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Advances in Data Analysis and Classification
https://doi.org/10.1007/s11634-021-00469-0
REGULAR ARTICLE
Quantile composite-based path modeling: algorithms,
properties and applications
Pasquale Dolce1·Cristina Davino2·Domenico Vistocco3
Received: 6 September 2020 / Revised: 5 September 2021 / Accepted: 22 September 2021
© The Author(s) 2021
Abstract
Composite-based path modeling aims to study the relationships among a set of
constructs, that is a representation of theoretical concepts. Such constructs are opera-
tionalized as composites (i.e. linear combinations of observed or manifest variables).
The traditional partial least squares approach to composite-based path modeling
focuses on the conditional means of the response distributions, being based on ordinary
least squares regressions. Several are the cases where limiting to the mean could not
reveal interesting effects at other locations of the outcome variables. Among these:
when response variables are highly skewed, distributions have heavy tails and the
analysis is concerned also about the tail part, heteroscedastic variances of the errors
is present, distributions are characterized by outliers and other extreme data. In such
cases, the quantile approach to path modeling is a valuable tool to complement the
traditional approach, analyzing the entire distribution of outcome variables. Previous
research has already shown the benefits of Quantile Composite-based Path Model-
ing but the methodological properties of the method have never been investigated.
This paper offers a complete description of Quantile Composite-based Path Model-
ing, illustrating in details the method, the algorithms, the partial optimization criteria
along with the machinery for validating and assessing the models. The asymptotic
properties of the method are investigated through a simulation study. Moreover, an
application on chronic kidney disease in diabetic patients is used to provide guidelines
BCristina Davino
cristina.davino@unina.it
Pasquale Dolce
pasquale.dolce@unina.it
Domenico Vistocco
domenico.vistocco@unina.it
1Department of Public Health, University of Naples Federico II, Naples, Italy
2Department of Economics and Statistics, University of Naples Federico II, Naples, Italy
3Department of Political Science, University of Naples Federico II, Naples, Italy
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P. Dolce et al.
for the interpretation of results and to show the potentialities of the method to detect
heterogeneity in the variable relationships.
Keywords Composite-based path modeling ·OLS regression ·Quantile regression
Mathematics Subject Classification 62-07: (Statistics) Data analysis ·62H99:
(Statistics) Multivariate analysis, forse ·62G08: (Statistics) Nonparametric
regression ·62P10: (Statistics) Applications to biology and medical sciences
1 Introduction
Several are the approaches to study the relationships among different constructs and
between each construct and its corresponding observed or manifest variables (MVs). In
most common models, each block of MVs measures a construct, and prior knowledge is
used to define the theoretical model. Two are the main parameters in this type of model:
the path coefficients and the loadings. Path coefficients represent the relationships
between constructs while loadings measure the relationship between constructs and
the corresponding MVs.
Covariance structure analysis (Jöreskog 1978) and Partial Least Square Path
Modeling (PLS-PM) (Esposito Vinzi et al. 2010;Hairetal.2017) are the two main-
stream approaches. Even if they are commonly considered as alternative, they belong to
two different families of statistical methods. Covariance structure analysis, essentially
used in factor-based Structural Equation Modeling (SEM), exploits the covariance
matrix of MVs to estimate the model parameters. PLS-PM instead summarizes each
block of MVs in a component, or composite, namely an exact linear combination of the
MVs, focusing on the explained variance of MVs (Wold 1982,1985). Each composite
is a proxy of the construct associated to the correspondent block. For the aforesaid
reasons, PLS-PM is commonly referred to as a component-based, composite-based
or variance-based approach. Herman Wold, who proposed the PLS-PM, referred to
this approach as “Soft Modeling”. The name indicates that the method requires “soft”
distributional assumptions, in contrast to the estimation method for factor-based mod-
els, which requires strong assumptions on the error distributions (thus the name “hard
modeling”) (Wold 1975,1982; Tenenhaus et al. 2005;Chin1998).
PLS-PM exploits least square regression to estimate the model coefficients and
therefore focuses on the conditional mean of the response variables. Many are the
cases where the analysis of the average alone could produce an incomplete view of the
complex structure of the relationships among variables. When heteroscedastic vari-
ances of least square regression residuals occurs, and/or when response variables are
highly skewed, the study of the conditional distribution at locations different from the
mean can complement the classical approach and provide a richer picture of the inves-
tigated phenomenon. Following this idea, Quantile Composited-based Path Modeling
(QC-PM), proposed by Davino and Esposito Vinzi (2016), exploits quantile regres-
sion (Koenker and Basset 1978) to look beyond the average. It is a valuable tool to
study the relationships among variables, and to model location, scale and shape of the
responses. QC-PM can be used to complement PLS-PM to investigate whether the
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effects of explanatory constructs change over the entire distribution of the response
constructs. It is worth emphasizing that PLS-PM and QC-PM are not competing meth-
ods, and therefore their comparison in terms of performance is of little interest. The
two methods have different objectives: PLS-PM focuses on the conditional means
of the dependent variables providing an instant summary, while QC-PM explores
relationships among variables outside of conditional mean. Finding that estimated
coefficients vary across the conditional quantiles does not imply that the PLS-PM
results are invalid.
Previous research (Davino et al. 2016,2017,2018,2020; Davino and Esposito
Vinzi 2016) was oriented to show the relative advantages of QC-PM when the interest
is in the effects of explanatory constructs on the entire distribution of the response
constructs, and to assist in the interpretation of results and in their use combined with
PLS-PM results.
The aim of this paper is to provide a complete and organic description of QC-PM,
since its methodological properties have never been investigated. This goal is pursued
through several innovative contributions introduced in the paper: a clear and detailed
explanation of the method introducing also the case of one block and two blocks, an
improvement of the method allowing to handle the measurement invariance issue, an
application with artificial data that allows to highlight the potential of the method.
More specifically, in order to clarify the characteristics of the method, a step-by-step
description of the algorithms, the partial optimization criteria, and the formalization
of the models are provided. A relevant part of the present work is devoted to studying
the properties of QC-PM, which have never been investigated before.
An analytical discussion of the properties of the involved estimators is a daunting
task, due to the complexity of composite-based path modeling. This is a fertile ground
for the use of simulation studies, which provide information on the performance of
the method in terms of bias, efficiency and robustness of the estimates (Paxton et al.
2001). In particular, we exploit a Monte Carlo simulation design generating data
from a composite-based population and considering a set of different scenarios. This
allows us to assess the effects of several drivers: correlation within the blocks of MVs,
correlation among constructs, effect of heavy-tails and skewness of MV distributions,
effect of sample size.
A discussion on the asymptotic properties of the method is offered, along with
empirical evidence on the behaviour of the estimators in terms of bias and efficiency.
An innovative contribution to the estimation of the outer model is also provided to
fulfil the measurement invariance required to compare path coefficients estimates over
quantiles.
Moreover, an application on Chronic Kidney Disease (CKD) in diabetic patients is
provided to show benefit of using QC-PM as a supplement to PLS-PM. In particular,
we apply QC-PM on data already used in a research that proposed a quantile approach
to factor based SEM (Wang et al. 2016). Because the original data were not available,
they have been artificially derived mimicking the model, the relations among variables
and the estimates obtained in the original study (Wang et al. 2016). As the artificial
data are generated from a scenario where relationships among variables change with
quantiles, the application highlights the potentialities of the method in detecting the
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heterogeneity in the variable relationships and stressing its complementary role with
the traditional methods for composite-based path modeling.
The paper is organized as follows: Sect. 2describes QC-PM in detail, formalizing
the estimation process starting from the simplest case of one block of MVs and moving
until the general path model for multi-block data. Section 3illustrates the assessment
measures of QC-PM in terms of goodness of fit and statistical significance of the esti-
mated coefficients. Section 4shows the simulation design and the main results, while
the applicative potentialities of QC-PM, along with guidelines for the interpretation
of results, are provided through the study on the artificial data set concerning CKD
in diabetic patients in Sect. 5. Finally, a summary of the proposal and an outline for
future developments are given in Sect. 6.
2 QC-PM: quantile composite-based path modeling
QC-PM is strongly related to PLS-PM. Therefore the modeling and estimation pro-
cedures have much in common, the same holds for their properties. The theoretical
foundations are framed in the iterative algorithm proposed by Wold (1966a,b), the
Nonlinear estimation by Iterative PArtial Least Squares (NIPALS) algorithm, an alter-
native algorithm for implementing principal component analysis. More broadly, Partial
Least Squares (PLS) refers to a set of iterative alternating Ordinary Least Squares
(OLS) algorithms, extending the NIPALS algorithm to implement a large number of
multivariate statistical techniques (Esposito Vinzi and Russolillo 2013), depending on
the involved MVs. For example, in case of one block of MVs, PLS provides principal
component analysis. In case of two blocks of MVs, multivariate regularized regression
can be obtained (PLS regression). In case of multi-block data, PLS algorithm produces
PLS-PM (Lohmöller 1989).
The numerical solutions of all these methods are obtained through an iterative
algorithm, which is the first stage of the procedure. The basic idea of the PLS iterative
algorithm, which computes the weights used to define the composites, is to partition
the set of model parameters to be estimated in subsets. At each step of the algorithm,
one subset of parameters is considered known and held fixed, while the other subset
is estimated (Lohmöller 1989). A least squares criterion is adopted to estimate the
parameters in each step. The name PLS comes from the use of OLS to face with the
least square criterion at each step. For example, in case of multi-block data, namely
PLS-PM, the procedure comes down to a set of simple and multiple OLS regressions,
and Pearson correlation computation.
The QC-PM algorithm follow exactly the same steps of the PLS-PM algorithm, but
replacing simple and multiple OLS regression with their Quantile Regression (QR)
counterparts (Koenker 2005). The same for classical Pearson correlation, which is
replaced with quantile correlation (Li et al. 2014). As well as PLS-PM, QC-PM is
based on a two-stage procedure. The first stage aims at computing the outer weights
by an iterative procedure (these weights are then used for computing the composites).
In the second stage, model parameters (loadings and path coefficients) are estimated
through regression analysis using the composites. As for PLS-PM, partial criteria are
optimized at each step.
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The use of quantile based tools in all phases of the algorithm shifts the focus
to the entire conditional distributions of the involved response variables, allowing
to estimate partial conditional quantiles. Through the use of different conditional
quantiles, the whole response distribution can be inspected. Therefore QC-PM is a
valuable complement to PLS-PM, as much as quantiles are a complement to the
average. For the sake of illustration, next subsections present QC-PM for the simplest
model (one block of MVs), for two blocks of MVs, and for multi-block data (the
general model), respectively.
The presentation of the QC-PM algorithm will follow the same steps and the same
approach generally used to present the PLS-PM algorithm (Lohmöller 1989; Tenen-
haus et al. 2005; Esposito Vinzi and Russolillo 2013). A basic knowledge of QR is
assumed. Appendix provides a basic introduction of QR idea and goals. For a more
detailed description of QR, please refer to Koenker and Basset (1978), Davino et al.
(2013) and Furno and Vistocco (2018).
2.1 Quantile path modeling for one block of manifest variables
The simplest model involves one block of MVs and a construct. The relationships
between them are depicted in Fig. 1. MVs are denoted with X={xip}and represented
through rectangles. It is worth to recall that i=1,...,nrefers to the observations, n
denoting their number, and p=1,...,Prefers to the MVs, Pdenoting the number
of MVs in each block. The corresponding construct is labeled with ξ={ξi}and is
placed in oval or circle. This diagram is called path diagram.
The relationships among MVs and construct in the path diagram can be translated
into a system of simultaneous equations. In particular, for each considered quantile
θ∈(0,1), the link between each MV xpand ξis defined through the following
equation:
xip =αp(θ)+λp(θ)ξi+ip (θ),(1)
Fig. 1 A path diagram for an hypothetical one-block model
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where αpis a location parameter, λpis the loading coefficient, capturing the effect
of ξon xp, and p={ip}is the error term vector. The only assumption is that the
generic θth conditional quantile of xpcan be expressed as:
Qθxp|ξ=αp(θ)+λp(θ)ξ(2)
which implies that the θth quantile of the error term p(θ)is equal to zero and p(θ)
is independent of ξ. No assumptions on the error distribution are required.
The model captures the common variation among the MVs in Xthat depends
on the construct ξ. The use of QR allows to encompass the effect of ξon the whole
conditional distribution of each xp, effect that might not be equal at different locations,
i.e. quantiles. In fact, the construct could be a weak factor at some location of the MV
distributions, exerting a stronger effect at other conditional quantiles. In other cases
the effect could be almost uniform along the entire distribution. The quantile model
offers the opportunity to investigate the possible different situations.
Parameters are estimated through the classical PLS algorithm for one block of
variables, replacing OLS regression with QR at each step of the procedure. This
corresponds to iteratively optimize a quantile partial criterion. The algorithm consists
of three main steps. In the initialization step, arbitrary values are set for the outer
weights ˆ
wp, namely the coefficients used to compute the composite ˆ
ξ, which expresses
the construct as a linear combination of the MVs. Starting from such initialization
values, a first approximation of the composite is computed as a linear combination
of the MVs in X,ˆ
ξ(0)=P
p=1ˆw(0)
pxp. Then, a loop starts and at each sth iteration
(s=0,1,2,...), each MV xpis regressed on the composite, minimizing the following
quantile loss function:
ˆw(s)
p(θ)=argmin
wp(θ)
n
i=1
ρθxip −αp(θ)−ξ(s−1)
iwp(θ),(3)
where ρθ(.)is the check function, which asymmetrically weights positive and negative
residuals, namely:
ρθ(r)=θrif r>0
(θ−1)rif r≤0.(4)
The outer weights are then iteratively computed until convergence through simple QR
models, where each MV is the response variable and the composite is the regressor.
In each step the estimated weights are normalized and used to update the ˆ
ξ, obtained
as a linear combination of the response MVs xp.
By weighting the MVs considering their quantile covariation with the construct, for
each quantile θthe proposed algorithm chooses a linear combination of MVs that is a
consistent quantile composite. Once convergence is reached or the maximum number
of iterations is achieved, loadings are estimated by means of simple QRs over the
corresponding scores. The pseudo–code of the QC-PM iterative algorithm for the case
of one block of MVs is provided in Algorithm 1.
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Algorithm 1 The QC-PM algorithm for the case of one block of MVs
1: for each quantile θdo
STEP 1: Initialization
2: s←0iteration counter
3: Choose arbitrary outer weights ˆw(s)
p(θ)(p=1, ..., P)
4: Compute ˆ
ξ(s)(θ) =P
p=1ˆw(s)
p(θ)xp
STEP 2: Iteration
5: repeat
6: s←s+1increment the iteration counter
7: for all manifest variables xpdo
8: Compute ˆw(s)
p(θ)solving the quantile regressions:
xip =α(s)
p(θ)+w(s)
p(θ)ˆ
ξ(s−1)
i+ip (θ)(i=1,...,n;p=1,...,P)
9: end for
10: Normalize the weights:
ˆw(s)
p(θ)=ˆw(s)
p(θ)
Xˆ
w(s)(θ)where ˆ
w(s)(θ)={ˆw(s)
p(θ)}
11: Compute ˆ
ξ(s)(θ):
ˆ
ξ(s)(θ)=
P
p=1
ˆw(s)
p(θ)xp
12: until ˆ
w(s)(θ)≈ˆ
w(s−1)(θ)
STEP 3: Estimation
13: for all manifest variables xpdo
14: Estimate ˆ
λp(θ)solving the quantile regressions:
xip =αp(θ)+λp(θ)ˆ
ξ(s)
i(θ)+ip (θ)(i=1,...,n;p=1,...,P)
15: end for
16: end for
2.2 Quantile path modeling for two blocks of manifest variables
Figure 2depicts the path diagram for an hypothetical two-block model. In such a
case, let X={xip}and Y={yij}denote the two blocks of MVs. In particular,
the former consists of the explanatory MVs and the latter of the response MVs. We
denote with Pthe number of explanatory MVs, as in the previous subsection, and
with Jthe number of response MVs, yjbeing the generic response MV. Moreover,
ξ={ξi}is the construct representing the explanatory block, and η={ηi}the construct
representing the dependent block. The general model consists of two sub-models: the
inner model and the outer model. The inner model refers to the relationships between
the constructs, the outer model between each construct and its block of MVs.
By referring to the outer model, for each quantile θ∈(0,1),theMVsxpin
the explanatory block, and the MVs yjin the dependent block, are related to their
correspondent constructs through the following system of bilinear equations:
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P. Dolce et al.
Fig. 2 A path diagram for an hypothetical two-block model
xip =αxp (θ)+λxp (θ)ξi+ip (θ),(5)
yij =αyj (θ)+λyj (θ)ηi+ωij (θ).(6)
where αxp and αyj are location parameters, λxp is the loading coefficient capturing
the effect of ξon xp,λyj is the loading coefficient capturing the effect of ηon yj,
while ={ip}and ω={ωij}are the error terms. The usual assumptions on the error
terms already mentioned above are required.
The inner model specifies the dependence relationships between the two constructs.
The dependent construct ηis linked to the explanatory construct ξby the following
model:
ηi(θ)=β0(θ)+β1(θ)ξi+ζi(θ),(7)
where β1is the so-called path coefficient capturing the effects of ξon the dependent
construct η, and ζ={ζi}is the inner error variable.
The procedure for the estimation of the model parameters requires a multi-step
algorithm and follows the same structure of the PLS algorithm for two blocks of MVs,
defined for example in Esposito Vinzi and Russolillo (2013), where partial criteria are
optimized iteratively. The pseudo code of the QC-PM algorithm for the case of two
blocks of MVs is detailed in Algorithm 2.
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Quantile composite-based path modeling
Algorithm 2 The QC–PM algorithm for the case of two blocks of MVs
1: for each quantile θdo
STEP 1: Initialization
2: s←0iteration counter
3: Choose arbitrary outer weights ˆw(s)
xp (θ)(p=1, ..., P)
4: Compute ˆ
ξ(s)(θ)=P
p=1ˆw(s)
xp (θ)xp
STEP 2: Iteration
5: repeat
6: s←s+1increment the iteration counter
7: dependent block
8: for all manifest variables yjof the dependent block do
9: Compute ˆw(s)
j(θ)solving the quantile regressions:
yij =α(s)
j(θ)+w(s)
j(θ)ˆ
ξ(s−1)
i+ij (θ)(i=1,...,n;j=1,...,J)
10: end for
11: Normalize the weights:
ˆw(s)
yj (θ)=
ˆw(s)
yj (θ)
Yˆ
w(s)
y(θ)
where ˆ
w(s)
y(θ)={ˆw(s)
yj (θ)}
12: Compute ˆ
η(s)(θ):
ˆ
η(s)(θ)=
J
j=1
ˆw(s)
yj (θ)yj
13: explanatory block
14: for all manifest variables xpof the explanatory block do
15: Compute ˆw(s)
p(θ)solving the quantile regressions:
xip =α(s)
p(θ)+w(s)
p(θ)ˆη(s)
i(θ)+ip (θ)(i=1,...,n;p=1,...,P)
16: end for
17: Normalize the weights:
ˆw(s)
xp (θ)=ˆw(s)
xp (θ)
Xˆ
w(s)
x(θ)
where ˆ
w(s)
x(θ)={ˆw(s)
xp (θ)}
18: Compute ˆ
ξ(s)
(θ):
ˆ
ξ(s)(θ)=
P
p=1
ˆw(s)
xp (θ)xp
19: until ˆ
w(s)
x(θ),ˆ
w(s)
y(θ)≈ˆ
w(s−1)
x(θ),ˆ
w(s−1)
y(θ)
STEP 3: Estimation
20: for all manifest variables xpdo explanatory block
21: Estimate ˆ
λxp (θ)solving the quantile regressions:
xip =αp(θ)+λp(θ)ˆ
ξ(s)
i(θ)+ip (θ)(i=1,...,n;p=1,...,P)
22: end for
23: for all manifest variables yjdo dependent block
24: Estimate ˆ
λyj (θ)solving the quantile regressions:
yij =αj(θ)+λj(θ)ˆη(s)
i(θ)+ij (θ)(i=1,...,n;j=1,...,J)
25: end for
26: Estimate the path coefficient β1(θ)solving the quantile regressions: path coefficient
ηi=β0(θ)+β1(θ)ˆ
ξi(θ)+ζi(θ)
27: end for
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P. Dolce et al.
In the initialization step of the algorithm, arbitrary values are set for the outer
weights ˆ
wxp to compute a first approximation of the composite as a linear combination
of the MVs in X,ˆ
ξ(0)=P
p=1ˆw(0)
xp xp. Then, the iterative algorithm step proceeds
over two phases. At each sth iteration (s=0,1,2,...), the response MVs yjare
regressed on the approximation of the composite ˆ
ξ(s−1), minimizing the following
quantile loss function:
ˆw(s)
yj (θ)=argmin
wyj(θ)
n
i=1
ρθyij −αp(θ)−ˆ
ξ(s−1)
iwyj (θ),(8)
where ρθ(.)is the check function defined as above.
In the second phase, the estimated ˆw(s)
yj (θ),for j=1,..., J, are used to compute
the composite ˆ
η(s)through a linear combination of the response MVs yj, and then the
explanatory MVs xxp, are regressed on the obtained linear combination, minimizing
the following quantile loss function:
ˆw(s)
xp (θ)=argmin
wxp(θ)
n
i=1
ρθxip −αp(θ)−ˆη(s)
iwxp (θ).(9)
Finally, an updated approximation of the composite ˆ
ξ(s)is obtained as a linear
combination of the explanatory MVs xp, using the weigths ˆw(s)
xp (θ).
These two phases are iteratively repeated until convergence of the outer vectors,
wx(θ)={wxp (θ)}and wy(θ)={wyj (θ)}, is achieved. Then loadings and path
coefficients are estimated through quantile regression.
QC-PM algorithm returns, for each quantile θ, a linear combination of the explana-
tory MVs xxp by weighting the corresponding MVs on the basis of their quantile
covariation with the linear combination of the response MVs yj.
2.3 Quantile path modeling for multi-block data
Figure 3depicts a path model for multi-block data using the case of three blocks. The
general model for Kblocks follows the same logic. Let us assume that Pvariables
are collected in a table Xof data partitioned in Kblocks: X=[X1,X2...,XK].Let
Xk={xip
k}be a generic block of MVs, where i=1,...,n, with ndenoting the
number of observations, pk=1,..., Pk, with Pkbeing the number of MVs in the kth
block. We denote by ξk={ξik}and xpk={xip
k}the LV and a generic MV of the kth
block, respectively.
A construct that never appears as a dependent variable in the model is called exoge-
nous, while the endogenous constructs play only the role of dependent variables or of
both dependent and explanatory variables. In Fig. 3, for example, ξ1is an exogenous
construct and ξ2and ξ3are endogenous constructs.
As for the case of two blocks of MVs, the general model consists of the inner model
and the outer model. For each quantile θ∈(0,1), in the outer model it is assumed
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Quantile composite-based path modeling
Fig. 3 A path model with three blocks of MVs
that each MV xpkis related to its own construct through the following equations:
xip
k=αpk(θ)+λpk(θ)ξik +ip
k(θ),(10)
where αpkis the location parameter, ξk={ξik}is the construct representing the kth
block of MVs, λpkis the loading coefficient, capturing the effect of ξkon xpkand
k={ip
k}is the error term vector, using the usual above mentioned assumption on
the errors.
The inner model captures and specifies the dependence relationships among con-
structs. A generic endogenous construct, ξk, is linked to the related explanatory
constructs, ξk,k∈Jk, where Jk={k:ξkis predicted by ξk}, by:
ξik(θ)=βk0(θ)+
k∈Jk
βkk(θ)ξik +ζik(θ),(11)
where βkkis the path coefficient capturing the effects of ξkon the dependent construct
ξ
k, and ζ
k={ζik}is the inner error variable vector, with the usual assumption on the
errors.
A description of the general QC-PM algorithm is provided in Algorithm 3.
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P. Dolce et al.
Algorithm 3 The general QC-PM algorithm for the case of multi-block data
1: for each quantile θdo
STEP 1: Initialization
2: s←0iteration counter
3: Choose arbitrary outer weights ˆw(s)
pk(θ)(pk=1,...,Pk;k=1,...,K)
4: Compute ˆ
ξ(s)
k(θ)=Pk
pk=1ˆw(s)
pk(θ)xpk
STEP 2: Iteration
5: repeat
6: s←s+1increment the iteration counter
7: inner approximation phase (k∈Jk)
8: Compute quantile correlation:
τ(θ)(s)
kk=qcor (θ)ˆ
ξ(s−1)
k(θ),ˆ
ξ(s−1)
k(θ)
9: Compute inner scores as,
ˆ
ξ(s)
k(θ)=
k∈Jk
τ(θ)(s)
kkˆ
ξ(s−1)
k(θ),where Jk={k:ξkis predicted by ξk}
ˆ
ξ(s)
k(θ)=
k∈Jk
τ(θ)(s)
kkˆ
ξ(s−1)
k(θ),where Jk={k:ξkpredicts ξk}
10: outer approximation phase (k=1,...,K)
11: for all manifest variables xpkdo
12: Compute ˆw(s)
pk(θ)solving the quantile regressions:
xip
k=α(s)
pk(θ)+w(s)
pk(θ)ˆ
ξ(s)
i(θ)+ip
k(θ)(i=1,...,n;pk=1, ..., Pk)
13: end for
14: Normalize the weights:
ˆw(s)
pk(θ)=ˆw(s)
pk(θ)
Xˆ
w(s)
k(θ)
where ˆ
w(s)
k(θ)={ˆw(s)
pk(θ)}
15: Compute ˆ
ξ(s)
k(θ)=Pk
pk=1ˆw(s)
pk(θ)xpk
16: until ˆ
w(s)≈ˆ
w(s−1)
STEP 3: Estimation
17: for all manifest variables xpkdo estimation of loadings
18: Estimate ˆ
λxp (θ)solving the quantile regressions:
xip =αp(θ)+λp(θ)ˆ
ξ(s)
i(θ)+ip (θ)(i=1,...,n;p=1,...,P)
19: end for
20: Estimate βkk(θ), solving the quantile regressions: estimation of path coefficients
ξik(θ)=βk0(θ)+
k∈Jk
βkk(θ)ξik (θ)+ζik(θ)
21: end for
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Quantile composite-based path modeling
The weight vector w(θ)={wpk(θ)},(pk=1,...,Pk;k=1,...,K),used
to define the composites, is computed by an iterative algorithm that proceeds over
two phases, so-called inner and outer approximation phases, iteratively repeated until
convergence of the outer vectors w(θ) is achieved (i.e., the change of the outer weights
from one iteration to the next is smaller than a predefined tolerance).
In the inner phase, composites are approximated as weighted aggregates of the
adjacent composites: two composites are adjacent if there exists a link in the inner
model connecting the corresponding constructs, that is, an arrow going from one
construct to the other in the path diagram, independently of the direction. The inner
weights are defined as the values of the quantile correlation between the composites
obtained at the previous step. According to the PLS-PM terminology, this mode to
compute inner weights is called factorial inner scheme. Another scheme can be also
applied, called centroid scheme, where the inner weights are computed as the signs of
the quantile correlation between the composites (Tenenhaus et al. 2005). These two
schemes generally provides very close results, but factorial scheme is more advisable
when correlation between composites is close to zero. In this case, correlation may
oscillate from small negative to small positive values during the iteration cycles, and
factorial scheme is more advisable because it takes into account the strength of the
correlation, instead of just the sign. It is worth to note that unlike Pearson correlation,
quantile correlation is not a symmetric measure (Li et al. 2014), hence it is necessary
to specify the role played by the involved constructs in each equation (i.e., explanatory
or dependent one) for the calculation of the inner weights.
In the outer estimation phase, composites are approximated through a normal-
ized weighted aggregate of the corresponding MVs. Outer weights are computed
through simple quantile regressions, where each MV is regressed on the corre-
sponding inner approximation composite. Then, the weights are normalized so that
var[Xkwk(θ)]=1. According to the PLS-PM terminology (Tenenhaus et al. 2005),
this mode to compute outer weights is called Mode A. The so-called Mode B is also fea-
sible in QC-PM, computing the outer weights as regression coefficients in the quantile
multiple regression of the inner approximation composite on its own MVs. Basically,
Mode B takes account of collinearity among MVs of the some blocks, while Mode A
ignores this collinearity.
In the QC-PM iterative procedure, at each sth iteration (s=0,1,2,...),thefol-
lowing partial criterion are then optimized:
ˆ
wpk(θ)=argmin
wpk(θ)
n
i=1
ρθxip
k−αp(θ)−ξ(s−1)
ik wpk(θ)(12)
where ρθ(.)is the check function defined as above.
When Mode B is applied, the following criterion is instead minimized:
ˆ
w(s)
k(θ)=argmin
wk(θ)
n
i=1
ρθξ(s−1)
ik (θ) −αp(θ)−Xikwk(θ)(13)
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P. Dolce et al.
When convergence is achieved, loadings and path coefficients are estimated through
QR.
As a matter of fact, QC-PM provides, for each quantile of interest, a set of outer
weights, loadings and path coefficients, offering a more complete picture of the rela-
tionships among variables both in the outer model and in the inner model.
The algorithm provides quantile-based composites, and it is useful to deal with
heterogeneity both in the structural model and in the measurement model. In such a
case the interest is in evaluating how weights and composites vary across quantile.
However, if the interest is in comparing estimated models over quantiles, the measure-
ment invariance (Henseler et al. 2016) has to be fulfilled. If weights, and consequently
composites, change over quantiles, a proper comparison among path coefficients esti-
mated at different quantiles is indeed not reliable, because the same concept may not
be measured across quantiles. To this end, a test on the weights defined as a vari-
ant of the Wald test described in Koenker and Basset (1982) can be exploited. The
null hypothesis of the test states that the weights are identical. In case of significant
differences among weights, or in case there is the requirement to keep the weights
fixed, a new variant of QC-PM can be implemented simply setting the quantile to the
median in the iterative procedure. The use of the median in the iterative procedure
(step 2) of the algorithm is in line with the approach proposed in Wang et al. (2016)
for factor-based SEM. In such an approach the quantile varies only in step 3, to obtain
quantile-dependent path coefficients. The median approach can be generalized to the
whole iterative process to provide measurement invariance.
3 Model assessment and validation
Once the algorithm converges and estimates for loadings and path coefficients are
obtained, there are many tools for assessing both the inner and outer model. Results,
namely loadings and path coefficients, can also be validated from an inferential point
of view (Davino et al. 2016).
Goodness of fit measures most commonly used in PLS-PM cannot be directly
adapted to QC-PM. Moving from OLS to QR requires indeed amendment. The intro-
duction of an effective goodness of fit approach in QR is still an open issue in the
scientific literature (Koenker and Machado 1999; He and Zhu 2003). This does make
it odd to directly compare OLS and QR, even considering that the two methods opti-
mize different criteria. Therefore, a direct comparison between PLS-PM and QC-PM
is not possible.
Starting with the inner model, the coefficient of determination R2of the endoge-
nous constructs (Esposito Vinzi et al. 2010) is the criterion mostly used in PLS-PM.
Considering that QR loss function is not based on a least squares criterion but rather
on a least absolute deviation criterion in terms of weighted residuals, the use of R2in
QC-PM goes against the underlying rationale of the method. This issue is particularly
relevant since most of the assessment indexes in PLS-PM are based on the multiple
linear determination coefficient or squared Pearson correlation coefficient. It is against
this background we employ the pseudo-R2proposed by Koenker and Machado (1999),
so to have a measure that simulates the role and interpretation of the R2for QC-PM
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Quantile composite-based path modeling
assessment. It is important, however, to bear in mind that the pseudo–R2is designed
differently.
QC-PM estimates a set of parameters for each conditional quantile θof interest and,
consequently, it requires a set of assessment measures for each estimated model. In
particular, for each θ,thepseudo–R2compares the residual absolute sum of weighted
differences using the selected model (RASW) with the total absolute sum of weighted
differences using a model with the only intercept (TAS W ). RASW corresponds to the
residual sum of squares in classical regression, TASW to the total sum of squares of
the dependent variable. Pseudo–R2aims to evaluate if the full model (i.e. the model
with the regressors) is better in terms of residuals the “restricted” (the model with
the only intercept). More precisely, the pseudo–R2is calculated as one minus the
ratio between RASW and TASW. In essence, pseudo-R2can be considered as a local
measure of goodness of fit for a particular quantile as it measures the contribute of
the selected regressors to the explanation of the dependent variable with respect to the
trivial model without regressors. With an R2,pseudo–R2values range between 0 and
1: the more it is close to 1, the more the model with regressors can be considered a
good model (i.e., the θth conditional quantile function is significantly altered by the
effect of the covariates). If on one hand the pseudo–R2will always be smaller than
the R2and a direct comparison with R2in PLS-PM is not feasible, on the other hand
pseudo–R2is useful to the end of identifying locations in the distribution of outcome
variable where model may show a better/worse fit (for example, if the model fits in the
tail, there’s not guarantee that it fits well anywhere else) (Kováˇc and Želinský 2013).
For the sake of generality, we consider below the case of multi-block QC-PM. Once
convergence is reached and composites are obtained, several QRs are carried out in the
inner part of the model, according to the number of considered quantiles. Such QRs
estimate the path coefficients linking endogenous and exogenous constructs. As stated
in Sect. 2.3, a generic endogenous construct, ξk, is linked to the related explanatory
constructs, ξk,k∈Jk, where Jk={k:ξkis predicted by ξk}. For the convenience
of the reader, we report again Eq. (11) that describes this relationship:
ξik(θ)=βk0(θ)+
k∈Jk
βkk(θ)ξik (θ)+ζik(θ).(14)
Since ˆ
ζk(θ) represents the residuals of the model explaining the kth endogenous
construct, for each considered quantile θ,RASW is the corresponding minimizer:
RASW
k(θ)=
ˆ
ζk(θ)≥0
θ
ˆ
ζk(θ)
+
ˆ
ζk(θ)<0
(1−θ)
ˆ
ζk(θ)
,(15)
where positive and negative residuals are asymmetrically weighted, respectively with
weights equal to θand (1−θ).TheTASW is instead:
TASW
k(θ)=
ξk≥θ
θ|ξk−θ|+
ξk<θ
(1−θ)|ξk−θ|.(16)
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P. Dolce et al.
Therefore, the obtained pseudo–R2can be computed as follows:
pseudo−R2
k(θ)=1−RASW
k(θ)
TASW
k(θ).(17)
The pseudo—R2ranges between 0 and 1, since RASW (θ)is always less than
or equal TASW(θ). It indicates, for each considered quantile, whether the presence
of the covariates influences the correspondent conditional quantile of the response
variable. It is worth noticing that the pseudo-R2is not a symmetric measure, assuming a
different value when the role of the variables is reversed. The index, computed for each
inner equation, measures the amount of variability of a given endogenous construct
explained by its explanatory constructs. The average of all the pseudo—R2indexes
provides a synthesis of the evaluations regarding the inner model.
As regards to the outer model, the assessment is carried out considering the relations
between each construct and its own MVs and the estimate of the error term vector ˆk.
The pseudo—R2can be used to assess convergent validity for each outer model,
applying for each block the average of the pseudo—R2indexes of the related MVs,
and can be used for assessing the quality of the whole outer models computing a
weighted average of all measures over all the blocks, using the number of MVs for
each block as weights. In particular, a pseudo−R2
pk(θ ) is computed on the basis of
Eq. (10) considering the kth block, for each MV and for each considered quantile
θ. This measure, called Communalitypk (θ), with p=1,...,P,k=1,...,K,
indicates how much of the MV’ variance can be explained by the corresponding
component. The communality of the block kresults:
Communalityk(θ) =1
Pk
Pk
pk=1
pseudo−R2
pk (θ).(18)
The quality of the whole outer model is finally obtained through the average of the
Communality indexes of all the blocks.
It should be noted that, as described in Sect. 2.3, if the quantile in the iterative
procedure is set equal to the median to solve the measurement invariance issue, the
assessment of the outer model is limited to the quantile θ=0.5.
Another measure of assessment is provided by the Redundancy index, which is
defined only for the endogenous block. Please note that low levels of redundancy does
not necessarily mean that the structural model is poorly specified. This index only
combines the evaluations of both the inner model and the outer model (Lohmöller
1989;Hairetal.2011), thus can be used as a measure of assessment of the global
model, but specific measures for the two sub-models are also needed.
Redundancy can be computed for each endogenous MV or for the whole block, as
an average of the redundancies of its MVs. For each MV of the endogenous block xpk,
Redundancy is computed multiplying its Communality measure by the pseudo—R2
obtained in the corresponding inner model:
Redundancypk (θ) =Communalitypk(θ ) ×pseudo−R2
k(θ) (19)
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Quantile composite-based path modeling
The overall Redundancy of the block kis obtained averaging the measures associated
to the MVs of the endogenous block:
Redundancyk(θ) =
Pk
pk=1
Redundancypk (θ)
Pk
.(20)
Possible variants could exploit different goodness of fit measures available in the
quantile framework as well as the amendment of some assessment indexes proposed
in PLS-PM literature (Benitez et al. 2020;Hairetal.2020,2017,2019; Amato et al.
2004).
Two main approaches can be used to evaluate the statistical significance of the
coefficients related to the different quantiles. The first approach exploits the asymptotic
normal distribution of QR estimators (Koenker and Basset 1978). Such estimators
are indeed asymptotically normal, with variance–covariance matrix depending on the
model assumptions. Independent and identically distributed errors, independent and
not identically distributed errors, and dependent errors determine obvious differences
in the variance–covariance matrix. The alternative resorts to bootstrap theory (Efron
and Tibshirani 1993), commonly used both in PLS-PM and QR. Bootstrap permits
to estimate the standard errors of the coefficients using a distribution free approach.
QR literature counts several bootstrap procedures, the xy-pair method (Parzen et al.
1994) being the simplest and widespread solution. It is also known as design matrix
bootstrap. Bootstrap standard errors are exploited to compute confidence intervals
and to perform hypothesis tests. Resampling methods are also useful in case of small
samples. For example, a jackknife approach could be used to estimate the standard
errors of the coefficients. Statistical tests could be also easily introduced in QC-PM to
test if coefficients at different quantiles can be considered statistically different (Gould
1997). We will not expand on the details here. Readers who are interested can consults
Davino et al. (2013) and Furno and Vistocco (2018) for a thorough explanation and
all bibliographic indications on inference in QR.
4 A simulation study
A Monte Carlo simulation study has been designed to investigate the performance of
QC-PM considering different scenarios. As already stated above, QC-PM and PLS-
PM are complementary rather than alternative approaches. Therefore, it is advisable
to use both methods in real data application: in many cases, indeed, the focus on con-
ditional mean is not sufficient and a more comprehensive look at the entire conditional
distribution is necessary.
Nevertheless, since there is not a real competing method to QC-PM, namely a
composite-based approach focusing on conditional quantile, we chose to investigate
properties of the QC-PM estimators comparing them with PLS-PM results. It is well
known that PLS-PM produces consistent estimates for composite-based model param-
eters and performs well in the considered scenarios.
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P. Dolce et al.
4.1 Simulation design and data generation
We operate in the context proposed by Schlittgen et al. (2020), generating data from
composite-based populations using the cbsem R package (Schlittgen 2019). Deter-
mination of the covariance matrix in the procedure proposed by Schlittgen et al.
(2020) can be derived considering three scenarios, named formative–formative (ff),
formative–reflective (fr) and reflective–reflective (rr). We used the scenario rr, where
outer weights are not required. The procedure requires path coefficients, loadings and
variances and covariances of exogenous constructs. The parameters must be chosen
such that sets of weights can be found to fulfill the equation defining the covariance
matrix (see the Vignette from the cbsem R package for further details) (Schlittgen
2019).
We set the relationships in the model assuming the theoretical path model repre-
sented in Fig. 3and then we simulated data for the given values of the parameters.
The postulated inner model is:
ξ2=β20 +β21ξ1+ζ2
ξ3=β30 +β31ξ1+β32ξ2+ζ3.
The outer model can be instead written as:
x11 =α11 +λ11ξ1+11
x21 =α21 +λ21ξ1+21
x31 =α31 +λ31ξ1+31
x12 =α12 +λ12ξ2+12
x22 =α22 +λ22ξ2+22
x32 =α32 +λ32ξ2+32
x13 =α13 +λ13ξ3+13
x23 =α23 +λ23ξ3+23
x33 =α33 +λ33ξ3+33.
The simulation study considered different scenarios both in the outer and in the
inner part of the model. Moreover, the effect of sample size and non-normality dis-
tributions were also considered. The cases of homogeneous blocks (no differences
among loadings) and heterogeneous blocks (large differences among loadings) were
used to assess the outer model. For the inner model, the effect of different correlation
levels between constructs was investigated.
That, in short, are the design-factors we considered for the simulation study: sample
size, homogeneity of blocks, size effect and variable distributions. In particular, we
used the following levels for each design-factor.
Sample sizes. We set n∈{50,100,200,300,400,500,1000}.Thevaluen=50
allows us to investigate the performance of the method in case of application with
small sample size. The other values used for nare instead typically encountered
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Quantile composite-based path modeling
Table 1 Levels of the design-factors considered in the simulation design
in research applications, the largest values being useful to study the asymptotic
properties of QC-PM estimators.
Loadings.Wesetλpk=0.9(p=1,2,3;k=1,2,3)for homogeneous blocks
and loadings λ1k=0.9,λ
2k=0.6,λ
3k=0.3(k=1,2,3)for heterogeneous
blocks, these last values to reflect very large differences among loadings.
Path coefficients. We set, for all inner relationships, βkk∈{0.2,0.3,0.4,0.5}to
take into account different levels of correlations among constructs.
Skewness and Kurtosis. We set both equal to 0 for normal distribution, while
skewness =2 and kurtosis =6 were used for mimicking exponential distributions.
The total number of scenarios obtained from the combination of the above described
levels of the design-factors is equal to 112 (7 sample sizes ×2 loadings ×4 path
coefficients ×2 skewness and kurtosis). For each considered scenario, we generated
500 replications.
A synthesis schema of the simulation design is offered in Table 1.
The R software environment (R Core Team 2020) for statistical computing were
used to generate and analyze data.
Data with non-normal distribution were generated using the technique described in
Vale and Maurelli (1983), who extended the method proposed by Fleishman (1978).
The performance of QC-PM was assessed considering the Relative Bias (RBias)
and the Root Mean Square Error (RMSE) of the estimates on the basis of the 500
replications. RBias was computed as:
RBias =1
S
S
s=1
(ˆ
θs−θ)
θs=1,2,...,500
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P. Dolce et al.
where Srepresents the number of replications in the simulation, ˆ
θsis the estimate
for the generic replication, and θis the corresponding population parameter. Instead,
RMSE was computed as:
1
S
S
s=1
(ˆ
θs−θ)2s=1,2,...,500
Clearly, because MSE =Var(ˆ
θ) +bias(ˆ
θ)2,RMSE entails information on both
bias and variability of the estimates.
4.2 Simulation results
In presenting simulation results, we choose to focus on the effect of sample size on
the bias and efficiency, and, consequently, on the consistency of the estimates. Since
the number of considered scenarios (112) is too large, in the following we focus only
on the more interesting and enlightening scenarios. In particular, we present results
for all the considered levels of loadings in the three blocks of variables for PLS-PM
and for QC-PM at quantile θ=0.5 (measurement model was indeed restricted to the
median regression model).
Additionally, since there are interesting differences between homogeneous blocks
and heterogeneous blocks, we reported results for both the cases.
For the inner model, we present results for all the three path coefficient estimates
obtained through PLS-PM and QC-PM at quantiles θ∈{0.25,0.5,0.75}, but only for
βkk=0.3, a small/moderate effect. Indeed, as known in literature (Tenenhaus 2008),
results are similar as correlations between composites increases.
The following subsections details results according to the three considered factors,
that is sample size, level of heterogeneity within blocks and degree of skew-
ness/kurtosis of the distribution. The resulting scenarios are organized in three groups:
–Group 1, focusing at the effect of sample size,
–Group 2, focusing at the effect of the level of heterogeneity within blocks,
–Group 3, focusing at the effect of the degree of skewness/kurtosis of the distribu-
tion.
The effect of sample size
The first set of considered scenarios (from hereinafter Group 1) allows us to focus
only on the effect of sample size, neutralising, as far as possible the effect of the other
factors: outer blocks are considered homogeneous with high correlations among the
MVs (all λvalues set equal to 0.9), data generated from normal distributions and path
coefficients set equal to 0.3. These settings result in seven scenarios for Group 1.
Figure 4shows the distribution of loadings for such scenarios for the outcome block
(Block 3). The results for the other two blocks are not shown since they do not differ
much from those of Block 3. The two columns of Fig. 4depict the results for PLS-PM
and QC-PM (recall that we set θ=0.5 for the outer model). The seven scenarios of the
group, corresponding to the different values of the sample size n, are represented on
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Quantile composite-based path modeling
Fig. 4 Diminutive distribution charts of the loadings of scenarios belonging to Group 1 with
n∈{50,100,200,300,400,500,1000}, homogeneous blocks, path coefficients equal to 0.3 and Normal
distributed data
the horizontal axis. Finally, the rows refer to the coefficients λ13,λ23,λ33 associated
with the MVs of the block.
Data are represented through diminutive distribution charts (Rudis 2019), a variant
of boxplots aimed to visualize distribution characteristics: each box ranges from the
10th percentile to the 90th percentile, the triangle indicates the mean value of the
distribution and the circle the median. Both PLS-PM and QC-PM show distributions
that converge to the true parameter value (horizontal line at the value 0.9) as nincreases.
Moreover, the variability of the estimates is rather small, although for QC-PM slightly
higher. Regarding bias, it is interesting to note, that for the quantile model, the 10th
percentile has a smaller distance from the true parameter than the PLS-PM, whatever
the value of n.
Table 2reports the values of RBias and RMSE for each loading distribution. In
particular, the table shows the average values of RBias and RMSE for each block
(columns) and each scenario (rows). The table is row-partitioned according to the
model. The values of RBias and RMSE for the loading estimates do not change sub-
stantially between QC-PM and PLS-PM when blocks are homogeneous and variables
are normally distributed, with large sample sizes (at least 300 observations). At the
lowest considered sample size (n=50), QC-PM always shows higher bias compared
to PLS-PM and such behavior is also confirmed combining bias with variability of the
estimates (see RMSE columns).
The distribution of the path coefficients (Fig. 5) is also affected by sample size and
shows less marked differences between the two methods (the comparison between
Figs. 4and 5must be done with caution because the two figures have different vertical
scales). As in Fig. 4, each diminutive distribution chart refers to a scenario (horizontal
axis) but here the three rows refer to the path coefficient (β21,β31,β32). Each vertical
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P. Dolce et al.
Table 2 Values of RBias and RMSE for loadings of the scenarios belonging to Group 1 with n∈
{50,100,200,300,400,500,1000}, homogeneous blocks, path coefficients equal to 0.3 and Normal dis-
tributed data (levels of the varying factor, sample size, are in the first column)
RBias RMSE
Block X1 Block X2 Block X3 Block X1 Block X2 Block X3
PLS-PM
n=50 0.035 0.036 0.037 0.040 0.042 0.039
n=100 0.039 0.039 0.038 0.038 0.038 0.037
n=200 0.038 0.038 0.038 0.036 0.036 0.035
n=300 0.038 0.038 0.038 0.035 0.035 0.035
n=400 0.038 0.038 0.038 0.035 0.035 0.035
n=500 0.039 0.038 0.038 0.035 0.035 0.035
n=100 0.038 0.038 0.038 0.035 0.035 0.035
QC-PM
n=50 0.043 0.039 0.044 0.066 0.082 0.066
n=100 0.040 0.043 0.042 0.056 0.052 0.050
n=200 0.041 0.039 0.041 0.044 0.042 0.043
n=300 0.039 0.039 0.040 0.040 0.040 0.040
n=400 0.039 0.039 0.039 0.039 0.039 0.039
n=500 0.040 0.039 0.039 0.039 0.038 0.038
n=1000 0.039 0.039 0.039 0.036 0.036 0.036
Results for PLS-PM and QC-PM estimated at θ=0.5 are reported each scenario
Fig. 5 Diminutive distribution charts of the path coefficients of scenarios belonging to Group 1 with
n∈{50,100,200,300,400,500,1000}, homogeneous blocks, path coefficients equal to 0.3 and Normal
distributed data
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Quantile composite-based path modeling
Table 3 Values of RBias and RMSE for the path coefficients of scenarios belonging to Group 1 with
n∈{50,100,200,300,400,500,1000}], homogeneous blocks, path coefficients equal to 0.3 and Normal
distributed data (the levels of the varying factor, the sample size, are in the first column)
RBias RMSE
β21 β31 β32 β21 β31 β32
PLS-PM
n=50 −0.031 −0.007 −0.016 0.120 0.121 0.122
n=100 −0.039 −0.037 −0.053 0.092 0.091 0.088
n=200 −0.062 −0.047 −0.068 0.067 0.066 0.069
n=300 −0.074 −0.046 −0.062 0.058 0.051 0.055
n=400 −0.068 −0.055 −0.049 0.050 0.047 0.048
n=500 −0.075 −0.062 −0.052 0.047 0.044 0.044
n=1000 −0.066 −0.053 −0.065 0.035 0.033 0.035
QC-PM θ=0.25
n=50 0.008 −0.015 0.026 0.163 0.160 0.163
n=100 −0.033 −0.033 −0.053 0.123 0.122 0.123
n=200 −0.057 −0.061 −0.057 0.090 0.089 0.094
n=300 −0.074 −0.033 −0.058 0.079 0.072 0.074
n=400 −0.083 −0.064 −0.054 0.071 0.065 0.062
n=500 −0.066 −0.058 −0.059 0.063 0.058 0.060
n=1000 −0.063 −0.050 −0.072 0.043 0.043 0.048
QC-PM θ=0.5
n=50 0.014 0.020 0.021 0.160 0.150 0.154
n=100 −0.024 −0.028 −0.025 0.118 0.111 0.115
n=200 −0.059 −0.043 −0.069 0.083 0.079 0.084
n=300 −0.066 −0.030 −0.066 0.071 0.064 0.070
n=400 −0.062 −0.050 −0.045 0.060 0.059 0.060
n=500 −0.082 −0.059 −0.051 0.060 0.053 0.055
n=1000 −0.066 −0.052 −0.063 0.043 0.039 0.042
QC-PM θ=0.75
n=50 −0.023 0.018 0.013 0.152 0.161 0.161
n=100 −0.025 −0.044 −0.053 0.128 0.123 0.119
n=200 −0.087 −0.027 −0.084 0.096 0.088 0.093
n=300 −0.072 −0.054 −0.063 0.077 0.074 0.071
n=400 −0.069 −0.048 −0.060 0.068 0.063 0.065
n=500 −0.074 −0.066 −0.055 0.063 0.061 0.058
n=1000 −0.073 −0.057 −0.063 0.047 0.043 0.045
block refers to a model: PLS-PM and QC-PM for θ∈{0.25,0.5,0.75}. It is interesting,
in the case of the path coefficients, the correspondence of the mean and median values
of the estimates with the true value of the parameter (horizontal line at 0.3).
Table 3shows, for each horizontal block, the results obtained from the correspond-
ing model in terms of RBias and RMSE values for the path coefficient estimates, in the
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P. Dolce et al.
Fig. 6 Diminutive distribution charts of the loadings of scenarios belonging to Group 2 with n∈
{50,100,200,300,400,500,1000}, heterogeneous blocks, path coefficients equal to 0.3 and Normal dis-
tributed data
seven scenarios. PLS-PM and QC-PM perform very similar both in terms of bias and
efficiency. In general, as sample size increases, RMSE decreases for all path coefficient
estimates. The RMSE is slightly higher in QC-PM at quantile θ=0.25 and θ=0.75.
The effect of the level of heterogeneity
The convergence of estimates as nincreases is also confirmed in the case of hetero-
geneous blocks. Figure 6shows the diminutive distribution charts of the loadings of
the scenarios belonging to the second group of scenarios, Group 2, still encompassing
normal distributions and with path coefficients equal to 0.3 but with MVs differently
correlated to each construct (0.9, 0.6 and 0.3).
In this group of scenarios, unlike Group 1, the distribution of loadings gets closer
to the true population parameter especially for λ13. However, the heterogeneity of the
blocks has a distortive effect on the estimates of the path coefficients (Fig. 7) both in
case of PLS-PM and QC-PM. Tables 4and 5confirm decreasing bias and variability
as nincreases. In this case but both RBias and RMSE are always higher than the
homogeneous case.
The effect of the degree of skewness/kurtosis
The third group of scenarios, Group 3, worth to be mentioned aims to show the effect
of an asymmetric distribution in the data generation process. Also this group includes
seven scenarios (varying the sample size), with homogeneous loadings (equal to 0.9)
and path coefficients equal to 0.3. Data are here generated by an Exponential distribu-
tion. Convergence is confirmed for this group as well, but there are differences between
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Fig. 7 Diminutive distribution charts of the path coefficients of scenarios belonging to Group 2 with
n∈{50,100,200,300,400,500,1000}, heterogeneous blocks, path coefficients equal to 0.3 and Normal
distributed data
Table 4 Values for the RBias and RMSE for the loadings (averages for each block) of the scenarios belonging
to Group 2 with n∈{50,100,200,300,400,500,1000}, heterogeneous blocks, path coefficients equal to
0.3 and Normal distributed data (the levels of the varying factor, the sample size, are in the first column)
RBias RMSE
Block X1 Block X2 Block X3 Block X1 Block X2 Block X3
PLS-PM
n=50 0.307 0.295 0.282 0.208 0.205 0.186
n=100 0.265 0.282 0.262 0.160 0.164 0.153
n=200 0.274 0.252 0.272 0.141 0.138 0.138
n=300 0.274 0.274 0.279 0.134 0.133 0.133
n=400 0.272 0.273 0.279 0.130 0.129 0.129
n=500 0.282 0.275 0.281 0.129 0.129 0.128
n=1000 0.280 0.283 0.284 0.124 0.125 0.124
QC-PM
n=50 0.304 0.297 0.296 0.248 0.247 0.230
n=100 0.258 0.277 0.256 0.188 0.193 0.178
n=200 0.270 0.244 0.261 0.155 0.153 0.151
n=300 0.266 0.271 0.272 0.146 0.145 0.144
n=400 0.267 0.268 0.275 0.138 0.138 0.137
n=500 0.276 0.277 0.276 0.136 0.136 0.134
n=1000 0.277 0.282 0.283 0.128 0.129 0.128
For each scenario, results for PLS-PM and QC-PM estimated at θ=0.5areshown
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Table 5 Path coefficients RBias and RMSE values of scenarios belonging to Group 2 with
n=[50,100,200,300,400,500,1000], heterogeneous blocks, path coefficients equal to 0.3 and Nor-
mal distributed data (the levels of the varying factor, the sample size, are in the first column)
RBias RMSE
β21 β31 β32 β21 β31 β32
PLS-PM
n=50 −0.015 −0.064 −0.061 0.117 0.117 0.120
n=100 −0.124 −0.130 −0.131 0.095 0.095 0.100
n=200 −0.186 −0.145 −0.167 0.084 0.075 0.080
n=300 −0.186 −0.191 −0.170 0.075 0.076 0.073
n=400 −0.213 −0.176 −0.180 0.078 0.070 0.070
n=500 −0.207 −0.188 −0.170 0.074 0.069 0.065
n=1000 −0.217 −0.172 −0.183 0.070 0.059 0.061
QC-PM θ=0.25
n=50 −0.009 0.010 −0.047 0.167 0.167 0.165
n=100 −0.137 −0.120 −0.119 0.129 0.128 0.133
n=200 −0.197 −0.145 −0.166 0.111 0.098 0.104
n=300 −0.183 −0.194 −0.182 0.092 0.092 0.091
n=400 −0.226 −0.183 −0.174 0.092 0.084 0.081
n=500 −0.220 −0.198 −0.169 0.087 0.080 0.076
n=1000 −0.218 −0.175 −0.183 0.076 0.065 0.067
QC-PM θ=0.5
n=50 −0.012 0.019 −0.063 0.148 0.158 0.156
n=100 −0.108 −0.097 −0.131 0.122 0.126 0.128
n=200 −0.190 −0.142 −0.164 0.100 0.091 0.096
n=300 −0.179 −0.194 −0.171 0.085 0.088 0.085
n=400 −0.216 −0.178 −0.183 0.087 0.081 0.080
n=500 −0.210 −0.196 −0.166 0.082 0.080 0.073
n=1000 −0.213 −0.173 −0.182 0.074 0.063 0.065
QC-PM θ=0.75
n=50 −0.049 −0.004 −0.082 0.161 0.162 0.167
n=100 −0.133 −0.133 −0.149 0.128 0.131 0.133
n=200 −0.196 −0.152 −0.186 0.106 0.100 0.103
n=3000 −0.194 −0.198 −0.175 0.095 0.095 0.092
n=4000 −0.219 −0.174 −0.185 0.092 0.083 0.083
n=500 −0.211 −0.182 −0.168 0.085 0.078 0.076
n=1000 −0.219 −0.175 −0.187 0.077 0.067 0.068
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Fig. 8 Diminutive distribution charts of the loadings of scenarios belonging to Group 3 with n∈
{50,100,200,300,400,500,1000}, homogeneous blocks, path coefficients equal to 0.3 and exponential
distributed data
the PLS-PM and QC-PM. As regards loadings (Fig. 8), the distributions are always
more variable in QC-PM than in PLS-PM. Nevertheless QC-PM always manages to
capture the true parameter value within the 90% of central values. Considering the
average values of RBias and RMSE in all blocks (Table 6), better performance of QC-
PM in terms of efficiency and unbiasedness is confirmed especially for larger sample
sizes. Looking at the distribution of path coefficient estimates (Fig. 9), we note the
ability of QC-PM to capture the positive skewness of the distribution used to generate
the data: the variability of the estimates is smaller for θ= 0.25 and larger in the right
tail (θ= 0.75) and the parameter is overestimated at the lowest quantile and underes-
timated at the highest quantile. For θ= 0.5 both methods provide unbiased estimates.
The RBias values in Table 7confirm the reduction in the bias as nincreases. More
complex is the interpretation of the RMSE values, which combine bias and variability:
the estimates at quantile 0.25, for example, are more biased but less variable than those
at quantile 0.5, so the RMSE is affected by a kind of trade-off between variability and
bias.
5 An application on Chronic Kidney Disease in diabetic patients
QC-PM potentialities are described through an artificial dataset which simulates a
study on CKD in diabetic patients. The original study was proposed by Wang et al.
(2016) who used real data to examine the potential risk factors of CKD through a
quantile approach to factor-based SEM. In particual, data were generated mimicking
the model and estimates obtained by Wang et al. (2016), since the original data were
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Fig. 9 Diminutive distribution charts of the path coefficients of Group 3 of scenarios belonging to
Group 3 with n∈{50,100,200,300,400,500,1000}, homogeneous blocks, path coefficients equal to
0.3 and exponential distributed data
Table 6 Values of RBias and RMSE for the loading (averages for each block) of the scenarios belonging to
Group 3 with n∈{50,100,200,300,400,500,1000}, homogeneous blocks, path coefficients equal to 0.3
and Exponential distributed data (the levels of the varying factor, the sample size, are in the first column)
BIAS RMSE
Block X1 Block X2 Block X3 Block X1 Block X2 Block X3
PLS-PM
n=50 0.033 0.034 0.034 0.051 0.047 0.045
n=100 0.037 0.039 0.037 0.039 0.040 0.039
n=200 0.038 0.038 0.037 0.037 0.037 0.036
n=300 0.038 0.038 0.038 0.036 0.036 0.036
n=400 0.038 0.038 0.038 0.036 0.036 0.036
n=500 0.039 0.038 0.038 0.036 0.035 0.035
n=1000 0.039 0.038 0.038 0.035 0.035 0.035
QC-PM
n=50 0.032 0.031 0.032 0.081 0.086 0.085
n=100 0.030 0.030 0.032 0.058 0.058 0.059
n=200 0.029 0.029 0.029 0.047 0.046 0.046
n=300 0.029 0.030 0.030 0.040 0.040 0.041
n=400 0.029 0.029 0.030 0.037 0.037 0.037
n=500 0.031 0.029 0.030 0.036 0.035 0.036
n=1000 0.030 0.029 0.029 0.032 0.031 0.031
For each scenario, results for PLS-PM and QC-PM estimated at θ=0.5areshown
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Table 7 Values of RBias and RMSE for the Path coefficients of the scenarios belonging to Group 3 with n∈
{50,100,200,300,400,500,1000}, homogeneous blocks, path coefficients equal to 0.3 and Exponential
distributed data (the levels of the varying factor, the sample size, are in the first column)
BIAS RMSE
β21 β31 β32 β21 β31 β32
PLS-PM
n=50 −0.007 −0.032 0.012 0.139 0.148 0.144
n=100 −0.035 −0.028 −0.061 0.108 0.111 0.114
n=200 −0.073 −0.040 −0.057 0.080 0.079 0.082
n=300 −0.055 −0.054 −0.067 0.066 0.067 0.069
n=400 −0.070 −0.054 −0.045 0.058 0.057 0.058
n=500 −0.055 −0.054 −0.055 0.053 0.053 0.053
n=1000 −0.070 −0.055 −0.051 0.041 0.039 0.038
QC-PM θ=0.25
n=50 −0.349 −0.376 −0.348 0.163 0.163 0.160
n=100 −0.406 −0.403 −0.395 0.150 0.149 0.150
n=200 −0.430 −0.432 −0.410 0.144 0.144 0.139
n=300 −0.436 −0.430 −0.426 0.141 0.140 0.138
n=400 −0.443 −0.430 −0.435 0.139 0.136 0.137
n=50 −0.428 −0.435 −0.437 0.134 0.137 0.136
n=1000 −0.449 −0.436 −0.429 0.137 0.134 0.132
QC-PM θ=0.50
n=50 0.039 −0.001 0.049 0.175 0.169 0.164
n=100 −0.010 −0.013 −0.052 0.133 0.124 0.129
n=200 −0.045 −0.031 −0.048 0.092 0.085 0.088
n=300 −0.035 −0.065 −0.064 0.080 0.075 0.076
n=400 −0.067 −0.055 −0.059 0.066 0.065 0.065
n=500 −0.048 −0.052 −0.062 0.062 0.059 0.061
n=1000 −0.071 −0.055 −0.059 0.047 0.042 0.042
QC-PM θ=0.75
n=50 0.446 0.303 0.401 0.277 0.247 0.273
n=100 0.358 0.349 0.273 0.217 0.210 0.193
n=200 0.351 0.347 0.300 0.171 0.169 0.159
n=300 0.363 0.322 0.295 0.154 0.146 0.143
n=400 0.335 0.307 0.329 0.140 0.130 0.133
n=500 0.370 0.327 0.304 0.139 0.128 0.124
n=1000 0.357 0.330 0.315 0.124 0.115 0.111
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Fig. 10 Theoretical model for Chronic Kidney Disease data, following Wang et al. (2016)
not available. Even if we use artificial data, the involved variables and their relations
are in line with the study by Wang et al., allowing a clear practical interpretation of
results. Both studies are quantile based, even if Wang et al. exploited factor-based
SEM while we focus on composite-based path modeling.
The main objective of this section is to show QC-PM in action, stressing its comple-
mentarity with the traditional methods for composite-based path modeling (PLS-PM),
which focus only on conditional means. The advantage of using artificial data allows
us to obtain a scenario where relationships among variables change with quantiles
(i.e., there are different relations considering the different parts of the dependent vari-
able distributions). Our main objective was not recovering parameters, but evaluate if
QC-PM is able to detect this heterogeneity in the variable relationships.
5.1 Data description
This application aims to study the effect of some risk factors on CKD. We started
from the original path model in Wang et al. (2016) and removed the non significant
predictors. In particular, the study investigates Type 2 diabetic patients who might have
experienced CKD. Data consist of 300 patients. Diagnosis and staging of CKD were
based on urinary albumin-creatinine ratio (ACR) and estimated glomerular filtration
rate (eGFR). These two variables are the MVs of the outcome block named Kidney
disease. The considered risk factors were Blood pressure and Lipid. The former was
measured by systolic blood pressure (SBP) and diastolic blood pressure (DBP), while
the latter by total cholesterol (TC), high-density lipoprotein (HDL), and triglycerides
(TG). Therefore, the inner model underpinning our design and subsequent analyses
consists of two exogenous constructs, Blood pressure (ξ1) and Lipid (ξ2), and one
endogenous construct, Kidney disease (ξ3). Figure 10 depicts the corresponding path
diagram.
Data generation process exploits the classical covariance-based approach for SEM.
As above mentioned, the results of the original study by Wang et al. (2016) represent the
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Fig. 11 Theoretical models for the simulated data: patients with low (a) and high (b) severity of kidney
disease
starting point for the generation of artificial data. Therefore, the parameters of the SEM
are set to the values of the model estimated in that study. In particular, Blood pressure
was positively correlated with the severity of Kidney disease, and the correlation was
stronger for higher quantiles. Lipid was found to be positively correlated with Kidney
disease and, also for this variable, the correlation was stronger for higher quantiles.
The resulting variance–covariance matrix characterizes the multivariate distribution
used to generate data. The generation process was carried out using the software EQS
6.1 (Bentler 2006), computation and analysis using R (R Core Team 2020).
Heterogeneity in the inner model was introduced in the artificial data assuming that
the exogenous constructs exert a differenteffect on the different parts of the endogenous
construct distribution. It results that the path coefficients differ across quantiles. In
order to generate data with these features, we supposed that two different populations
exist, and for each population the model parameters are different. In particular we
divided the patients in two groups. The first group was represented by patients with low
severity of kidney disease, and thus the relationship between kidney disease and each of
the two exogenous constructs is weaker (Fig. 11a). The second group was represented
by patients with high severity of kidney disease: in such a case, the relationship between
kidney disease and each exogenous constructs is stronger (Fig. 11b). In order to focus
only on heterogeneity in the inner model, as in Wang et al. (2016), the loadings between
constructs and the corresponding MVs were set all equal to 1 for both the populations.
The simulation procedure was articulated in the following three steps:
1. data were generated from a multivariate normal population, X∼N(0,), where
is the population covariance matrix using the values in Fig. 11a for the parameters
of the model. The sample size was set equal to 300. For each of the two MVs of
Kidney disease block (ACR and eGFR), we removed the observations higher than
the quantile 0.6 of the same MV. In other words, once the observations were sorted
in non decreasing order with respect to the values on each MV, we kept the first 60%
of observations, i.e. the first 180 observations. Then, the MVs of the endogenous
block were transformed in order to have realistic values ranging from 1 to 6 for
ACR, and from 50 to 90 for eGFR;
2. data were generated from a multivariate normal population, X∼N(0,), where
is the population covariance matrix using the values in Fig. 11b for the param-
eters of the model. The sample size was set again equal to 300. For each of the
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Table 8 Check for block unidimensionality and internal consistency
MVs Eig. 1st Eig. 2nd C.alpha DH.rho
Blood pressure 2 1.73 0.266 0.847 0.99
Lipid 3 2.34 0.359 0.860 0.99
CKD 2 1.79 0.205 0.886 0.99
two MVs of the Kidney disease block, we kept the 40% of central observations
around the MV mean (100 units), namely 20% on the left-neighborhood of the
mean and the other 20% on its right-neighborhood. The two resultings MVs were
transformed so to have values ranging from 6 to 10 for ACR, and from 90 to 120
for eGFR;
3. the two data sets generated at the previous steps were stacked obtaining an unique
data set with sample size 300. Note that the MVs of the exogenous blocks in the
two models come from the same population, while obviously the same does not
hold for the MVs of the endogenous block.
According to such data generation process, we expect that QC-PM provides esti-
mates for the parameters of model (a) for quantiles smaller than 0.6, and estimates for
the parameters of model (b) for quantiles larger than 0.6.
5.2 Results
This section describes a complete application of QC-PM, from the preliminary analysis
to the evaluation of the goodness of fit. The aim is to illustrate the potential of the
method along with the guidelines for the interpretation of the results.
An initial inspection of unidimensionality and internal consistency of blocks was
performed. To check unidimensionality, we carried out a principal component analysis
for each block of MVs. If a block is unidimensional, the first eigenvalue is expected
to be the only one greater than 1 and much higher than the second one.
The internal consistency of each block of MVs was instead evaluated through the
Cronbach’s α. Such index assumes equal population covariances among the indicators
of one block, and such assumption is likely not met in empirical research. However, this
index can be used as a lower bound for reliability (Benitez et al. 2020). We also consider
Dijkstra–Henseler’s ρ(Dijkstra and Henseler 2015) to evaluate composite reliability.
Table 8shows that all the blocks are unidimensional and internally consistent. The
method used to obtain the artificial data in Sect. 5provides equal loadings and therefore
the values of Dijkstra–Henseler’s ρ(DH.rho) are all equal to 0.99.
Model parameters were estimated through PLS-PM and QC-PM by setting the
quantile in the iterative procedure to the median and considering a dense grid of
quantiles in the inner model. The two panels in Fig. 12 show the different QC-PM
path coefficient estimates across quantiles and the PLS-PM path coefficient estimates:
Fig. 12a depicts the path coefficient connecting Blood pressure to Kidney disease,
while Fig. 12b refers to Lipid. In particular, quantiles are represented on the horizontal
axis and coefficients on the vertical axis. The horizontal solid lines represent the PLS-
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0.20.40.60.8
0.1 0.2 0.3 0.4 0.5 0.6
Blood pressure
quantile
path coefficients
(a)
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0.2 0.4 0.6 0.8
0.1 0.2 0.3 0.4 0.5 0.6
Lipid
quantile
path coefficients
(b)
Fig. 12 Path coefficient estimates (y-axis) across quantiles (x-axis) linking Blood pressure (a)andLipid
(b)toKidney disease. The horizontal solid lines represent the PLS-PM estimates and the vertical dotted
line drawn at quantile 0.6 refer to the threshold used in the data generation process
Table 9 Path coefficients and
corresponding standard error
from a classical PLS-PM (first
row) and from a QC-PM applied
on the inner model for a selected
set of quantiles
(θ∈{0.25,0.50,0.75})
Quantile Blood pressure Lipid
(±standard error) (±standard error)
PLSPM 0.320* (±0.051) 0.426* (±0.048)
0.25 0.189* (±0.062) 0.263* (±0.096)
0.50 0.291* (±0.090) 0.547* (±0.101)
0.75 0.416* (±0.051) 0.588* (±0.053)
*p<0.05
PM estimates while the broken lines represent the QC-PM estimates over quantiles.
The vertical dotted line drawn at quantile 0.6 in each figure refers to the threshold used
in the data generation process (we expect that for quantiles smaller than 0.6, QC-PM
produces estimates for the parameters specified in the model shown in the Fig. 11a,
while for quantiles larger than 0.6, QC-PM produces estimates for the parameters
specified in the model shown in the Fig. 11b). Figure 12 shows the ability of the QC-
PM to detect the structure underlying the simulated data (Fig. 11). QC-PM was indeed
able to distinguish the different effects in the different parts of the Kidney disease
distribution: both path coefficients increase with quantiles and results are consistent
with the true values specified in the population models shown in Fig. 11. Table 9reports
the path coefficient estimates (±standard errors) obtained using PLS-PM (first row)
and QC-PM for the quantiles θ∈{0.25,0.50,0.75}. Standard errors of estimates were
obtained using bootstrap. All path coefficients were statistically significant. On the
whole, except for coefficients at θ=0.75, PLS-PM estimates are slightly more efficient
than QC-PM ones. This is in line with theory: just like mean is more efficient than
median, OLS regression estimates are usually more efficient than QR estimates. Both
Blood pressure and Lipid have a positive impact on Kidney disease, which increases for
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patients with higher levels of CKD. This positive and increasing effect is well-known
in literature: hypertension, high presence of cholesterol, lipoprotein and triglycerides
are all considered leading causes of CKD (Bakris and Ritz 2009).
The assessment of the models is carried out using the measures introduced in Sect.
3. It is worth to recall again that a direct comparison of the measures of fit of the two
methods is not appropriate, since the two methods optimize different criteria. Hence,
the objective is neither to compare PLS-PM and QC-PM results nor to identify the
best model. Instead, we aim to illustrate how to use the measures define above for the
assessment of QC-PM results.
With respect to the inner model, PLS-PM produces an R2equal to 0.214, while
QC-PM provides pseudo−R2values increasing from lower to higher quantiles (0.051,
0.123, 0.185), for the quantiles θ∈{0.25,0.50,0.75}. This result was expected and
coherent with the data structure, as relationships among constructs increase with quan-
tiles. The assessment of the outer model is carried out in two steps. Table 10 shows,
for each block, the communality values related to each MV and to the whole block
(in bold) both for PLS-PM and QC-PM. For the latter, obviously, estimates refer only
to the median because, as specified in Sect. 2.3, quantiles are allowed to vary only in
the inner model. Overall, the communality of blocks is satisfactory. From the average
communality of each block (last row in each block—values in bold), each construct
explains much of the variability of its own MVs. Considering the individual commu-
nality of each MV, we did not find much differences, coherently with the way data
were generated (i.e., all loadings are equal). The global communalities are satisfactory
showing a good fit of the outer model.
Finally, Redundancy values are reported in Table 11, PLS-PM on the first column
and QC-PM on the subsequent columns. Results reveal a low ability of predictor
constructs to explain the variability of the outcome MVs for low quantiles, while
redundancies achieved almost moderate levels for high quantiles and for PLS-PM
(Latan and Ramli 2013; Latan and Ghozali 2015).
Table 10 Communalities for
PLS-PM and QC-PM (θ= 0.50) Construct MV PLSPM QCPM at θ=0.5
Blood pressure SBP 0.907 0.575
DBP 0.821 0.683
0.864 0.629
Lipid TC 0.780 0.490
HDL 0.809 0.584
TG 0.745 0.534
0.778 0.536
CKD ACR 0.903 0.713
eGFR 0.892 0.692
0.897 0.703
Global 0.846 0.624
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Table 11 Redundancy measures
for PLS-PM and QC-PM applied
on a selected set of quantiles
(θ∈{0.25,0.50,0.75})
QC-PM
PLS-PM 0.25 0.5 0.75
ACR 0.193 0.036 0.088 0.132
eGFR 0.191 0.035 0.85 0.128
CKD 0.192 0.036 0.086 0.130
6 Conclusions and insights for future works
The original proposal of QC-PM was presented for the first time at the 8th Interna-
tional Symposium on PLS and Related Methods (PLS’14) which took place in 2014
in Paris (www.pls14.org). The method was proposed with the aim to extend classical
least squares methods for conditional mean to the estimation of conditional quantile
functions in the context of composite-based path modeling. QC-PM complements the
well-known and consolidated PLS-PM by exploring heterogeneous effects of explana-
tory constructs over the entire conditional distributions of the response constructs.
The present paper has formalized QC-PM and the iterative procedure for parameter
estimation, starting from the simplest case of one block of MVs and moving until the
general path model for multi-block data. In addition, a methodological variation in the
estimation phase of the outer model is also proposed. The applicative potentialities of
QC-PM, along with guidelines for the interpretation of results, were provided through
the analysis of an artificial data set on CKD in diabetic patients. The example highlights
how QC-PM can complement traditional methods for composite-based path modeling
in presence of heterogeneity in the relationships among variables. The properties of
the method across different scenarios were investigated through a simulation study.
The simulation design took into account the factors that typically affect the results
of composite-based path modeling methods: sample size, strength of the relationship
within the blocks (homogeneous vs heterogeneous blocks), different levels of correla-
tions between constructs and shape of distributions in the outcome blocks. Data were
generated from composite-based populations. The comparison among the different
scenarios was carried out in terms of RBias and RMSE of estimates obtained from
500 replications for each scenario. Several similarities between QC-PM and PLS-PM
emerged comparing the performance of the two methods in all generated scenarios.
Nevertheless, some differences were identified. However, it is worth to recall that the
spirit of the simulation study is to show the properties of QC-PM rather than to pro-
vide a comparison with PLS-PM. In fact, QC-PM and PLS-PM are not alternative but
complementary methods.
Simulations point out similar results for QC-PM and PLS-PM, both in terms of
bias and RMSE. This confirms our insight to consider QC-PM as a supplementary
method to PLS-PM, with similar features but able to assess relationships between
variables in different parts of the distribution. However, it is noted that variability of the
QC-PM estimates is always greater even though the bias is smaller (the true population
parameter is always within the 90% range of the central values for large samples). Even
if the convergence of estimates is confirmed as the sample size increases in the case of
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heterogeneous blocks, both RBias and RMSE are always higher than the homogeneous
case. The new element that emerges in the case of an asymmetric distribution is the
ability of QC-PM to capture the positive skewness of the distribution used to generate
the data. The variability of the path coefficient estimates is smaller for θ= 0.25 and
larger in the right tail (θ= 0.75) and the parameter is overestimated at the lowest
quantile and underestimated at the highest quantile.
From a methodological point of view, a promising extension of QC-PM will accom-
modate the case of observed or unobserved heterogeneity among observations. In the
PLS-PM literature several contributions allows to treat both kind of heterogeneity
(Sarstedt et al. 2016,2011b; Lamberti et al. 2016; Sarstedt et al. 2011a; Esposito
Vinzi et al. 2008). In the QR literature, Davino and Vistocco (2018) proposed an
innovative approach to identify group effects through a quantile regression model.
Future studies will be devoted to combine these approaches into QC-PM. Moreover,
since a recent work by Davino et al. (2020) exploited the ability of QC-PM for in-
sample prediction, future research will further evaluate the proposed approach from
an out-of-sample prediction perspective.
A further development, albeit a minor one, will consider the implementation of
another way of calculating outer weights based on a measure of quantile correlation.
Several contributions in the literature extends the first proposal of quantile correlation
(Li et al. 2014) introducing different alternatives to measure the linear correlation
between any two random variables for a given quantile (Tang et al. 2021;Xuetal.
2020). The introduction of a descriptive measure such as quantile correlation into
the process of calculating outer weights would have an interesting computational
advantage over traditional modes requiring the estimation of regression models.
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CARE Agreement.
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Appendix
Quantile Regression was proposed by Koenker and Basset (1978) as a complementary
and robust approach to classical regression analysis. In their seminal paper, the authors
remind that “in statistical parlance the term robustness has come to connote a certain
resilience of statistical procedures to deviations from the assumptions of hypothetical
models” and that the need for robust statistics alternative to least squares estimation
dates back to the nineteenth century.
Just one year earlier, Mosteller and Tukey (1977) sensed the need to identify more
robust regression methods by stating that “What the regression curve does is give a
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Quantile composite-based path modeling
7 8 9 10 11 12 13 14
1000 2000 3000 4000 5000 6000
x
y
Slope
quantiles
coefficients
250 350 450 550
0.1 0.25 0.5 0.75 0.9
Fig. 13 Left-hand side: Scatterplot with simulated data, ordinary least square regression line (solid line)
and quantile lines (dashed lines). Right-hand side: representation of QR slopes, dashed horizontal line
corresponding to OLS slope
grand summary for the averages of the distributions corresponding to the set of X’s.
We could go further and compute several different regression curves corresponding to
the various percentage points of the distributions and thus get a more complete picture
of the set. Ordinarily this is not done, and so regression often gives a rather incomplete
picture. Just as the mean gives an incomplete picture of a single distribution, so the
regression curve gives a correspondingly incomplete picture for a set of distributions”.
Quantile Regression (QR) was introduced precisely for the purpose of going beyond
the study of average effects in a regression model and to provide a description of the
whole conditional distribution of a response variable in terms of a set of regressors. QR
can be exploited in case of location, scale and shape shifts on the dependent variable but
also when a monotone transformation of the response and/or the explanatory variables
is advisable.
In order to show QR in action, we introduce an example dealing with a sample of
n=10,000 observations generated from the following model:
y=1+2x+(1+x)e
x∼N(10;1)
e∼N(−1+20x);ex/3.
The scatterplot in Fig. 13 (left-hand side) shows the typical fan-like shape, sign that
the amount of variability over the expected value of the dependent variable for a given
value of the explanatory variable xis not the same at every level of x, but varies
systematically with the level of x.
Estimating a least-squares regression model (solid line in Fig. 13) would not fully
capture the relationship between the two variables. In such a case, it is more interest-
ing to explore the effect of the regressor at different parts of the distribution of the
dependent variable.
123
P. Dolce et al.
A QR model for a given conditional quantile and for the ith observation can be
formulated as follows:
yi=xiβ(θ) +i(θ )
Qθ(ˆ
y|X)=Xˆ
β(θ)
where Xis the regressor matrix and xia row of this matrix, ythe vector containing the
dependent variable, 0 <θ <1 is a generic quantile, Qθ(.|.) denotes the conditional
quantile function for the θth quantile and is the error term such that Qθ((θ )|X)=0.
QR offers a complete view of a response variable providing a method for mod-
elling the rates of changes at multiple points (conditional quantiles) of its distribution
(Koenker 2005; Davino et al. 2013) without requiring assumptions on the errors.
Although different functional forms can be used, we will deal here only with simple
linear regression models.
For each quantile, a regression line is estimated and, as a consequence, the estimated
values of the response variable conditioned to given values of the regressors, provides
the conditional quantiles of the dependent variable.
Going back to the example with simulated data, five QR lines have been estimated
considering the following set of θvalues: 0.1, 0.25, 0.5, 0.75, 0.9. The dashed lines in
Fig. 13 (left-hand side) are obtained by estimating the effect of xon the selectd con-
ditional quantiles of y. They confirm the positive impact of the regressor but showing
that this contribution increases as we move on the distribution of the yvariable.
In Fig. 13 (right-hand side), the slopes of the estimated quantile lines are graphically
represented. The horizontal axis displays the different quantiles while the effect of the
regressor is represented on the vertical axis. The dashed line parallel to the horizontal
axis corresponds to the ordinary least squares coefficient. This graphical representation
allows to visually catch the different effect of the regressor on the yvariable.
The parameter estimates in QR linear models have the same interpretation as those
of any other linear model. The intercept measures the value of the dependent variable
setting to zero all the regressors. Each slope coefficient ˆ
βi(θ) =∂Qθ(y|X)
∂xican be
interpreted as the rate of change of the θth conditional quantile of the dependent
variable per unit change in the value of the ith regressor, holding constant the other
regressors.
The conditional quantile estimator is obtained as a generalisation of the uncondi-
tional quantile estimator:
ˆ
β(θ) =argmin
β
Eρθ(y−Xβ)(21)
where ρθ(.) is an asymmetric absolute loss function which uses an unbalanced weight-
ing system (weight equal to (1-θ) for the sum of negative deviations and weight equal
to θfor the sum of positive deviations).
The most widespread algorithm for finding QR estimates is the one proposed in
Koenker and d’Orey (1987) as a variant of the Barrodale and Roberts (1974)simplex
algorithm. Although it is theoretically possible to extract infinite quantiles, a finite
number is numerically distinct in practice. This is known as quantile process. A fairly
123
Quantile composite-based path modeling
accurate approximation of the whole quantile process can be obtained using a dense
grid of equally spaced quantiles in the unit interval (0, 1) (Davino et al. 2013).
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