Efficient computations with the likelihood ratio distribution

What is the probability that the likelihood ratio exceeds a threshold t, if a specified hypothesis is true? This question is asked, for instance, when performing power calculations for kinship testing, when computing true and false positive rates for familial searching and when computing the power of discrimination of a complex mixture. Answering this question is not straightforward, since there is are a huge number of possible genotypic combinations to consider. Different solutions are found in the literature. Several authors estimate the threshold exceedance probability using simulation. Corradi and Ricciardi [1] propose a discrete approximation to the likelihood ratio distribution which yields a lower and upper bound on the probability. Nothnagel et al. [2] use the normal distribution as an approximation to the likelihood ratio distribution. Dørum et al. [3] introduce an algorithm that can be used for exact computation, but this algorithm is computationally intensive, unless the threshold t is very large. We present three new approaches to the problem. Firstly, we show how importance sampling can be used to make the simulation approach significantly more efficient. Importance sampling is a statistical technique that turns out to work well in the current context. Secondly, we present a novel algorithm for computing exceedance probabilities. The algorithm is exact, fast and can handle relatively large problems. Thirdly, we introduce an approach that combines the novel algorithm with the discrete approximation of Corradi and Ricciardi. This last approach can be applied to very large problems and yields a lower and upper bound on the exceedance probability. The use of the different approaches is illustrated with examples from forensic genetics, such as kinship testing, familial searching and mixture interpretation. The algorithms are implemented in an R-package called DNAprofiles, which is freely available from CRAN.