A nonlinear PLS path modeling based on monotonic B-spline transformations

INTRODUCTION PLS path modeling is widely used in marketing applications. It is based on linear equations. However, in practical applications, many relations cannot be regarded as linear. For example, the relations between satisfaction and its attributes are nonlinear (Mittal et al., 1998). In this paper, we present a two step approach in order to include nonlinear relationships between manifest and latent variables. We use a certain type of data transformation, often called optimal scaling, with monotonic Bspline optimal transformations of the manifest variables prior to PLS path modeling. Combining optimal scaling methods and structural equation models is not new; LISREL users already perform these kinds of variables transformation using maximum likelihood estimation (Meijernick, 1995). De Leeuw (1988) advocated a two-step approach and Hwang and Takane (2002) presented an alternative approach to PLS path modeling which facilitates the use of optimal scaling but nothing has been made in the PLS path modeling framework. This can be explained by the lack of global optimization criterion. In this paper, we decided to stay in a general context with no global criterion. We focused on the outer model thinking in term of nonlinear principal components analysis (nonlinear PCA, Young et al., 1978). PLS path modeling is based on the estimation of latent variables as composite variables of their own manifest variables. They are created as linear combination in the space of their own manifest variables which makes it relevant to use nonlinear PCA. In the practical applications; we present a comparison between classical PLS path modeling and B-spline transformed PLS path modeling on customer satisfaction data.