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MULTIGROUP ANALYSIS IN
PARTIAL LEAST SQUARES (PLS)
PATH MODELING: ALTERNATIVE
METHODS AND EMPIRICAL
RESULTS
Marko Sarstedt, Jo
¨rg Henseler and
Christian M. Ringle
ABSTRACT
Purpose – Partial least squares (PLS) path modeling has become a
pivotal empirical research method in international marketing. Owing to
group comparisons’ important role in research on international market-
ing, we provide researchers with recommendations on how to conduct
multigroup analyses in PLS path modeling.
Methodology/approach – We review available multigroup analysis
methods in PLS path modeling and introduce a novel confidence set
approach. A characterization of each method’s strengths and limitations
and a comparison of their outcomes by means of an empirical example
extend the existing knowledge of multigroup analysis methods. Moreover,
we provide an omnibus test of group differences (OTG), which allows
testing the differences across more than two groups.
Measurement and Research Methods in International Marketing
Advances in International Marketing, Volume 22, 195–218
Copyright r2011 by Emerald Group Publishing Limited
All rights of reproduction in any form reserved
ISSN: 1474-7979/doi:10.1108/S1474-7979(2011)0000022012
195
Findings – The empirical comparison results suggest that Keil et al.’s
(2000) parametric approach can generally be considered more liberal in
terms of rendering a certain difference significant. Conversely, the novel
confidence set approach and Henseler’s (2007) approach are more
conservative.
Originality/value of paper – This study is the first to deliver an in-depth
analysis and a comparison of the available procedures with which to
statistically assess differences between group-specific parameters in PLS
path modeling. Moreover, we offer two important methodological
extensions of existing research (i.e., the confidence set approach and
OTG). This contribution is particularly valuable for international
marketing researchers, as it offers recommendations regarding empirical
applications and paves the way for future research studies aimed at
comparing the approaches’ properties on the basis of simulated data.
INTRODUCTION
Studies on international marketing have frequently made use of partial
least squares (PLS) path modeling (Hair, Ringle, & Sarstedt, 2011;Hair,
Sarstedt, Ringle, & Mena, 2012;Lohmo
¨ller, 1989;Wold, 1975, 1982)to
empirically test theoretical models (for an overview, see Henseler, Ringle, &
Sinkovics, 2009). As part of international marketing researchers’ toolbox,
PLS path modeling has become a pivotal instrument for estimating
and analyzing complex path relationships between latent variables. This
method belongs to a family of alternating least squares algorithms that
extend principal component analysis and canonical correlation analysis to
estimate (mainly linear) relationships between latent variables (Lohmo
¨ller,
1989).
As with any other statistical method, PLS path modeling applications are
usually based on the assumption that the analyzed data stem from a single
population (i.e., a unique global model represents all the observations well).
However, in many real-world applications, such as in international
marketing, this assumption of homogeneity is unrealistic, because indivi-
duals are likely to be heterogeneous in their perceptions and evaluations of
latent constructs (e.g., Jedidi, Jagpal, & DeSarbo, 1997;Sarstedt & Ringle,
2010). This notion holds specifically for research on international market-
ing, which often analyzes differences in parameters in respect of different
subpopulations such as countries and cultures (Brettel, Engelen,
MARKO SARSTEDT ET AL.196
Heinemann, & Vadhanasindhu, 2008;Graham, Mintu, & Rodgers, 1994;
Grewal, Chakravarty, Ding, & Liechty, 2008;Rodrı
´guez & Wilson, 2002).
Although several studies explicitly broach the issue of group-specific effects
in their research questions, ignoring population heterogeneity – when
performing PLS path modeling on an aggregate data level – can seriously
bias the results and, thereby, yield inaccurate management conclusions
(Sarstedt, Schwaiger, & Ringle, 2009).
Although cross-national or cross-cultural differences are related to
observed heterogeneity, there can also be unobserved heterogeneity that
cannot be attributed to one (or more) pre-specified variable(s). Similar to
ignoring observed heterogeneity, unobserved heterogeneity is a serious
problem in respect of interpreting PLS path modeling results if it is not
considered in the analysis. Various response-based segmentation approaches
have recently been developed to deal with unobserved heterogeneity. These
segmentation approaches generalize, for example, genetic algorithm (Ringle,
Sarstedt, & Schlittgen, 2010), and typological regression approaches
(Esposito Vinzi, Ringle, Squillacciotti, & Trinchera, 2007;Esposito Vinzi,
Trinchera, Squillacciotti, & Tenenhaus, 2008) to PLS path modeling. Finite
mixture PLS (FIMIX-PLS; Sarstedt & Ringle, 2010; Hahn, Johnson,
Herrmann, & Huber, 2002;Sarstedt, Becker, Ringle, & Schwaiger, 2011)is
currently regarded the primary approach of all these segmentation
techniques, and has become mandatory for evaluating PLS path modeling
results (Sarstedt, 2008; Hair et al., 2012). Hair et al. (2011, p. 147), for
example, point out that ‘‘using this technique, researchers can either confirm
that their results are not distorted by unobserved heterogeneity or they can
identify thus far neglected variables that describe the uncovered data
segments.’’ Although these response-based segmentation approaches rely on
different statistical concepts, they all share the same final analysis step: A
comparison of the PLS parameter estimates across the identified latent
segments (e.g., Rigdon, Ringle, & Sarstedt, 2010;Ringle, Sarstedt, & Mooi,
2010). Therefore, no matter whether heterogeneity is observed or
unobserved, there is a need for PLS-based approaches to multigroup
analysis.
Despite its obvious importance for the international marketing discipline,
research on multigroup analysis is a rather new field. Only a small number
of methodologically oriented articles have to date been dedicated to the
discussion of available approaches (e.g., Chin & Dibbern, 2010;Rigdon
et al., 2010). Researchers’ discussions, for example, on internet forums like
http://www.smartpls.de, show that there is a strong need to clarify how
multigroup analysis can be carried out within a PLS path modeling
Multigroup Analysis in PLS Path Modeling 197
framework. Given this background, the purpose of this chapter is to
illustrate the use of multigroup analysis procedures in PLS path modeling.
Specifically, we describe available multigroup analysis approaches,
comment on their strengths and limitations, and illustrate their use by
means of an empirical example. We also propose a novel nonparametric
approach based on a comparison of bootstrap confidence intervals. This
method has been designed as a more conservative approach to PLS
multigroup analysis.
Prior approaches to PLS multigroup analysis are restricted in that they
only allow testing the differences in two groups’ parameters. However,
researchers in international marketing and other cross-cultural research
fields frequently encounter situations in which they would like to compare
more than two groups. A naive approach would be to conduct all possible
pairwise group comparisons, which would, however, quickly boost the
familywise error rate beyond any prespecified acceptable Type-I error level
(Mooi & Sarstedt, 2011). To overcome this problem, we introduce a
permutation-based analysis of variance approach, which maintains the
familywise error rate, does not rely on distributional assumptions, and
exhibits an acceptable level of statistical power.
MULTIGROUP ANALYSIS IN PLS PATH
MODELING
Conceptually, the comparison of group-specific effects entails the con-
sideration of a categorical moderator variable which, in line with Baron and
Kenny (1986, p. 1174), ‘‘affects the direction and/or strength of the relation
between an independent or predictor variable and a dependent or criterion
variable.’’ Following this concept, group effects are nothing more than a
variable’s moderating effect whereby the categorical moderator variable
expresses each observation’s group membership (Henseler et al., 2009). As a
consequence, multigroup analysis is generally regarded as a special case
of modeling continuous moderating effects (Henseler & Chin, 2010;
Henseler & Fassott, 2010). Fig. 1 illustrates the categorical moderator
variable concept graphically. Here, x
1
to x
3
represent (reflective) indicator
variables of an exogenous latent variable x,y
1
to y
3
represent (reflective)
indicator variables of an endogenous latent variable Z, and yis the
parameter of the relationship between xand Z. Lastly, mrepresents a
categorical moderating variable, which potentially exerts an influence on all
MARKO SARSTEDT ET AL.198
model relations. Researchers are usually interested in analyzing group
effects related to structural model relations. More precisely, a population
parameter yis hypothesized as different across two subpopulations (i.e., y
(1)
and y
(2)
), which are expressed by different modalities in m.
A primary concern when comparing model estimates across groups is
ensuring that the construct measures are invariant across the groups.
Amongst other criteria, as described by Steenkamp and Baumgartner
(1998), this entails, for example, that the estimates satisfy the requirement of
measurement invariance. With reference to Fig. 1, this requirement implies
that the moderator variable’s effect is restricted to the parameter yand does
not entail group-related differences in the item loadings.
Three approaches to multigroup analysis have been proposed within a PLS
path modeling framework thus far. The first approach, introduced by Keil
et al. (2000), involves estimating model parameters for each group separately,
and using the standard errors obtained from bootstrapping as the input for a
parametric test. This method is generally labeled the parametric approach
(Henseler, 2007). Chin (2003b) proposed and further described a distribution-
free data permutation test (Chin & Dibbern, 2010;Dibbern & Chin, 2005),
because the parametric approach’s distributional assumptions do not fit PLS
path modeling’s distribution-free character. This test seeks to scale the
observed differences between groups by comparing these differences to those
between groups randomly assembled from the data. Henseler (2007)
proposed and described another nonparametric procedure, which directly
compares group-specific bootstrap estimates from each bootstrap sample (see
also Henseler et al., 2009).
x
1
x
2
x
3
y
1
y
2
y
3
ηξ θ
m
Fig. 1. Moderator Modeling Framework.
Multigroup Analysis in PLS Path Modeling 199
Parametric Approach
The parametric approach was initially applied by Keil et al. (2000) (see also
Chin, 2000) and depicts a modified version of the two independent samples
t-test. As such, this approach requires the data (i.e., the PLS estimations of a
certain path coefficient across all bootstrapping subsamples) to be normally
distributed, which runs contrary to PLS path modeling’s distribution-free
character. Consequently, researchers should run a Kolmogorov–Smirnov
test with Lilliefors correction – or, in the case of small sample sizes below 50,
the Shapiro–Wilk test – to assess whether the data follow a normal
distribution (Mooi & Sarstedt, 2011). In addition to carrying out these tests,
researchers should also visually inspect the theoretical and empirical
probability distributions by means of q–qplots (Chambers, Cleveland,
Kleiner, & Tukey, 1983).
Executing the parametric test requires researchers to first run the standard
PLS path modeling algorithm for each group, followed by the bootstrapping
procedure (e.g., Hair et al., 2011; Henseler et al., 2009) to obtain the
standard errors of the group-specific parameter estimates (Keil et al., 2000).
The choice of test statistic depends on whether the parameter estimates’
standard deviations differ significantly across the groups, which can be
assessed by means of Levene’s test. If the parameter estimates’ standard
deviations are equal, the test statistic is computed as follows (Keil et al.,
2000; the equation provided by these authors has a flaw that we corrected):
t¼
~
yð1Þ~
yð2Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ððnð1Þ1Þ2=ðnð1Þþnð2Þ2ÞÞ se2
yð1Þþððnð2Þ1Þ2=ðnð1Þþnð2Þ2ÞÞ se2
yð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1=nð1ÞÞþð1=nð2ÞÞ
p(1)
Here, ~
yð1Þ(~
yð2Þ) denote the original parameter estimate for a path
relationship in group one (two), n
(1)
(n
(2)
) the number of observations in
group one (two), and seyð1Þ(seyð2Þ) the path coefficient’s standard error in
group one (two) obtained from the bootstrapping procedure. Moreover, t
represents the empirical t-value that must be larger than the critical value
from a t-distribution with n
(1)
+n
(2)
2 degrees of freedom.
1
In cases where
Levene’s test indicates that the standard errors are unequal, the test statistic
takes the following form (Chin, 2000):
t¼
~
yð1Þ~
yð2Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ððnð1Þ1Þ=nð1ÞÞse2
yð1Þþððnð2Þ1Þ=nð2ÞÞse2
yð2Þ
q(2)
MARKO SARSTEDT ET AL.200
This test statistic is asymptotically t-distributed and the degrees of
freedom (df) are determined by means of the Welch–Satterthwaite
equation. The equation below was derived by Nitzl (2010) for use in
combination with bootstrapping (note that the first draft by Chin (2000) is
not entirely correct):
df ¼ðnð1Þ1Þ=nð1Þse2
yð1Þþðnð2Þ1Þ=nð2Þse2
yð2Þ2
ðnð1Þ1Þ=nð1Þ2Þse4
yð1Þþðnð2Þ1Þ=nð2Þ2se4
yð2Þ
2
(3)
Permutation-Based Approach
The permutation-based approach was developed by Chin (2003b) and
subsequently further described by Chin and Dibbern (2010), as well as
Dibbern and Chin (2005). Analogous to Edgington and Onghena (2007), the
permutation-based test procedure builds on the observations’ random
assignment to groups. The procedure is as follows:
1. Run the PLS path modeling algorithm separately for each group.
2. Randomly permute the data; that is, the observations are randomly
exchanged between the two groups. More precisely, n
(1)
observations are
drawn without replacement and assigned to the first group; all remaining
observations are assigned to the second group. Thus, in each permutation
run uðu2f1;...;UgÞ, the group-specific sample size remains constant
ði:e:; nð1Þ
u¼nð1Þand nð2Þ
u¼nð2Þ;8uÞ. In accordance with commonly sug-
gested rules of thumb for bootstrapping sample sizes (Hair et al., 2012),
the minimum number of permutation runs should be 5,000.
3. Run the PLS path modeling algorithm for each group per permutation
run uto obtain the group-specific parameter estimates ~
yð1Þ
uand ~
yð2Þ
u.
4. Compute the differences in the permutation run-specific parameter
estimates du¼~
yð1Þ
u~
yð2Þ
u.
5. Test the null hypothesis that the population parameters are equal across
the two groups ðH0:yð1Þ¼yð2ÞÞ.
By not relying on distributional assumptions, the permutation-based
approach overcomes a key disadvantage of the parametric approach and,
thus, fits the PLS path modeling method’s characteristics. However, the
permutation-based approach requires group-specific sample sizes to be fairly
similar (Chin & Dibbern, 2010), which is a central limitation.
Multigroup Analysis in PLS Path Modeling 201
Henseler’s PLS Multigroup Analysis
From a procedural perspective, the approach proposed by Henseler (2007)
closely resembles the parametric approach. Initially, the subsamples are
exposed to separate bootstrap analyses, and the bootstrap outcomes serve as a
basis for testing the potential group differences. However, Henseler’s (2007)
approach differs in the way the bootstrap estimates are used to assess the
robustness of the group-specific parameter estimates. Instead of relying on
distributional assumptions, the new approach evaluates the bootstrap
outcomes’ observed distribution. Given two subsamples with different
parameter estimates ~
yð1Þand ~
yð2Þ, groups can be indexed – without any loss
of generality – so that ~
yð1Þ4~
yð2Þ. In order to assess the significance of a group
effect, the conditional probability pðyð1Þyð2Þj~
yð1Þ;~
yð2Þ;CDFðyð1ÞÞ;CDFðyð2ÞÞÞ
has to be determined on the basis of the group-specific parameter estimates
~
yðgÞðg2f1;2gÞ and the empirical cumulative distribution functions (CDFs).
In an initial step, the centered bootstrap estimates ( ~
yðgÞ
i) have to be
computed as follows:
~
yðgÞ
i¼~
yðgÞ
i1
BX
B
i¼1
~
yðgÞ
iþ~
yðgÞ(4)
where ~
yðgÞ
irepresents the bootstrap estimate in group gðg2f1;2gÞ and
bootstrap sample iði2f1;...;BgÞ. By using the Heaviside step function
H(x
), as defined by
HðxÞ¼1þsgnðxÞ
2(5)
and the bootstrap estimates as discrete manifestations of the CDFs, the
conditional probability is computed as follows:
pðyð1Þyð2Þj~
yð1Þ;~
yð2Þ;CDFðyð1ÞÞ;CDFðyð2ÞÞÞ ¼ 1
B2X
B
i¼1X
B
j¼1
H~
yð2Þ
j~
yð1Þ
i
(6)
The idea behind Henseler’s (2007) approach is simple. Each centered
bootstrap estimate of the second group is compared with each centered
bootstrap of the first group across all the bootstrap samples. The number of
positive differences divided by the total number of comparisons (i.e., B
2
)
indicates the probability that the second group’s population parameter will
be greater than that of the first group.
MARKO SARSTEDT ET AL.202
Henseler’s (2007) approach does not build on any distributional
assumptions and is simple to apply by using the bootstrap outputs
generated by established PLS path modeling software packages such as
SmartPLS (Ringle, Wende, & Will, 2005) and PLS-graph (Chin, 2003a).
Researchers can easily make the final calculations with available spreadsheet
software applications. However, Henseler’s (2007) approach only allows
testing the one-sided hypotheses. As the bootstrap-based distribution is not
necessarily symmetric, it cannot be used to test two-sided hypotheses.
Nonparametric Confidence Set Approach
As an answer to prior methods’ deficiencies, we propose the confidence set
approach, which builds conceptually on Keil et al.’s (2000) parametric test. Keil
et al.’s (2000) approach is a modified version of the two independent samples t-
test, which accounts for the fact that the standard deviation is obtained through
bootstrapping. As such, the test indirectly compares two bootstrap confidence
intervals, assuming that the data are normally distributed.
In accordance with this test, researchers can directly compare the group-
specific bootstrap confidence intervals, regardless of whether the data are
normally distributed or not. The procedure is as follows:
1. Run the PLS path modeling algorithm separately for each group.
2. Construct the bias-corrected a%-bootstrap confidence intervals (pre-
ferably 95% in order to avoid Type-II error inflation) for groups one and
two, ð~
yð1Þ
low;~
yð1Þ
up Þ,andð~
yð2Þ
low;~
yð2Þ
up Þ:
3. If the parameter estimate for a path relationship of group one ~
yð1Þsided
falls within the corresponding confidence interval of group two ð~
yð2Þ
low;~
yð2Þ
up Þ,
or if the parameter estimate of group two ~
yð2Þfalls within the
corresponding confidence interval of group one ð~
yð1Þ
low;~
yð1Þ
up Þ, it can be
assumed that there are no significant differences between the group-
specific path coefficients with regard to a significance level a. Conversely,
if there is no overlap, one can assume that group-specific path coefficients
are significantly different.
An important element of the confidence set approach is the bootstrap
confidence interval. Several methods for constructing bootstrap confidence
intervals have been proposed in the literature (e.g., Davison & Hinkley,
1997;Efron & Tibshirani, 1993). An obvious way to construct a confidence
interval for a parameter based on bootstrap estimates is to use a set of B
bootstrap samples x
iði2f1;...;BgÞ and calculate the bootstrap-specific
Multigroup Analysis in PLS Path Modeling 203
parameters ~
y
i. Similar to random subsampling, it is presumed that an
interval containing 90% of the ~
y
iis a 90% confidence interval for yif the
estimates are sorted in ascending sequence. Although this so-called
percentile method (Efron, 1981) is appealing due to its easy implementation,
prior research has shown that – in the case of small samples (especially
regarding asymmetric distributions) – the percentile method does not work
well (Chernick, 2008). In addition, this method has a clear tendency to
underestimate the upper confidence limit, leading to severe under-coverage
(Shi, 1992).
The double bootstrap is an alternative approach which generally provides
more accurate bootstrap confidence intervals (i.e., bootstrap the bootstrap;
McCullough & Vinod, 1998). Articles on double bootstrap methods appear
regularly in the statistical literature (e.g., Davidson & MacKinnon, 2007;
McKnight, McKean, & Huitema, 2000), but this technique has not yet
found its way into methodological research on PLS path modeling. The
double bootstrap’s basic principle is to take resamples from each bootstrap
resample; that is, for each element of x
i¼ðx
1;x
2;...;x
BÞ(i.e., the first-level
bootstrap), further resamples x
ij ¼ðx
11;...;x
1M;...;x
B1;...;x
BM Þðj2
f1;...;MgÞ are drawn from the second level. Both types of bootstrap
samples are used to estimate path coefficients on the two levels; that is, ~
y
i
(first level) and ~
y
ij (second level). Fig. 2 illustrates the general concept.
However, this approach is computationally demanding. Specifically, the
second-level bootstrap generates Mbootstrap samples for each first-level
bootstrap, leading to an overall number of BMþBbootstrap samples.
For example, following Hair et al.’s (2011) recommendation to use at least
5,000 bootstrap samples would require drawing more than 25 10
6
bootstrap samples.
Fig. 2. The Double Bootstrap Method.
MARKO SARSTEDT ET AL.204
Based on this principle, Shi (1992) proposed an accurate and efficient
double bootstrap method to estimate bootstrap confidence intervals. In this
method, the bootstrap distribution is estimated using
Q
i¼1
MX
M
j¼1
Hð~
y
ij ~
yÞ(7)
where Q
i2f0;1gis random under the empirical distribution ~
Fi. Its values
are sorted in ascending sequence ðQ
ð1ÞQ
ð2ÞQ
ðBÞÞand are used to
determine the lower and upper confidence limits:
ð~
yðgÞ
low;~
yðgÞ
up Þ¼ð
~
yðgÞ
l
½;~
yðgÞ
u
½
Þ(8)
where [ ] is a nearest integer function with the arguments given by
l¼ðBþ1ÞQða=2Þ;and (9)
u¼ðBþ1ÞQð1a=2Þ(10)
Since estimating the bootstrap confidence interval (Efron & Tibshirani,
1993) entails potential systematic errors, Davison and Hinkley (1997)
proposed a bias correction, which should be considered when constructing
the interval. The use of bias-corrected confidence intervals was introduced
to PLS path modeling in the context of the confirmatory tetrad analysis
(Gudergan, Ringle, Wende, & Will, 2008) and bootstrapping-based
significance testing (Henseler et al., 2009). The bias correction is as follows:
bias ¼1
BX
B
i¼1
~
y
i~
y1
BM X
B
i¼1X
M
j¼1
~
y
ij 2
BX
B
i¼1
~
y
iþ~
y
!
¼3
BX
B
i¼1
~
y
i1
BM X
B
i¼1X
M
j¼1
~
y
ij 2~
y
(11)
This bias correction is used to estimate the confidence interval’s lower and
upper limits:
ð~
yðgÞ
low;~
yðgÞ
up Þ¼ð
~
yðgÞ
½lbias;~
yðgÞ
½ubiasÞ(12)
Although Shi’s (1992) method for estimating double bootstrap-based
confidence intervals has proven to be accurate in various data constellations,
the improvement in accuracy comes at the expense of computational demand.
Multigroup Analysis in PLS Path Modeling 205
MULTIGROUP ANALYSIS WITH MORE THAN TWO
GROUPS
All previously presented approaches to group comparison in PLS path modeling
have in common that they test the difference in the parameters between two
groups. As previously mentioned, researchers in international marketing and
other cross-cultural research fields frequently encounter situations in which they
would like to compare more than two groups. As soon as there are more than two
groups, two questions arise: Does a parameter differ between groups? And, if so,
between which groups does it differ? Although the second question can be
answered by means of pairwise group comparisons, the first question demands
more attention. Again, as mentioned, a naive approach would be to conduct all
possible pairwise group comparisons, which would lead to the well-known
multiple testing problem; that is, the familywise error rate quickly exceeds any
prespecified acceptable Type-I error level.
There are, however, several ways of controlling the familywise error rate. A
standard remedy is the Bonferroni correction, which aims at retaining the
familywise error rate by dividing each comparison’s error-rate by the overall
number of comparisons. The Bonferroni correction tends to be conservative;
that is, it sacrifices statistical power for the sake of a predefined level of Type-I
error. An alternative would be to conduct an ANOVA (i.e., an overall F-test),
comparing the different groups’ bootstrap outputs. However, using an
ANOVA would mean relying on distributional assumptions (e.g., Hair,
Black, Babin, & Anderson, 2010; Mooi & Sarstedt, 2011), which Chin and
Dibbern (2010) criticize. An optimal test for the differences between multiple
groups in a PLS path modeling framework should (1) maintain the familywise
error rate, (2) deliver an acceptable level of statistical power, and (3) not rely
on distributional assumptions. Another desirable feature is that such a test
should be available in PLS path modeling software packages. In this section,
we propose such an omnibus test of group differences (OTG).
Our OTG approach uses bootstrapping, permutation, and random
selection’s asymptotic properties. The underlying idea of this nonparametric
OTG dates back to Pitman (1938) – although the concrete implementation is
inspired by Bortz, Lienert, and Boehnke (2003), who proposed a
‘‘randomized ANOVA’’ method. Their method tests the hypothesis that G
samples are drawn from populations with identical means. Applied to PLS
path modeling, the OTG approach consists of the following steps:
1. The first step encompasses groupwise bootstrapping. Per group, a large
number of bootstrap samples are drawn and estimated in order to obtain
MARKO SARSTEDT ET AL.206
an empirical distribution of the group-specific model parameters. The
number of bootstrap samples should be equal across the groups. The
presentation of the bootstrap estimates may be structured as shown in
Table 1.
2. The bootstrap results of the previous step facilitate the variance ratio’s
computation. Analogous to a one-way ANOVA (e.g., Mooi & Sarstedt,
2011), the variance explained by the grouping variable is evaluated
relatively to the overall variance:
FR¼s2
between
s2
within
¼GBð1=ðG1ÞÞ PG
g¼1ð
Ag
AÞ2
1=ðB1ÞPG
g¼1PB
i¼1ð~
yðgÞ
i
AgÞ2(13)
In this equation, ~
yðgÞ
iis the parameter estimate from the i
th
bootstrap
sample (i=1,y,B) of group g(g=1,y,G),
Agthe average over the
bootstrap parameter estimates of group g, and
Athe grand mean of all
the bootstrap values.
3. This permutation step uses the previously generated bootstrap estimates
(e.g., as displayed in Table 1). The elements of the first row – the
outcomes of the first bootstrap estimation in each group – can be
permuted in G! different ways, whereby each permutation has the same
likelihood of occurrence. If this idea is extended to all Brows, this results
in (G!)
B
permutations. Since the test outcomes are independent of the
group index, there are only (G!)
B1
different permutations.
For many bootstrap samples, the associated number of permutations
becomes extremely high (e.g., in the case of B=5,000 bootstraps and
G=3 groups, (3!)
4,999
=9.508 10
3,889
permutations are required). Such
extensive computations are not feasible within a reasonable time.
Table 1. Arranging the Groupwise Bootstrap Estimates of a Specific
Model Parameter.
Bootstrap Estimation Groups
12yG
1~
yð1Þ
1~
yð2Þ
1
y~
yðGÞ
1
2~
yð1Þ
2~
yð2Þ
2
y~
yðGÞ
2
yyyyy
B~
yð1Þ
B~
yð2Þ
B
y~
yðGÞ
B
Multigroup Analysis in PLS Path Modeling 207
Consequently, we draw on the random selection (i.e., Monte Carlo)
concept. A reasonably high number of permutations (e.g., 5,000) are
sufficient to obtain an outcome that approximates the results for (G!)
B1
different permutations. Subsequently, the variance ratio F
R
can be
computed for each randomly selected permutation (e.g., one obtains
5,000 F
R
values).
4. The error probability is computed in the final step. As is usual with
regard to randomization tests, one has to examine whether the empirical
F
R
value from Step 1 is among the a% largest values of the empirical F
R
value distribution obtained from the previous step. The error probability
pcan be determined as follows:
p¼1
UX
U
u¼1
HðFRFRuÞ(14)
In this equation, H() is again the Heaviside step function, Udenotes the
number of permutations, and FRuthe empirical F
R
-value obtained in
permutation run u.
The proposed OTG approach offers a possibility to control the
familywise error rate. This approach does not rely on distributional
assumptions, nor is it as conservative as the Bonferroni correction. The
OTG approach can be applied to the regular bootstrap output of
standard PLS path modeling software implementations, such as
SmartPLS (Ringle et al., 2005).
2
EMPIRICAL EXAMPLE
Overview
In this section, we use a well-established PLS path model and empirical
data to illustrate and compare the different multigroup analysis
approaches. The selected PLS path model draws on prior studies by
Homburg and Rudolph (1997), as well as by Festge and Schwaiger (2007),
and examines the effects of customer satisfaction drivers on customer
loyalty in industrial markets.
3
Since the focus of this section is not centered
on the substantive model as such, but on an illustration of the multigroup
analysis approaches, we only provide a brief description of the data and
model set-up.
MARKO SARSTEDT ET AL.208
Measures and Data
The data originate from a survey – by means of standardized mail
questionnaires – of a major industrial firm’s customers in three countries
(Germany, n=65; the United Kingdom, n=115; and France, n=170). All the
respondents rated their satisfaction with the different performance features
related to the firm’s products and services. Our model includes the following
three performance features, which have been shown to significantly affect
customer satisfaction in industrial markets (e.g., Festge & Schwaiger, 2007;
Sarstedt et al., 2009): (1) satisfaction with products, (2) satisfaction with
services, and (3) satisfaction with pricing. The corresponding construct
measures were adapted from Homburg and Rudolph (1997), as well as from
Festge and Schwaiger (2007), using a five-item scale to measure satisfaction
with products, and two three-item scales to measure satisfaction with services
and pricing. Loyalty was measured with three well-known items (intention to
repurchase, word-of-mouth recommendation, and intention to remain a
customer in the long run) from prior research (Zeithaml, Berry, &
Parasuraman, 1996). We used reflective indicator variables measured on
seven-point Likert-type scales.
Results
Principal components analysis supports the scales’ unidimensionality. In
addition, we computed coefficient bvalues which range from 0.59
(satisfaction with services; German subsample) to 0.83 (satisfaction with
products; German subsample), and are thus above the commonly suggested
threshold of 0.50 (Revelle, 1979). Subsequent PLS path model analyses reveal
that all measures meet the commonly suggested criteria for measurement
model assessment as described, for example, by Chin (1998),Henseler et al.
(2009), and Hair et al. (2012). Specifically, the analyses per country show that
all indicators exhibit loadings above 0.70, and that the constructs’ average
variance extracted (AVE) values are above 0.50. Likewise, all constructs
achieve high composite reliability values of 0.80 and higher (Table 2).
We used two approaches to assess the constructs’ discriminant validity.
First, we examined the indicators’ cross loading, which revealed that no
indicator loads higher on an opposing construct (Hair et al., 2012). Second,
we applied the Fornell and Larcker (1981) criterion and tested whether each
construct’s AVE is greater than its squared correlation with the remaining
constructs. Both analyses clearly indicate that the constructs exhibit
Multigroup Analysis in PLS Path Modeling 209
discriminant validity. Overall, these results provide clear support for the
measures’ reliability and convergent validity.
Table 2 shows the results of the structural model evaluation. The bootstrap
analyses using 5,000 samples and a number of cases equal to the country-
specific sample size (using the individual sign change option) show that all the
satisfaction features – with the exception of satisfaction with services in the
German subsample – have a significant (pr0.10) effect on customer loyalty. A
comparison of the country-specific path coefficients reveals several differences
in the effects. For example, whereas satisfaction with products has the
strongest effect on loyalty in the German subsample, it has a much weaker
effect in the UK subsample. Instead, satisfaction with prices exerts the
strongest influence on loyalty in the UK subsample. In respect of the French
subsample, the effects are somewhat balanced across the three satisfaction
types. However, the question emerges whether these numeric differences
between country-specific path coefficients are statistically significant.
In a first step, we applied the OTG approach to assess if the path
coefficients are equal across the three groups. The analysis reveals that in
respect of all three structural model relations, the null hypothesis that the
three path coefficients are equal across the three groups can be rejected.
Table 2. Country-Specific Results.
Germany United Kingdom France
Latent variables
Satisfaction with services Composite reliability 0.829 0.860 0.848
AVE 0.619 0.672 0.650
Satisfaction with products Composite reliability 0.910 0.889 0.895
AVE 0.669 0.616 0.630
Satisfaction with prices Composite reliability 0.829 0.833 0.895
AVE 0.619 0.625 0.623
Loyalty Composite reliability 0.869 0.836 0.846
AVE 0.689 0.631 0.646
n65 115 170
Path relationships
Satisfaction with services-Loyalty 0.040 0.238
***
0.195
***
Satisfaction with products-Loyalty 0.669
***
0.130
*
0.289
***
Satisfaction with prices-Loyalty 0.163
*
0.500
***
0.398
***
R
2
0.690 0.600 0.609
Notes:
*
Significance at 0.10,
**
significance at 0.05,
***
significance at 0.01.
MARKO SARSTEDT ET AL.210
Specifically, the analysis yields F
R
values of 579.93 (Services-Loyalty),
3,393.36 (Products-Loyalty), and 1,504.48 (Prices-Loyalty), rendering all
differences significant at pr0.01. These results suggest that, in respect of all
three relationships, at least one path coefficient differs from the remaining
two across the three countries.
Table 3 shows the differences in three comparisons’ path coefficient
estimates (Germany vs. the United Kingdom, Germany vs. France, and the
United Kingdom vs. France), and provides the results of multigroup
comparisons based on the parametric approach, the permutation test, and
Henseler’s (2007) approach. The analysis shows that, generally, the
multigroup comparison test results correspond very closely. However,
differences emerge in respect of Keil et al.’s (2000) parametric tests,
which, in most cases, yields higher t-values than the permutation test. For
example, in the comparison of the German and the UK subsamples, Keil
et al.’s (2000) test renders the relationship between satisfaction with services
and loyalty significant (pr0.10), whereas this result does not occur in the
permutation test. Consequently, the parametric approach can generally be
considered more liberal in terms of rendering a certain difference significant.
Conversely, Henseler’s (2007) approach appears to be rather conservative in
this respect. Although the approach indicates several significant differences,
one has to bear in mind that it only allows testing a one-sided hypothesis.
Comparing its results with, for example, the critical t-values of a one-sided
Table 3. Multigroup Comparison Test Results.
Relationship Comparison 9diff9t
Parametric
t
Permutation
p
Henseler
Services-Loyalty Germany vs. United Kingdom 0.198 1.930
*
1.632 0.095
Germany vs. France 0.155 1.530 1.351 0.130
United Kingdom vs. France 0.043 0.410 0.441 0.363
Products-Loyalty Germany vs. United Kingdom 0.539 4.285
***
3.285
***
0.005
Germany vs. France 0.270 2.662
***
2.614
***
0.013
United Kingdom vs. France 0.159 1.503 1.367 0.107
Prices-Loyalty Germany vs. United Kingdom 0.338 2.156
**
2.052
**
0.021
Germany vs. France 0.235 1.967
**
1.802
*
0.063
United Kingdom vs. France 0.102 0.930 0.959 0.193
Notes:
*
Significant at 0.10,
**
significant at 0.05,
***
significant at 0.01.
Results for Henseler (2007) eligible for a one-sided test.
Multigroup Analysis in PLS Path Modeling 211
parametric test (e.g., 1.28 for a=0.10), clearly shows that Henseler’s (2007)
approach reveals fewer significant effects.
Table 4 shows the bias-corrected 95% confidence intervals according to
Shi’s (1992) approach, as well as the corresponding multigroup analysis
results. Again, if the parameter estimate for a path relationship of one group
(Table 2) does not fall within the corresponding confidence interval of
another group (Table 4) and vice versa, there exists no overlap and we can
assume that the group-specific path coefficients are significantly different
with regard to a significance level a.
Comparing the confidence set approach’s results with those of prior tests
shows that the former is more conservative than Keil et al.’s (2000) test.
Whereas the parametric approach indicates a significant difference (pr0.05)
between the German and French subsamples in terms of the satisfaction
with prices and loyalty relationship, this is not the case with the confidence
set approach. Overall, in terms of significant differences, the approach
closely resembles the permutation test’s results.
Table 4. Bias-corrected 95% Confidence Intervals (Shi 1992) and
Multigroup Comparison Results.
Relationship Confidence Intervals Comparison Significance
Germany United
Kingdom
France
Services-
Loyalty
[0.206,0.250] [0.035,0.380] [0.065,0.325] Germany vs. United
Kingdom
Nsig.
Germany vs. France Nsig.
United Kingdom vs.
France
Nsig.
Products-
Loyalty
[0.329,0.991] [0.021,0.275] [0.115,0.469] Germany vs. United
Kingdom
Sig.
Germany vs. France Sig.
United Kingdom
vs. France
Nsig.
Prices-
Loyalty
[0.158,0.447] [0.303,0.658] [0.239,0.551] Germany vs. United
Kingdom
Sig.
Germany vs. France Nsig.
United Kingdom vs.
France
Nsig.
Notes: Sig. denotes a significant difference at 0.05; Nsig. denotes a nonsignificant difference at 0.05.
MARKO SARSTEDT ET AL.212
SUMMARY, CONCLUSIONS, AND FUTURE
RESEARCH
PLS path modeling is a key multivariate analysis method for empirical
research in international marketing (e.g., Henseler et al., 2009), and
multigroup analyses are of primary interest in this field (e.g., Hoffmann,
Mai, & Smirnova, 2011). This research contributes to the literature on PLS
path modeling in several ways: First, we present and compare the
procedures available for multigroup analysis in PLS path modeling. Second,
we introduce the novel nonparametric confidence set approach based on the
comparison of parameter estimates and bootstrap confidence intervals.
Third, we address the issue of simultaneously comparing more than two
groups by providing a permutation-based analysis of variance approach
that maintains the familywise error rate, does not rely on distributional
assumptions, and exhibits an acceptable level of statistical power.
The results of the empirical example suggest that Keil et al.’s (2000)
parametric approach is the most liberal of the procedures as it, compared to
the permutation test, generally yields higher t-values. Furthermore, Keil
et al.’s (2000) approach renders more differences significant than the novel
confidence set approach does. The confidence set approach, just like
Henseler’s (2007) procedure, appears to be very conservative, as they indicate
fewer significant differences vis-a
`-vis alternative multigroup comparison tests.
International marketing research often deals with relatively small sample
sizes and a relatively large number of groups (i.e., data from different cultures
or countries). Our novel confidence set approach specifically provides
researchers with certain advantageous functions in these kinds of situations.
The confidence set approach is nonparametric, can handle relatively small
sizes, and is more conservative than the other approaches and, thus, is less
prone to Type-II errors. These aspects are particularly relevant when
conducting multigroup analysis in international marketing research.
Overall, our findings suggest that if researchers need to compare more
than two groups (e.g., countries or cultures), they should first conduct the
OTG in order to test the hypothesis that a model parameter differs across
groups. If this hypothesis is supported, or if there are only two groups,
researchers should subsequently apply the novel confidence set approach to
multigroup analysis with regard to comparing two groups of data.
Obviously, our empirical illustration using satisfaction data can only be a
first step toward understanding the different multigroup analysis
approaches’ adequacy. For example, with regard to the confidence set
Multigroup Analysis in PLS Path Modeling 213
approach’s conservative performance, it is unclear if the corresponding path
relationships are truly identical in the population, or if the approach’s
potential lack of statistical power biases the outcome. The approaches may
perform differently, depending on the model set-up and sample at hand. It is
therefore necessary to compare the approaches’ point estimation accuracy,
and their statistical power in systematically changed data constellations by
conducting a Monte Carlo experiment. In a related context, Qureshi and
Compeau (2009) evaluate the ability of variance and covariance-based
approaches to structural equation modeling to detect between-group
differences and to accurately estimate the moderating effects’ strength.
However, the authors only consider Keil et al.’s (2000) parametric
approach, rather than comparing the performance of different approaches
within a PLS path modeling framework.
Another important issue of, and avenue for future research on, multi-
group comparisons is exploring ways to test for measurement invariance
(e.g., Steenkamp & Baumgartner, 1998;Vandenberg & Lance, 2000)ina
PLS path modeling context. If measurement invariance cannot be
established, the differences in path coefficients cannot be fully attributed
to true relationships, because respondents from different groups might have
systematically interpreted a given measure in conceptually different ways.
Although measurement invariance should be added to the well-established
criteria reliability, homogeneity, and validity when performing multigroup
analysis, prior research on PLS path modeling has largely neglected this
issue. Haenlein and Kaplan (2011) proposed an approach to control for
gamma change, which occurs when the construct’s domain (i.e., its meaning)
differs in each group. Specifically, the authors propose a combination of
Box’s M test and ordinary least squares regressions, which can help assess
this bias’s magnitude and, hence, support researchers when they have to
decide whether parameter estimates can be trusted or not. However, Rigdon
et al. (2010, p. 269) provide a different perspective on measurement
invariance in PLS path modeling, stating that ‘‘an insistence on measure-
ment invariance across groups carries its own assumption that the impact of
group membership is limited to the structural parameters of the structural
model. In many cases, this assumption is questionable or even implausible,
and researchers should consider group membership effects on both
structural and measurement parameters.’’ Furthermore, the authors point
out that PLS path modeling is a method based on approximation and
designed for situations with a less firmly established theoretical base (Wold,
1982). Therefore, researchers should interpret the results from PLS path
modeling involving multiple groups with the necessary caution.
MARKO SARSTEDT ET AL.214
NOTES
1. Sarstedt and Wilczynski (2009) describe a complementary approach for paired
samples.
2. A code file for R (R-Development-Core-Team, 2011), which performs the
approach, can be obtained from the second author upon request.
3. Sarstedt et al. (2009) applied the FIMIX-PLS method (Hahn et al., 2002;
Rigdon et al., 2010;Ringle, Wende, & Will, 2010;Ringle, Sarstedt, & Mooi, 2010;
Sarstedt et al., 2011;Sarstedt & Ringle, 2010) to the original study by Festge and
Schwaiger (2007) to uncover unobserved heterogeneity.
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