Care and Handling of Univariate Outliers in the General Linear Model to Detect Spuriosity—A Bayesian Approach

Abstract

We deal with the situation covered by the univariate general linear model, that is, it is intended that n observations be generated in accordance with the usual model y = Xβ + ε however, it is feared that k of the observations are spurious, that is, not generated in the manner intended, so that for an unknown set of k distinct integers, say (i 1, … ik ), a subset of the first n integers, we have, specifically, that ytj , = x tj ′, β + a j , + ∊ tj , where in general x t ′ denotes the t-th row of X, and where (a 1, …, a k ), so called shift parameters, are such that – ∞ < a j , < ∞. In this paper, we discuss the posterior distribution of β, when indeed it is assumed a priori that any given set of k observations has uniform probability I/( n k ) of being spurious. The properties of the posterior of β are discussed, and the results used in an example using data generated from a response surface design. Ad hoc procedures are discussed for gaining information on k, when k is unknown. These ad hoc procedures are illustrated using a famous set of data from Darwin