# Permutation

## Abstract

The permutation algorithm allows to test if pre-defined data groups have statistically significant differences in their group-specific parameter estimates (e.g., outer weights, outer loadings and path coefficients). It also supports the MICOM procedure for analyzing measurement invariance.

## Description

The purpose of using the permutation routine in SmartPLS is twofold:

(1) It allows conducting a PLS-SEM multigroup analysis (Hair et al., 2024; Sarstedt, Henseler, and Ringle, 2011) as suggested by Dibbern and Chin (2005) and Chin and Dibbern (2010). Thereby, you can decide if group-specific PLS-SEM results have statistically significant differences.

(2) It allows conducting the PLS-SEM measurement invariance assessment as suggested by Henseler, Ringle, and Sarstedt's (2015) MICOM routine. Thereby, you can substantiate that significant differences in the group-specific PLS-SEM results do not stem from differences in the constructs (e.g., customer loyalty) across groups.

The results report of the permutation routine in SmartPLS includes the outcomes of the PLS multigroup analysis (using the permutation test) and the MICOM results for assessing measurement invariance.

## Permutation Settings in SmartPLS

### Select Groups

The selected groups will be assessed for significant differences in the parameter estimates (e.g., outer weights, outer loadings and path coefficients) and measurement invariance (MICOM).

The data group selected under Group A will be compared against the data group selected under Group B.

Note: If the combo box for group selection is empty, you need to double click on the dataset (see SmartPLS project window). Then, use the available options to generate groups of data for the PLS-SEM multi-group analysis.

### Permutations

Permutations are created with observations randomly drawn from the original set of data (without replacement). More precisely,

*n*observations are drawn without replacement and assigned to Group A; all remaining observations are assigned to Group B. It is important to note that*n*equals the number of observations of Group A in the original dataset. The remaining data that are assigned to Group B also have the same number of observations that Group B has in the original dataset. Consequently, in each permutation run, the group-specific sample sizes remain constant and equal the size of each group in the original dataset.To ensure stability of the results, the number of permutations should be large. For a quick initial assessment, one may choose a smaller number of permutation subsamples (e.g., 500 or 1,000). For the final results preparation, however, one should use a large number of permutations (e.g., 5,000).

Note: Larger numbers of permutations increase the computation time.

The default value is 1,000.

### Test Type

When using the permutation procedure's results to create confidence intervals, it is important to determine, if you wish conducting a one-tailed or two-tailed (or one-sided or two-sided) significance test. This selection also affects p value computation.

The default value is two tailed.

### Significance Level

This option allows you to specify the significance level that is used for the confidence interval computations.

The default values is 0.05.

### Do Parallel Processing

Permutation will be performed on multiple processors (if your computer offers more than one core). Using parallel computing considerably reduces computation time.

The default setting uses parallel processing.

## References

- Chin, W. W., and Dibbern, J. (2010).
**A Permutation Based Procedure for Multi-Group PLS Analysis: Results of Tests of Differences on Simulated Data and a Cross Cultural Analysis of the Sourcing of Information System Services between Germany and the USA**, 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, 171-193. - Hair, J. F., Sarstedt, M., Ringle, C. M., & Gudergan, S. P. (2024).
**Advanced Issues in Partial Least Squares Structural Equation Modeling (PLS-SEM)**, 2nd Ed., Thousand Oaks, CA: Sage. - Henseler, J., Ringle, C. M., and Sarstedt, M. (2015).
**Testing Measurement Invariance of Composites Using Partial Least Squares**,*International Marketing Review*, 33(3), 405-431. - Sarstedt, M., Henseler, J., and Ringle, C. M. (2011).
**Multi-Group Analysis in Partial Least Squares (PLS) Path Modeling: Alternative Methods and Empirical Results**,*Advances in International Marketing*, 22, 195-218. - Edgington, E., and Onghena, P. (2007).
**Randomization Tests**, 4th Ed., Chapman & Hall: London. - More literature ...

# Cite correctly

## Please always cite the use of SmartPLS!

Ringle, Christian M., Wende, Sven, & Becker, Jan-Michael. (2024). SmartPLS 4. BÃ¶nningstedt: SmartPLS. Retrieved from https://www.smartpls.com