#### Partial least squares path modeling using ordinal categorical indicators

Partial least squares path modeling using ordinal categorical indicators
Florian Schuberth 0 1 2
J o¨rg Henseler 0 1 2
Theo K. Dijkstra 0 1 2
0 Faculty of Economics and Business, University of Groningen , Nettelbosje 2, P.O. Box 800, 9747 AE Groningen , The Netherlands
1 Faculty of Engineering Technology, University of Twente , P.O. Box 217, 7500 AE Enschede , The Netherlands
2 Faculty of Business Management and Economics, University of Wu ̈rzburg , Sanderring 2, 97070 Wu ̈rzburg , Germany
This article introduces a new consistent variance-based estimator called ordinal consistent partial least squares (OrdPLSc). OrdPLSc completes the family of variancebased estimators consisting of PLS, PLSc, and OrdPLS and permits to estimate structural equation models of composites and common factors if some or all indicators are measured on an ordinal categorical scale. A Monte Carlo simulation (N ¼ 500) with different population models shows that OrdPLSc provides almost unbiased estimates. If all constructs are modeled as common factors, OrdPLSc yields estimates close to those of its covariance-based counterpart, WLSMV, but is less efficient. If some constructs are modeled as composites, OrdPLSc is virtually without competition.
Structural equation models categorical indicators; Common factors; Consistent partial least squares; Ordinal; Composites; Polychoric correlation
1 Introduction
Structural equation modeling (SEM) has become an established method in the fields of
business and social sciences. Its capacity to model constructs, to take into account various
forms of measurement error, and to test entire theories makes it a prime candidate for
encountering a variety of research issues.
For SEM two types of estimators need to be differentiated: covariance- and
variancebased estimators. Covariance-based parameter estimates are obtained by minimizing the
distance between the empirical variance-covariance matrix of the indicators and its
theoretical counterpart. Variance-based estimators, on the contrary, create proxies for
constructs first and subsequently estimate model parameters based on theses proxies. While
covariance-based methods are preferred if the model contains constructs modeled as
common factors, variance-based estimators are favoured if the underlying model consists
of constructs modeled as composites, in particular, when the composites are endogenous in
the structural model.
Among variance-based estimators, partial least squares (PLS) path modeling is regarded
as the ‘‘most fully developed and general system’’
(McDonald 1996, p. 240)
and it was
even called a ‘‘silver bullet’’
(Hair et al. 2011)
. The use of PLS path modeling is prevalent
in many fields, e.g., information systems research
(Marcoulides and Saunders 2006)
or
marketing
(Hair et al. 2012)
. Because of its capability to model both factors and
composites,1 the latest version of PLS, known as consistent PLS, is a vigorous method for
estimation and is acknowledged by researchers across different disciplines. Common
factors can be used to model constructs of behavioral research such as attitudes or
personality traits, whereas composites can be applied to model strong concepts
(H o¨o¨ k and
Lo¨ wgren 2012)
, i.e., the abstraction of artefacts such as management instruments,
innovations, or information systems. Consequently, PLS path modeling is a preferred statistical
tool for success factor studies
(Albers 2010)
.
Recently, a lot of development has been done in the field of PLS path modeling. For
example, a new criterion for discriminant validity based on heterotrait-monotrait ratio of
common factor correlations
(Henseler et al. 2015)
, the standardized root mean square
residual (SRMR) as a measure of overall model fit (Henseler et al. 2014), and
bootstrapbased tests for overall model fit
(Dijkstra and Henseler 2015a)
were introduced. Since PLS
creates composites as proxies for all kinds of constructs, its estimates suffer from
attenuation and are biased in case of an underlying common factor model
(Schneeweiss 1993)
.
Therefore, a consistent PLS (PLSc) version was developed which corrects for this bias to
consistently estimate SEMs with common factors
(Dijkstra and Henseler 2015b)
. All these
developments are based on the PLS algorithm and therefore on ordinary least squares
(OLS) regression analysis implicitly assuming that all indicators are continuous.
Since numerous studies are based on data collected by questionnaires, the indicators
used are rarely measured on a metric scale. Hence, in many situations researchers are faced
with data measured on ordinal categorical scales, e.g., in marketing research, in particular
customer satisfaction surveys
(Hair et al. 2012; Coelho and Esteves 2007)
.
It is well known in the PLS path modeling literature as well as in other fields that
treating categorical variables as continuous can lead to biased estimates and therefore to
invalid inferences and erroneous conclusions. Lohmo¨ ller recognizes that the ‘‘½. . . standard
procedures cannot be used for the categorical and ordinal-scaled variables ½::: ’’ (Lohm o¨ller
1 For a comparison of constructs modeled as composites or common factors, see
Rigdon (2012)
.
2013, Chap. 4). Also
Hair et al. (2012)
mention that PLS is often used with categorical
indicators but that their use in a procedure like PLS which uses OLS as estimator can be
problematic. Several approaches to address this issue in the context of PLS are provided by
the literature, e.g., ordinal PLS (OrdPLS) an innovative approach to deal with ordinal
categorical indicators in a psychometric way
(Cantaluppi 2012; Cantaluppi and Boari
2016)
. As OrdPLS is based on the traditional PLS algorithm, its use is limited to models
where all constructs are modeled as composites. However, researchers often deal with
models containing constructs modeled as common factors instead of composites
(Ringle
et al. 2012; Hair et al. 2012)
. So, there is a real need for improving methods like OrdPLS
to be able to deal with common factors, composites and ordinal categorical indicators.
We provide such a development and contribute to the literature an extension of OrdPLS
called ordinal consistent partial least squares (OrdPLSc). It combines the advantages of
both, OrdPLS and PLSc. Hence, OrdPLSc is an estimator which enables researchers to
consistently estimate structural equation models including not only composites, but
common factors and ordinal categorical indicators too. Figure 1 contrasts the properties of
traditional PLS, PLSc, OrdPLS, and OrdPLSc with respect to dealing with common factors
and taking into account the scale of ordinal categorical indicators.
We run a Monte Carlo simulation to investigate the performance of OrdPLSc in
different conditions and compare it as benchmark to means and variance adjusted weighted
least squares (WLSMV). The latter approach is a consistent covariance-based estimator
which is typically used for structural equation models with common factors in case of
ordinal categorical indicators. Moreover, we show how traditional PLS, PLSc, and OrdPLS
behave for different kinds of models and show how PLS and PLSc are affected when the
scale of ordinal categorical indicators is ignored.
Corrects for attenuation in case of common factors No
OrdPLS
(Cantaluppi, 2012)
Yes
OrdPLSc
(this paper)
fo ro
le ta
ca ic
s d
e in
th la
fro irc
o
t g
un te
cco lca
a a
ek ind
a r
T o
Yes
No
The remainder of the paper is organized as follows: The next section shows the
development from PLS to PLSc and provides a reformulation of these two procedures in
terms of indicators correlation matrices. In Sect. 3 we give a literature review of existing
approaches dealing with categorical indicators in the framework of PLS, in particular we
present the idea of the OrdPLS approach. In Sect. 4 we introduce ordinal consistent PLS
(OrdPLSc) to the literature. In the following Sect. 5, we present the design of our Monte
Carlo simulation, which is conducted to examine the performance of OrdPLSc and
different other estimators under several conditions. We present these findings in Sect. 6. The
article closes with the discussion of the results in Sect. 7. An Appendix covers the figure of
the threshold parameter distribution.
2 The development from PLS path modeling to consistent PLS path modeling
PLS was developed by
Wold (1975)
for the analysis of high dimensional data in a
lowstructure environment and has undergone various extensions and modifications. It is an
approach similar to generalized canonical correlation analysis (GCCA), and in addition
able to emulate several of
Kettenring (1971)
’s techniques for the canonical analysis of
several sets of variables
(Tenenhaus et al. 2005)
.
In its most modern appearance known as consistent PLS (PLSc)
(Dijkstra and Henseler
2015a, b)
, it can be understood as a well-developed SEM method. It is capable to estimate
recursive and non-recursive structural models with constructs modeled as composites and
common factors. Both obtain the outer weights and the final stand-ins for the constructs by
the classical PLS algorithm. While traditional PLS simply relies on OLS to estimate the
model parameters, its extended version, PLSc, uses two-stage least squares (2SLS) to
consistently estimate even non-recursive path models. Furthermore, PLSc is able to handle
both constructs modeled as composites and as common factors by using a post-correction
for attenuation for correlations between common factors, and common factors and
indicators.
The classical common factor model assumes that the variance of a block of indicators
ðx1; . . .; xK Þ is completely explained by the underlying common factor (n in the large
circle) and by their random errors ð 1; . . .; K Þ, see Fig. 2a. Hence, the indicators reflect the
underlying common factor (reflective measurement model). This sort of indicator is also
known as effect indicators
(Bollen and Bauldry 2011)
. Common factors are usually used in
behavioral research.
As Fig. 2b depicts, composites (n in the hexagon) are formed as linear combinations of
their belonging indicators ðx1; . . .; xK Þ. Since the indicators form the composite, they are
related to composite-formative measurement models.2 Furthermore, the composite model
does not put any restrictions on the covariances of the indicators belonging to one block,
hence, it relaxes the assumption that all covariation between the indicators has to be
explained by the common factor. Composites are often used as proxies for scientific
concepts of interest
(Ketterlinus et al. 1989; Maraun and Halpin 2008; Tenenhaus 2008;
Rigdon 2012)
.
2 In general, the literature provides two definitions of a formative measurement model: (i) the (composite)
indicators which completely determine composite (Fornell and Bookstein 1982), and (ii) the (causal)
indicators which do not completely explain the underlying latent variable. See
Bollen and Bauldry (2011)
for a more detailed description.
1
x1
· · ·
· · ·
· · ·
· · ·
K
xK
k
xk
ξ
x1
· · ·
· · ·
xK
xk
ξ
(a) Common factor model
(b) Composite model
For the derivation of OrdPLS(c) it is crucial to describe the well-known PLS algorithm
(Wold 1975)
and its extension to PLSc in terms of indicator covariances or correlations,
respectively. Since in PLS no distinction between exogenous and endogenous constructs is
made, we use the following notation: g is a ðJ 1Þ vector containing all modeled
constructs which are connected by the structural model, whether they are modeled as common
factors or as composites. The ðK 1Þ vector x contains all indicators which measure the
common factors or build the composites, respectively.
2.1 Partial least squares
For a sample of size n, all observations of the K indicators are stacked in a data matrix X of
dimension ðn KÞ. For simplicity, the Kj indicators belonging to one common factor or
one composite gj are grouped to form block j with j ¼ 1; . . .; J. Observations of block j are
stacked in the data matrix Xj of dimension ðn KjÞ with PjJ¼1 Kj ¼ K. Furthermore, we
assume that each indicator is standardized, as is customary in PLS, to have mean zero and
unit variance, such that the indicators’ sample covariance matrix S equals the sample
correlation matrix.
The PLS estimation procedure consists of three parts. In the first part, for each block j
initial arbitrary outer weights w^jð0Þ ðKj 1Þ are chosen which satisfy the following
condition: w^jð0Þ0Sjjw^jð0Þ ¼ 1 where the Kj Kj matrix Sjj contains the sample correlations of the
indicators of block j. This condition holds for all outer weights in each iteration i and can
be achieved by using the scaling factor ðw^jðiÞ0Sjjw^jðiÞÞ 21 for the outer weights w^ðiÞ in each
j
iteration.
In the second part, the iterative PLS algorithm starts with step one, the outer
approximation of gj:
g^jðiÞ ¼ Xjw^jðiÞ
with
w^ðiÞ0Sjjw^jðiÞ ¼ 1;
j
ð1Þ
where g^jðiÞ is a column vector of length n. Since outer weights are scaled, all outer proxies
also have mean zero and unit variance.
In the second step, the inner proxy of gj is calculated as a linear combination of inner
weights and outer proxies of gj0 :
J
X
j0¼1
g~ðiÞ
j ¼
ejðji0Þg^jð0iÞ;
where g~jðiÞ is again a column vector of length n. The inner weight ejj0 defines how the inner
proxy g~j is built. Three different schemes for the calculation of ejj0 are commonly used:
centroid
(Wold 1982)
, factorial (Lohmo¨ ller 2013), and path weighting. However, all
schemes yield essentially the same results
(Noonan and Wold 1982)
, hence, we only
consider the centroid scheme.3 The inner weights are chosen according to the signs of the
correlations between the outer proxies
i
ejðj0Þ ¼
(
signðw^jðiÞ0Sjj0 w^jð0iÞÞ;
0;
for j 6¼ j0 and if construct j and j0 are adjacent
otherwise;
where adjacent refers to the constructs j and j0 directly connected by the structural model.
All inner proxies g~ðiÞ are again scaled to have unit variance.
j
In the third and last step of the iterative part, new outer weights are calculated. This can
be done in three ways: mode A, mode B, and mode C. For mode A, estimated outer weights
of block j equal the estimated coefficients of a multivariate regression from the indicators
of block j on its related inner proxy. Due to standardization, the new estimated outer
weights w^ðiþ1Þ equal the correlations between the inner proxy and its related indicators:
j
J
X
j0¼1
w^ðiþ1Þ
j
/
Sjj0 w^ðiÞejðji0Þ
j0
with
w^jðiþ1Þ0Sjjw^jðiþ1Þ ¼ 1:
In contrast, for mode B, the new outer weights equal the estimated coefficients of a
regression from the inner proxy on its connected indicators:
J
w^jðiþ1Þ / Sjj 1 X
j0¼1
Sjj0 w^ðiÞejðji0Þ
j0
with
w^jðiþ1Þ0Sjjw^jðiþ1Þ ¼ 1:
Mode C, also known as MIMIC mode, is a mixture of mode A and mode B and is not
considered here.4
As the traditional PLS algorithm has no single optimization criteria to be minimized, the
new outer weights w^ðiþ1Þ are checked for significant changes compared to the outer weights
j
from the previous iteration step w^jðiÞ. If there is a significant change in the weights, the
algorithm starts again at step one by building new outer proxies with the new outer
weights, otherwise it stops.
In the last part, the obtained stable outer weights w^j are used to build final composite
stand-ins for both common factors and composites: g^j ¼ Xjw^j. For constructs which are
modeled as common factors, the factor loadings are estimated by OLS in accordance with
3 For more details on the other schemes, see
Tenenhaus et al. (2005)
.
4 A consistent version of mode C, for any of its 2J - 2 implementations, can be obtained by using the
properties of mode A and mode B, see
Dijkstra (1981
, 1985, Chap. 2, par. 5.2), but since mode C is
intermediate between the other modes, adding mode C does not really contribute to a further understanding.
ð2Þ
ð3Þ
ð4Þ
ð5Þ
the measurement model. In contrast, for constructs which are modeled as composites the
final weights equal the stable weights from the last iteration. Finally, path coefficients are
estimated by OLS with respect to the structural model.
2.2 Consistent PLS
PLS is based on composites, which implies that estimates are biased if constructs are
modeled as common factors.5 In general, a composite model has larger absolute inter
composite correlations compared to the absolute inter common factor correlations of a
model with the same structure but where all constructs are modeled as common factors.
However, a transformation of the model-implied correlation matrix of a composite model
into the model-implied correlation matrix of a common factor model can be achieved by a
correction for attenuation
(Cohen et al. 2013, Chap. 2.10)
. Consistent PLS (PLSc) uses this
correction to obtain consistent estimates for models containing common factors
(Dijkstra
and Henseler 2015a, b)
. The correction requires that each common factor is measured by at
least two indicators and uses the proportionality between the population outer weights and
the population factor loadings, wj ¼ cjkj. The estimated correction factor for block j
satisfies the following condition
plimðc^jÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffi
k0jRjjkj;
where kj is a column vector of length Kj containing the population loadings of common
factor gj and Rjj is the Kj Kj population correlation matrix of the indicators of block j.6
The correction factor c^j can be obtained as
2 w^0jðSjj
c^j ¼ w^0jðw^jw^0j
diagðSjjÞÞw^j
diagðw^jw^0jÞÞw^j
Moreover, PLSc is able to consistently estimate the path coefficients of recursive and
nonrecursive models7 using OLS or 2SLS according to the structural model. Since all variables
are standardized, the estimated path coefficients are based on the correlations between the
columns of g^ ðn JÞ. The correlation between the common factors j and j0 is consistently
estimated by:
ð6Þ
ð7Þ
ð8Þ
ð9Þ
5 Both, common factors as well as composites are legit ways of construct modeling, see
Rigdon (2012)
.
6 The use of mode B for common factors is not considered here. For a consistent version of PLS using mode
B see
Dijkstra (1981
, 2011).
7 PLSc relaxes the assumptions of the basic design
(Wold 1982)
where non-recursive structural models are
not allowed.
:
Using the corrected correlation of Eq. (10) for the estimation of the structural model, one
obtains consistently estimated path coefficients between the common factors.8 For
constructs which are modeled as composites no correction of the correlation is required
because, by construction, they are not affected by attenuation. In case construct j is
modeled as a common factor and construct j0 as a composite, the consistently estimated
correlation is obtained as
w^0jSjj0 w^j0
ccorðgj; gj0 Þ ¼ c^jðw^0jw^jÞ :
ð10Þ
ð11Þ
3 The development from PLS to ordinal PLS
Since incorrectly handling ordinal categorical variables as continuous can lead to biased
inferences and therefore to erroneous conclusions, the literature provides several
approaches to deal with discrete indicators: dichotomize the ordinal categorical indicator, a
mixture of PLS and correspondence analysis (CA), Partial Maximum Likelihood PLS
(PML-PLS), and non-metric PLS (NM-PLS).
Common practice in PLS is to replace a categorical indicator by a dummy matrix which
is known as dichotomizing. Since each categorical indicator is replaced by s 1 dummy
variables, where s is the number of observed categories, s 1 outer weights are obtained
for the original variable. This contradicts the idea of treating an indicator as a whole.
Betzin and Henseler (2005)
use correspondence analysis to quantify ex-ante categorical
indicators. As the quantified indicators are obtained, PLS is used to estimate the model
parameters. As a result, individual weights are obtained for each category of the categorical
indicator. Again, this has the drawback that no single outer weight for a categorical
indicator is calculated.
Partial Maximum Likelihood Partial Least Squares (PML-PLS)
(Jakobowicz and
Derquenne 2007)
is a modified version of the original PLS algorithm. It is a combination of
PLS and generalized linear models designed to deal with indicators of any scale. For
categorical indicators, individual outer weights are computed for each category by
ANOVA. Based on those, one ’global’ weight per categorical indicator is calculated.
However, statistical properties like the proportionality of outer weights to factor loadings
are unknown for the global weights and further investigation is needed. Moreover, the
authors note that PML-PLS ‘‘is especially advantageous in the case of nominal or binary
variables’’
(Jakobowicz and Derquenne 2007)
but we focus on ordinal categorical
indicators.
The last approach, non-metric partial least squares (NM-PLS) extends PLS by an
alternating least squares optimal scaling (ALSOS) algorithm to quantify qualitative
indicators and gain outer weights
(Russolillo 2012)
. ALSOS is a procedure which quantifies
qualitative variables by preserving properties of the original measurement scales and
8 For more details, e.g., the consistent estimation of non-recursive models and the correction for nonlinear
structural equation models see
Dijkstra (1983
, 1981, 1985, 2010, 2011),
Dijkstra and Schermelleh-Engel
(2014)
.
optimizes an objective optimization criteria by alternating least squares
(Young 1981)
. In
the case of NM-PLS, the categorical indicator is quantified in a way that the correlation
between the inner proxy and the quantified categorical indicator is maximized. As a result
for each indicator one outer weight is obtained as in traditional PLS for continuous
indicators.
However, the evaluation of the presented approaches is based on empirical studies and,
to our knowledge, no simulation studies have been conducted to investigate their statistical
properties. For an extension to PLSc in order to deal with common factors, it is necessary
that the outer weights are proportional to the factor loadings. Moreover, the modified PLS
procedures are often applied to common factor models which represents a misspecified
model in the context of PLS. Hence, an assessment of their statistical properties is hardly
possible and we decided not to pursue any of the previously mentioned methods.
3.1 Ordinal PLS
A promising approach to deal with ordinal categorical indicators is ordinal PLS (OrdPLS9)
(Cantaluppi 2012)
. It is a modified procedure for handling ordinal categorical variables in a
classical psychometric way. In Sect. 2 we showed that all parameters can be obtained by
the use of the correlation matrix S. Traditional PLS uses the Bravais-Pearson (BP)
correlation matrix which requires all indicators to be continuous for consistency. The
observation of an ordinal categorical variable is a qualitative measure, yet it is often coded
as numeric and therefore mistakenly treated as quantitative by researchers. This routinely
happens in applications with binary and ordinal categorical indicators which results in
biased BP correlation estimates
(Quiroga 1992; O’Brien and Homer 1987; Wylie 1976;
Carroll 1961)
. To fix this, OrdPLS uses a consistent correlation matrix as input for the
traditional PLS algorithm. An advantage of OrdPLS over the approaches previously
introduced is its transparent way of dealing with ordinal categorical variables. Moreover,
the original PLS algorithm remains untouched and it is just provided by a consistent
correlation matrix as input for the algorithm.
Since OrdPLS does not correct for attenuation, it shows the same drawbacks as PLS if
common factors are included in the model. Nevertheless, we consider OrdPLS as a
powerful extension of PLS when applied under appropriate circumstances, i.e., for models
with only composites. Furthermore, it is straightforward to extend by PLSc, to overcome
its drawback for common factor models, see Sect. 4. In the following subsection we
present Pearson’s considerations of ordinal categorical variables to provide a better
understanding of the polychoric and polyserial correlation.
3.2 Ordinal categorical variables according to Pearson
Pearson (1900
, 1913) considers an ordinal categorical variable as a crude measure of an
underlying continuous random variable, while
Yule (1900)
assumes categorical variables
being inherently discrete. In this paper we follow Pearson’s idea: an observed ordinal
categorical indicator x is the result of a polytomized standard normally distributed random
variable x :
9 OrdPLS was originally called OPLS
(Cantaluppi 2012)
. An anonymous reviewer suggested to use a
different name in order to avoid confounding with O-PLS
(Trygg and Wold 2002)
. We came to an
agreement with Cantaluppi to speak of OrdPLS in the future. We thank the anonymous reviewer for
suggesting such a disambiguation.
Fig. 3 Pearson’s idea of an
ordinal categorical variable
. . .
τ1
τi−1
τi
τs−1
x ¼ xi if
si 1
x \si i ¼ 1; . . .; s
where the threshold parameters s0; . . .; ss determine the observed categories. The first and
last threshold are fixed: s0 ¼ 1 and ss ¼ 1. Moreover, thresholds are assumed to be
strictly increasing: s0\s1\. . .\ss.
Figure 3 depicts the idea of an underlying continuous variable: For indicator x category
xi is observed if the realisation of the underlying continuous variable x is in between si 1
and si.
3.3 Polychoric and polyserial correlation
Since an ordinal categorical variable is determined by an underlying continuous variable, it
is more appropriate to consider the correlation between these underlying quantitative
continuous variables for evaluating the linear relationship of interest. This is achieved by
using the polychoric or the polyserial correlation
(Drasgow 1988)
. To present the
principles of the polychoric correlation, we consider two ordinal categorical variables x1 and x2
with consecutive categories i ¼ 1; . . .; s and j ¼ 1; . . .; r. They are constructed in the way
presented in Eq. (12). The two underlying continuous variables x1 and x2 are assumed to be
jointly bivariate standard normally distributed with correlation q. The correlation between
x1 and x2 can be estimated by maximum likelihood using the following log-likelihood
function:
ln L ¼ lnðcÞ þ
nij lnðpijÞ;
Xs Xr
i¼1 j¼1
where lnðcÞ is a constant term, nij denotes the observed joint absolute frequency of x1 ¼ i
and x2 ¼ j, and pij is the probability that category i and j are observed jointly. Due to the
joint normality assumption, pij is obtained as:
pij ¼ U2ðsx1i ; sx2j ; qÞ
U2ðsx1i ; sx2j 1 ; qÞ
U2ðsx1i 1 ; sx2j ; qÞ þ U2ðsx1i 1 ; sx2j 1 ; qÞ;
ð14Þ
ð12Þ
ð13Þ
where U2 is the cumulative distribution function of the bivariate standard normal
distribution. The parameters sx1i , sx2j , and q are chosen to maximize the function ln L. In order to
reduce computational burden, a two-step procedure can be used
(Olsson 1979)
. In the first
step, threshold parameters are estimated separately as quantiles of cumulative marginal
frequencies, i.e., s^x1i ¼ U 1ðpiÞ where pi equals the cumulative marginal relative frequency
up to category i and the function U 1 represents the quantile function of the standard
normal distribution (analogous for x2). Second, given the estimated threshold parameters,
Eq. (13) is maximized with respect to q. In case of a continuous and an ordinal categorical
variable, the correlation between the two continuous variables is obtained by the polyserial
correlation
(Olsson et al. 1982)
. For more than two variables, a multivariate version is used
to estimate the correlations
(Poon and Lee 1987)
. Moreover, a less computational intensive
two-step approach can be used for the multivariate version
(Lee and Poon 1987)
. OrdPLS
as well as OrdPLSc makes use of the polychoric and polyserial correlation when ordinal
categorical indicators are part of the model.
4 Ordinal consistent partial least squares
We introduce a new approach which deals with common factors, composites, and ordinal
categorical indicators. It is called ordinal consistent partial least squares (OrdPLSc) and is
a combination of OrdPLS and PLSc. It uses the polychoric correlation, a consistent
correlation matrix in case of ordinal categorical indicators, as input for the PLS algorithm and
corrects for attenuation if common factors are included in the model. Since the use of the
polychoric correlation matrix does not affect the original PLS algorithm, the
proportionality property of the outer weights is maintained and the correction of attenuation can be
applied to the inter-composite correlation matrix in the same manner as in PLSc. Figure 4
illustrates commonalities and differences of the three previously presented PLS approaches
and OrdPLSc.
The role of an ordinal categorical indicator x, more precisely its underlying continuous
variable x , is influenced by its position in the model. As Fig. 5a displays, when the ordinal
categorical indicator belongs to a common factor, its outcome is indirectly influenced by
the underlying common factor and a measurement error through the underlying
continuous variable x . An ordinal categorical indicator that is part of a composite, see Fig. 5b, is
simply a crude measure of an underlying continuous variable (represented by a double
headed arrow) which actually builds the composite along with other indicators belonging
to this block.
To ignore the nature of the ordinal categorical indicators may cause serious problems.
First, in common factor models the correlation between the indicator and its underlying
factor is underestimated
(Quiroga 1992; O’Brien and Homer 1987)
, which leads to biased
estimates. Second, in the case of a composite, disregarding the scale of the ordinal
categorical indicator leads to biased estimates, too. This is well known as the
error-in-variables problem
(see, e.g., Wooldridge 2012, Chap. 15)
.
Determining polychoric correlations
Determining
polychoric
correlations
PLS
algorithm
PLS
algorithm
PLS
algorithm
PLS
algorithm
OLS/2SLS
OLS
OLS
OLS/2SLS
5 Monte Carlo simulation
In order to investigate the performance of OrdPLSc under various conditions and to
compare it with PLSc, OrdPLS, and PLS for structural equation models containing ordinal
categorical indicators, we ran a Monte Carlo simulation. In particular, we considered their
unbiasedness and their efficiency, the most important properties of an estimator.
Furthermore, we studied the bias of PLS and OrdPLS estimates for common factor models
with ordinal categorical indicators. Also for PLSc, which is known to be a consistent
estimator in the framework of continuous indicators
(Dijkstra and Henseler 2015a)
, we
examined the behavior when ordinal categorical variables are used instead of continuous ones.
We conducted a Monte Carlo simulation with 1000 multivariate standard normally
distributed samples with 500 observations each. The continuous indicators were
categorized in the way presented in Sect. 3.2. We only considered consecutive categories, i.e.,
1; 2; . . .; s. To compare all estimators in a fair way, inadmissible solutions10 were removed
and replaced by proper estimations before evaluation.
We considered the following experimental conditions: two population models (a model
with three common factors and a model with one common factor and two composites), four
different numbers of categories (2, 3, 5, and, 7 categories), and five different distributions
of the ordinal categorical indicators (symmetric, moderate asymmetric, extreme
asymmetric, alternating moderate asymmetric, and alternating extreme asymmetric). Each
condition was estimated by OrdPLSc, PLSc, OrdPLS, and PLS. As a benchmark
comparison for the pure common factor model we also estimated the model by WLSMV, a
consistent covariance-based three stage least squares estimator
(Muthe´n 1984; Lee et al.
1990b)
, which is considered the golden standard for common factor models with ordinal
categorical indicators.11
5.1 Two population models
Starting point were two kinds of models: one model with only common factors and one
model with one common factor and two composites. The pure common factor model was
chosen to compare OrdPLSc to its covariance-based counterpart WLSMV. In designing the
path structure of the models, we chose a structure used several times in the literature
(Hwang et al. 2010; Henseler 2012; Henseler and Sarstedt 2013)
.
5.1.1 Population model with only common factors
First we considered a pure common factor model with the following population structural
equations
g1 ¼ c1n1 þ f1
g2 ¼ c2n1 þ b21g1 þ f2;
ð15Þ
ð16Þ
where c1 ¼ 0:6; c2 ¼ 0:0; b21 ¼ 0:6; varðf1Þ ¼ 0:64; varðf2Þ ¼ 0:64, and covðf1; f2Þ ¼ 0.
As Fig. 6 depicts, each common factor was reflectively measured by three indicators with
factor loadings 0.8, 0.7, 0.6 for n, 0.7, 0.7, 0.7 for g1, and 0.5, 0.7, 0.9 for g2.
All measurement errors and structural residuals were mutually independent as well as
all common factors were assumed to be independent of the measurement errors. Therefore,
the indicators population correlation matrix is given by:
10 Inadmissible solutions are estimations with absolute factor loadings larger than 1, non positive-definite
construct correlation matrix, or estimations which have not converged.
11 The mixed model with an endogenous composite cannot be estimated by WLSMV because of
identification problems. Moreover, as in OrdPLS and OrdPLSc, covariance-based estimators for categorical
indicators are typically based on polychoric correlation, see Lee et al.
(1990a, 1992)
,
De Leon (2005
),
Liu
(2007)
,
Katsikatsou et al. (2012)
.
δ23
y23
δ22
y22
η2
5.1.2 Population model with two composites and one common factor
Second, we considered a model with the identical structural model used for the model with
three common factors, but two of the constructs were modeled as composites instead of
common factors. Figure 7 depicts the population model in terms of common and composite
factors. We deliberately chose this representation of the composites and not the one used in
Fig. 2 to clarify the construction of the population correlation matrix of the indicators.
Here n and g1 are constructs modeled as composites. Since the relationship between a
composite and its indicators can be expressed by composite loadings (Fig. 7) or weights,
we also reported the weights: the composites were formed by their connected indicators:
n ¼ x0wx where w0x ¼ ð0:3; 0:5; 0:6Þ and g1 ¼ y10wy1 where w0y1 ¼ ð0:4; 0:5; 0:5Þ. The
common factor g2 was again measured by three indicators with the following loadings: 0.5,
0.7, and 0.9.
1
x1
λx1 = .4
−0.32
−0.12
−0.24
2
x2
The population correlation matrix of the indicators has the following form:
(18)
5.2 Number of categories
We considered four different numbers of indicator categories: 2, 3, 5, and 7. An increasing
number of categories diminishes the bias of the BP correlation
(O’Brien and Homer 1987)
.
Hence, we expect a decreasing difference between PLS and OrdPLS as well as PLSc and
OrdPLSc as the number of categories increases.
5.3 Threshold parameter distribution
We investigated differently skewed ordinal categorical indicators by varying threshold
parameter distributions for each number of categories. We considered threshold
distributions used in the literature before
(Rhemtulla et al. 2012)
: symmetric, moderately
asymmetric, extremely asymmetric, alternating moderately asymmetric, and alternating
extremely asymmetric distributed threshold parameters. In the alternating asymmetric
threshold distribution scenario, the same thresholds were used, but the direction of
asymmetry was reversed for the indicators x2, y11, y13, and y22.12 Since BP correlations are
more downward biased for more asymmetrical threshold distributions
(Bollen and Barb
1981; Faber 1988; Holgado-Tello et al. 2010)
and even more for alternating skewed
indicators
(Olsson 1980)
, we expect an increasing difference between OrdPLSc and PLSc
estimates as well as OrdPLS and PLS estimates from the symmetrical to the alternating
extreme threshold distribution.
5.4 Data generation and analysis
All simulations were conducted within the R (version 3.2.2) statistical programming
environment
(R Core Team 2015)
. Multivariate standard normally distributed data sets
were drawn using the mvrnorm function of the MASS package
(Venables and Ripley 2002)
.
To obtain PLS and PLSc estimates, we primarily used functions provided by the matrixpls
package
(Ro¨ nkk o¨ 2015)
, which allows the use of the empirical correlation matrix as input
for PLS and PLSc. A slightly modified version of those functions was also used for
OrdPLS and OrdPLSc. The modified version is provided by the authors upon request.
Since matrixpls is still under development we also partly verified our results obtained with
ADANCO
(Henseler and Dijkstra 2015)
. The polychoric correlation was calculated by the
polychoric function from the psych package
(Revelle 2015)
using the two-step approach.13
WLSMV estimation was carried out using the lavaan package
(Rosseel 2012)
.
6 Results
This section shows the results of our study.14 In the following, we summarize our findings
in terms of bias with respect to the quality of the parameter estimates for the model
containing only common factors and the mixed model. The bias is the deviation of the
estimated parameter mean across all Monte Carlo simulation runs from its population
counterpart
1
Bias ¼ 1000
1000
X ^
hi
i¼1
h
ð19Þ
where h represents the population parameter and h^ is the estimated parameter. The bias
statistic provides information about the estimators’ unbiasedness and is used as one
performance measure to compare OrdPLSc estimates with estimates from approaches
commonly applied. Moreover, we assessed the estimators’ efficiency in terms of average
12 For an exact description of the threshold parameter distribution, see the Appendix.
13 If the polychoric correlation matrix was not positive definite an eigenvector smoothing was done to
assure its positive definiteness. Moreover, we followed the recommendation of
Savalei (2011)
and used the
‘ADD’ approach (0.5) for empty cells in the case of two categories and the ‘NONE’ approach else. The
same was done for WLSMV.
14 The complete results are provided in the supplementary material.
standard deviation across all Monte Carlo simulation runs. We finish by summarizing
inadmissible results, i.e., Heywood cases.
In general, for the moderate asymmetric and the alternating moderate asymmetric
threshold parameter distribution the estimators led to similar results. The same was
observed for extremely and alternating extremely distributed thresholds. For latter
conditions, all estimators showed a poorer performance, which confirmed our expectations.
6.1 Bias of the parameter estimates
7 2
Number of categories
3
5
7
Estimator: OrdPLSc
WLSMV
PLSc
OrdPLS
PLS
Fig. 8 Model with only common factors: bias for b and c2
7 2
Number of categories
3
5
7
Estimator: OrdPLSc
WLSMV
PLSc
OrdPLS
PLS
slightly biased estimates were obtained. This bias diminished with an increasing number of
categories.
In contrast, PLSc path coefficient estimates behaved surprisingly well in most of the
conditions. The estimated path coefficients were biased for extremely asymmetrically
distributed threshold parameters and even more biased for the alternating extreme
threshold parameter distribution. The population zero-path c2 was approximately unbiased
in almost every condition except for alternating extremely distributed threshold parameters
with 2 categories. This bias declined with an increase in the number of categories. In
contrast, factor loading estimates were downward-biased in all conditions but the bias
dramatically declined as the number of categories increased. However, the bias was still
present for 7 categories.
We obtained different results for OrdPLS which led to a fairly constant bias in all
conditions unaffected by the number of categories. In particular, the estimated path
coefficients b^ and c^2 were downward-biased while the estimated zero-path coefficient c^1
was upward-biased. Factor loading estimates were all upward biased, except for the
β
γ2
7 2
Number of categories
3
5
7
Estimator: OrdPLSc
PLSc
OrdPLS
PLS
estimates of the largest factor loading ky23 ¼ 0:9 which were only slightly biased. This bias
was largely unaffected by the number of categories.
PLS produced the most biased path coefficient estimates for c1 and b1. While the bias of
OrdPLS was fairly constant in all conditions, the bias of the PLS estimates converged to
the bias of the OrdPLS estimates with an increasing number of categories. A similar
pattern was observed for PLS factor loading estimates. For 2 categories, factor loading
estimates were slightly biased, but the bias became more pronounced and converged to the
bias of the OrdPLS factor loading estimates as the number of categories increased.
Next, we examined the estimates obtained for the mixed population model. Again, for
the sake of simplicity, Figs. 10 and 11 only depict the bias of the estimates for the path
coefficients b(=0.6) and c2(=0), for the factor loading ky22 (=0.7) and for the weight
wy12 (=0.5) of the model with two composites and one common factor.
The OrdPLSc estimator led to almost unbiased path coefficient, factor loading, and
weight estimates under the considered conditions. Only for an alternating extreme
asymmetric threshold parameter distribution, path coefficient estimates were clearly biased for
two categories. However, this bias disappeared for more than two categories.
0.1
0.0
−0.1
−0.2
0.1
0.0
−0.1
−0.2
0.1
s 0.0
a
iB−0.1
−0.2
7 2
Number of categories
3
5
7
Estimator: OrdPLSc
PLSc
OrdPLS
PLS
Fig. 11 Mixed model: bias for ky22 and wy12
OrdPLS led to very similar results compared to OrdPLSc for estimates affected only by
composites (c^1 and weights). The estimated zero-path c^2 was also unbiased except for
alternating extremely skewed indicators with 2 categories, while the path coefficient
estimate b^ which is only affected by a common factor was constantly biased. Factor
loading estimates were again all upward-biased under almost every threshold parameter
distributions. This bias was neither affected by the number of categories nor by the
threshold parameter distribution.
In contrast, PLSc path coefficient estimates were all biased. This bias was more
pronounced by the asymmetry of the threshold parameter distribution. Moreover, factor
loadings were underestimated. In general, factor loading estimates showed a very similar
behavior as the estimated factor loadings from the model with only common factors. Most
weight estimates were only slightly biased, but estimates for wx2 and wy1 showed a clear
bias. All biases decreased and PLSc estimates converged to the OrdPLSc estimates as the
number of categories increased.
PLS produced almost the same biased estimates for path coefficient c1 and the weights
as PLSc. The other path coefficients were also biasedly estimated under all conditions.
While this bias decreased with an increasing number of categories, the upward-biased
factor loading estimates became even more biased for an increasing number of categories.
Again, average PLS factor loading estimates tended to converge to OrdPLS average factor
loading estimates.
6.2 Efficiency
Apart from unbiasdness, an estimator’s efficiency is of interest to assess its quality.
Therefore, we evaluated the standard deviations of the standardized path coefficient, loading, and
weight estimates. In general, all standard deviations decreased with an increasing number of
categories, but increased for more asymmetric threshold parameter distributions.
Considering the pure common factor model, WLSMV was always more efficient than
OrdPLSc. Since comparing estimators efficiency is only meaningful for unbiased or
slightly biased estimates, the other results for the pure common factor model are not
evaluated.
Also the estimates for the composite model became more efficient with an increasing
number of categories. For estimated parameters between composites only, PLS and PLSc
as well as OrdPLS and OrdPLSc produced almost the same standard errors. Estimated
parameters connected with at least one common factor showed larger standard deviations
for OrdPLSc than OrdPLS. In most cases, path coefficient and weight estimates were less
efficient for OrdPLS than PLS, while factor loadings were more efficiently estimated by
OrdPLS.
6.3 Inadmissible solutions
We finish the results part by comparing the inadmissible solutions. Inadmissible solutions
are results with absolute factor loadings greater than one, a non positive semi-definite
construct correlation matrix, or results where the estimation algorithm did not converge.
Figure 12 depicts the relative frequencies of inadmissible results.
PLS, OrdPLS, and PLSc produced almost no inadmissible solutions for both kind of
models. In contrast, OrdPLSc and WLSMV produced a few inadmissible solutions under
every condition. The total number of inadmissible results increased for more skewed
distributed indicators. The most inadmissible results were produced for alternating
extremely distributed threshold parameters.
A similar pattern was observed for inadmissible results during the bootstrap procedure.
PLS and OrdPLS again produced no improper solutions. In general, the number of
inadmissible results during the bootstrap procedure increased for PLSc with an increasing
number of categories, while it decreased for OrdPLSc and WLSMV.
7 Discussion
The first goal of our study was to propose a variance-based estimator for structural
equation models that is able to consistently estimate models with common factors,
composites, and ordinal categorical indicators. We developed OrdPLSc combining the
approaches and thus favorable characteristics of OrdPLS and PLSc.
Our results confirmed that OrdPLSc fulfills its intended purpose. For a sample size of
500 observations, OrdPLSc factor loading, weight, as well as path coefficient estimates
were almost unbiased under every condition. As the combination of the polychoric
Model with only common factors
Mixed model
S
y
m
m
itrce
M
o
d
tre
a
e
E
x
tre
m
e
correlation and PLSc led to larger standard errors of parameter estimates, OrdPLSc
produced a few improper solutions in terms of absolute factor loadings larger than 1. The
number of inadmissible solutions was mainly driven by the estimates of the largest factor
loading ky23 . However, the number of inadmissible solutions were in an acceptable range.
Compared to WLSMV, OrdPLSc produced very similar estimates but with larger standard
errors and a few more inadmissible solutions. However, OrdPLSc outperformed PLS,
OrdPLS, and PLSc in terms of bias for both models, which makes OrdPLSc to be the
dominant approach under the considered variance-based estimators if ordinal categorical
indicators are included in the model. In case of model parameters which are not connected
to a common factor, OrdPLSc and OrdPLS as well as PLSc and PLS produced almost the
same estimates and standard errors. This is not surprising, as no correction for attenuation
is needed, which is the only difference between OrdPLSc and OrdPLS, and PLSc and PLS,
respectively.
Second, we investigated the behavior of PLSc, OrdPLS, and PLS in different scenarios
using ordinal categorical indicators. Although PLSc uses the BP correlation and therefore
does not account for the scale of ordinal categorical indicators, it was surprisingly accurate
in estimating the path coefficient of the model with only common factors in most
conditions. This could be due to the use of identical threshold parameters for the indicators,
but further research is needed.15 Furthermore, PLSc behaved as expected, factor loadings
were underestimated and the bias increased for more asymmetric threshold parameter
distribution, which is due to the downward-bias of the BP correlation. This bias declined as
the number of categories increased because the bias of the BP correlation decreased.
Therefore, the use of PLSc for models with both common factors and composites is
appropriate but only for indicators with a large number of categories. In our simulation
study, 7 categories were not enough for the bias to disappear completely.
Moreover, our findings support the results of
Cantaluppi (2012)
that OrdPLS path
coefficient estimates are less biased than PLS estimates in the pure common factor model.
Although it takes into account the scale of ordinal categorical indicators, the problem of
attenuation remains unaddressed which led to downward-biased estimated path coefficients
and upward-biased estimated factor loadings. As this bias was almost unaffected by the
number of categories and the indicators’ distribution, OrdPLS estimates were constantly
biased. However, OrdPLS accurately estimated the model parameters which were not
connected to common factors because no correction for attenuation is needed. Therefore,
OrdPLS is an appropriate estimator for models containing only constructs modeled as
composites.
Traditional PLS suffers from two shortcomings: no correction for attenuation in case of
common factors and not accounting for the scale of ordinal categorical indicators. For a
small number of categories the bias of attenuation and the bias of the BP correlation
cancelled out, which led to only slightly biased factor loading estimates. When the number
of categories increased, the bias of the BP correlation decreased and PLS factor loading
estimates became more and more inaccurate and converged to the OrdPLS estimates,
which do not suffer from the bias of the BP correlation. Therefore, PLS should be
cautiously used for models containing common factors regardless whether ordinal categorical
indicators are included or not.
Since OrdPLSc uses the polychoric correlation which assumes normality for the latent
variables underlying each ordinal categorical indicator, it cannot be declared anymore as
an approach which is free of distributional assumptions. However, the assumption of joint
normality of the underlying unobservable variables can be relaxed, as the polychoric
correlation produces fairly unbiased correlation estimates for elliptically symmetric
distributed variables
(Kukuk 1999)
. Furthermore, due to the nature of the ordinal categorical
indicators, point estimates of factor scores or composite scores should not be directly
calculated from their observations. To overcome this shortcoming procedures like the
mode estimation, median estimation, or mean estimation can be used
(Cantaluppi 2012)
.
This issue currently limits the use of OrdPLSc for prediction.16
In our simulation study, we only considered situations where all indicators were
measured on an ordinal categorical scale. In empirical research practice, continuous indicators
are often included in the model. In such a situation, the polyserial or BP correlation should
be used, too, to estimate the population correlation matrix. Future research should
investigate the behavior of OrdPLSc for models containing a mixture of ordinal categorical and
continuous indicators. As a study is limited to its design, we further recommend to
15 Since the BP correlation is about to be proportional biased
(for a certain range of correlations, see Kukuk
1991)
, bias cancels out for path coefficients and only affects factor loading and correction factor estimates.
Results may change for indicators with a different number of categories and different threshold parameter
distribution.
16 This issue is subject of current research by Florian Schuberth and Gabriele Cantaluppi.
investigate the behavior of OrdPLSc, in particular, for small sample sizes. In more general,
we recommend to investigate the use of the polychoric correlation in other variance-based
estimators which can be expressed in terms of indicators correlation matrix, e.g.,
generalized structural component analysis
(Hwang and Takane 2014)
.
Acknowledgment Jo¨rg Henseler acknowledges a financial interest in ADANCO and its distributor,
Composite Modeling.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
Appendix
0.8
0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
0.0
0.8
) 0.6
x
=X0.4
(
P0.2
0.0
0.8
0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
0.0
2
3
5
7
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