Exclusion probabilities and likelihood ratios with

Authors: KlaasJan Slooten Thore Egeland

Publish Date: 2015/07/10

Volume: 130, Issue: 1, Pages: 39-57

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Abstract

The statistical evidence obtained from mixed DNA profiles can be summarised in several ways in forensic casework including the likelihood ratio LR and the Random Man Not Excluded RMNE probability The literature has seen a discussion of the advantages and disadvantages of likelihood ratios and exclusion probabilities and part of our aim is to bring some clarification to this debate In a previous paper we proved that there is a general mathematical relationship between these statistics RMNE can be expressed as a certain average of the LR implying that the expected value of the LR when applied to an actual contributor to the mixture is at least equal to the inverse of the RMNE While the mentioned paper presented applications for kinship problems the current paper demonstrates the relevance for mixture cases and for this purpose we prove some new general properties We also demonstrate how to use the distribution of the likelihood ratio for donors of a mixture to obtain estimates for exceedance probabilities of the LR for nondonors of which the RMNE is a special case corresponding to L R0 In order to derive these results we need to view the likelihood ratio as a random variable In this paper we describe how such a randomization can be achieved The RMNE is usually invoked only for mixtures without dropout In mixtures artefacts like dropout and dropin are commonly encountered and we address this situation too illustrating our results with a basic but widely implemented model a socalled binary model The precise definitions modelling and interpretation of the required concepts of dropout and dropin are not entirely obvious and we attempt to clarify them here in a general likelihood framework for a binary modelWe explain some of the mathematical expressions based on a simple example without dropout or dropin There are two contributors to a mixture and the question is whether a person S has contributed corresponding to H p or not corresponding to H d We assume the contributors to be unrelated and the suspect to be either a contributor or unrelated to the contributors This means for example that Pmathcal M=M mid mathcal S=g mathcal S neq mathcal C 1=Pmathcal M=M since the genotype of a noncontributor does not influence the mixture’s likelihood because the noncontributor is unrelated to the contributors Similarly Pmathcal C i=g=f g for both contributors We work with the hypotheses as random variables according to Eqs 24 and 25 meaning that we also regard the mixture itself as random

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