Quantile composite-based path modeling: algorithms, properties and applications

The simplest model involves one block of MVs and a construct. The relationships between them are depicted in Fig. 1. MVs are denoted with \(\mathbf{X = \{x_{ip}\}}\) and represented through rectangles. It is worth to recall that \(i = 1, \ldots , n\) refers to the observations, n denoting their number, and \(p = 1, \ldots , P\) refers to the MVs, P denoting the number of MVs in each block. The corresponding construct is labeled with \(\varvec{\xi } = \{\xi _{i}\}\) 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 \(\theta \in (0,1)\), the link between each MV \(\mathbf{x} _{p}\) and \(\xi \) is defined through the following equation:where \(\alpha _{p}\) is a location parameter, \(\lambda _{p}\) is the loading coefficient, capturing the effect of \(\varvec{\xi }\) on \(\mathbf{x} _{p}\), and \(\varvec{\epsilon }_p=\{\epsilon _{ip}\}\) is the error term vector. The only assumption is that the generic \(\theta \)th conditional quantile of \(\mathbf{x} _{p}\) can be expressed as:which implies that the \(\theta \)th quantile of the error term \(\varvec{\epsilon }_p\left( \theta \right) \) is equal to zero and \(\varvec{\epsilon }_p\left( \theta \right) \) is independent of \(\varvec{\xi }\). No assumptions on the error distribution are required.

The model captures the common variation among the MVs in \(\mathbf{X} \) that depends on the construct \(\varvec{\xi }\). The use of QR allows to encompass the effect of \(\varvec{\xi }\) on the whole conditional distribution of each \({\mathbf {x}}_{p}\), 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 \(\hat{\mathbf{w }}_{p}\), namely the coefficients used to compute the composite \(\varvec{\hat{\xi }}\), 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 \(\mathbf{X} \), \(\varvec{\hat{\xi }}^{(0)}\!\!= \sum _{p=1}^{P}{{\hat{w}}}^{(0)}_p\mathbf{x} _{p}\). Then, a loop starts and at each sth iteration \((s=0,1,2, \ldots )\), each MV \(\mathbf{x} _{p}\) is regressed on the composite, minimizing the following quantile loss function: where \(\rho _{\theta } \left( .\right) \) is the check function, which asymmetrically weights positive and negative residuals, namely: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 \(\varvec{\hat{\xi }}\), obtained as a linear combination of the response MVs \(\mathbf{x} _{p}\).

By weighting the MVs considering their quantile covariation with the construct, for each quantile \(\theta \) the 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.

Figure 2 depicts the path diagram for an hypothetical two-block model. In such a case, let \(\mathbf{X} =\{x_{ip}\}\) and \(\mathbf{Y} = \{y_{ij}\}\) 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 P the number of explanatory MVs, as in the previous subsection, and with J the number of response MVs, \({\varvec{y}}_{j}\) being the generic response MV. Moreover, \(\varvec{\xi }=\{\xi _{i}\}\) is the construct representing the explanatory block, and \(\varvec{\eta }=\{\eta _{i}\}\) 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 \(\theta \in (0,1)\), the MVs \(\mathbf{x} _{p}\) in the explanatory block, and the MVs \(\mathbf{y} _{j}\) in the dependent block, are related to their correspondent constructs through the following system of bilinear equations:where \(\alpha _{xp}\) and \(\alpha _{yj}\) are location parameters, \(\lambda _{xp}\) is the loading coefficient capturing the effect of \(\varvec{\xi }\) on \(\mathbf{x} _{p}\), \(\lambda _{yj}\) is the loading coefficient capturing the effect of \(\varvec{\eta }\) on \(\mathbf{y} _{j}\), while \(\varvec{\epsilon }=\{\epsilon _{ip}\}\) and \(\varvec{\omega }=\{\omega _{ij}\}\) 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 \(\varvec{\eta }\) is linked to the explanatory construct \(\varvec{\xi }\) by the following model:where \(\beta _1\) is the so-called path coefficient capturing the effects of \(\varvec{\xi }\) on the dependent construct \(\varvec{\eta }\), and \(\varvec{\zeta }=\{\zeta _{i}\}\) 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.

In the initialization step of the algorithm, arbitrary values are set for the outer weights \(\hat{\mathbf{w }}_{xp}\) to compute a first approximation of the composite as a linear combination of the MVs in \(\mathbf{X} \), \(\varvec{\hat{\xi }}^{(0)}=\sum _{p=1}^{P}{{\hat{w}}}^{(0)}_{xp}{} \mathbf{x} _{p}\). Then, the iterative algorithm step proceeds over two phases. At each sth iteration \((s=0,1,2,\ldots )\), the response MVs \(\mathbf{y} _{j}\) are regressed on the approximation of the composite \(\varvec{\hat{\xi }}^{(s-1)}\), minimizing the following quantile loss function:where \(\rho _{\theta } \left( .\right) \) is the check function defined as above.

In the second phase, the estimated \(\hat{w}^{(s)}_{yj}\left( \theta \right) \), for \(j = 1, \ldots , J\), are used to compute the composite \(\varvec{\hat{\eta }^{(s)}}\) through a linear combination of the response MVs \(\mathbf{y} _{j}\), and then the explanatory MVs \(\mathbf{x} _{xp}\), are regressed on the obtained linear combination, minimizing the following quantile loss function:Finally, an updated approximation of the composite \(\varvec{\hat{\xi }}^{(s)}\) is obtained as a linear combination of the explanatory MVs \(\mathbf{x} _{p}\), using the weigths \(\hat{w}^{(s)}_{xp}\left( \theta \right) \).

These two phases are iteratively repeated until convergence of the outer vectors, \({\varvec{w}}_{x}\left( \theta \right) =\{w_{xp}\left( \theta \right) \}\) and \({\varvec{w}}_{y}\left( \theta \right) =\{w_{yj}\left( \theta \right) \}\), is achieved. Then loadings and path coefficients are estimated through quantile regression.

QC-PM algorithm returns, for each quantile \(\theta \), a linear combination of the explanatory MVs \(\mathbf{x} _{xp}\) by weighting the corresponding MVs on the basis of their quantile covariation with the linear combination of the response MVs \(\mathbf{y} _{j}\).

Figure 3 depicts a path model for multi-block data using the case of three blocks. The general model for K blocks follows the same logic. Let us assume that P variables are collected in a table \(\mathbf{X} \) of data partitioned in K blocks: \(\mathbf{X} = [\mathbf{X} _1,\mathbf{X} _2 \ldots , \mathbf{X} _K]\). Let \(\mathbf{X} _k = \{x_{ip_k}\}\) be a generic block of MVs, where \(i = 1, \ldots , n\), with n denoting the number of observations, \(p_k = 1, \ldots , P_k\), with \(P_k\) being the number of MVs in the kth block. We denote by \(\varvec{\xi }_k = \{\xi _{ik}\}\) and \(\mathbf{x} _{p_k} = \{x_{ip_k}\}\) 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 exogenous, while the endogenous constructs play only the role of dependent variables or of both dependent and explanatory variables. In Fig. 3, for example, \(\varvec{\xi }_1\) is an exogenous construct and \(\varvec{\xi }_2\) and \(\varvec{\xi }_3\) are 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 \(\theta \in (0,1)\), in the outer model it is assumed that each MV \(\mathbf{x} _{p_k}\) is related to its own construct through the following equations:where \(\alpha _{p_k}\) is the location parameter, \(\varvec{\xi }_k=\{\xi _{ik}\}\) is the construct representing the kth block of MVs, \(\lambda _{p_k}\) is the loading coefficient, capturing the effect of \(\varvec{\xi }_k\) on \(\mathbf{x} _{p_k}\) and \({\epsilon _k}=\{\epsilon _{ip_k}\}\) is the error term vector, using the usual above mentioned assumption on the errors.

The inner model captures and specifies the dependence relationships among constructs. A generic endogenous construct, \(\varvec{\xi }_{k'}\), is linked to the related explanatory constructs, \({\varvec{\xi }}_{k}\), \(k\in {J_{k'}}\), where \(J_{k'}=\lbrace k: \varvec{\xi }_{k'} \text { is predicted by } \varvec{\xi }_{k} \rbrace \), by:where \(\beta _{k'k}\) is the path coefficient capturing the effects of \(\varvec{\xi _k}\) on the dependent construct \(\varvec{\xi _k'}\), and \(\varvec{\zeta _k'}=\{\zeta _{ik'}\}\) 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.

The weight vector \(\varvec{{w}}\left( \theta \right) =\{{w}_{p_k}\left( \theta \right) \}\), \((p_k=1,\ldots ,P_k; k=1,\ldots ,K)\), 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 \({\varvec{w} (\theta )}\) 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 normalized weighted aggregate of the corresponding MVs. Outer weights are computed through simple quantile regressions, where each MV is regressed on the corresponding inner approximation composite. Then, the weights are normalized so that var\([\mathbf{X} _k\mathbf{w} _k\left( \theta \right) ]=1\). 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 feasible 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 \((s=0,1,2,\ldots )\), the following partial criterion are then optimized:where \(\rho _{\theta } \left( .\right) \) is the check function defined as above.

When Mode B is applied, the following criterion is instead minimized: 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 relationships 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 measurement invariance (Henseler et al. 2016) has to be fulfilled. If weights, and consequently composites, change over quantiles, a proper comparison among path coefficients estimated 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 variant 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.

We operate in the context proposed by Schlittgen et al. (2020), generating data from composite-based populations using the cbsem R package (Schlittgen 2019). Determination 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 represented in Fig. 3 and then we simulated data for the given values of the parameters. The postulated inner model is:The outer model can be instead written as: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 distributions 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.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 \(\times \) 2 loadings \(\times \) 4 path coefficients \(\times \) 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:where S represents the number of replications in the simulation, \(\hat{\theta _s}\) is the estimate for the generic replication, and \(\theta \) is the corresponding population parameter. Instead, RMSE was computed as:Clearly, because \(MSE=Var(\hat{\theta })+bias(\hat{\theta })^2\), RMSE entails information on both bias and variability of the estimates.

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 \(\theta =0.5\) (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 \(\theta \in \{0.25, 0.5, 0.75\}\), but only for \(\beta _{k'k}=0.3\), 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 skewness/kurtosis of the distribution. The resulting scenarios are organized in three groups:

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 (\(\xi _1\)) and Lipid (\(\xi _2\)), and one endogenous construct, Kidney disease (\(\xi _3\)). 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 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 different effect 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: According to such data generation process, we expect that QC-PM provides estimates 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.

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 \(\alpha \). 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 \(\rho \) (Dijkstra and Henseler 2015) to evaluate composite reliability. Table 8 shows that all the blocks are unidimensional and internally consistent. The method used to obtain the artificial data in Sect. 5 provides equal loadings and therefore the values of Dijkstra–Henseler’s \(\rho \) (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-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 9 reports the path coefficient estimates (± standard errors) obtained using PLS-PM (first row) and QC-PM for the quantiles \(\theta \in \{0.25, 0.50, 0.75\}\). Standard errors of estimates were obtained using bootstrap. All path coefficients were statistically significant. On the whole, except for coefficients at \(\theta \)=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 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 \(R^2\) equal to 0.214, while QC-PM provides \(pseudo\!-\!R^2\) values increasing from lower to higher quantiles (0.051, 0.123, 0.185), for the quantiles \(\theta \in \{0.25, 0.50, 0.75\}\). This result was expected and coherent with the data structure, as relationships among constructs increase with quantiles. 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 communality 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).