Kinship cases with partially specified hypotheses

Forensic kinship testing is the statistical comparison of a set of hypothesised relationships, based on genetic marker data from the individuals in question and possibly other relatives. In most circumstances each hypothesis is completely specified in terms of a pedigree, but this is not always the case in more complex scenarios. For example, suppose that we are asked to test $H_1$: A is the grandmother of B, against $H_2$: A and B are unrelated, and that the data also includes a third individual whose relationship with the others is uncertain. There may then be multiple pedigrees consistent with each hypothesis, with the consequence that the standard likelihood ratio (LR) cannot be calculated unless prior probabilities are specified for all alternatives.

In response to these challenges we introduce a generalised likelihood ratio (GLR), defined as the ratio of the maximal likelihood of the data given $H_1$ to the maximal given $H_2$. This resembles a version of the LR test used in classical hypothesis testing, but differs in several aspects. Most importantly, in the forensic setting we usually consider discrete alternatives rather than continuous parameter spaces.

The properties of the GLR statistic are explored through real-life examples of kinship testing and disaster victim identification (DVI). In particular, we demonstrate how the GLR may help to resolve and report the results in complex DVI cases. As a final application we demonstrate how the GLR can be used to check correctness of pedigree data, an essential quality control step in projects involving genotypes from related individuals. Unlike the other examples, this one operates over a continuous parameter space, enabling tools from classical statistics to guide decision-making.