Multivariate linear path models
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Many investigators, especially in the fields of heath science and epidemiology, have been interested in providing a causal interpretation of the statistical relationships they find in the course of modeling a set of data. For quantitative variables, path modeling has been used to provide a causal interpretation for a given system of linear relationship. This is usually achieved by estimating and testing direct and indirect effect of endogenous variables on subsequent variables in a causal chain. However, the methods that are traditionally implemented are limited by requirement of a complete causal ordering of variables. In this dissertation, methodology is presented that extends the traditional univariate path model in the multivariate frame work, called multivariate linear path models. In multivariate linear path models, variables are defined as column vectors and path coefficients are defined as matrices of coefficients. A Calculus of Coefficients (COC) for multivariate path models is presented. That results in a partitioning of the matrix of total effects into the sum of a matrix of direct effects and all matrices of indirect effects through intermediate outcome vectors. The multivariate COC derived in this study extends that for the classical univariate path model to the multivariate case, where vectors of outcome variables replace single variables in the causal chain. A general methodology for inferences is developed that utilize Union-Intersection of Intersection-Union tests to test single indirect effects and bootstrap methods to testing matrices of indirect effects. The methods are applied to data from the Western New York Health Study to describe the effects of health behaviors such as diet, smoking, drinking, and exercise on an index of risk factors for cardio-metabolic disease. We partition the total effects of the health behavior variables into direct and indirect effects on a Cardio-Metabolic Risk Index (CMRI) through anthropometric variables and through composite blood measures that are interpreted to reflect chronic inflammation, endogenous steroid levels, anemia, and blood viscosity.