# Beyond linearity - a new Partial Least Squares - Path Modelling (PLS-PM) inner weighting scheme for detecting and approximating nonlinear structural relationships in Structural Equation Models

## TL;DR

A new method for PLS-PM is introduced to better detect and approximate nonlinear relationships in structural equation models.

## Contribution

A novel smooth weighting scheme is proposed to handle nonlinear structural relationships in PLS-PM.

## Key findings

- The new scheme can recover various nonlinear functional forms in SEM.
- It outperforms existing schemes for sample sizes larger than 75 units.
- Performance is robust even with error contamination in observed variables.

## Abstract

A new inner weighting scheme for Partial Least Squares – Path Modelling (PLS-PM) is proposed to detect and approximate nonlinear structural relationships in Structural Equation Models (SEM). PLS-PM is an iterative method used for the estimation of Structural Equation Models (SEM), a widely used analytical tool for assessing causal relationships between latent variables. However, PLS-PM struggles to address the structural nonlinear relationships. To address this limitation, a new PLS-PM inner weighting scheme, smooth weighting, is proposed as an additional option to the traditional centroid, factor, and path weighting schemes. A real marketing dataset is used to demonstrate the usefulness of the method for finding evidence of nonlinearity, and a simulated dataset is used to assess its ability to approximate underlying (unknown) nonlinear structural relationships. The results show that the proposed scheme can recover several nonlinear functional forms, outperforming existing inner weighting schemes for commonly used sample sizes (larger than 75 units), regardless of the level of error contamination in the observed manifest variables.