Fig 1.
Flowchart for the PLS-PM algorithm.
Fig 2.
Structural equation models used in examples 1 and 2.
(a) Structural equation model representing the causes and consequences of customer satisfaction (reduced version of the European Customer Satisfaction Index (ECSI) model). (b) Structural model implemented in the Monte Carlo simulation study.
Fig 3.
Structural model functional form of the relationships implemented in the Monte Carlo simulation study.
Table 1.
Population means and standard deviations of simulated latent variables of example 2 and sample counterparts based on the 10 000 generated observations).
Table 2.
R2 values of the centroid weighting and smooth weighting schemes. R2 values are based on correlations between estimated and predicted factor scores.
Fig 4.
Estimated partial relationships of the reduced ECSI model of Example I, assuming they are all nonlinear and approximated by a cubic regression spline with dimension of the basis K = 10.
The smooth weighting scheme estimated factor scores are represented as •. The solid blue line corresponds to the estimated relationship, and the area in light blue the corresponding 95% bootstrap credible intervals based on 500 replicates. The first derivative of the partial relationship is represented by the dashed blue line.
Fig 5.
Estimated partial relationships of the reduced ECSI model of Example I, assuming only the relationship between Customer Satisfaction and Customer Loyalty is nonlinear and approximated by a cubic regression spline with dimension of the basis K = 10.
The smooth weighting scheme estimated factor scores are represented as •. The solid blue line corresponds to the estimated relationship, and the area in light blue the corresponding 95% bootstrap credible intervals based on 500 replicates. The first derivative of the partial relationship is represented by the dashed blue line.
Fig 6.
Assessment of the PLS-PM and PLSs-PM algorithms convergence.
Each matrix depicts the confusion matrix of Converged and Non-converged iterations, for the same sample, by level of communality and sample size.
Fig 7.
Mean number of iterations to obtain convergence (out of the converged ones), by sample size and communality level.
The points in the plots mark, respectively sample sizes of 75, 100, 150, 250, 300, 500, 750 and 900 units. Althought PLS-PM using the centroid scheme outperforms PLSs-PM when the level of comunality is very low (25%), when the communality is 50% the latter outperforms the former when sample size is greater or equal than 500 units. And when the communality reaches 75% (which is not in general an hard assumption) it performs better when sample size is gretar or equal than 250 units.
Fig 8.
Percentage of samples (out of the converged ones) with with at least one negative outer weight, by sample size and communality level.
The percentage of generated outer weights decreases as both the samples size and the communality increase. However, PLSs-PM scheme outperforms PLS-PM in all cases, and it even generates zero negative outer weigths when the communality is 75% and samples size is greater or equal than 300 units.
Fig 9.
Absolute bias comparison of the five endogenous variables () for the eight different sample sizes (
) and three levels of communality (25%, 50% and 75%).
Fig 10.
Root mean square error comparison of the five endogenous variables () for the eight different sample sizes (
) and three levels of communality (25%, 50% and 75%).
Fig 11.
Estimated relationships between ξ and and
of the simulated data set of example II.
The population factor scores are represented as dots in light gray. The solid black line corresponds to the true relationship. The solid blue line corresponds to the relationship as estimated by PLS-PM algorithm. The solid red line and the light red shaded area corresponds to the relationship as estimated by PLSs-PM algorithm and the corresponding 95% credibility intervals.