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

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 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.

Appendices

Appendix A: Expect higher LR for linked markers

The claim of the title is consistent with the results, see Tables 1 and 3 and figures. We prove that $f(\rho )=E[LR(\mathcal {H}_{P})]$ decreases with ρ. In particular, f(0)≥f(0.5). The latter claim is reasonable as it is equivalent to stating that the likelihood ratios for the two markers are positively correlated, which is intuitively reasonable.

Suppose we work on two loci. We then will show that, for 0 < ρ < 1/2 we have

$$\frac{d}{d\rho}E_{\rho}[LR_{\mathcal{P}ed}(\mathcal{H}_{\mathcal{P}ed})] \leq 0, $$

where

$$\begin{array}{@{}rcl@{}} E_{\rho}[LR_{\mathcal{P}ed}(\mathcal{H}_{\mathcal{P}ed})]&=&\sum\limits_{i_{1},i_{2},j_{1},j_{2}=0}^{2} \kappa_{i_{1},i_{2}}(\rho)\kappa_{j_{1},j_{2}}(\rho)\\ &&\times E[LR^{(1)}_{i_{1}}(\mathcal{H}_{j_{1}})]E[LR^{(2)}_{i_{2}}(\mathcal{H}_{j_{2}})]. \end{array} $$

By (2.8), we have

$$\begin{array}{@{}rcl@{}} \frac{d}{d\rho}E_{\rho}[LR_{\mathcal{P}ed}(\mathcal{H}_{\mathcal{P}ed})]&=&\sum\limits_{i_{1},i_{2},j_{1},j_{2}=0}^{2} \frac{d}{d\rho}(\kappa_{i_{1},i_{2}}(\rho))\kappa_{j_{1},j_{2}}(\rho)\\ &&\times E[LR^{(1)}_{i_{1}}(\mathcal{H}_{j_{1}})]E[LR_{i_{2}}^{(2)}(\mathcal{H}_{j_{2}})] \end{array} $$

Since

$$\kappa_{i,0}(\rho)+\kappa_{i,1}(\rho)+\kappa_{i,2}(\rho)=P(I_{1}=i), $$

we have

$$\frac{d}{d\rho}(\kappa_{i,i}(\rho))=-\sum\limits_{j \neq i } \frac{d}{d\rho}\kappa_{i,j}(\rho). $$

For brevity we denote

$$ X_{i_{1}i_{2},j_{1}j_{2}}=E[LR^{(1)}_{i_{1}}(\mathcal{H}_{j_{1}})]E[LR_{i_{2}}^{(2)}(\mathcal{H}_{j_{2}})].$$

We then have

$$\begin{array}{@{}rcl@{}} \frac{d}{d\rho}E_{\rho}[LR_{\mathcal{P}ed}(\mathcal{H}_{\mathcal{P}ed})] &=& \frac{d}{d\rho}(\kappa_{0,1}(\rho))\\ &&\times\left( \sum\limits_{i,j=0}^{2}(X_{01,ij}-X_{00,ij})\kappa_{ij}(\rho){\vphantom{\sum\limits_{i,j=0}^{2}}}\right)\\ &&+ \frac{d}{d\rho}(\kappa_{0,2}(\rho))\\ &&\times\left( \sum\limits_{i,j=0}^{2}(X_{02,ij}-X_{00,ij})\kappa_{ij}(\rho){\vphantom{\sum\limits_{i,j=0}^{2}}}\right)\\ &&+ \frac{d}{d\rho}(\kappa_{1,0}(\rho))\\ &&\left( \sum\limits_{i,j=0}^{2}(X_{10,ij}-X_{11,ij})\kappa_{ij}(\rho){\vphantom{\sum\limits_{i,j=0}^{2}}}\right)\\ &&+ \frac{d}{d\rho}(\kappa_{1,2}(\rho))\\ &&\times\left( \sum\limits_{i,j=0}^{2}(X_{12,ij}-X_{11,ij})\kappa_{ij}(\rho){\vphantom{\sum\limits_{i,j=0}^{2}}}\right)\\ &&+ \frac{d}{d\rho}(\kappa_{2,0}(\rho))\\ \\&&\times\left( \sum\limits_{i,j=0}^{2}(X_{20,ij}-X_{22,ij})\kappa_{ij}(\rho){\vphantom{\sum\limits_{i,j=0}^{2}}}\right) \\&&+ \frac{d}{d\rho}(\kappa_{2,1}(\rho))\\ &&\times\left( \sum\limits_{i,j=0}^{2}(X_{21,ij}-X_{22,ij})\kappa_{ij}(\rho){\vphantom{\sum\limits_{i,j=0}^{2}}}\right) \end{array} $$

Now we use that κ _{i, j} (ρ) = κ _{j, i} (ρ) for all i, j. Then we get

$$\begin{array}{@{}rcl@{}} \frac{d}{d\rho}E_{\rho}[LR_{\mathcal{P}ed}(\mathcal{H}_{\mathcal{P}ed})] &=& \frac{d}{d\rho}(\kappa_{0,1}(\rho))\\ &&\times\left( \sum\limits_{i,j=0}^{2}(X_{01,ij}-X_{00,ij}\right.\\ &&\qquad\left.+X_{10,ij}-X_{11,ij})\kappa_{ij}(\rho){\vphantom{\sum\limits_{i,j=0}^{2}}}\right) \\&+& \frac{d}{d\rho}(\kappa_{0,2}(\rho))\\ &&\times\left( \sum\limits_{i,j=0}^{2}(X_{02,ij}-X_{00,ij}\right.\\ &&\qquad\left.+X_{20,ij}-X_{22,ij})\kappa_{ij}(\rho){\vphantom{\sum\limits_{i,j=0}^{2}}}\right) \\&+& \frac{d}{d\rho}(\kappa_{1,2}(\rho))\\ &&\times\left( \sum\limits_{i,j=0}^{2}\left( X_{12,ij}-X_{11,ij}\right.\right.\\ &&\qquad\left.\left.+X_{21,ij}-X_{22,ij}\right)\kappa_{ij}(\rho){\vphantom{\sum\limits_{i,j=0}^{2}}}\right) \end{array} $$

All terms in the sums are smaller than or equal to zero. For example, considering (i, j)=(1, 2) in the last term, we get

$$(X_{12,12}-X_{11,12}+X_{21,12}-X_{22,12})\kappa_{12}(\rho),$$

which is equal to

$$\begin{array}{@{}rcl@{}} && \kappa_{12}(\rho)(E[LR_{1}^{(1)}({\mathcal H}_{1})]- E[LR_{2}^{(1)}({\mathcal H}_{1})])(E[LR^{(2)}_{2}(\mathcal{H}_{2})]\\ &&-E[LR^{(2)}_{1}(\mathcal{H}_{2})]), \end{array} $$

which is seen to be non-positive since it is the product of κ ₁₂(ρ)≥0 with a negative and a positive term.

The result then follows since

$$\frac{d}{d\rho}(\kappa_{ij}(\rho)) \geq 0 \text{ if } i \neq j, $$

which is a consequence of the fact that increasing recombination probabilities monotonously increases the probabilities that two linked markers have different numbers of IBD pairs for 0≤ρ≤1/2.

Appendix B: Additional example

Example B.1 (Variance for first cousins)

Now $\kappa _{0}=\frac {3}{4}, \kappa _{1}=\frac {1}{4}$ and therefore we get in a fashion similar to Example 3.1

$$\begin{array}{@{}rcl@{}} E[LR(\mathcal{H}_{P})^{2}]&=&\frac{27}{64}E[(LR_{0}\cdot LR_{0})(\mathcal{H}_{0})]\\ &&+ \frac{9}{64}E[(LR_{0}\cdot LR_{0})(\mathcal{H}_{1})]\\ && +\frac{9}{64}E[(LR_{0}\cdot LR_{1})(\mathcal{H}_{0})]\\&&+ \frac{3}{64}E[(LR_{0}\cdot LR_{1})(\mathcal{H}_{1})]\\ &&+\frac{9}{64}E[(LR_{1}\cdot LR_{0})(\mathcal{H}_{0})]\\ &&+\frac{3}{64}E[(LR_{1}\cdot LR_{0})(\mathcal{H}_{1})]\\&&+\frac{3}{64}E[(LR_{1}\cdot LR_{1})(\mathcal{H}_{0})]\\ && +\frac{1}{64}E[(LR_{1}\cdot LR_{1})(\mathcal{H}_{1})] \\&=& \frac{27}{64}+\frac{9}{64}+\frac{9}{64}+\frac{3}{64}\frac{L+3}{4} +\frac{9}{64}\\ &&+\frac{3}{64}\frac{L+3}{4}+\frac{3}{64}\frac{L+3}{4}+\frac{1}{64}E[LR_{1}(\mathcal{H}_{1})^{2}] \\&=& \frac{23L+489}{512}+\frac{1}{1024}\sum\limits_{a<b}\frac{p_{a}}{p_{b}}+\frac{p_{b}}{p_{a}}. \end{array} $$