Exclusion probabilities and likelihood ratios with applications to mixtures

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 non-donors, of which the RMNE is a special case corresponding to LR>0. 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 drop-in are commonly encountered and we address this situation too, illustrating our results with a basic but widely implemented model, a so-called binary model. The precise definitions, modelling and interpretation of the required concepts of dropout and drop-in are not entirely obvious, and we attempt to clarify them here in a general likelihood framework for a binary model.