Int J Legal Med (1996) 109:218-219

SHORT COMMUNICATION

CH Brenner (2486 Hilgard Ave, Berkeley Ca 94709 USA, email)

R Fimmers,
MP Baur
(Institute for Medical Statistics, University of Bonn)

Likelihood Ratios for Mixed Stains

When the Number of Donors Cannot be Agreed

**Abstract: **Suppose that part of the prosecution’s evidence in some crime case is analysis of a
blood stain, and that the traits E discovered in the stain suggest multiple donors. Then the
prosecution will probably allege some specific inculpatory hypothesis H_{0} about the sources of the
stain, and Pr(E|H_{0}) can be calculated. It is desirable to use this as the numerator of a likelihood
ratio. However, in general the obvious denominator Pr(E|~H_{0}) cannot be calculated, so unless the
defense is sufficiently obliging as to stipulate to a specific choice among the potentially infinite
number of more or less exculpatory alternative hypotheses, the desired likelihood ratio can’t be
evaluated. We show that nonetheless, in most cases there is an adequate inequality.

**Discussion**

*Notation:*

E denotes some evidence, consisting for example of a handful of RFLP bands under some probe.

H_{0} is the prosecution’s explanation, such as that E comes from the suspect and the two victims.

H* _{i}*,

Put *e _{i}*=Pr(E|H

We have heard the claim that the probability *e*’ and therefore the
likelihood ratio *e*_{0}/*e*’ cannot be
computed, and therefore that several possibilities *e*_{0}/*e _{i}*,

That is, *e*’ depends on the allocations Pr(H* _{i}*|~H

However, it does not follow that a variety of calculations must be presented. Although the reasoning is simple we feel it is worthwhile setting out in advance rather than relying on the ability of the expert to think clearly under the pressure of cross-examination.

Even though there is no way to compute e’ itself we can usually provide a very useful upper
bound for it, and hence a lower bound for *e*_{0}/*e*’. If there is a largest
among the *e _{i}*,

Therefore even though the desired likelihood ratio *e*_{0}/*e*’ can’t be calculated explicitly, the
prosecution can simply say that it is at least *e*_{0}/*e*_{max}, which can be calculated.

**Limitation: **When the blood stain evidence consists of multiple bands in an RFLP typing, *e*_{max}will
surely exist (and will usually correspond to the minimum number of people sufficient to contribute
the observed number of bands). However, if every allele of some discrete-allele system is
represented -- which can easily happen with DQa for example -- then Pr(E|*i* contributors)
APPROACHES 1 as*i *increases and our method is not helpful. In such a case if any incrimination
is to be inferred from the evidence the onus is on the prosecutor to present effective arguments
limiting the plausible number of contributors.

To illustrate, suppose the sperm fraction in a multiple rape case shows the THO1 alleles 6, 9, 9.3,
and 10, with respective frequencies *a*=22%, *b*=17%, *c*=33%, and *d*=2%. The prosecution believes
that the 9, 9.3 suspect and one other man contributed. The defense’s best claim is that there were
*three* assailants not including the suspect. Under this interpretation the evidence is actually
exculpatory, with a likelihood ratio of

But if the prosecution can persuade the jury that the number of assailants is only two, then the likelihood ratio grows to

.

**Other situations: **Similar reasoning can also be used to sidestep some other kinds of ambiguity.
For example:

*unknown accomplice: *If the prosecution believes that the stain comes from the suspect and an
accomplice of unknown race, then there is no prejudice in calculating Pr(E|H_{0}) as if the
accomplice has the race that least commonly provides the unexplained traits.

*unknown race: *When the prosecution alleges that the stain comes from the suspect alone and the
alternatives are that it came from an unknown person, there is no need to present multiple
calculations corresponding to various races. There is no prejudice to the defense in assuming the
most probable race for the perpetrator when calculating the denominator of the likelihood ratio.

**Conclusions: **In summarizing blood stain evidence to the court it is a common practice to
present multiple calculations to the court, corresponding to varying assumptions. This practice
may (or may not) impress the court with the expert’s technical wizardry, but it usually has no
logical merit. If the likelihood ratios comparing the various prosecution versus defense hypotheses
are all large, an adequate and common sense approach is to present only the smallest ratio. The
prosecutor can then honestly and effectively argue that the evidence is *at least* so strong, without
needing to argue as to the *a priori* likely number or race of contributors, and without giving extra
data of uncertain and unexplained relevance.

1. For example this procedure was mandated in the murder trial of O.J. Simpson.