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PLS-MGA: A Non-Parametric Approach to

Partial Least Squares-based Multi-Group
Analysis

Conference Paper January 2010

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PLS-MGA A Non-Parametric Approach to
Partial Least Squares-based Multi-Group
Analysis

Jorg Henseler1

Institute for Management Research, Radboud University Nijmegen, Thomas van

Aquinostraat 3, 6525 GD Nijmegen, The Netherlands, j.henseler@fm.ru.nl

1 Introduction

For decades, researchers have applied partial least squares path modeling
(PLS, see Tenenhaus et al., 2005; Wold, 1982) to analyze complex relation-
ships between latent variables. In particular, PLS is appreciated in situations
of high complexity and when theoretical explanation is scarce (Chin, 1998).
Many fields of research have embraced the specific characteristics of PLS path
modeling, for instance behavioral sciences (c. f. Bass et al., 2003; Henningsson
et al., 2001) as well as many disciplines of business research, such as marketing
(c. f. Fornell, 1992; Ulaga and Eggert, 2006), strategy (c. f. Hulland, 1999), and
management information systems (c. f. Gefen and Straub, 1997; Chin et al.,
2003).
In many instances, researchers face a heterogeneity of observations, i. e. for
different sub-populations, different population parameters hold. This hetero-
geneity can result from different manifestations of an observed grouping vari-
ables or the assignment of cases to latent segments. For example, institutions
releasing national customer satisfaction indices may want to know whether
model parameters differ significantly between different industries (c. f. For-
nell, 1992). Another example would be cross-cultural research in general, in
which the culture or country plays the role of a grouping variable, thereby
defining the sub-populations. In case of latent segments, the grouping vari-
able is unknown a priori. While several PLS-based segmentation techniques
have been proposed (c. f. Esposito Vinzi et al., 2007), they all share the final
step of analysis: a comparison of PLS parameter estimates across groups (i.e.,
identified segments). Therefore, both in cases of observed and unobserved het-
erogeneity there is a need for PLS-based approaches to multi-group analysis.
The predominant approach to multi-group analysis was brought foreward
by Keil et al. (2000) and Chin (2000). These authors suggest to apply an un-
paired samples t-test to the group-specific model parameters using the stan-
dard deviations of the estimates resulting from bootstrapping. As Chin (2000)
2 Jorg Henseler

notes, the parametric assumptions of this approach constitute a major short-

coming. As PLS itself is distribution-free, it would be favorable to have a
non-parametric PLS-based approach to multi-group analysis.
The main contribution of this paper is to develop a non-parametric PLS-
based approach to multi-group analysis in order to overcome the shortcoming
of the current approach. The paper is structured as follows. Next to this
introductory section, the second section presents the existing approach and
elaborates upon its strengths and weaknesses. The third section develops the
new approach and describes its characteristics. The fourth section presents an
application of both the existing and the new PLS-based approach to multi-
group analysis to an example from marketing about the consumer switching
behavior in a liberalized electricity market. Finally, the fifth section discusses
the findings of this paper and highlights avenues for further research.

2 The Chin/Keil Approach to Multi-Group Analysis

In multi-group analysis, a population parameter is hypothesized to differ for

two or more subpopulations. At first, we limit our focus on the case of two
groups, and will generalize in the discussion.
Typically, multi-group analysis consists of two steps. In a first step, a
sample of each subpopulation is analyzed, resulting in groupwise parameter
estimates g . In a second step, the significance of the differences between
groups is evaluated.
Chin (2000) as well as Keil et al. (2000) propose to use an unpaired samples
t-test for comparing the parameter estimate of the first group, (1) , with the
parameter estimate of the second group, (2) . The test statistic is as follows
(see Chin, 2000):

(1) (2)
t= r (1)
2 2
(N (1) 1) (N (2) 1)
q
1 1
N (1) +N (2) 2
s(1) + N (1) +N (2) 2
s(2) N (1)
+ N (2)

This statistic follows a t-distribution with N (1) + N (2) 2 degrees of free-

dom. The subsample-specific path coefficients are denoted as , the sizes of
the subsamples as N , and the path coefficient standard errors as resulting
from bootstrapping as s. Instead of bootstrapping, sometimes jackknifing is
applied (e.g. Keil et al., 2000).
The t-statistic as provided by Equation 1 is known to perform reasonably
well if the two empirical bootstrap distributions are not too far away from
normal and/or the two variances s2(1) and s2(2) are not too different from
one another. If the variances of the empirical bootstrap distributions are as-
sumed different, Chin (2000) proposes to apply a Smith-Satterthwaite test.
The modified test statistic becomes (see Chin, 2000):
 PLS-MGA 3

(1) (2)
t= q (2)
s2(1) + s2(2)

Also this statistic follows a t-distribution. The number of the degrees of free-
dom for the t-statistic is determined by means of the Welch-Satterthwaite
equation (Satterthwaite, 1946; Welch, 1947)1 :
 2 2
s (1) s2(2)

N (1)
+
N (2)
(t)  2 2  2 2 (3)
s (1) s (2)
1 1
N (1) 1

N (1)
+ N (2) 1

N (2)

The outcome of this formula is a real value and must be rounded to the next
integer in order to obtain the number of degrees of freedom.

3 A new PLS-based approach to multi-group analysis

It is obvious that the forementioned approaches to group comparisons with
their inherent distributional assumptions do not fit PLS path modeling, which
is generally regarded as being distribution-free. Taking into account this crit-
icism against the available approaches, this paper presents an alternative
approach to PLS-based group comparisons that does not rely on distribu-
tional assumptions. The working principle of the novel PLS multi-group anal-
ysis (PLS-MGA) approach is as follows: Just like within the parametric ap-
proaches, the data is divided into subsamples according to the level of the
grouping variable, and the PLS path model is estimated for each subsample.
Moreover, each subsample becomes subject to a separate bootstrap analysis.
The novelty of the new approach to PLS-based multi-group analysis lies in
the way in which the bootstrap estimates are used to assess the robustness
of the subsample estimates. More specifically, instead of relying on distribu-
tional assumptions, the new approach evaluates the observed distribution of
the bootstrap outcomes. It is the aim of this section to determine the proba-
bility of a difference in group-specific population parameters given the group
specific estimates and the empirical cumulative distribution functions (CDFs).
Let (g) (g {1, 2}) be the group-specific estimates. Without loss of general-
ity, let us assume that (1) >(2) . In order to assess the significance of a group

effect, we are looking for P (1) (2) | (1) , (2) , CDF((1) ), CDF((2) ) .
(g)
Let J be the number of bootstrap samples, and j (j {1, . . . , J}) the
bootstrap estimates. In general, the mean of the bootstrap estimates differs
from the group-specific estimate, i. e. the empirical distribution of (g) does
not have (g) as its central value. In order to overcome this, we can determine
(g)

the centered bootstrap estimates j as:
1
Note that the formula proposed by Chin (2000) is incorrect.
4 Jorg Henseler
J
(g) (g) 1 X (g)
+ (g) .

g, j : j = j (4)
J i=1 i

Making use of the bootstrap estimates as discrete manifestations of the CDFs

we can calculate
   
(1) (2)
P (1) (2) | (1) , (2) , CDF((1) ), CDF((2) ) = P i

j (5)

Using the Heaviside step function H(x) as defined by

1 + sgn(x)
H(x) = , (6)
2
Equation 5 transforms to
J J
  1 X X  (2) (1)

P (1) (2) | (1) , (2) , CDF((1) ), CDF((2) ) = 2

H j j .
J i=1 j=1
(7)
Equation 7 is the core of the new PLS-based approach to multi-group analysis.
The idea behind it is simple: Each centered bootstrap estimate of the second
group is compared with each centered bootstrap of the first group. The number
of positive differences divided by the total number of comparisons (i.e., J 2 )
indicates how probable it is in the population that the parameter of the second
group is greater than the parameter of the first group.

4 A marketing example
We illustrate the use of both the existing and the new PLS-based approach
to multi-group analysis on the basis of an example from marketing. More
specifically, we investigate the customer switching behavior in a liberalized
energy market. According to prior studies as well as relationship marketing
theory (c. f. de Ruyter et al., 1998; Jones et al., 2000), customers are less likely
to switch their current energy provider if they are satisfied or if they perceive
high switching costs. From the Elaboration Likelihood Model it can be derived
that consumer behavior differs depending on the level of involvement (Bloemer
and Kasper, 1995; Petty and Cacioppo, 1981).
A cross-sectional study among consumers was conducted in order to test
the proposed hypotheses. The data at hands stems from computer-assisted
telephone interviews with 659 consumers. 334 consumers indicated to be
highly involved in buying electricity, while 325 consumers said to have a
low involvement. Customer satisfaction, switching costs, and customer switch-
ing intention were measured by multiple items using mainly five-point Likert
scales.
We create a PLS path model as depicted in Figure 1. This model captures
the two direct effects of customer satisfaction and perceived switching costs
 PLS-MGA 5

on customer loyalty. In order to account for the moderating effect of involve-

ment, we estimate the model separately once for the group of highly involved
consumers and once for the group of consumers having low involvement. Fig-
ure 1 also reports the standardized path coefficients per group as estimated
by means of the PLS software XLSTAT-PLSPM (Addinsoft SARL, 2007).

1 = .4422 (High Involvement)

Customer 1 = .2635 (Low Involvement)
Satisfaction

Switching
Intention

Perceived
2 = .2794 (High Involvement)
Switching Costs
2 = .2910 (Low Involvement)

R2 = .2669 (High Involvement)

R2 = .1994 (Low Involvement)

Fig. 1. Structural model with groupwise parameter estimates (standardized PLS

path coefficients).

Moreover, we conduct bootstrap resampling analyses with 500 bootstrap

samples per group. Based on the estimates, the bootstrap estimates and their
standard deviations, we calculated the p-values for group differences in the
effects of customer satisfaction and perceived switching costs on switching in-
tention. Table 1 contrasts the results of the different PLS-based approaches
to multi-group analysis, i. e. the parametric test with equal variances assumed
(homoskedastic), the parametric test with equal variances not assumed (het-
eroskedastic), and the non-parametric PLS-MGA.
Overall, the results show that the different PLS-based approaches to multi-
group analysis come to very similar conclusions. All approaches coincide in
finding a significant difference in strength of the effect of customer satisfac-
tion on switching intention, using a significance level of .05. This means, for
consumers having high involvement, the level of customer satisfaction is a
stronger predictor of switching behavior than for consumers having low in-
volvement. Moreover, all approaches coincide in rejecting a group effect in the
impact of perceived switching costs on switching intention. Despite the gen-
eral convergence of findings, there seem to be notable differences in statistical
6 Jorg Henseler

Hypothesis Statistical Test p-value (one-sided)

Customer Satisfaction parametric, homoskedastic .0123
parametric, heteroskedastic .0056
Switching intention PLS-MGA (non-parametric) .0056
Perceived Switching Costs parametric, homoskedastic .4308
parametric, heteroskedastic .4428
Switching intention PLS-MGA (non-parametric) .5528

Table 1. Comparison of statistical tests on group differences.

power between the approaches. For instance, both the parametric test with
equal variances not assumed and PLS-MGA are able to detect the group effect
on a .01 significance level, whereas the parametric test with equal variances
assumed is not.

5 Discussion

It was the aim of this contribution to introduce a non-parametric approach

to PLS-based multi-group analysis. The new approach, PLS-MGA, does not
require any distributional assumptions. Moreover, it is simple to apply in that
it uses the bootstrap outputs that are generated by the prevailing PLS im-
plementations such as XLSTAT-PLSPM (Addinsoft SARL, 2007), SmartPLS
(Ringle et al., 2007), PLS-Graph (Soft Modeling, Inc., 2002), or SPAD-PLS
(Test & Go, 2006).
Technically, the new approach to PLS-based multi-group analysis, PLS-
MGA, is purely derived from bootstrapping in combination with a rank sum
test, which makes it conceptually sound. Still, its use has only been illustrated
by means of one numerical example. Future research should conduct Monte
Carlo simulations on PLS-MGA in order to obtain a better understanding
of its characteristics, such as for instance its statistical power under various
levels of sample size, effect size, construct reliability, and error distributions.
Another opportunity for further research would be to extend PLS-MGA
to analyze more than two groups at a time. In particular, an adaptation of
the Kruskal-Wallis test (Kruskal and Wallis, 1952) to PLS-based multi-group
analysis seems promising.
Finally, PLS-based multi-group analysis has been limited to the evalua-
tion of the structural model so far, including this article. However, PLS path
modeling does not put any constraints on the measurement model so that
measurement variance could be an alternative explanation for group differ-
ences. Up to now, no PLS-based approaches for examining measurement in-
variance across groups have been proposed yet. Given its ease and robustness,
PLS-MGA may also be the point of departure for the examination of group
differences in measurement models.
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Keywords

partial least squares path modeling; PLS; group comparison; multi-group anal-
ysis; customer switching behavior

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