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.