Structural Equation Modeling, 17:82–109, 2010

Copyright © Taylor & Francis Group, LLC

ISSN: 1070-5511 print/1532-8007 online
DOI: 10.1080/10705510903439003

A Comparison of Approaches for the Analysis

of Interaction Effects Between Latent Variables
Using Partial Least Squares Path Modeling
Jörg Henseler
Institute for Management Research
Radboud University Nijmegen

Wynne W. Chin
Department of Decision and Information Sciences
University of Houston

In social and business sciences, the importance of the analysis of interaction effects between
manifest as well as latent variables steadily increases. Researchers using partial least squares (PLS)
to analyze interaction effects between latent variables need an overview of the available approaches
as well as their suitability. This article presents 4 PLS-based approaches: a product indicator
approach (Chin, Marcolin, & Newsted, 2003), a 2-stage approach (Chin et al., 2003; Henseler &
Fassott, in press), a hybrid approach (Wold, 1982), and an orthogonalizing approach (Little, Bovaird,
& Widaman, 2006), and contrasts them using data related to a technology acceptance model. By
means of a more extensive Monte Carlo experiment, the different approaches are compared in
terms of their point estimate accuracy, their statistical power, and their prediction accuracy. Based
on the results of the experiment, the use of the orthogonalizing approach is recommendable under
most circumstances. Only if the orthogonalizing approach does not find a significant interaction
effect, the 2-stage approach should be additionally used for significance test, because it has a higher
statistical power. For prediction accuracy, the orthogonalizing and the product indicator approach
provide a significantly and substantially more accurate prediction than the other two approaches.
Among these two, the orthogonalizing approach should be used in case of small sample size and
few indicators per construct. If the sample size or the number of indicators per construct is medium
to large, the product indicator approach should be used.

Along with the development of scientific disciplines, the complexity of hypothesized relation-
ships has steadily increased (Cortina, 1993). Moreover, social, psychological, and administrative

Correspondence should be addressed to Jörg Henseler, Institute for Management Research, Nijmegen School of
Management, Radboud University Nijmegen, Thomas van Aquinostraat 1, 6525 GD Nijmegen, The Netherlands.
E-mail: j.henseler@fm.ru.nl

82
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 83

theories are understood to depend on environmental factors, as for instance expressed by

contingency theory (see, e.g., Woodward, 1958). Thus, besides the examination of direct effects
between constructs, researchers are more and more interested in interaction effects (Carte
& Russell, 2003). Researchers who want to use structural equation modeling (SEM) to test
complex models with interaction effects of latent variables can basically choose between two
techniques: covariance-based SEM, and partial least squares (PLS) path modeling. Although
there is a growing body of literature on how interaction effects can be modeled by means of
covariance-based SEM (cf. reviews of Li et al., 1998; Marsh, Wen, & Hau, 2004; Lee, Song, &
Poon, 2004; Schumacker & Marcoulides, 1998), there is a lack of comprehensive studies that
compare existing approaches for the analysis of interaction effects between so-called latent
variables using PLS path modeling. Although several approaches have been suggested (cf.
Chin, Marcolin, & Newsted, 2003; Henseler & Fassott, in press; Wold, 1982), it remains
unclear what the particular strengths and weaknesses of each approach are. When choosing
among different approaches for the analysis of interaction effects using PLS path modeling,
a researcher addresses three questions to the different approaches: To what extent can each
approach recover the true parameter value of the interaction effect and the single effects,
respectively? How powerful or conservative is each approach? Finally, if the main objective of
the model is to predict rather than to test a theory—which is a typical scenario of PLS path
modeling—which approach has the highest prediction accuracy?
The objective of this article is to answer these questions, and to develop a guideline for
researchers who would like to apply PLS path modeling to analyze interaction effects between
latent variables. The article begins with a detailed presentation of the three approaches that have
been suggested so far: the so-called product indicator approach (Chin, Marcolin, & Newsted,
1996, 2003), a two-stage approach (Chin et al., 2003; Henseler & Fassott, in press), and a hybrid
approach that is based on an initial proposal by Wold (1982). Moreover, an additional approach
for SEM in general, recently suggested by Little, Bovaird and Widaman (2006), is adapted
to PLS path modeling. These four approaches are compared in terms of their performance
(i.e., point estimate accuracy, power, and prediction accuracy). To illustrate the differences
in estimation outcomes, we begin by reanalyzing the data of a technology acceptance study
originally conducted by Chin et al. (2003). Additionally, we use a Monte Carlo simulation
to systematically examine the behavior of each of these approaches. We then compare and
contrast results, draw respective conclusions, and give recommendations for how interaction
effects can optimally be modeled by means of PLS path modeling.

APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS

BETWEEN LATENT VARIABLES USING PLS PATH MODELING

Interaction effects (also called moderating effects) are evoked by variables, the variation of
which influences the strength or the direction of a relationship between an independent and
a dependent variable (Baron & Kenny, 1986, p. 1174). Such moderator variables can be
categorical or metric in nature. Typically, the effect of categorical moderator variables is tested
by means of group comparisons. For this purpose, observations are grouped according to their
value of the categorical moderator variable. Subsequently, analyses are conducted, and the
84 HENSELER AND CHIN

outcomes are compared across groups. Alternatively, in the case of metric moderator variables,
the product of two variables is used to represent the interaction effect. For a structural model,
the regression equation would have the following form:

˜ D “0 C “1
Ÿ C “2
 C “3
Ÿ
 C © (1)

Here, ˜ is the endogenous variable that shall be explained by the exogenous variable Ÿ, the
moderator variable , and the interaction of the two. The “s represent the regression parameters,
where “0 stands for the constant. The unexplained variance is captured by the error term ©.
Note that ˜, Ÿ, and  are latent variables, and thus are supposed to be measured with error.
Equation 1 can be rearranged into a different form, representing a regression of ˜ on Ÿ having
the constant as well as the slope of the exogenous variable Ÿ depending on the level of the
latent moderator variable  (cf. Jaccard & Turrisi, 2003, p. 17):

˜ D .“0 C “2
/ C .“1 C “3
/
Ÿ C © (2)

This form provides intuitive appeal to the interpretation of interaction effects: An increase
in the moderator variable  of 1 SD implies a change of the effect of Ÿ on ˜ by “3 . For
instance, if  is standardized and increased from 0 to 1, the slope of Ÿ changes from “1 to
“1 C “3 .
In the literature related to PLS path modeling, three approaches for the analysis of interaction
effects between variables have been presented so far. First, Chin et al. (1996, 2003) developed
the so-called product indicator approach. Second, Henseler and Fassott (in press) and Chin et al.
(2003) suggested using a two-stage approach under certain circumstances. Third, based on an
initial proposal by Wold (1982), the inventor of PLS, a hybrid approach can be constructed.
Finally, we adapt an orthogonalizing approach suggested by Little et al. (2006) for modeling
interactions among latent variables to the effect that it can be used with PLS path modeling.
We now describe all four approaches in detail.

The Product Indicator Approach

Busemeyer and Jones (1983) as well as Kenny and Judd (1984) introduced an initial approach
for the use of SEM methodology to study interaction effects among latent variables. Recently,
Marsh et al. (2004) refined this approach and postulated using the unconstrained product of
indicators, thereby delivering a full-fledged specification of structural equation models with
interactions. All of these authors suggested building product terms using the indicators of the
latent independent variable and the indicators of the latent moderator variable. These product
terms serve as indicators of the interaction term in the structural model. Chin et al. (1996,
2003) were the first to transfer this approach to PLS path modeling. First, they introduced a
new latent variable, the latent interaction term. Further, they suggested creating the so-called
product indicators pij ; that is, all possible pairwise products of the centered indicators of the
exogenous variable (xi ) and of the moderator variable (mj ):

pij D xi
mj 8i; j: (3)

 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 85

The product indicators pij become the indicators of the latent interaction term. If the
exogenous latent variable Ÿ has I indicators and the latent moderator variable  has J
indicators, then the latent interaction variable will have I
J product indicators. Figure 1
shows a simple example of the product indicator approach. Note that Chin et al. (2003)
recommended using the centered original indicators to produce the product indicators. Although
such a proceeding does not necessarily diminish the multicollinearity resulting from building
the product term (see Echambadi & Hess, 2007; contrary to Cohen, 1978; Cronbach, 1987), it
does facilitate the interpretation of the interaction model results.
Although this approach has been considerably difficult to implement in a covariance-based
SEM context, it was found to be easily implementable in PLS path modeling. One question
that is particularly raised in SEM is whether all possible indicator products should be formed
and assigned to the interaction term. Jöreskog and Yang (1996) showed that one product
term is sufficient to estimate the moderating effect. Jonsson (1998) used several but not all
product terms to get a better estimate of the interaction term’s standard error. However, this
coincides with a stronger bias of the estimates themselves (Jonsson, 1998). As PLS path
modeling does not rely on distributional assumptions and therefore does not require any
estimate of parameter standard errors, the correct estimation of the interaction term’s path
coefficient is to be prioritized against the estimation of its standard error. Furthermore, PLS
path modeling relies on consistency at large (cf. Schneeweiss, 1993); that is, a large number
of indicators per latent variable are needed to get unbiased estimates of the latent variable.
Hence, a limitation of the number of product indicators, as discussed by Jöreskog and Yang
(1996) in the case of covariance-based SEM, would not be worthwhile. Moreover, due to the
character of the PLS estimation, a reduction of product indicators does not greatly facilitate
the speed of the estimation process. Thus, the approach by Chin et al. (2003) seems quite
promising.
Usually, the path coefficient of the interaction term (i.e., the latent variable with the product
indicators) is immediately used to quantify the interaction effect and interpreted like the

FIGURE 1 Product indicator approach.

86 HENSELER AND CHIN

parameter estimate of a product term in multiple regression (the “3 in Equation 1). We would
like to emphasize that this is not feasible,1 due to a special characteristic of PLS. PLS calculates
path coefficients from standardized latent variable scores; that is, for the structural model
Equation (1) is not estimated, but a regression in which all predictors, including the interaction
term, are standardized. The regression equation of the structural model as estimated by PLS
takes the following form:

.Ÿ
/ .Ÿ
/
˜ D “1
Ÿ C “2
 C “3
C© (4)
s.Ÿ
/

In general, if the product of two standardized variables does not equal the standardized
product of these variables, the value “3 will be different from the value “3 in Equation 1. To
make “3 interpretable, we suggest adjusting the standard deviation of the interaction term’s
latent variable score prior to calculating the structural model regression with the interaction
term. Concretely, the latent variable scores of the interaction term should be multiplied by
the weighted average of the standard deviations of the product indicators, using the respective
loadings of the product indicators as weights. As neither Chin et al. (2003) nor any other
researchers applying the product indicator approach (e.g., Pavlou & Gefen, 2005) raise the
issue of standardized interaction terms, it remains unclear whether an adjustment has ever
been made. The fact that PLS internally works with standardized latent factor scores implies
that the constant of the structural model equation equals zero. In contrast to covariance-based
SEM (see Marsh et al., 2004), the PLS product indicator approach does not require explicit
modeling of the means structure. However, if the indicator means and standard deviations are
meaningful, an external post hoc calculation of nonstandardized model parameters seems to be
worthwhile.

The Two-Stage Approach

The idea of the two-stage approach was initially suggested by Chin et al. (2003) and elaborated
by Henseler and Fassott (in press). These authors recognized that if the exogenous variable or
the moderator variable are formative, the pairwise multiplication of indicators is not feasible.
“Since formative indicators are not assumed to reflect the same underlying construct (i.e., can be
independent of one another and measuring different factors), the product indicators between two
sets of formative indicators will not necessarily tap into the same underlying interaction effect”
(Chin et al., 2003, Appendix D). Instead of using the product indicator approach, Henseler and
Fassott (in press) similarly articulated the two-stage PLS approach for estimating moderating
effects in particular when formative constructs are involved. The two-stage approach makes
use of PLS path modeling’s advantage of explicitly estimating latent variable scores. The two
stages are built up as follows:

1 This phenomenon can best be observed at an interaction model with one indicator per latent variable. Usually,

the path coefficient “3 as estimated by the product indicator approach will not equal the regression parameter “3 of
the product term in a multiple regression between the indicators unless indicators are standardized.
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 87

Stage 1: In the first stage, the main effect PLS path model is run to obtain estimates for
the latent variable scores. The latent variable scores are calculated and saved for further
analysis.
Stage 2: In the second stage, the interaction term Ÿ
 is built up as the element-wise
product of the latent variable scores of the exogenous variable Ÿ and the moderator
variable . This interaction term as well as the latent variable scores of Ÿ and  are used
as independent variables in a multiple linear regression on the latent variable scores of
the endogenous variable ˜.

Figure 2 illustrates the two-stage approach. Note that although the latent variable scores of
Ÿ and  are standardized, the interaction term is not—and should not be. If the interaction
term were standardized, it would become difficult to quantify an interaction effect, because an
interpretation as illustrated at the basis of Equation 2 would not be feasible any more.
Although the latent variable scores are estimated in the first stage, they are used in the
second stage to determine the coefficients of the regression function in the form of Equation 1.
The second stage can be realized by multiple linear regression or be implemented within PLS
path modeling by means of single-indicator measurement models. Although Chin et al. (2003)

FIGURE 2 Two-stage approach.

88 HENSELER AND CHIN

as well as Henseler and Fassott (in press) limited the usage of the two-stage approach to cases
when the exogenous or the moderator variable or both are formative, this limitation is not
mandatory. It can also be applied to models with interaction effects among latent variables
with reflective measurement models. However, a clear disadvantage of the two-stage approach
is that the moderating effect is not taken into account when estimating the latent variable scores.
The fact that the two-stage approach is a limited-information approach was a key reason for
Chin et al. (2003) to prefer the product indicator approach.

The Hybrid Approach

Wold (1982), the inventor of PLS, presented a simple device for the PLS estimation of path
models with nonlinearities in the structural model. Although he considered in depth only a
model with a quadratic term, the approach is generalizable to other nonlinear relations between
latent variables, in particular encompassing interaction effects. Wold’s approach combines the
advantages of the two prior methods. First, it takes the structural model into account when
estimating the latent variable scores. Second, the interaction term’s scores are guaranteed to
coincide with the product of the interacting latent variables’ scores. Wold’s approach combines
elements of the two-stage approach and the product indicator approach, making it a hybrid
approach. Like in the two-stage approach, the element-wise product of the latent variable
scores of the independent and the moderator variable serves as an interaction term; also as in
the product indicator approach, the interaction term is updated during the algorithm runtime
and used to estimate the latent variable scores. To illustrate the working principle of the hybrid
approach, we draw on Tenenhaus, Vinzi, Chatelin, and Lauro’s (2005) description of the PLS
algorithm, and extend it where necessary (in italics). The PLS algorithm delivers estimates for
the latent variable scores by means of an iterative process that basically consists of four steps:

1. Outer estimation of the latent variable scores: Outer estimates of the latent variables, ŸO on
and ˜on , are calculated as linear combinations of their respective indicators. These outer
_

estimates are standardized (i.e., M D 0, SD D 1). The weights of the linear combinations
result from Step 4 of the previous iteration. When the algorithm is initialized, and
no weights are yet available, any arbitrary nontrivial linear combination of indicators
can serve as an outer estimation of a latent variable. The first step is extended so that
after computing the outer estimates of all latent variables, an interaction term proxy is
calculated as the element-wise product of the respective outer estimates.
2. Estimation of the inner weights: For each latent variable, inner weights are calculated
to reflect how strongly the other latent variables are connected to it. There are three
schemes available for determining the inner weights. Wold (1982) originally proposed
the centroid scheme. Later, Lohmöller (1989) developed the factor weighting scheme
and the path weighting scheme. The centroid scheme uses the sign of the correlations
between a latent variable2 and its adjacent latent variables, and the factor scheme uses
the very correlations. The path weighting scheme pays tribute to the arrow orientations
in the path model. The weights of those latent variables that explain the focal latent
variable are set to the regression coefficients yielded from a regression of the focal latent

2 More precisely, the outer estimate.

 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 89

variable on its explaining latent variables. The weights of those latent variables that are
explained by the focal latent variable are determined as in the factor weighting scheme.
Independent of the weighting scheme, a weight of zero is assigned to all nonadjacent
latent variables. In the second step, inner weights for the interaction term proxy are also
determined.
3. Inner estimation of the latent variable scores: Inner estimates of the latent variables, ŸO in
and ˜in , are calculated as linear combinations of the outer estimates of their respective
_

adjacent latent variables, using the previously determined inner weights. In this step,
the interaction term proxy is also used to estimate the latent variable scores of the
endogenous variable.
4. Estimation of the outer weights: The outer weights are calculated either as the covariances
between the inner estimates of each latent variable and its indicators (in Mode A) or as
the regression weights resulting from the ordinary least squares regression of the inner
estimates of each latent variable on its indicators (in Mode B). In this step, there are no
changes necessary for the hybrid approach, because the interaction term does not have
any indicators assigned.

These four steps are iterated until the change in outer weights between two iterations goes
below a predefined limit. The algorithm terminates after Step 1, delivering estimates of latent
variable scores for all latent variables including the interaction term. The path coefficients result
from a regression of the endogenous variable’s scores on the other variables’ scores (including
the interaction term).
As shown, the hybrid approach requires a modification of the PLS algorithm. However, as
up to now it has not been implemented in any of the leading PLS software distributions,3 the
approach has not been available for applied research yet.

The Orthogonalizing Approach

Little et al. (2006) recently suggested an orthogonalizing approach for modeling interactions
among latent variables. Although this approach was applied to covariance-based SEM, it
is easily transferable to PLS path modeling. Basically, the orthogonalizing approach is an
extension of the use of residual centering for moderated multiple regressions as described by
Lance (1988). The underlying idea of residual centering is that “[i]deally, an interaction term
is uncorrelated with (orthogonal to) its first-order effect terms” (Little et al., 2006, p. 499).
One way to achieve this goal is mean-centering the variables that are multiplied to form the
interaction term. However, more often than not, a certain degree of correlation between the
interaction term and the original variables remains. To eliminate this remaining correlation,
residual centering can be used:

Residual centering, as originally suggested by Lance (1988), is essentially a two-stage OLS

procedure in which a product term [: : : ] is regressed onto its respective first-order effect[s]. The

3 We considered the following software distributions (author in parenthesis): LVPLS (Lohmöller, 1984) including

later graphical extensions, PLS-Graph (Chin, 1993–2003), SmartPLS (Ringle et al., 2005), and SPAD-PLS (DECISIA,
2003).
90 HENSELER AND CHIN

residuals of this regression are then used to represent the interaction [: : : ] effect. The variance
of this new orthogonalized interaction term contains the unique variance that fully represents
the interaction effect, independent of the first-order effect variance (as well as general error or
unreliability). (Little et al., 2006, p. 500)

As a consequence of the orthogonality of the interaction term, the parameter estimates of

the single effects in a model with interaction term are identical to the parameter estimates
of the direct effects in a model without interaction. Furthermore, residual centering yields a
regression coefficient for the orthogonalized cross-product term that can directly be interpreted
as the effect of the interaction on the dependent variable (Lance, 1988, p. 164) and thus
replace the assessment of the increase in the coefficient of determination due to the inclusion
of the interaction term. Little et al. (2006) seized these advantages to SEM by introducing
a modification to the product indicator approach. Like in the latter one, product indicators
are first created as element-wise products of the indicators of the independent and the mod-
erator variables. For two latent variables, Ÿ and , with two indicators each, say x1 and x2 ,
and m1 and m2 , respectively, the following preliminary product indicators p1 to p4 will be
created:

p11 D x1
m1 (5)

p12 D x1
m2 (6)

p21 D x2
m1 (7)

p22 D x2
m2 (8)

Each of the four preliminary product indicators is then regressed on all indicators of the
exogenous and the moderator variable:

p11 D b0;11 C b1;11x1 C b2;11 x2 C b3;11 m1 C b4;11 m2 C e11 (9)

p12 D b0;12 C b1;12x1 C b2;12 x2 C b3;12 m1 C b4;12 m2 C e12 (10)

p21 D b0;21 C b1;21x1 C b2;21 x2 C b3;21 m1 C b4;21 m2 C e21 (11)

p22 D b0;22 C b1;22x1 C b2;22 x2 C b3;22 m1 C b4;22 m2 C e22 (12)

The residuals of these regressions, in this case e11 to e22 , are then used as indicators of the
interaction term, in analogy to the product indicator approach. That way, it is ensured that
the indicators of the interaction term do not share any variance with any of the indicators of
the exogenous as well as the moderator variable. From the fact that PLS calculates the latent
variable scores as linear combinations of the respective indicators, it can be derived that the
interaction term is orthogonal to its constituting latent variables.
The orthogonalizing approach as described by Little et al. (2006) has a correlated error
structure that is required to provide unbiased estimates. In contrast to covariance-based SEM,
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 91

PLS path modeling does not and cannot impose constraints onto error covariances. Releasing
constraints is therefore neither necessary nor possible.4

SOFTWARE IMPLEMENTATION

To illustrate the four approaches, and to compare them in terms of their performance, it was
necessary to use adequate PLS software. Although three of the four approaches—the product
indicator approach, the two-stage approach, and the orthogonalizing approach—could have
been executed by means of available software, one approach, the hybrid approach, needs a
manipulation of the standard PLS algorithm. As none of the available PLS software packages
allows for manipulations of the PLS algorithm itself,5 we created our own implementation of
the PLS algorithm applying the algorithm in vector form following the detailed description
of Tenenhaus et al. (2005). As an extension to the PLS algorithm, the hybrid approach was
implemented as described in the previous section. We used R 2.3.1 (R Development Core Team,
2006) as programming language. Besides the thorough compliance with the algebraic terms as
formulated by Tenenhaus et al. (2005), the correctness of the PLS code was further verified
by contrasting the results of two data analyses conducted with our implementation against
the results of the PLS path modeling implementations PLS-Graph 3.0 (Chin, 1993–2003) and
SmartPLS 2.0 M2 (Ringle, Wende, & Will, 2005). Except for obvious rounding inaccuracies,
the results were identical. Further evidence of the correctness of our PLS implementation is
provided by reanalyzing the data that underlay the second study of Chin et al. (2003), which
is illustrated in the following section.

ANALYZING AN EXAMPLE FROM MANAGEMENT INFORMATION

SYSTEMS RESEARCH: A MODIFIED TECHNOLOGY
ACCEPTANCE MODEL

To illustrate the different approaches for the analysis of interaction effects between latent
variables using PLS path modeling, we chose a particular technology acceptance model (TAM),
mainly for two reasons: First, PLS has become a method of choice for statistical analysis in
TAM studies (cf. Gefen & Straub, 1997). Second, moderating effects play a prominent role in
the TAM literature (cf. Bhattacherjee & Sanford, 2006; Brown & Venkatesh, 2005). Drawing
on Davis (1989), Chin et al. (2003) identified perceived usefulness and enjoyment as direct
antecedents of information technology adoption intentions. Moreover, they tested an interaction
effect of enjoyment on the perceived usefulness–intention relation by means of the PLS product
indicator approach. The structural model can be expressed by the following equation:
adoption intention D “1
usefulness C “2
enjoyment
(13)
C “3
usefulness
enjoyment C ©

4 We thank an anonymous reviewer for this observation.

5 SmartPLS, as Java-based software, principally allows for plug-ins. However, this functionality had not yet been
provided by the programmers when this article was written.
92 HENSELER AND CHIN

TABLE 1
Different Model Results for the Technology Acceptance Model

Parameter Estimate “
(Bootstrap t Value)

Perceived
Analyzer Approach Usefulness Enjoyment Interaction R2 f2

Chin, Marcolin, & Main effects model 0.517 0.269 0.465

Newsted (2003) Product indicator 0.449 0.227 0.209 0.500 0.070
Own calculations Main effects model 0.5165 0.2687 0.4649
(8.2575) (5.3682)
Product indicator 0.4486 0.2262 0.2092 0.4995 0.0691
(7.4094) (5.2924) ( 4.1534)
Two-stage 0.4447 0.2269 0.1413 0.5003 0.0708
(7.2649) (5.2639) ( 4.3531)
Hybrid 0.4446 0.2267 0.1413 0.5001 0.0704
(7.0779) (4.9759) ( 4.1218)
Orthogonalizing 0.5165 0.2686 0.1848 0.4988 0.0676
(8.8435) (5.8844) ( 1.9472)
Orthogonalizing (adjusted) 0.5165 0.2686 0.1537 0.4988 0.0676
Product indicator (adjusted) 0.4470 0.2257 0.1516 0.4994 0.0689

We reanalyzed the data from Chin et al.6 using all four approaches for the analysis of
interaction effects between latent variables using PLS path modeling. Furthermore, we adjusted
the parameter estimates of the product indicator approach as well as the orthogonalizing
approach. Table 1 presents the outcomes we gained by means of the different approaches
and contrasts them to the original results of Chin et al. (2003).
In a first step, we estimated the main effects model. We came to the same results as Chin et al.
(2003). Perceived usefulness (parameter estimate 0.5165) and enjoyment (parameter estimate
0.2687) explain 46.49% of the variance in adoption intention. In a second step, we tested
a perceived usefulness–enjoyment interaction by means of the four approaches. The product
indicator approach reproduced Chin et al.’s (2003) estimates for the single effects (“1 D 0:4486
and “2 D 0:2622) as well as for the interaction effect (“3 D 0:2092), increasing the explained
variance to 0.4995. The results of the two-stage approach and the hybrid approach are almost
identical to those of the product indicator approach, with one exception: the interaction term’s
path coefficient for both approaches. Both the two-stage approach and the hybrid approach yield
an estimate for “3 of 0.1413, which is a decrease in absolute terms of more than 32%. Overall,

6 “The data were obtained from a single organization that had recently installed electronic mail. A total of 60

questions relating to a recent introduction of electronic mail were presented. Of the 575 questionnaires distributed, 250
usable responses were analyzed representing 43.5 percent of those surveyed. On average, the respondents had been
using electronic mail for 9 months, sent 2.53 messages per day (s.d. D 2.36) and received 4.79 messages per day
(s.d. D 3.49). Respondents were on average 39 years old (s.d. D 9.28) and had worked for the company an average
of 11 years (s.d. D 6.9). Sixty percent of the respondents were male. The respondents came from various levels in the
organization, 13 percent were managers, 12 percent were engineers, 38 percent were technicians, and the remaining
37 percent were clerical workers” (Chin et al., 2003, online appendix, p. 9).
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 93

the results of the two-stage approach and the hybrid approach differ only marginally. In contrast,
the results of the orthogonalizing approach differ substantially. Obviously, the parameters of
the single effects equal those of the main effects model. The parameter of the interaction effect
has an intermediate value. For all approaches, bootstrapping with 500 bootstrap resamples was
performed. Whereas for the single effects, the bootstrap t values do not differ strongly across
approaches and warrant a p < :001 significance level, the interaction effect bootstrap t value of
the orthogonalizing approach is much lower than the respective value of the other approaches,
and does not signal significance at a 5% level.
Furthermore, Table 1 presents the results of the product indicator approach with the inter-
action term’s standard deviation adjusted prior to calculating the structural model regression.
The same was done for the orthogonalizing approach. The adjustments resulted in smaller path
coefficients of the interaction effect, coming closer to the values of the two-stage and the hybrid
approach. At the same time, the other path coefficients as well as the proportion of explained
variance remain almost unchanged.
As the results reveal, there are some differences across the four approaches. However, it
remains unclear which of the approaches delivers estimates most proximate to the true under-
lying population parameters, and in particular, whether the interaction effect can be considered
significant on a predefined significance level like, for example, ’ D :05. Furthermore, there
might or might not be a ranking of the approaches in terms of their prediction accuracy.
Although in this study, there were almost no differences in variance explanation (R2 s varied
for only 0.3%), this does not necessarily imply that in general, all four approaches have the
same prediction accuracy. To find generalizable patterns and to investigate the appropriateness
of the four approaches presented, we conducted a Monte Carlo simulation.

A MONTE CARLO EXPERIMENT

The goal of this experiment is to elucidate the performance of the different approaches for
the analysis of interaction effects between latent variables using PLS path modeling. We
compare the point estimate accuracy, the power, and the prediction accuracy of the four
considered approaches at different numbers of indicators per construct and different numbers
of observations. The steps of the Monte Carlo experiment are as follows: First, we define
an underlying true model and determine the factor attributes. Second, we generate random
data that emerge from the model parameters. Third, given the random data, we let each PLS
approach estimate the model under each factor combination.
The choice of the underlying model is crucial for the simulation outcomes. We define an
underlying true model that is as simple as possible, consisting of one exogenous latent variable,
one latent moderator variable, and one endogenous latent variable. As true path coefficients,
0.3 for the moderated path (“1 ), 0.5 for the second single effect (“2 ), and 0.3 for the interaction
effect (“3 ) were chosen. The simulation model is depicted in Figure 3. We opted for a mixed
design, in which the four different approaches are fixed factors, and the number of indicators
per latent variable as well as the number of observations serve as random factors. As possible
numbers of indicators per latent variable, six different levels were used: 2, 4, 6, 8, 10, or 12
indicators. As representative numbers of observations for PLS path models with interaction
94 HENSELER AND CHIN

FIGURE 3 Simulation model.

effects, five different levels were selected: 50, 100, 150, 200, and 500 observations.7 We chose
a full-factorial design to have the possibility of capturing eventual interaction effects between
the factors. Hence, 30 (six levels of indicators times five levels of observations) conditions
emerged.
Under each of the 30 conditions, 500 Monte Carlo runs were intended. For each run,
standard-normal latent variable scores were created for the exogenous, the endogenous, and
the moderator variable, building the basis for the calculation of indicator scores. Loadings
of all indicators were set to 0.7. Not only does the value of 0.7 allow for comparisons with
existing Monte Carlo studies (mainly Chin et al., 2003), it also makes the measurement models
touch conventional acceptance thresholds of reliability and validity, where internal consistency
(reliability) as represented by Cronbach’s alpha ranges from 0.658 (for 2 indicators) to 0.920
(for 12 indicators) and Dillon-Goldstein’s rho ranges from 0.794 to 0.993 respectively, for
indicator reliability of 0.49 (meeting Hulland’s [1999] threshold of 0.4), and convergence
validity as represented by an average variance extracted of 0.49 (coming close to Fornell and
Larcker’s [1981] threshold of 0.50). The values xij of the i th indicator of each latent variable
Ÿj were created as a linear combination of the respective latent variable scores and a normal-
distributed random vector:
p
xij D 0:7
Ÿj C 1 0:72
N.0; 1/ (14)

Furthermore, all indicators xij were standardized, having a mean of zero and a standard deviation
of one. As additional input for the product indicator approach, the product indicators were
calculated following Equation 3. For the orthogonalizing approach, like in Equations 9 to 12,
regressions were applied and their residuals saved as indicators of the interaction term.
For each run in each condition, all four approaches for the analysis of interaction effects
between latent variables using PLS path modeling were used to estimate the model. We selected
the path weighting scheme as inner weighting scheme, because it is the only scheme that

7 Chin, Marcolin, and Newsted (2003) also investigated the case of 20 observations. However, taking into account

that in the orthogonalizing approach, the indicators of the interaction term are regressed on all indicators of the
exogenous variables, several conditions would have led to singularities. For instance, having 10 indicators per construct,
a regression with 20 independent variables would have to be estimated by means of 20 observations.
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 95

takes into account the causal order of the constructs (Lohmöller, 1989). Furthermore, within
all approaches, all latent variables were estimated with Mode A, which usually represents
reflective measurement models (cf. Chin, 1998). Each estimation was accompanied by 500
bootstrap calculations to assess the significance of the estimates, thereby following Mooney
and Duval’s (1993) recommendation of 500 resamples when applying bootstrapping to estimate
a parameter using a single sample. For each bootstrap sample, the product indicators as well
as the orthogonalized product indicators were recalculated.8 The following PLS estimation
outcomes were measured for each run:

 Path coefficient estimates for the single effects and the interaction effect.
 Bootstrap t values for all effects.
 The squared correlation between the predicted latent variable scores of the endogenous
variable and its true scores.

In the following sections, we report and discuss the simulation outcomes for parameter accuracy,
statistical power, and prediction accuracy.

Parameter Accuracy
To compare the different parameters, we examined to what extent the parameter estimates
deviated from the true values. At first, we assessed the mean relative bias (MRB). The MRB
is the mean over the deviations from the true value, and is algebraically defined as (Reinartz,
Echambadi, & Chin, 2002, p. 237):

1 X XOi Xi
t
MRB D (15)
t i D1 Xi

Positive MRBs indicate an overestimation of the true parameter, negative MRBs an un-
derestimation. Table 2 gives an overview of the MRB of each approach under all conditions
for the three path coefficients. The overall picture gives mixed evidence: Whereas the two-
stage approach and the orthogonalizing approach yield the most accurate estimates of the
single effects, the product indicator approach together with the orthogonalizing approach
perform best for the estimation of the interaction effect. To verify this rather qualitative
evaluation, we conducted a series of analyses of variance (ANOVAs).9 Box’s test of equality
of covariance matrices was significant, F .714; 365748823/ D 136:889, p < :001, indicating
that the covariance matrices vary among conditions. However, potential biases in significance
are remedied by means of the balanced design of our experiment.
Table 3 contains the F tests resulting from the three ANOVAs. Although there are no
significant three-way interactions among the independent variables, there are significant two-

8 We thank an anonymous reviewer for this hint. It must be stated, though, that until now, existing implementations

of the bootstrap in PLS software do not allow inclusion of this recalculation.

9 Although we encountered significant sphericity, with Bartlett’s test indicating a ¦2 .3/ of 1135.473, we decided

against a multivariate analysis of variance (MANOVA), because MANOVA does not allow for random factors, and—
albeit significant—correlations between the relative biases of the three path coefficients were well below 0.1, thus not
substantial.
 TABLE 2
Mean Relative Bias of the Path Coefficients

No. of Indicators
No. of
Parameter Observations Approach 2 4 6 8 10 12

“1 50 Two-stage 0.0816 0.0256 0.0040 0.0076 0.0154 0.0113

Product indicator adjusted 0.0910 0.0477 0.0274 0.0279 0.0228 0.0221
Hybrid 0.0823 0.0290 0.0061 0.0032 0.0129 0.0098
Orthogonalizing 0.0849 0.0278 0.0067 0.0066 0.0182 0.0203
100 Two-stage 0.0880 0.0439 0.0201 0.0090 0.0056 0.0043
Product indicator adjusted 0.0931 0.0520 0.0306 0.0225 0.0183 0.0176
Hybrid 0.0890 0.0452 0.0215 0.0105 0.0071 0.0059
Orthogonalizing 0.0870 0.0397 0.0205 0.0072 0.0040 0.0018
150 Two-stage 0.0898 0.0480 0.0308 0.0194 0.0179 0.0059
Product indicator adjusted 0.0921 0.0529 0.0362 0.0261 0.0251 0.0130
Hybrid 0.0901 0.0488 0.0317 0.0203 0.0189 0.0068
Orthogonalizing 0.0913 0.0474 0.0297 0.0182 0.0155 0.0054
200 Two-stage 0.1010 0.0478 0.0343 0.0287 0.0190 0.0166
Product indicator adjusted 0.1029 0.0511 0.0384 0.0329 0.0238 0.0214
Hybrid 0.1014 0.0485 0.0349 0.0294 0.0196 0.0174
Orthogonalizing 0.0995 0.0474 0.0330 0.0276 0.0180 0.0170
500 Two-stage 0.1018 0.0572 0.0369 0.0270 0.0240 0.0213
Product indicator adjusted 0.1025 0.0583 0.0383 0.0286 0.0257 0.0232
Hybrid 0.1020 0.0574 0.0372 0.0273 0.0243 0.0216
Orthogonalizing 0.1007 0.0565 0.0361 0.0268 0.0235 0.0210
“2 50 Two-stage 0.1496 0.0722 0.0394 0.0308 0.0198 0.0092
Product indicator adjusted 0.1650 0.1046 0.0863 0.0841 0.0755 0.0670
Hybrid 0.1509 0.0747 0.0428 0.0345 0.0236 0.0131
Orthogonalizing 0.1545 0.0720 0.0419 0.0387 0.0218 0.0091
100 Two-stage 0.1597 0.0862 0.0588 0.0392 0.0348 0.0238
Product indicator adjusted 0.1658 0.0991 0.0779 0.0625 0.0570 0.0469
Hybrid 0.1604 0.0876 0.0610 0.0413 0.0370 0.0264
Orthogonalizing 0.1617 0.0899 0.0599 0.0433 0.0356 0.0240
150 Two-stage 0.1648 0.0935 0.0676 0.0478 0.0346 0.0333
Product indicator adjusted 0.1697 0.1020 0.0787 0.0602 0.0482 0.0472
Hybrid 0.1655 0.0946 0.0690 0.0494 0.0363 0.0349
Orthogonalizing 0.1638 0.0932 0.0666 0.0501 0.0353 0.0331
200 Two-stage 0.1657 0.0954 0.0639 0.0488 0.0375 0.0310
Product indicator adjusted 0.1685 0.1015 0.0714 0.0575 0.0472 0.0407
Hybrid 0.1660 0.0962 0.0649 0.0499 0.0386 0.0322
Orthogonalizing 0.1653 0.0946 0.0643 0.0496 0.0377 0.0308
500 Two-stage 0.1690 0.1014 0.0692 0.0540 0.0451 0.0370
Product indicator adjusted 0.1700 0.1036 0.0720 0.0570 0.0485 0.0404
Hybrid 0.1693 0.1018 0.0696 0.0545 0.0456 0.0375
Orthogonalizing 0.1683 0.1025 0.0700 0.0554 0.0439 0.0378
“3 50 Two-stage 0.1687 0.1017 0.0907 0.0776 0.0587 0.0617
Product indicator adjusted 0.1255 0.0233 0.0035 0.0417 0.0342 0.0458
Hybrid 0.1626 0.0947 0.0818 0.0685 0.0490 0.0510
Orthogonalizing 0.1094 0.0033 0.0197 0.0204 0.0271 0.0075
100 Two-stage 0.1489 0.0887 0.0726 0.0659 0.0456 0.0370
Product indicator adjusted 0.1135 0.0350 0.0151 0.0032 0.0170 0.0243
Hybrid 0.1466 0.0836 0.0676 0.0604 0.0400 0.0312
Orthogonalizing 0.1107 0.0276 0.0108 0.0056 0.0091 0.0122
150 Two-stage 0.1437 0.0959 0.0663 0.0545 0.0450 0.0430
Product indicator adjusted 0.1139 0.0549 0.0243 0.0096 0.0000 0.0001
Hybrid 0.1414 0.0928 0.0628 0.0505 0.0410 0.0390
Orthogonalizing 0.1138 0.0553 0.0266 0.0186 0.0134 0.0118
200 Two-stage 0.1452 0.0959 0.0719 0.0539 0.0456 0.0393
Product indicator adjusted 0.1261 0.0658 0.0396 0.0216 0.0123 0.0046
Hybrid 0.1435 0.0934 0.0692 0.0509 0.0426 0.0361
Orthogonalizing 0.1240 0.0670 0.0426 0.0286 0.0224 0.0171
500 Two-stage 0.1396 0.0894 0.0664 0.0497 0.0435 0.0354
Product indicator adjusted 0.1323 0.0773 0.0540 0.0375 0.0309 0.0222
Hybrid 0.1390 0.0883 0.0652 0.0485 0.0422 0.0342
Orthogonalizing 0.1325 0.0787 0.0562 0.0412 0.0353 0.0274

Note. Best performing approach per condition is marked in bold.

96
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 97

TABLE 3
F Test Over the Relative Biases

Mean Degrees of
Square Freedom
Relative Bias
of the Effect Determinant MSHyp MSErr df Hyp df Err F Significance

RB(“1 ) Intercept 73.285 11.142 1 6 6.577 .042

Approach 0.359 0.081 3 14 4.422 .022
Indicators 9.982 0.025 5 26 396.500 .000
Observations 1.179 0.091 4 17 12.996 .000
Approach  Indicators 0.009 0.003 15 60 2.839 .002
Approach  Observations 0.075 0.003 12 60 22.451 .000
Indicators  Observations 0.019 0.003 20 60 5.689 .000
3-way interaction 0.003 0.006 60 59,878 0.530 .999

RB(“2 ) Intercept 330.221 24.129 1 5 13.686 .013

Approach 0.915 0.197 3 14 4.644 .018
Indicators 23.794 0.031 5 22 772.843 .000
Observations 0.348 0.188 4 13 1.852 .179
Approach  Indicators 0.023 0.005 15 60 4.341 .000
Approach  Observations 0.180 0.005 12 60 34.529 .000
Indicators  Observations 0.013 0.005 20 60 2.576 .003
3-way interaction 0.005 0.005 60 59,878 1.131 .228

RB(“3 ) Intercept 182.556 19.006 1 5 9.605 .024

Approach 8.754 0.860 3 14 10.184 .001
Indicators 18.308 0.127 5 26 144.285 .000
Observations 0.765 0.852 4 14 0.898 .492
Approach  Indicators 0.075 0.016 15 60 4.732 .000
Approach  Observations 0.800 0.016 12 60 50.501 .000
Indicators  Observations 0.068 0.016 20 60 4.274 .000
3-way interaction 0.016 0.014 60 59,878 1.167 .177

way interactions; that is, the influence of the approach used is not independent from the
number of indicators and the number of observations. However, contrasting the mean square
of the direct effects with that of the interaction effects, the contribution of the interaction
effects is rather small. Furthermore, profile plots were generated to detect possible crossover
interactions. Figure 4 contains six pithy profile plots visualizing the shaded cells of Table 2.
These exemplary profile plots reveal that crossover interactions play only a minor role. We thus
proceed to the assessment of direct effects. As Table 3 reveals, all factors have significant direct
effects on the relative bias of the estimate. To identify differences among the four approaches,
we conducted post hoc tests over the approach factor. We applied Tamhane T2, Dunnett T3,
and Dunnett C, because they do not assume covariances to be equal across cells. All three
tests provide a coherent pattern (see Table 4): Concerning the relative bias of the single effects
estimates, the product indicator approach provides significantly worse results than the other
approaches, which together build a rather homogenous subset. Concerning the relative bias of
the interaction effect estimates, the product indicator and the orthogonalizing approach do not
differ significantly, and the hybrid approach and the two-stage approach deliver significantly
more inaccurate estimates. Thus, for the estimation of both the single effects and the interaction
effect, the orthogonalizing approach belongs to the best performing subset.
98 HENSELER AND CHIN

(1a) Estimated Marginal Means of relative bias of single effect of Ÿ at number of observations D 50

(1b) Estimated Marginal Means of relative bias of single effect of Ÿ at number of indicators D 4

FIGURE 4 Estimated Marginal Means of relative biases with cross-over interactions. (continued )

Statistical Power
A researcher intending to make a conclusion about the existence of an interaction effect would
want to avoid two errors: (a) concluding that there is an interaction effect although in reality
there is none (Type I error), and (b) concluding that there is no interaction effect although
there is one in reality (Type II error). To avoid Type I errors, one uses a predefined significance
criterion (e.g., ’ D :05) when rejecting the null hypothesis; to avoid Type II errors, one has
to apply a statistical test with satisfactory statistical power. “The power of a statistical test of
a null hypothesis is the probability that it will lead to the rejection of the null hypothesis, i.e.,
the probability that it will result in the conclusion that the phenomenon exists” (Cohen, 1988,
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 99

(2a) Estimated Marginal Means of relative bias of single effect of  at number of observations D 50

(2b) Estimated Marginal Means of relative bias of single effect of  at number of indicators D 4

FIGURE 4 (Continued ).

p. 4). Often, a power of one minus four times the significance level is advocated, thus 80%
for a significance criterion of .05, implying that a Type I error is regarded as four times as
serious as a Type II error. The power of a statistical test depends on several factors, namely the
statistical significance criterion used in the test, the effect size in the population, the sample
size, and the measurement reliability. In the Monte Carlo experiment, we kept the effect size
in the population constant—the interaction effect of the true model as presented in Figure
3 has an effect size of about 0.136. Moreover, we used a constant significance criterion of
.05 throughout the experiment. We evaluated the bootstrap t values, and estimated the power
of each approach per experimental condition as the proportion of the Monte Carlo runs that
yielded a significant interaction effect.
100 HENSELER AND CHIN

(3a) Estimated Marginal Means of relative bias of interaction effect Ÿ   at number of observations D 50

(3b) Estimated Marginal Means of relative bias of interaction effect Ÿ   at number of indicators D 4

FIGURE 4 (Continued ).

Figure 5 depicts the estimated marginal means of the power of each approach when the
model has 2, 4, 6, 8, 10, or 12 indicators per latent variable, and there are 50, 100, 150, 200,
and 500 observations. On the face of Figure 5, the two-stage approach and the hybrid approach
excel under almost all conditions. Only if there are very few indicators the orthogonalizing
approach seems to be the most powerful of the approaches. To corroborate these findings,
an ANOVA was conducted, similar to the one for parameter accuracy. Again, the covariance
matrices across the 30 experimental conditions varied significantly, as documented by Levene’s
test, F .119; 59857/ D 581:818, p < :001, which again is not harmful to the between-subject
tests because of the balanced design. The results of the tests of between-subject effects are
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 101

TABLE 4
Multiple Comparisons Over the Relative Biases

Significance of Significance of Significance of

Differences in RB(Ÿ) Differences in RB() Differences in RB(interaction)
Approaches Under
Comparison Tamhane Dunnett T3 Games-Howell Tamhane Dunnett T3 Games-Howell Tamhane Dunnett T3 Games-Howell

Two-stage vs. .000 .010 .000 .000 .010 .000 .000 .000 .000
product indicator
Two-stage vs. hybrid .847 .848 .686 .490 .495 .369 .000 .010 .000
Two-stage vs. .897 .898 .748 .845 .847 .684 .000 .008 .000
orthogonalizing
Product indicator vs. .000 .010 .000 .000 .010 .000 .000 .000 .000
hybrid
Product indicator vs. .000 .009 .000 .000 .010 .000 .209 .208 .162
orthogonalizing
Hybrid vs. .199 .206 .155 .997 .997 .960 .000 .008 .000
orthogonalizing

TABLE 5
F Test Over the Statistical Power

Degrees of
Mean Square Freedom
Partial
Determinant MSHyp MSErr df Hyp df Err F Significance ˜2

Intercept 28,943.379 1,088.428 1 5 26.592 .004 .843

Approach 83.507 8.462 3 19 9.869 .000 .615
Indicators 127.015 10.602 5 27 11.980 .000 .687
Observations 969.876 14.104 4 29 68.767 .000 .905
Approach  Indicators 2.821 0.682 15 60 4.135 .000 .508
Approach  Observations 6.323 0.682 12 60 9.266 .000 .650
Indicators  Observations 8.463 0.682 20 60 12.403 .000 .805
3-way interaction 0.682 0.127 60 59,857 5.357 .000 .005

presented in Table 5. All three two-way interactions as well as the three-way interaction of
approach, number of indicators, and number of observations are significant. These interactions
become most obvious in plate (1a) of Figure 5.
Despite the crossover interaction, all direct effects are significant as well. In particular, the
statistical power differs per approach, F .3; 19/ D 9:869, p < :001. To further examine these
differences, we look at the post hoc tests presented in Table 6. As we cannot assume cell
covariances to be equal, we behold the Tamhane T2, Dunnett T3, and Games-Howell tests,
which all lead to coherent conclusions. The two-stage approach and the hybrid approach can be
considered as a homogenous subset, which has the highest statistical power. The orthogonalizing
approach has a significantly weaker statistical power, but is still significantly more powerful
than the product indicator approach.

Prediction Accuracy
A researcher who wants to include interaction effects in a model for prediction purposes would
be interested in the different approaches’ ability to predict an endogenous latent variable. To
102 HENSELER AND CHIN

FIGURE 5 Statistical power of the four approaches to detect the interaction effect.

examine the prediction accuracy, we look at the proportion of the true endogenous variable’s
variance that can be explained by each approach. Again, we consider the 30 predefined
conditions. Table 7 exhibits the average (over 500 Monte Carlo samples) squared correlations
.r 2 / between the predicted and the true values of the endogenous latent variable. It can
quickly be seen that depending on the number of indicators and observations, the highest
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 103

TABLE 6
Multiple Comparisons Over the Statistical Power

Significance of Differences in
Interaction Term’s Statistical Power

Approaches Under Comparison Tamhane T2 Dunnett T3 Games-Howell

Two-stage vs. product indicator .000 .009 .000

Two-stage vs. hybrid .994 .994 .944
Two-stage vs. orthogonalizing .001 .011 .001
Product indicator vs. hybrid .000 .009 .000
Product indicator vs. orthogonalizing .000 .009 .000
Hybrid vs. orthogonalizing .007 .017 .007

TABLE 7
Squared Correlations (r 2 ) Between the Predicted and the True Values
of the Endogenous Latent Variable

No. of Indicators
No. of
Observations Approach 2 4 6 8 10 12 Average

50 Two-stage .2874 .3776 .4104 .4175 .4438 .4550 .4221

Product indicator adjusted .3015 .4087 .4539 .4750 .4995 .5134
Hybrid .2874 .3777 .4107 .4178 .4442 .4559
Orthogonalizing .3094 .4208 .4672 .4820 .5042 .5088

100 Two-stage .2744 .3547 .3799 .3956 .4131 .4230 .3850

Product indicator adjusted .2825 .3740 .4047 .4245 .4441 .4533
Hybrid .2743 .3547 .3799 .3957 .4132 .4231
Orthogonalizing .2854 .3770 .4053 .4224 .4387 .4453

150 Two-stage .2742 .3420 .3708 .3888 .4025 .4094 .3719

Product indicator adjusted .2801 .3552 .3873 .4081 .4227 .4303
Hybrid .2742 .3421 .3709 .3888 .4026 .4095
Orthogonalizing .2813 .3562 .3864 .4035 .4162 .4228

200 Two-stage .2655 .3387 .3704 .3823 .3967 .4035 .3650

Product indicator adjusted .2697 .3486 .3832 .3962 .4117 .4193
Hybrid .2656 .3387 .3704 .3823 .3968 .4036
Orthogonalizing .2707 .3490 .3822 .3940 .4072 .4131

500 Two-stage .2669 .3310 .3624 .3768 .3890 .3962 .3558

Product indicator adjusted .2686 .3353 .3674 .3823 .3948 .4022
Hybrid .2669 .3310 .3624 .3768 .3890 .3962
Orthogonalizing .2689 .3351 .3666 .3809 .3929 .3996

Average .2777 .3574 .3896 .4046 .4211 .4292 .3799

Note. The best performing approach per condition is marked in bold.

104 HENSELER AND CHIN

prediction accuracy is achieved either by the orthogonalizing or the product indicator ap-
proach.
Again, we chose a balanced mixed factorial design with the approach as a fixed factor and
the number of indicators and observations as random factors. From the ANOVA, all interaction
terms emanate as significant. Whereas the two-way interactions—the interaction of approach
with number of indicators, F .15; 60000/ D 6:003, p < :001, ˜2p D 0:600; the interaction
of number of indicators with number of observations, F .12; 60000/ D 43:048, p < :001,
˜2p D 0:896; and the interaction of approach with number of observations, F .20; 60000/ D
19:577, p < :001, ˜2p D 0:867, are significant and substantial, the three-way interaction
of approach with number of indicators and number of observations, F .60; 59878/ D 1:464,
p D :011, ˜2p D 0:001, lacks substantiality. Furthermore, we found a significant main effect of
the PLS interaction approach, F .3; 14721/ D 6:341, p D :006, ˜2p D 0:564. The number of
indicators, F .5; 27995/ D 198:944, p < :001, ˜2p D 0:973, and the number of observations,
F .4; 21875/ D 20:157, p < :001, ˜2p D 0:787, turned out to be significant determinants
of prediction accuracy as well. As post hoc analyses, Tamhane’s T2 and Tukey’s honestly
significant difference (HSD) test came to very proximate results. Furthermore, Tukey’s HSD
revealed two homogenous subsets of approaches. The first subset contains the two-stage and
the hybrid approach, both showing an average explained variance of 37%. The second subset
consists of the orthogonalizing and the product indicator approach, both explaining about 39%
of the true endogenous variable’s variance. In conclusion, the orthogonalizing and the product
indicator approach provide a significantly and substantially more accurate prediction than the
other two approaches. Among these two, the orthogonalizing approach should be used in case
of small sample size and few indicators per construct. If the sample size or the number of
indicators per construct is medium to large, the product indicator approach should be used.10
Looking at the two-stage and the hybrid approach we find again that there is no notable
difference between them. Interestingly, the advantages of the orthogonalizing and the product
indicator approach over the other two approaches in terms of prediction accuracy could not be
validly derived from the empirical R2 values of the TAM example presented earlier.

RECOMMENDATIONS

Based on the results of the Monte Carlo simulation, it is possible to give recommendations to
researchers who want to analyze interaction effects between latent variables by means of PLS
path modeling. The differences in estimation outcomes depending on the selected approach
make it necessary for a researcher to make a well-based decision on which approach to use
for the modeling of interaction effects by means of PLS path modeling. As the outcomes
reveal, none of the approaches excels in all criteria. Instead, each approach has a number of
strengths and weaknesses, which make it suitable for one application but less for another. The
choice of approaches should therefore mainly be based on the researcher’s objectives. Is the
model aimed at detecting interaction effects; that is, shall the question be answered whether the
interaction delivers a significant additional explanation of the endogenous variable (first case)?

10 If prediction is the only purpose of the moderated path analysis, any adjustment of the interaction term’s standard

deviation is arbitrary, and can thus be ignored.

 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 105

Or is the model meant for finding an estimate for the true parameter of an interaction effect,
thus describing the relations (second case)? Or is it the objective of the interaction model to
give a better prediction of the endogenous latent variable (third case)?
In the first case, when a researcher is mainly interested in the significance of an interaction
effect, an approach that needs a minimal amount of observations given a particular significance
level (Type-I-error), power (Type-II-error), and effect size is preferable. As the Monte Carlo
simulation revealed, both the two-stage and the hybrid approach have a high level of statistical
power compared with the orthogonalizing and especially the product indicator approach. Only
in the case of few indicators and few observations, the orthogonalizing approach seems to be
advantageous. Recognizing the easy use of the two-stage approach on the one hand, and the
lack of freely available software implementation of the hybrid approach on the other hand,
it appears recommendable to apply the two-stage approach to assess the significance of an
interaction effect.
In the second case, when the model is meant for finding an estimate for the true parameter
of an interaction effect, at first the product indicator approach might catch one’s eye, because it
provides the least biased estimates for the interaction effect for medium to large sample sizes.
Still, the orthogonalizing approach is not significantly worse than the product indicator approach
in point accuracy, and delivers the most accurate estimates of the interaction effects for small
sample sizes. However, one must recognize that the product indicator approach’s higher point
accuracy of interaction effects comes at a cost, namely the downward biased estimation of the
single effects. As the orthogonalizing approach does not share this disadvantage, the general
use of the orthogonalizing approach to estimate the path coefficients of interaction effects
should be advocated. As long as both the exogenous and the moderator variable are centered,
and the interaction term is either nonstandardized or ex post adjusted, the path coefficient of
the interaction term tells how much the path coefficient of the moderated relation changes for
an observation whose value of the moderator variable is 1 SD above zero. Going back to our
technology acceptance example and using the estimates from the orthogonalizing approach (see
Table 1), the adoption intention of an individual perceiving enjoyment 1 SD higher than the
average is influenced by 0.3628 (D 0:5165 0:1537) times perceived usefulness and 0.2686
times enjoyment. It must be noted, though, that whereas the path coefficient of the interaction
effect might serve as a first entry to interpretation, Carte and Russell (2003) emphasized that
the interaction’s regression coefficient should not form the basis for assessing the strength of
the interaction effect. Instead, Cohen’s (1988) f 2 effect size measure for hierarchical multiple
regression can be applied. It is defined as:
2 2
Rincl Rexcl
f2 D 2
(16)
1 Rincl

2
where Rexcl is the variance accounted for by the independent and the moderator variable as
2
such, and Rincl is the combined variance accounted for by the independent and the moderator
variable and their interaction.11 By convention, f 2 effect sizes of 0.02, 0.15, and 0.35 are
regarded as small, medium, and large, respectively (Cohen, 1988).

11 Note that Chin et al. (2003, p. 211) mistakenly labeled R 2 2

excl instead of Rincl in the denominator of this formula,
thereby provoking an underestimation of f 2 .
106 HENSELER AND CHIN

TABLE 8
Recommendations for the Use of the Approaches

Condition

Few Indicators Many Indicators

Few Many Few Many

Objective Approach Observations Observations Observations Observations

Explanation Product indicator o C

Two-stage o CC C CC
Hybrid o CC C CC
Orthogonalizing C CC
Description of Product indicator o o o o
single effects Two-stage C o C o
Hybrid C o C o
Orthogonalizing C o C o
Description of Product indicator C o C C
interaction effect Two-stage o o
Hybrid o o
Orthogonalizing CC o CC o
Prediction Product indicator C C CC C
Two-stage o o C C
Hybrid o o C C
Orthogonalizing CC C CC C

Note. CC D highly recommendable; C D recommendable; o D acceptable; D not recommendable.

In the third case, when a researcher wants to achieve as precise a prediction as possible
of the endogenous latent variable, the situation is clearer. Either the product indicator or the
orthogonalizing approach should be chosen. Both yield higher prediction accuracy than the
other two approaches.
Table 8 sums up the partial recommendations that can be derived from the Monte Carlo
experiment. Overall, we recommend using the orthogonalizing approach. Among the four
presented approaches, it delivers the best point accurate estimates for interaction effect as well
as for the single effects. Moreover, it has a high prediction accuracy, which is of focal interest
for studies using PLS path models mainly for predication purposes, like customer satisfaction
indexes (e.g., Fornell, 1992) and many technology acceptance studies. The major disadvantage
of the orthogonalizing approach is its somewhat lower statistical power compared to other
approaches. This is unproblematic as long as an interaction effect is found to be significant.
However, if an interaction effect has been found to be nonsignificant by the orthogonalizing
approach, there remains some ambiguity in the reason for that finding: Is it because there is
no interaction effect in reality, or is it because the approach did not have enough statistical
power to find it? In such a case, we propose using additionally the more powerful two-stage
approach to test whether an interaction effect is significant or not.

LIMITATIONS AND FURTHER RESEARCH

It was our aim to compare the suitability of several PLS-based approaches for the analysis of
interaction effects between latent variables. As we limited our study to PLS-based approaches,
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 107

other SEM techniques like LISREL or regressions of summated scales were not considered. For
direct effects, a comparison of biases in PLS estimates with biases in estimates of covariance-
structure-based SEM has already been carried out elsewhere (cf. Cassel, Hackl, & Westlund,
1999). However, it might be fruitful to extend such research to incorporate PLS and LISREL
approaches alike to model interaction effects.
Not only beyond PLS but also within the domain of PLS path modeling, there are promising
terrains for extensions of our research. Being aware that our research is focused on the analysis
of interaction effects of continuous (latent) variables, we recognize that there are at least
two fields that are strongly related: multigroup analysis and polynomial terms. Multigroup
comparisons are a special case of analysis of moderated relationships, and interaction effects
themselves are a special case of nonlinear effects.
Multigroup comparisons can be regarded as analyses of interaction effects having a categor-
ical moderator variable. In principle, all categorical moderator variables could be transformed
into dummy variables, and used in combination with each of the four presented approaches.
However, it remains unclear how well the four approaches perform compared with approaches
that are customized for multigroup analyses, as, for example, the one by Dibbern and Chin
(2005). Another important issue in multigroup comparisons is measurement invariance. In our
study, we assumed the measurement models to be invariant across different levels of the latent
moderator variable. Future research could also address this issue and examine the behavior of
the four approaches in the light of moderated measurement models.
In many cases, particularly when the independent variable correlates highly with the moder-
ator variable, interaction effects might be confounded with quadratic effects. Carte and Russell
(2003) strongly recommended adding quadratic terms in regression equations to avoid biased
estimates of the interaction effect. In our analyses, we focused on the interaction term alone,
and did not include quadratic terms. Although quadratic terms have already been included in
PLS path models (cf. Pavlou & Gefen, 2005), it remains unclear for researchers how this should
be done. Originally, all four approaches for the analysis of interaction effects between latent
variables using PLS path modeling have been designed to cope with nonlinear effects in general.
Thus, it appears likely that the findings of our study can also be generalized to other nonlinear
effects including quadratic and cubic effects. However, final evidence of the behavior of such ex-
tensions of PLS path models should be given by means of customized Monte Carlo simulations.
As we only used reflective measurement models throughout our contribution, it also re-
mains unclear whether our findings about the PLS-based analysis of interaction effects can
be generalized to PLS path models and formative measurement models. Whereas the product
indicator approach and its advancement, the orthogonalizing approach, are restricted to reflec-
tive measurement models, both the hybrid and the two-stage approach can handle formative
measurement models. In light of the increasing popularity of formative measurement models,
mainly in business success factor research, there is a need for approaches that can analyze
interactions among formative constructs. Furthermore, our item measures were all set at the
same reliability. How each method stacks up under conditions of heterogeneous items should
be tested in the future.
Finally, it is surprising that the hybrid approach, which has its roots in Wold’s (1982)
original proposal for the inclusion of nonlinear effects, does not excel in any of the examined
categories (i.e., parameter accuracy, statistical power, and prediction accuracy). More evidence
on its performance should be gathered to decide whether its implementation and its use in
business and social sciences are of any value at all.
108 HENSELER AND CHIN

REFERENCES

Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research:
Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173–
1182.
Bhattacherjee, A., & Sanford, C. (2006). Influence processes for information technology acceptance: An elaboration
likelihood model. MIS Quarterly, 30, 805–826.
Brown, S. A., & Venkatesh, V. (2005). Model of adoption of technology in households: A baseline model test and
extension incorporating household life cycle. MIS Quarterly, 29, 399–427.
Busemeyer, J. B., & Jones, L. E. (1983). Analysis of multiplicative combination rules when causal variables are
measured with error. Psychological Bulletin, 93, 549–562.
Carte, T. A., & Russell, C. J. (2003). In pursuit of moderation: Nine common errors and their solution. MIS Quarterly,
27, 479–501.
Cassel, C., Hackl, P., & Westlund, A. (1999). Robustness of partial least-squares method of estimating latent variable
quality structures. Journal of Applied Statistics, 26, 435–446.
Chin, W. W. (1993-2003). PLS-Graph. Houston, TX: University of Houston.
Chin, W. W. (1998). Issues and opinion on structural equation modeling. MIS Quarterly, 22, 1–12.
Chin, W. W., Marcolin, B. L., & Newsted, P. R. (1996). A partial least squares latent variable modeling approach for
measuring interaction effects: Results from a Monte Carlo simulation study and voice mail emotion/adoption study.
In J. I. DeGross, S. L. Jarvenpaa, & A. Srinivasan (Eds.), Proceedings of the Seventeenth International Conference
on Information Systems (pp. 21–41). Cleveland, OH: Association for Information Systems.
Chin, W. W., Marcolin, B. L., & Newsted, P. R. (2003). A partial least squares latent variable modeling approach for
measuring interaction effects: Results from a Monte Carlo simulation study and an electronic-mail emotion/adoption
study. Information Systems Research, 14, 189–217.
Cohen, J. (1978). Partialed products are interactions; Partialed powers are curve components. Psychological Bulletin,
85, 858–866.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum
Associates, Inc.
Cortina, J. M. (1993). Interaction, nonlinearity, and multicollinearity: Implications for multiple regression. Journal of
Management, 19, 915–922.
Cronbach, L. J. (1987). Statistical tests for moderator variables: Flaws in analyses recently proposed. Psychological
Bulletin, 102, 414–417.
Davis, F. D. (1989). Perceived usefulness, perceived ease of use, and user acceptance of information technology. MIS
Quarterly, 13, 319–340.
DECISIA. (2003). SPAD version 5.6.2 CS. Levallois-Perret, France: Author.
Dibbern, J., & Chin, W. W. (2005). Multi-group comparison: Testing a PLS model on the sourcing of application
software services across Germany and the USA using a permutation based algorithm. In F. W. Bliemel, A. Eggert,
G. Fassott, & J. Henseler (Eds.), Handbuch PLS-Pfadmodellierung: Methoden, Anwendung, Praxisbeispiele (pp.
135–139). Stuttgart, Germany: Schäffer-Poeschel Verlag.
Echambadi, R., & Hess, J. (2007). Mean-centering does not alleviate collinearity problems in moderated multiple
regression. Marketing Science, 26, 438–445.
Fornell, C. (1992). A national customer satisfaction barometer: The Swedish experience, Journal of Marketing, 56,
6–21.
Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measure-
ment error. Journal of Marketing Research, 18, 39–50.
Gefen, D., & Straub, D. W. (1997). Gender differences in the perception and use of e-mail: An extension to the
technology acceptance model. MIS Quarterly, 21, 389–400.
Henseler, J., & Fassott, G. (in press). Testing moderating effects in PLS path models: An illustration of available
procedures. In V. E. Vinzi, W. W. Chin, J. Henseler, & H. Wang (Eds.), Handbook of partial least squares path
modeling. Heidelberg, Germany: Springer.
Hulland, J. (1999). Use of partial least squares (PLS) in strategic management research: A review of four recent
studies. Strategic Management Journal, 20, 195–204.
Jaccard, J., & Turrisi, R. (2003). Interaction effects in multiple regression (2nd ed.). Thousand Oaks, CA: Sage.
 APPROACHES FOR THE ANALYSIS OF INTERACTION EFFECTS 109

Jonsson, F. Y. (1998). Nonlinear structural equation models: The Kenny-Judd model with interaction effects. In R. E.
Schumacker & G. A. Marcoulides (Eds.), Interaction and nonlinear effects in structural equation modeling (pp.
17–42). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Jöreskog, K. G., & Yang, F. (1996). Nonlinear structural relation models: The Kenny–Judd model with interaction
effects. In G. A. Marcoulides & R. E. Schumacker (Eds.), Advanced structural equation modeling (pp. 57–88).
Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Kenny, D. A., & Judd, C. M. (1984). Estimating the nonlinear and interaction effects of latent variables. Psychological
Bulletin, 96, 201–210.
Lance, C. E. (1988). Residual centering, exploratory and confirmatory moderator analysis, and decomposition of effects
in path models containing interactions. Applied Psychological Measurement, 12, 163–175.
Lee, S. Y., Song, X. Y., & Poon, W. Y. (2004). Comparison of approaches in estimating interaction and quadratic
effects of latent variables. Multivariate Behavioral Research, 39, 37–67.
Li, F., Harmer, P., Duncan, T. E., Duncan, S. C., Acock, A., & Boles, S. (1998). Approaches to testing interaction
effects using structural equation modeling methodology. Multivariate Behavioral Research, 33, 1–39.
Little, T. D., Bovaird, J. A., & Widaman, K. F. (2006). On the merits of orthogonalizing powered and product terms:
Implications for modeling interactions among latent variables. Structural Equation Modeling, 13, 497–519.
Lohmöller, J.-B. (1984). LVPLS 1.6 program manual: Latent variables path analysis with partial least-squares
estimation. Köln, Germany: Universität zu Köln, Zentralarchiv für Empirische Sozialforschung.
Lohmöller, J.-B. (1989). Latent variables path modeling with partial least squares. Heidelberg, Germany: Physica-
Verlag.
Marsh, H. W., Wen, Z., & Hau, K.-T. (2004). Structural equation models of latent interactions: Evaluation of alternative
estimation strategies and indicator construction. Psychological Methods, 9, 275–300.
Mooney, C., & Duval, R. (1993). Bootstrapping: A nonparametric approach to statistical inference. Newbury Park,
CA: Sage.
Pavlou, P. A., & Gefen, D. (2005). Psychological contract violation in online marketplaces: Antecedents, consequences,
and moderating role. Information Systems Research, 16, 372–399.
R Development Core Team. (2006). R: A language and environment for statistical computing. Vienna, Austria: R
Foundation for Statistical Computing.
Reinartz, W. J., Echambadi, R., & Chin, W. W. (2002). Generating nonnormal data for simulation of structural equation
models using Mattson’s method. Multivariate Behavioral Research, 37, 227–244.
Ringle, C. M., Wende, S., & Will, A. (2005). SmartPLS 2.0 (beta). Hamburg, Germany: University of Hamburg.
Schneeweiss, H. (1993). Consistency at large in models with latent variables. In D. Haagen, D. J. Bartholomew, & M.
Deistler (Eds.), Statistical modelling and latent variables (pp. 299–318). Amsterdam: Elsevier Science.
Schumacker, R. E., & Marcoulides, G. A. (Eds.). (1998). Interaction and nonlinear effects in structural equation
modeling. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Tenenhaus, M., Vinzi, V. E., Chatelin, Y. M., & Lauro, C. (2005). PLS path modeling. Computational Statistics and
Data Analysis, 48, 159–205.
Wold, H. (1982). Soft modeling: The basic design and some extensions. In K. G. Jöreskog & H. Wold (Eds.), Systems
under indirect observation: Causality, structure, prediction (Vol. 2, pp. 1–54). Amsterdam: North-Holland.
Woodward, J. (1958). Management and technology. London: Her Majesty’s Stationery Office.