The factor of two issue in mixed DNA profiles

A commonly used idea in forensic fields is known as the 'hierarchy of propositions'. DNA analysts commonly report at the sub-source level in the hierarchy. This means that they simply comment on the probability of the evidence for the given propositions that consider contributors that lead to a DNA profile and not on the source of specific biological components, not the activity that led to the transfer or the offence that is reported to have occurred. However DNA analysts also commonly report at a level even lower than the sub-source level. In this 'sub-sub-source' level only reference comparisons to components of a mixture are reported. The difference between the sub-source level and sub-sub-source level is the difference between comparing an individual to a mixture as a whole, or comparing them to only one component of a mixture. This idea has been expressed in the past as the 'two trace' problem or the 'factor of two' problem. With the advent of expert systems that can provide a measure of weight of evidence in the form of a likelihood ratio (LR) for any mixture, resolvable or not, the distinction between these two levels becomes more important. In this paper we explore how the LR can be constructed to report correctly at the sub-source level, by taking contributor orders and genotype set orders into account. We include worked examples of the LR calculation to help explain this confusing issue.