Path Analysis and Structural Equation Models

Abstract

The simplest path model we can think of is a multiple linear regression with m predictors and a single response Y. It can be fully estimated through a path model framework, but there is not really a statistical reason to do so. Therefore, in this section we start our elaborations with multivariate regression path models. Note that multivariate regression should not be confounded with multiple regression. In multivariate regression we have, in general, multiple predictors and multiple, potentially correlated responses. Let us collect these response variables in an n × q matrix Y with n as the sample size and q as the number of response variables. The multivariate regression model can be written as Y = XB + E. (3.1) The design matrix X is of dimension n × (m + 1) (since we have a unit vector for the intercept), the parameter matrix B is correspondingly (m + 1) × q, and the matrix of error terms E is n × q. There are several ways of computing multivariate regression models: One approach is to stay within a classical linear model context, estimate a multiple regression model for each response separately, and then compute various measures known from MANOVA (multivariate analysis of variance), subject to inference. A second approach is to fit the model through a path-analytic framework. A third approach is to use mixed-effects models and specify a random effect for the response variables. In this section we focus on the first two approaches: classical MANOVA approach and path modeling. Details regarding the third approach can be found in Berridge and Crouchley (2011). Let us start with MANOVA. We use the generalized prejudice dataset from Bergh et al. (2016), already presented in Sect. 2.4.4. This time we consider two indicators from the agreeableness scale (A1, A2) and two indicators from the openness scale © Springer International Publishing AG, part of Springer Nature 2018 P. Mair, Modern Psychometrics with R, Use R!, https://doi.