For a long time LVPLS 1.8 [95] was the only available software for PLS Path Modeling. The DOS-based program includes two different modules
for estimating path models. The LVPLSC method analyzes the covariance matrix of the observed variables, whereas the LVPLSX module is able to process raw data. In order to specify the input file an external editor is necessary. The input specification requires that the program parameters are defined at specific positions in the file. The results are reported in a plain text file. The program offers blindfolding and jackknifing as resam- pling methods in cases where raw data has been analyzed. When analyzing covariance/correlation matrices, resampling techniques cannot be applied [102]. Over the years other PLS path modeling software have been devel- oped.
The list includes SmartPLS [130], XLSTAT-PLSPM [35] in co-operation with Addinsoft France, (http://www.xlstat.com/en/products/xlstat-plspm/) - and the plspm package [146]. SmartPLS and XLSTAT-PLSPM are closed source and plspm is licensed under the General Public License (GPL≥2). All differences in model parameters due to the used software were in line with the predefined tolerance for the outer weights.
SmartPLS. SmartPLS is a stand alone software specialized for PLS path
models. It is built on a Java Eclipse platform making it operating system independent. The model is specified via drag and drop by drawing the structural model for the LVs and by assigning the indicators to the LVs. Data files of various formats can be uploaded. After fitting a model, coeffi- cients are added to the plot. More detailed output is provided in plain text, in the LATEX and HTML formats. The graph representing the model can be exported to PNG. Besides bootstrapping and blindfolding methods it supports the specification of interaction effects. A special feature of Smart- PLS is the finite mixture routine (FIMIX), a method to deal with unobserved heterogeneity [131];[149];[148].
XLSTAT-PLSPM. XLSTAT [1] is a modular statistical software relying on
Microsoft Excel for the input of data and the display of results, but the com- putations are performed using autonomous software components.
XLSTAT-PLSPMis integrated in XLSTAT as a module for the estimation
Department of Mathematics and Statistics of the University of Naples in Italy and Addinsoft in France and implements all the methodological fea- tures and most recent findings of the PLEASURE (Partial LEAst Squares strUctural Relationship Estimation) technology by Esposito Vinzi et al. [35]. Special features of XLSTAT-PLSPM are multi-group comparisons [18] and the REBUS segmentation approach [38] for the treatment of unobserved heterogeneity.
plspm in R. The plspm package implements the PLS method with em-
phasis on structural equation models in R. The fitting method ’plspm.fit’ re- turns a list including all the estimated parameters and almost all the statis- tics associated with PLS path models. The print method gives an overview of the following list elements: the outer model, inner model, scaled LVs, LVs for scaled = FALSE, outer weights, loadings, path coefficients matrix, R2, outer correlations, inner model summary, total effects, unidimension-
ality, goodness-of-fit, bootstrap results (only if activated) and data matrix. For the treatment of observed heterogeneity, pathmox and rebus.pls [145] are provided as a companion package [102].
Some developments in PLS -
PM for the building of
Composite Indicators
3.1
Introduction
The PLS-PM approach has enjoyed increasing popularity as a key multi- variate analysis method in various research disciplines in order to build a system of Composite Indicators. The model allows you to estimate causal relationships, defined according to a theoretical model linking two or more latent complex concepts, each measured through a number of observable indicators. The basic idea is that the complexity inside a system can be studied by taking into account the entirety of the causal relationships among LVs, each measured by several MVs. Nowadays, complex phenomena such as Development, Progress, Poverty, Social Inequality, Welfare and Quality of Life require, to be measured, the combination of different dimensions, to be considered together as the proxy of the phenomenon.This combination can be obtained by applying methodologies based on CIs [100]. As is well known, the main feature of a CI is that it summarizes this type of complex and multidimensional issue.
In building a CI, we are interested in (i) including elementary indicators on a non numerical scale, (ordinal and nominal data); (ii) including some kind
of CI relationship (logical, hierarchical, temporal or spatial); (iii) defining the roles of the EIs (MVs) as mediator and moderator variables; and (iv) defining the roles of the CIs (LVs) in the inner model (mediator and mo- derator LVs). For instance, when computing a CI, it could be interesting to consider demographic variables, such as religion or gender, categorical variables defining states, such as type of government. It would be inte- resting to know what is the role of these variables, is if they have a mo- derator or mediator effect, and if a consideration of these effects change the estimation of the LVs. Moreover, applications of SEMs are usually based on the assumption that the analyzed data stem from a single population, so that a unique global model represents all the observations effectively. However, in many real world applications, this assumption of homogene- ity is unrealistic. In modeling the real world, it is reasonable to expect that different classes showing heterogeneous behaviors may exist in the observed set of units. This is true also in CI frameworks. As a matter of fact, in developing a system of CIs, it is reasonable to suppose that diffe- rent models, i.e. different systems of weightings, should be applied in or- der to take into account differences among the units. Furthermore, in these frameworks also, it is of great importance to obtain clusters of units that are homogenous with regard to the weights to be applied in computing the CIs. For this reason, many improvements, in order to extend the classic al- gorithm of PLS-PM to the treatment of particular data, have been made, in particular to non-metric data, mediator and moderator data and hierarchi- cal data. Furthermore, several clustering techniques have been developed in PLS-PM to look for latent classes.
In the following sections these developments are presented.
Next, a chapter will be included developed on dealing with a hierarchical model.