Estimating the number of contributors to a DNA profile

Abstract

The broad topic of this paper is the evaluation of DNA evidence in criminal cases. More specifically, we deal with mixture evidence which refers to cases where there are, or could be, several contributors to a biological stain based on, e.g., blood or semen. The present paper adresses DNA mixtures based on single nucleotide polymorphism (SNP) markers, i.e., diallelic markers. Based on STR analysis, it is in most cases easy to identify the presence of a mixture since three or four bands will show up with a high probability for at least one locus. Obviously, this will not be the case for diallelic markers and interpreting mixtures will be a great challenge. We address this problem by first approaching the more general problem of estimating the number of contributors to a stain. In addition we discuss how the markers should be selected and how many are required.

Appendix

Consider the likelihood L(x) given in Eq. 3. Observe that the likelihood increases in x if all z _i =2 and decreases in x if all z _i <2. Assume that not all z _i equals 2. Then we show below that L(x) always has a single maximum for some x≥1.

Maximizing L(x) is equivalent to maximizing

$${\matrix{ {{f{\left( x \right)}} \hfill} & { = \hfill} & {{\log \;L{\left[ x \right]}} \hfill} \cr {{} \hfill} & { = \hfill} & {{{\sum\limits_{i = 1}^N {I{\left( {z_{i} = 0} \right)}2x\log p_{i} + I{\left( {z_{i} = 1} \right)}2x} }\log {\left( {1 - p_{i} } \right)}} \hfill} \cr {{} \hfill} & { = \hfill} & {{Cx + {\sum\limits_{i = 1}^N I }{\left( {z_{i} = 2} \right)}\log {\left( {g_{i} {\left( x \right)}} \right)},} \hfill} \cr } + I{\left( {z_{i} = 2} \right)}\log {\left( {1 - p^{{2x}}_{i} - {\left( {1 - p_{i} } \right)}^{{2x}} } \right)}}$$

where C<0 and

$${g_{i} {\left( x \right)} = 1 - p^{{2x}}_{i} - {\left( {1 - p_{i} } \right)}^{{2x}} }$$

We see by direct computation that ${g^{{''}}_{i} {\left( x \right)} &lt; 0}$ for all x>0. Defining h _i (x)=log(g _i (x)), it follows that ${h^{{''}}_{i} {\left( x \right)} &lt; 0}$ for all x>0, and thus that f(x)<0 for all x>0. Further, we get that lim _x→∞ g _i (x)=1, that lim _x→∞ h _i (x)=0, and that lim _x→∞ f(x)=–∞. These two facts about f show together that f, and thus L, has a unique maximum for some x≥0. For discrete x, one or two consecutive positive integers maximize L.