The likelihood ratio as a random variable for linked markers in kinship analysis

Int J Legal Med (2016) 130:14451456 DOI 10.1007/s00414-016-1416-2

ORIGINAL ARTICLE

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Web End = The likelihood ratio as a random variable for linked markers in kinship analysis

Thore Egeland1 Klaas Slooten2,3

Received: 21 March 2016 / Accepted: 7 July 2016 / Published online: 13 August 2016 Springer-Verlag Berlin Heidelberg 2016

Abstract The likelihood ratio is the fundamental quantity that summarizes the evidence in forensic cases. Therefore, it is important to understand the theoretical properties of this statistic. This paper is the last in a series of three, and the first to study linked markers. We show that for all non-inbred pairwise kinship comparisons, the expected likelihood ratio in favor of a type of relatedness depends on the allele frequencies only via the number of alleles, also for linked markers, and also if the true relationship is another one than is tested for by the likelihood ratio. Exact expressions for the expectation and variance are derived for all these cases. Furthermore, we show that the expected likelihood ratio is a non-increasing function if the recombination rate increases between 0 and 0.5 when the actual relationship is the one investigated by the LR. Besides being of theoretical interest, exact expressions such as obtained here can be used for software validation as they allow to verify the correctness up to arbitrary precision. The paper also presents results and advice of practical importance. For example, we argue that the logarithm of the likelihood ratio

[envelopeback] Thore Egelandmailto:[email protected]

Web End [email protected]

Klaas Slootenmailto:[email protected]

Web End [email protected]

1 Norwegian University of Life Sciences, 1432 Aas, Norway

2 Netherlands Forensic Institute, P.O. Box 24044, 2490 AA The Hague, The Netherlands

3 Department of Mathematics, VU University, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

behaves in a fundamentally different way than the likelihood ratio itself in terms of expectation and variance, in agreement with its interpretation as weight of evidence. Equipped with the results presented and freely available software, one may check calculations and software and also do power calculations.

Keywords Kinship analysis Paternity testing linked

markers Likelihood ratios - weight of evidence

Introduction

As the number of markers used by the forensic community increases, considerations of linked markers become increasingly important. Weight of evidence calculations for...