariant parameters are bolded and parenthesized. See Table 2 for a description of country numbers
The maximum likelihood alignment approach to testing for approximate measurement invariance: A paradigmatic cross-cultural application
547
noninvariance to be exceedingly low at 1.85%. Turning to the
intercepts, despite evidence of noninvariance related to 65 of these
parameters, their overall percentage of 17.2% is still substantially
lower than the recommended 25% cutpoint noted above. In total,
then, we feel con
f
dent in the trustworthiness of the latent mean
estimates and their comparisons across 27 countries as reported
in Table 5.
Alignment
f
t results.
In contrast to CFA for which goodness-
of-
f
t statistics are well known in the determination of well-
f
tting
models, the alignment method provides no such
f
t indexes.
Rather, given that this method assumes there is a pattern of
only
approximate
invariance in the data, analyses focus on the
f
tting functions in determination of the simplest model having
the largest amount of noninvariance.
Summarized in Table 4 are
the
f
tting functions of both the factor loading and intercept for
each item in the Family Values Scale. These values are provided
when the Technical 8 option is listed on the Output command
of the input
f
le and represent the contribution made by each of
these parameters to the
f
nal simplicity function. The far right
column represents the total contribution to the
f
tting function
by each item. In reviewing these values in Table 4, we see that
Item FVS14 contributed the least to the
f
tting functions of both
the factor loadings and intercepts thereby resulting in the lowest
overall contribution to the
f
tting function (-310.449). This result
can be interpreted as an indication that this item exhibited the
least amount of noninvariance (For an explanation of negative
f
t
function values, see Asparouhov & Muthén, 2014, Footnote 2).
Indeed, a review of the factor loading and intercept results reported
in Table 3, reveals FVS14 to be completely invariant across all 27
countries. However, of substantial interest here is why Item FVS5,
for which both the loadings and intercepts were also invariant
across the 27 countries, should result in a somewhat larger total
f
t function contribution of -411.886? One possible explanation
of this discrepancy could be that the largest degree of invariance
deviations for Item FVS5 are associated with the smallest groups
for which signi
f
cance is not as easy to achieve (T. Asparouhov,
personal communication, December 6, 2016).
The
R
2
value shown in Table 4 can be found in the computer
output following the alignment optimization process. For both
the factor loadings and item intercepts, this value represents
the explained variance/invariance index. As such, the
R
2
value
indicates the variation of these parameters across groups in the
con
f
gural model that can be explained by variation in the factor
means and variances across groups. According to Asparouhov
and Muthén (2014), a value close to 1.00 implies a high degree of
invariance, whereas a value close to 0.0 suggests a low degree of
invariance. Turning to Table 4, we see once again, however, that
whereas this fact holds true for Item FVS14, this is not the case
for Item FVS5 despite the fact that for both items, both the factor
loadings and intercepts were found to be invariant across the 27
countries. Again, this discrepancy within the same factor can be
reasonably attributed to the small sample size as noted earlier.
Factor mean results.
Factor mean values, as estimated by the
f
xed alignment method for each of the 27 countries are presented
in Table 5. Arranged in an ordered listing ranging from high to low,
the factor mean for each country is accompanied by identi
f
cation
of countries having factor means that are statistically signi
f
cantly
different (
p
< .05). These results are now detailed separately for
each of the two factors.
Factor 1: Family Roles Hierarchy
In examining this
f
rst factor, we begin by focusing on only
the
f
rst seven countries (Saudi Arabia, Algeria, Pakistan, Nigeria,
Georgia, Indonesia, and India) for at least four reasons. First,
led by Saudi Arabia with a factor mean of 1.085, these are the
countries having the highest mean values pertinent to the Family
Roles Hierarchy factor. Second, for each of these initial seven
countries, there are 20 other countries that have signi
f
cantly (
p
< .05) smaller factor means. Third, these 20 countries remain
exactly the same and in the same rank order for each of these
f
rst seven countries. Fourth, the initial seven countries precede
Country 29 (Iran), which served as the reference country for
the
f
xed alignment analyses. Of interest from a substantive
Table 4
ML alignment: Alignment
f
t statistics for the Family Values Scale across 27 countries
Factor loadings
Intercepts
Loadings + Intercepts
Factor 1
Factor 2
Item
Fit function
contribution
R
2
Fit function
contribution
R
2
Fit function
contribution
R
2
Total contribution
FVS1
FVS3
FVS4
FVS6
FVS15
FVS18
-178.360
-183.537
-180.733
-151.112
-182.657
-162.471
0.767
0.326
0.462
0.757
0.507
0.515
-174.779
-271.486
-319.854
-213.937
-235.396
-215.733
0.931
0.784
0.741
0.905
0.887
0.900
-353.139
-455.023
-500.587
-365.049
-418.053
-378.204
FVS2
FVS5
FVS8
FVS9
FVS10
FVS11
FVS12
FVS14
-186.008
-199.738
-178.396
-224.957
-196.183
-168.805
-179.279
-155.172
0.462
0.414
0.658
0.279
0.310
0.476
0.553
0.698
-230.949
-212.148
-296.795
-290.976
-215.119
-271.925
-213.986
-155.277
0.658
0.508
0.607
0.450
0.630
0.698
0.765
0.907
-416.957
-411.886
-475.191
-515.933
-411.302
-440.730
-393.265
-310.449
Barbara M. Byrne and Fons J.R. van de Vijver
548
perspective, are the countries comprising this group of 20 for
which the latent factor mean is statistically signi
f
cantly different
from those of the seven initially listed countries; their ordered
listing is as follows: Iran (29), South Korea (10), Ghana (28),
Bulgaria (26), Mexico (11), Hong Kong (7), Chile (25), Ukraine
(23), Brazil (8), Cyprus (5), United States (14), Japan (17), Greece
(1), France (19), Turkey (15), United Kingdom (3), Germany (2),
Canada (13), Netherlands (4), and Spain (20). Given no statistically
Table 5
ML alignment: Family Values Scale: Factor mean comparisons across 27 countries
a
Ranking
Country
Mean
value
Countries with signi
f
cantly smaller factor mean
Factor 1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Saudi Arabia (24)
Algeria (21)
Pakistan (27)
Nigeria (12)
Georgia (22)
Indonesia (16)
India (30)
Iran (29)
South Korea (10)
Ghana (28)
Bulgaria (26)
Mexico (11)
Hong Kong (7)
Chile (25)
Ukraine (23)
Brazil (8)
Cyprus (5)
United States (14)
Japan (17)
Greece (1)
France (19)
Turkey (15)
United Kingdom (3)
Germany (2)
Canada (13)
Netherlands (4)
Spain (20)
1.085
1.027
0.950
0.919
0.664
0.647
0.635
0.000
-0.095
-0.731
-1.408
-1.441
-1.570
-1.576
-1.883
-1.928
-1.977
-2.155
-2.231
-2.234
-2.419
-2.510
-2.594
-2.937
-2.996
-3.662
-3.932
29 10 28 26 11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
29 10 28 26 11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
29 10 28 26 11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
29 10 28 26 11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
29 10 28 26 11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
29 10 28 26 11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
29 10 28 26 11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
28 26 11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
28 26 11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
11 7 25 23 8 5 14 17 1 19 15 3 2 13 4 20
14 17 1 19 15 3 2 13 4 20
8 5 14 17 1 19 15 3 2 13 4 20
14 17 1 19 15 3 2 13 4 20
14 17 1 19 15 3 2 13 4 20
3 2 13 4 20
15 3 2 13 4 20
15 3 2 13 4 20
2 13 4 20
2 13 4 20
2 13 4 20
4 20
13 4 20
4 20
4 20
4
Factor 2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Algeria (21)
Nigeria (12)
Ghana (28)
Iran (29)
Saudi Arabia (24)
Indonesia (16)
Georgia (22)
Pakistan (27)
Cyprus (5)
India (30)
Ukraine (23)
Brazil (8)
Canada (13)
Chile (25)
Bulgaria (26)
Mexico (11)
United States (14)
Turkey (15)
United Kingdom (3)
Hong Kong (7)
Greece (1)
Spain (20)
France (19)
South Korea (10)
Germany (2)
Netherlands (4)
Japan (17)
0.326
0.085
0.047
0.000
-0.055
-0.062
-0.389
-0.458
-0.480
-0.493
-0.748
-0.781
-0.867
-0.924
-0.931
-0.947
-1.050
-1.253
-1.348
-1.391
-1.423
-1.428
-1.443
-1.566
-2.286
-2.298
-3.137
24 16 22 27 5 30 23 8 13 25 26 11 14 15 3 7 1 20 19 10 2 4 17
22 27 5 30 23 8 13 25 26 11 14 15 3 7 1 20 19 10 2 4 17
22 27 5 30 23 8 13 25 26 11 14 15 3 7 1 20 19 10 2 4 17
22 27 5 30 23 8 13 25 26 11 14 15 3 7 1 20 19 10 2 4 17
22 27 5 30 23 8 13 25 26 11 14 15 3 7 1 20 19 10 2 4 17
22 27 5 30 23 8 13 25 26 11 14 15 3 7 1 20 19 10 2 4 17
8 13 25 26 11 14 15 3 7 1 20 19 10 2 4 17
8 13 25 26 11 14 15 3 7 1 20 19 10 2 4 17
25 26 11 14 15 3 7 1 20 19 10 2 4 17
13 25 26 11 14 15 3 7 1 20 19 10 2 4 17
15 3 7 1 20 10 2 4 17
15 3 7 1 20 19 10 2 4 17
3 7 1 20 10 2 4 17
1 10 2 4 17
1 10 2 4 17
1 10 2 4 17
10 2 4 17
2 4 17
2 17
2 17
2 17
2 17
2 17
2 17
17
a
Parenthesized values represent country-assigned number within data
The maximum likelihood alignment approach to testing for approximate measurement invariance: A paradigmatic cross-cultural application
549
signi
f
cant difference between the latent means for Iran (29) and
South Korea (10), as indicated by the absence of the latter in the
list of countries having signi
f
cantly smaller means than Iran, it is
interesting to observe that the same countries ranging from Ghana
(28) to Spain (20) listed for the
f
rst seven countries, replicate as
well for Iran and South Korea. Beginning with Bulgaria (26), the
pattern of countries having signi
f
cantly smaller factor means is
more disjointed.
Factor 2: Family/Kin Relations
Although there are the same 21 countries for which their factor
means are signi
f
cantly lower than those for the
f
rst six countries
(Algeria, Nigeria, Ghana, Iran, Saudi Arabia, and Indonesia),
Algeria, with the highest factor mean value, stands out from
the rest in also showing the factor means for Saudi Arabia and
Indonesia to have a lower mean value.
The country means of the two factors are strongly correlated,
r
(27) = .76,
p
< .001, which means that countries with more traditional
family values have closer family and kin relationships. Further
validity evidence was gathered from correlating the country means
with relevant country-level social indicators. We found that level of
af
F
uence was correlated -.63 with the
f
rst factor (hierarchy) and
-.65 with the second factor (family/kin relationships),
N
= 27, both
p
s
< .001. Hofstede’s (2001) Power Distance (
N
= 22) showed positive
and signi
f
cant correlations of .66 and .61 (
p
< .001), respectively.
Hofstede’s Individualism (
N
= 22) was also signi
f
cantly (
p
< .05)
correlated with values -.62 and -.47, respectively. Schwartz’s (2012)
embeddedness factor (
N
= 12) showed a signi
f
cant correlation of
-.62 with the
f
rst factor and of .59 with Schwartz’s hierarchy factor
(both
p
s < .05). The second factor was not signi
f
cantly related
to the Schwartz factors. This patterning suggests cross-national
differences in both factors that are related to modernity and
westernization. Modernization tends to be associated with lower
scores on hierarchy and family/kin relationships.
These results are in line with
f
ndings based on raw scale
scores, reported by van de Vijver et al. (2006). This similarity is
not surprising. When we computed the correlations between the
aligned country means reported in Table 5 and the scale scores
(i.e., average item scores) used by van de Vijver et al. (2006), we
found a value of .99 for the
f
rst factor and a value of .98 for the
second factor (both
p
s < .001).
Discussion
Comparison of latent means across a large number of groups
is challenging and becomes increasingly so when such groups
are cross-cultural in nature and represent different countries. A
long-standing precondition for meaningful interpretation of these
comparisons is that the assessment scale is operating equivalently
across all groups. That is, testing of both the item factor loadings
and
i
tem
in
te
rcep
t
s
hav
e
shown
them
to
be
mea
su
remen
t
-
invariant across groups. Save for relying on tests based on partial
measurement invariance (Byrne et al., 1989), failure to satisfy this
precondition can preclude latent mean comparisons. For at least the
past 38 years, researchers have used the traditional CFA approach
to these tests for measurement invariance (Jöreskog, 1971) and
comparison of latent means (Sörbom, 1974). However, although
this methodological strategy works well when the number of
groups is small (2 or 3), it has been found to be problematic when
the number of groups is large (Asparouhov & Muthén, 2014; Byrne
& van de Vijver, 2010). Introduction of ESEM and the concept of
“approximate measurement invariance” in 2009 (Asparouhov &
Muthén) provided a new impetus to enabling tests for invariance
across a large number of groups. Followed by the introduction
of alignment optimization in 2014 (Asparouhov & Muthén),
these two methodological strategies in concert have greatly
expanded our procedures for testing measurement invariance and
subsequent comparison of latent means across a large number of
groups. In this article, we described and annotated the steps of
this new procedure, with the aim of making the procedure more
widely known and explaining its usefulness. We illustrated the
approach in a test of the measurement invariance and latent mean
differences related to the two-factor (Family Roles Hierarchy and
Family Kin Relations) Family Values Scale across 27 countries.
This data set was chosen as a previous study found that the
traditional CFA approach revealed many problems, such as a poor
f
t and dif
f
culties in identifying subsets of items or countries in
which the exact invariance model would hold (Byrne & Van de
Vijver, 2010).
The
f
ndings of the approximate invariance approach used in the
present study were very different. Our
f
ndings of noninvariance
for both the factor loadings (1.85%) and the intercepts (17.2%)
were well within the rule-of-thumb recommended cut-point
of
25%
proposed
by
Muthén
and
Asparouhov
(2014)
thereby
substantiating the trustworthiness of the alignment results.
Had the results exceeded the 25% limit, then a Monte Carlo
simulation study would have been needed to identify the sources
of noninvariance in more detail (Muthén & Asparouhov, 2014). It
is important to note that the alignment method revealed that only
a very small number of factor loadings challenged the invariance.
This
f
nding is easier to interpret than the
f
nding of a poor
f
t of
the measurement weights model of the conventional CFA model
(with an almost impossible job of
f
nding which items in which
countries are most challenging to invariance).
Of important interest, substantively, results pertinent to the
factor means are in line with earlier
f
ndings, which indicate
that family values are strongly related to modernity and the
accompanying focus on egalitarianism (as opposed to hierarchy)
and more individualism (as opposed to collectivism).
In closing out this article, we wish to express our enthusiastic
welcome and support of the alignment method in testing for
measurement invariance and latent mean differences when
the number of groups is large. For researchers whose interests
typically involve country comparisons, the automated nature
of the alignment analytic process can’t help but be particularly
bene
f
cial. Taken together, we consider the alignment method to
have several appealing features.
First
, and foremost, as mentioned
here, it enables tests for measurement invariance and latent
mean differences in large scale data, a feat not possible with the
CFA approach.
Second
, alignment allows for the estimation and
comparison of latent means despite the measurements not being
fully or partially invariant (Cieciuch et al., in press).
Third
, the
alignment method automates and substantially simpli
f
es these
comparative analyses.
Fourth
, given its capability to handle a
large number of groups, alignment can enable tests for invariance
in “sub-populations within countries and cohorts” (Munck et al.,
in press).
Fifth
, Munck et al. (in press) posit and illustrate how
alignment “is capable of producing re
f
ned scales and unbiased
statistical estimation of group means with signi
f
cance tests
Barbara M. Byrne and Fons J.R. van de Vijver
550
between pairs of group that adjust both for sampling errors and
missing data.”
Our focus in this paper was to take a nontechnical approach in
describing, explaining, and illustrating the alignment approach to
tests for invariance and latent mean differences. Along the way,
we cited key articles relevant to readers wishing more detailed and
technical information. We based our paradigmatic application on
an assessment scale having a two-factor structure. To the best of
our knowledge, this multifactorial example represents the
f
rst to
date reported in the literature. We hope that our selected example
data and walk-through of the steps involved in the application
of alignment will not only encourage other researchers to
venture forth in their use of this new and relatively sophisticated
methodology, but will also provide a springboard that makes their
initial venture less arduous.
Acknowledgements
We wish to thank Prof. James Georgas for use of these data
based on his Family Values Scale.
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