Structural Equation Modelling for small samples

Two complementary schools have come to the fore in the field of Structural Equation Modelling (SEM): covariance-based SEM and component-based SEM. The first approach developed around Karl Jöreskog. It can be considered as a generalisation of both principal component analysis and factor analysis to the case of several data tables connected by causal links. The second approach developed around Herman Wold under the name "PLS" (Partial Least Squares). More recently Hwang and Takane (2004) have proposed a new method named Generalized Structural Component Analysis. This second approach is a generalisation of principal component analysis (PCA) to the case of several data tables connected by causal links. Covariance-based SEM is usually used with an objective of model validation and needs a large sample (what is large varies from an author to another: more than 100 subjects and preferably more than 200 subjects are often mentioned). Component-based SEM is mainly used for score computation and can be carried out on very small samples. A research based on 6 subjects has been published by Tenenhaus, Pagès, Ambroisine & Guinot (2005) and will be used in this paper. In 1996, Roderick McDonald published a paper in which he showed how to carry out a PCA using the ULS (Unweighted Least Squares) criterion in the covariance-based SEM approach. He concluded from this that he could in fact use the covariance-based SEM approach to obtain results similar to those of the PLS approach, but with a precise optimisation criterion in place of an algorithm with not well known properties. In this research, we will explore the use of ULS-SEM and PLS on small samples. First experiences have already shown that score computation and bootstrap validation are very insensitive to the choice of the method. We will also study the very important contribution of these methods to multiblock analysis.