It was one of my brothers

Abstract

When DNA evidence is used to implicate a suspect, it may be of interest to know whether it is likely that the suspect's near relatives also share the suspect's DNA profile. In this study we discuss methods for evaluating the probability that at least one of a set of the suspect's full or half-siblings shares the suspect's DNA profile. We present three such methods: exact calculation, estimation via Monte Carlo simulations, and estimation by means of sandwiching the probability between an upper and a lower bound. We show that, under many circumstances, this upper bound itself provides an extremely quick and accurate estimate of the probability that at least one of the relatives matches the suspect's profile.

Appendix

The probability that both individuals in a pair of the suspect's relatives share the suspect's single-locus genotype can be found by summing over all possible genotypes among their parents in a manner similar to that described in Materials and methods. The results for various relative pairs are given as in the following equations, using the notation described previously and the convention that C _j =(1+jθ)(1+(j−1)θ)...(1+θ).

For the case in which both individuals in the pair are full siblings to ${\user1{{\wp }}}$, we derive:

$$\Pr {\left( {{\text{both match}}\left| {P = A_{1} A_{1} } \right.} \right)} = \frac{{8M_{{1,2}} {\left( {1 + 2\theta + M_{{1,3}} } \right)} + M_{{2,1}} M_{{2,0}} }}{{16C_{2} }}$$

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$$\Pr {\left( {{\text{both match}}\left| {P = A_{1} A_{2} } \right.} \right)} = {{\left[ {{\left( {1 + 2\theta } \right)}{\left( {4M_{{1,1}} + 4M_{{2,1}} + M_{{3,0}} } \right)} + 12M_{{1,1}} M_{{2,1}} } \right]}} \mathord{\left/ {\vphantom {{{\left[ {{\left( {1 + 2\theta } \right)}{\left( {4M_{{1,1}} + 4M_{{2,1}} + M_{{3,0}} } \right)} + 12M_{{1,1}} M_{{2,1}} } \right]}} {16C_{2} }}} \right. \kern-\nulldelimiterspace} {16C_{2} }$$

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The next equations hold when one individual is a full sibling and the other is a half-sibling to ${\user1{{\wp }}}$, or (same formula) when both individuals are half-siblings to the suspect but are full siblings to each other.

$$\Pr {\left( {{\text{both match}}\left| {P = A_{1} A_{1} } \right.} \right)} = \frac{{M_{{1,2}} {\left( {8M_{{1,4}} M_{{1,3}} + 6M_{{1,3}} M_{{2,0}} + M_{{2,1}} M_{{2,0}} } \right)}}}{{8C_{3} }}$$

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$$\Pr {\left( {{\text{both match}}\left| {P = A_{1} A_{2} } \right.} \right)} = {\left[ {{\left( {1 + 3\theta } \right)}{\left( {2M_{{1,2}} M_{{1,1}} + 2M_{{2,2}} M_{{2,1}} + 10M_{{1,1}} M_{{2,1}} + M_{{3,0}} {\left( {M_{{1,1}} + M_{{2,1}} } \right)}} \right)} + 6M_{{1,1}} M_{{2,1}} {\left( {M_{{1,2}} + M_{{2,2}} } \right)}} \right]}/16C_{3} .$$

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For pairs of individuals in which both individuals are half-siblings to the suspect and are half-siblings to each other, the following equations hold:

$$\Pr {\left( {{\text{both match}}\left| {P = A_{1} A_{1} } \right.} \right)} = \frac{{M_{{1,3}} M_{{1,2}} {\left( {3M_{{1,5}} + 1} \right)}}}{{4C_{3} }}$$

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$$Pr{\left( {{\text{both match}}\left| {P = A_{1} A_{2} } \right.} \right)} = {\left[ {{\left( {1 + 3\theta } \right)}{\left( {M_{{1,2}} M_{{1,1}} + M_{{2,2}} M_{{2,1}} } \right)} + 6M_{{1,1}} M_{{2,1}} {\left( {M_{{1,2}} + M_{{2,2}} } \right)}} \right]}/8C_{3} $$

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The remaining half-sibling case is the situation in which the two half-siblings are unrelated to each other (i.e., one is related to the suspect through the suspect's mother and the other is related through the father). The equations for this type of half-sib pair are as follows:

$$\begin{aligned} & \Pr {\left( {{\text{both match}}\left| {P = A_{1} A_{1} } \right.} \right)} = {\left[ {M_{{1,3}} M_{{1,2}} {\left( {4M_{{1,5}} M_{{1,4}} + 4M_{{1,4}} M_{{2,0}} + M_{{2,1}} M_{{2,0}} } \right)}} \right]}/4C_{4} \\ & \\ \end{aligned} $$

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$$\Pr {\left( {{\text{both match}}\left| {P = A_{1} A_{2} } \right.} \right)} = M_{{1,1}} M_{{2,1}} {\left[ {{\left( {1 + 4\theta } \right)}{\left( {3M_{{1,2}} + 3M_{{2,2}} + M_{{3,0}} } \right)} + 4M_{{1,2}} M_{{2,2}} } \right]}/4C_{4} .$$

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