PLS-SEM Algorithm
The partial least squares (PLS) path modeling method, also known as PLS structural equation modeling (PLS-SEM), was developed by Wold (1982) and further refined by Lohmöller (1989). At its core, the PLS-SEM algorithm is a sequence of regressions expressed in terms of weight vectors; the resulting outer weights satisfy fixed point equations once the algorithm has converged. Researchers use this algorithm to estimate latent variable (construct) scores from the indicator data and the specified path model, which in turn provide the basis for estimating every other relationship in the model. It is the core estimation procedure underlying all PLS-SEM analyses in SmartPLS.
How the PLS-SEM Algorithm Works
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 the ensuing convergence issues). The objective of the PLS-SEM algorithm is to find a stable set of outer/indicator weights that iteratively maximizes both the explained variance between connected composites and the explained variance in the indicators by their respective composites (when using Mode A, correlation weights); see also Hair et al. (2024).
The basic PLS algorithm, as presented by Lohmöller (1989), includes the following three stages:
| Stage | What happens |
|---|---|
| Stage 1 | Iterative estimation of latent variable scores and outer weights. After initializing the outer weights, compute step #4 once and then repeat steps #1 to #4: (1) inner weights, (2) inside approximation, (3) provisional outer weights, and (4) normalized outside approximation. Stop when the outer weights converge or the maximum number of iterations is reached. |
| Stage 2 | Estimation of path coefficients and loadings. |
| Stage 3 | Estimation of location parameters. |
The following code-based representation transcribes Lohmöller's basic PLS algorithm (1989, Table 2.1, p. 29), with the Mode A and Mode B assignment reconciled as explained below:
| Stage / step | Operation |
|---|---|
| Stage 1 |
Iterative estimation of weights and LV scores Starting at step #4, repeat steps #1 to #4 until convergence is obtained. |
| #1 | Inner weights |
| #2 | Inside approximation |
| #3 |
Outer weights: solve for the provisional weights
in the applicable regression.
Mode A block
Mode B block
|
| #4 | Outside approximation |
| Stage 2 | Estimation of path and loading coefficients. |
| Stage 3 | Estimation of location parameters. |
Source note on Mode A and Mode B: The labels beside the two regressions in Lohmöller's Table 2.1 appear to be interchanged. Equations (2.8) and (2.9) immediately above the table, together with the explanation on p. 30, define Mode A as the single regressions of the indicators on the inside approximation and Mode B as the multiple regression of the inside approximation on the indicators. The representation above follows those definitions.
Historical scope: Step #1 shows the centroid inner-weighting rule used in Lohmöller's basic algorithm. With other inner weighting schemes, the calculation of the inner weights changes, while the iterative structure of steps #1 to #4 remains.
Mathematical Notation Used in the Algorithm
Lohmöller uses Latin letters for observed quantities and estimates and systematic subscripts for constructs, indicators, and cases. During stages 1 and 2, the manifest and latent variable scores are treated as deviations from their means.
| Element | Meaning |
|---|---|
| j, i | Indices for latent variables (constructs) and their corresponding indicator blocks. |
| kj | Index of an indicator within block j; kj = 1, ..., Kj. |
| n | Observation or case index; n = 1, ..., N. |
| ykj,n | Observed value of indicator kj for case n. |
| Yj and Yjn | Current outside approximation of latent variable j, respectively its value for case n. |
| Ỹj and Ỹjn | Inside approximation of latent variable j, formed from the scores of adjacent latent variables. The tilde denotes an intermediate or conditional-expectation-based approximation. |
| vji | Inner weight connecting latent variables j and i. In the displayed centroid rule it is +1 or -1 according to the sign of their covariance, and 0 when they are not adjacent. |
| w̃kj | Provisional outer weight estimated in step #3. The normalized outer weight is wkj = fjw̃kj. |
| fj | Scaling factor in step #4, selected so that the updated latent variable score has unit variance: var(Yj) = 1. |
| djn, ekj,n | Regression residuals in the Mode B and Mode A outer-weight estimations, respectively. |
| cov(·; ·), sign(·) | Covariance and sign operators. The semicolon separates the two arguments of the covariance operator. |
| := | Assignment or redefinition: the expression on the left is updated using the expression on the right. |
| Adjacent | Two latent variables are directly connected in the specified path model. |
Hair et al. (2027), Henseler et al. (2012), and Lohmöller (1989) provide detailed explanations of how the basic PLS-SEM algorithm operates as implemented in SmartPLS.
PLS Algorithm Settings in SmartPLS
Weighting Scheme
PLS-SEM allows you to apply three structural model weighting schemes:
| Weighting scheme | Notes |
|---|---|
| Factor weighting scheme | Results differ little from the path weighting scheme. |
| Path weighting scheme (default) | The recommended approach. It provides the highest R² value for endogenous latent variables and is generally applicable to all kinds of PLS path model specifications and estimations. |
| Principal component analysis (PCA) | Uses the results of a PCA per construct to estimate the model. |
Note: SmartPLS no longer offers the centroid weighting scheme, which was available in SmartPLS 2 and 3.
Type of Results
This option lets you choose between standardized (default), unstandardized, and mean-centered PLS-SEM outcomes. These settings affect the construct scores provided by the PLS-SEM algorithm and the estimated coefficients (e.g., standardized coefficients in the structural model, or unstandardized coefficients with an intercept in the structural model).
Initial Outer Weights
- Standard: As the default (i.e., the SmartPLS setting), the initial outer weights are set to +1.
- Individual: SmartPLS lets you define individual initial outer weights for every indicator in the PLS path model. For example, an indicator that is assumed a priori to have a particularly strong positive relationship with the latent variable can be given an initial weight of +1, while the other indicators of the same measurement model receive 0.
Maximum Iterations
This option to change the settings for running the PLS-SEM algorithm is no longer available in SmartPLS 4. The permanent setting is 3,000 iterations. This parameter represents the maximum number of iterations used for calculating the PLS results. This number should be sufficiently large (e.g., 3,000 iterations). When checking a PLS-SEM result, make sure that the algorithm did not stop because the maximum number of iterations was reached, but because the stop criterion was met. Note: Selecting 0 for the maximum number of iterations lets you obtain results from the sum scores approach.
Stop Criterion
This option to change the settings for running the PLS-SEM algorithm is no longer available in SmartPLS 4. The permanent setting is 10-7. 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., 10-7).
Frequently Asked Questions
What is the PLS-SEM algorithm?
The PLS-SEM algorithm is the estimation procedure behind partial least squares structural equation modeling. It is a sequence of regressions in terms of weight vectors that iteratively estimates latent variable scores from the indicator data and the specified path model, which are then used to estimate all relationships in the model.
Who developed the PLS-SEM algorithm?
The method was developed by Wold (1982) and further refined by Lohmöller (1989), whose three-stage version of the algorithm is the one implemented in SmartPLS.
What are the three stages of the PLS-SEM algorithm?
Stage 1 iteratively estimates latent variable scores and outer weights. After the initial outside approximation (step #4), the algorithm repeats inner-weight estimation (step #1), inside approximation (step #2), provisional outer-weight estimation (step #3), and normalized outside approximation (step #4) until convergence. Stage 2 estimates path coefficients and loadings. Stage 3 estimates the location parameters.
Which weighting scheme should I use in SmartPLS?
The path weighting scheme is the default and recommended option because it provides the highest R² value for endogenous latent variables and applies to all kinds of PLS path model specifications. The factor weighting scheme produces very similar results. The PCA option instead estimates the model using per-construct PCA results.
Can I change the maximum number of iterations or the stop criterion?
No, not in SmartPLS 4. Both settings are fixed (3,000 iterations and a stop criterion of 10-7) and are no longer user-configurable. Setting the maximum number of iterations to 0 returns results from the sum scores approach instead of the iterative PLS algorithm.
Related SmartPLS Methods
References
- Hair, J. F., Hult, G. T. M., Ringle, C. M., & Sarstedt, M. (2027). A primer on partial least squares structural equation modeling (PLS-SEM) (4th ed.). Sage.
- Hair, J. F., Sarstedt, M., Ringle, C. M., Sharma, P. N., & Liengaard, B. D. (2024). Going beyond the untold facts in PLS-SEM and moving forward. European Journal of Marketing, 58(13), 81–106.
- Henseler, J., Ringle, C. M., & Sarstedt, M. (2012). Using partial least squares path modeling in international advertising research: Basic concepts and recent issues. In S. Okazaki (Ed.), Handbook of research in international advertising (pp. 252–276). Edward Elgar Publishing.
- Henseler, J., Ringle, C. M., & Sinkovics, R. R. (2009). The use of partial least squares path modeling in international marketing. In R. R. Sinkovics & P. N. Ghauri (Eds.), Advances in international marketing (pp. 277–320). Emerald.
- Lohmöller, J.-B. (1989). Latent variable path modeling with partial least squares. Physica.
- Wold, H. (1982). Soft modeling: The basic design and some extensions. In K. G. Jöreskog & H. Wold (Eds.), Systems under indirect observations: Part II (pp. 1–54). North-Holland.
- More literature ...
Cite correctly
Please always cite the use of SmartPLS!
Ringle, Christian M., Wende, Sven, & Becker, Jan-Michael. (2024). SmartPLS 4. Bönningstedt: SmartPLS. Retrieved from https://www.smartpls.com
