An approach to unified formulae for likelihood ratio calculation in pairwise kinship analysis

The likelihood ratio (LR) can be an efficient means of distinguishing various relationships in forensic fields. However, traditional list-based methods for derivation and presentation of LRs in distant or complex relationships hinder code editing and software programming. This paper proposes an approach for a unified formula for LRs, in which differences in participants' genotype combinations can be ignored for specific identification. This formula could reduce the difficulty of by-hand coding, as well as running time of large-sample-size simulation. The approach is first applied to a problem of kinship identification in which at least one of the participants is alleged to be inbred. This can be divided into two parts: i) the probability of different identical by descent (IBD) states according to the alleged kinship; and ii) the ratio of the probability that specific genotype combination can be detected assuming the alleged kinship exists between the two participants to the similar probability assuming that they are unrelated, for each state. For the probability, there are usually recognized results for common identification purposes. For the ratio, subscript letters representing IBD alleles of individual A's alleles are used to eliminate differences in genotype combinations between the two individuals and to obtain a unified formula for the ratio in each state. The unification is further simplified for identification cases in which it is alleged that both of the participants are outbred. Verification is performed to show that the results obtained with the unified and list-form formulae are equivalent. A series of unified formulae are derived for different identification purposes, based on which an R package named KINSIMU has been developed and evaluated for use in large-size simulations for kinship analysis. Comparison between the package with two existing tools indicated that the unified approach presented here is more convenient and time-saving with respect to the coding process for computer applications compared with the list-based approach, despite appearing more complicated. Moreover, the method of derivation could be extended to other identification problems, such as those with different hypothesis sets or those involving multiple individuals. The unified approach of LR calculation can be beneficial in kinship identification field.