Graphical Modeling of Spatial Health Data

Authored by: Adrian Dobra

Handbook of Spatial Epidemiology

Print publication date:  April  2016
Online publication date:  April  2016

Print ISBN: 9781482253016
eBook ISBN: 9781482253023
Adobe ISBN:

10.1201/b19470-36

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Abstract

Graphical models (Whittaker, 1990; Lauritzen, 1996) that encode multivariate independence and conditional independence relationships among observed variables X = (X 1, …, Xp ) have a widespread use in major scientific areas (e.g., biomedical and social sciences). In particular, a Gaussian graphical model (GGM) is obtained by setting off-diagonal elements of the precision matrix K = Σ −1 to zero of a p-dimensional multivariate normal model (Dempster, 1972). Employing a GGM instead of a multivariate normal model leads to a significant reduction in the number of parameters that need to be estimated if most elements of K are constrained to be zero and p is large. A pattern of zero constraints in K can be recorded as an undirected graph G = (V, E), where the set of vertices V = {1, 2, …, p} represent observed variables, while the set of edges EV × V link all the pairs of vertices that correspond to off-diagonal elements of K that have not been set to zero. The absence of an edge between X v 1 and X v 2 corresponds with the conditional independence of these two random variables given the rest and is denoted by X u 1 ╨   X u 2   |   X V \ { v 1 , v 2 } (Wermuth, 1976). This is called the pairwise Markov property relative to G, which in turn implies the local as well as the global Markov properties relative to G (Lauritzen, 1996). The local Markov property plays a key role since it gives the regression model induced by G on each variable Xv . More explicitly, consider the neighbors of v in G, that is, the set of vertices v′V such that (v, v′) ∈ E. We denote this set by bd G (v). The local Markov property relative to G says that X v ╨   X V \ { { v } ∪ bd G ( v ) } |   X bd G ( v ) . This is precisely the statement we make when we drop the variables { X v ′   :   v ′   :   ∈ V \ b d G ( v ) } from the regression of Xv on { X v ′   : v ′   ∈ V \ { v } } .

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