Estimating probability terms for the background presence of glass when considering activity in forensic casework

Coulson et al. [1] proposed methodology for the estimation of the P and S terms used in glass interpretation when assessing the value of the findings given activity level propositions. These terms arise in a model proposed by Evett [2], Evett and Buckleton [3], and are based on survey data. Specifically they proposed a model for estimating P_k, k = 0, 1, 2, … and S_n, n = 1, 2, where P_k is the probability of finding k distinct sources (or groups) of glass on a person of interest (POI), and S_n is the probability that the kth source consists of n fragments. In this article we make a number of extensions to the work of Coulson et al. [1]. Firstly we derive an estimate of the uncertainty in the parameter of the Coulson et al. model, and show how this may be used-for example, to compute an estimate of how the probabilities may vary or how to compare estimates resulting from different surveys. Secondly, we extend the model by allowing a more sensible modelling of the "excess" zeros (in the case of the P terms) and excess ones (in the case of the S terms). The methodology used to make these extensions relies on purely frequentist theory of estimation in keeping with the original work. A Bayesian approach to estimation will be the subject of future work. Additionally, we demonstrate the use of an R (R Core Team, [4]) package, called fitPS (Curran, [5]) which makes the methodology described in this article easy to implement in practice.