# PLS Algorithm

## Abstract

The PLS path modeling method was developed by Wold (1982). In essence, the PLS algorithm is a sequence of regressions in terms of weight vectors. The weight vectors obtained at convergence satisfy fixed point equations (see Dijkstra, 2010, for a general analysis of these equations).

## Description

The PLS path modeling method was developed by Wold (1992) and the PLS algorithm is essentially a sequence of regressions in terms of weight vectors (Henseler et al., 2009). The weight vectors obtained at convergence satisfy fixed point equations (see Dijkstra, 2010, for a general analysis of such equations and ensuing convergence issues). The basic PLS algorithm, as suggested by Lohmöller (1989), includes the following three stages:

Stage 1: Iterative estimation of latent variable scores consists of a 4-steps iterative procedure, which is repeated until convergence has been obtained (or the maximum number of iterations has been reached):

(1) outer approximation of the latent variable scores,
(2) estimation of the inner weights,
(3) inner approximation of the latent variable scores, and
(4) estimation of the outer weights.

Stage 3: Estimation of location parameters.

Hair et al. (2017) and Henseler et al. (2012) provide detailed explanations on how the basic PLS algorithm operates as it is implemented in SmartPLS 3.0 (Ringle et al., 2015).

## PLS Algorithm Settings in SmartPLS

### Weighting Scheme

PLS-SEM allows the user to apply three structural model weighting schemes:

(1) centroid weighting scheme,
(2) factor weighting scheme, and
(3) path weighting scheme (default).

While the results differ little for the alternative weighting schemes, path weighting is the recommended approach. This weighting scheme provides the highest R² value for endogenous latent variables and is generally applicable for all kinds of PLS path model specifications and estimations. Moreover, when the path model includes higher-order constructs (often called second-order models), researchers should usually not use the centroid weighting scheme.

### Maximum Iterations

This parameter represents the maximum number of iterations that will be used for calculating the PLS results. This number should be sufficiently large (e.g., 300 iterations). When checking the PLS-SEM result, one must make sure that the algorithm did not stop because the maximum number of iterations was reached but due to the stop criterion. Note: The selection of 0 for the maximum number of iterations allows you to obtain results of the sum scores approach.

### Stop Criterion

The PLS algorithm stops when the change in the outer weights between two consecutive iterations is smaller than this stop criterion value (or the maximum number of iterations is reached). This value should be sufficiently small (e.g., 10^-5^ or 10^-7^).

### Initial Outer Weights

As the default (i.e., the SmartPLS settings), the initial outer weights are set to +1. However, the following alternatives are available:

• Lohmöller Settings: Lohmöller suggested using +1 as initial outer weight for all indicators per measurement model except the last one, which uses an initial outer weight of -1. Thereby, the PLS-SEM algorithm converges faster. However, this kind of initialization can lead to counterintuitive signs of estimated PLS path coefficients in the measurement models and/or in the structural model.

• Individual Settings: SmartPLS to define individual initial outer weights for every indicator in the PLS path model. For example, are particularly important indicator can obtain a +1 (e.g., when the strong an positive relationship with the latent variable is assumed a prior), while the other indicators of the same measurement model obtain a 0.

## References

• Dijkstra, T. K. (2010). Latent Variables and Indices: Herman Wold’s Basic Design and Partial Least Squares, in Handbook of Partial Least Squares: Concepts, Methods and Applications (Springer Handbooks of Computational Statistics Series, vol. II), V. Esposito Vinzi, W. W. Chin, J. Henseler and H. Wang (eds.), Springer: Heidelberg, Dordrecht, London, New York, pp. 23-46.

• Hair, J. F., Hult, G. T. M., Ringle, C. M., and Sarstedt, M. (2017). A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM), 2^nd^ Ed., Sage: Thousand Oaks.

• Henseler, J., Ringle, C. M., and Sarstedt, M. (2012). Using Partial Least Squares Path Modeling in International Advertising Research: Basic Concepts and Recent Issues, in Handbook of Research in International Advertising, S. Okazaki (ed.), Edward Elgar Publishing: Cheltenham, pp. 252-276.

• Henseler, J., Ringle, C. M., and Sinkovics, R. R. (2009). The Use of Partial Least Squares Path Modeling in International Marketing, in Advances in International Marketing, R. R. Sinkovics and P. N. Ghauri (eds.), Emerald: Bingley, pp. 277-320.

• Lohmöller, J.-B. (1989). Latent Variable Path Modeling with Partial Least Squares, Physica: Heidelberg.

• Ringle, C. M., Wende, S., and Becker, J.-M. (2015). SmartPLS 3, SmartPLS GmbH: Boenningstedt.

• Wold, H. (1982). Soft Modeling: The Basic Design and Some Extensions, in Systems Under Indirect Observations: Part II, K. G. Jöreskog and H. Wold (eds.), North-Holland: Amsterdam, pp. 1-54.