## Contents: |

## Links: |

Charles H. Brenner, Ph.D. (home page)

E-mail: information or discussion

A computer program calculates likelihood ratios (LR) for the infinite class of mixed stain problems described by Weir et al (1997).

A feature of especial interest is that the appropriate algebraic expression for the LR is derived and displayed, not just the numeric value. As an example of the benefit, it can thereby be seen that in some cases so-called "conservative" allele frequencies would have an unfortunately anti-conservative effect in evaluating the evidential value of the genetic data.

The algorithm is well-adapted to systems with discrete alleles; RFLP systems can present special difficulties. If bands are close to one another or, even worse, the number of bands is not certain, then no program can substitute for human judgement and case-by-case analysis. However, the spreadsheet-like presentation of the program makes it a useful tool even in these ambiguous cases because various interpretations can conveniently be computed and compared.

**Keywords:** mixed stains, DNA profiles, likelihood ratios

Weir et al (1997) formulate a class of mixed stain situations
wherein a set *E* of stain bands or alleles is observed
in the evidence. Some of these are satisfactorily accounted for by
known or presumed contributors, and the remainder *U* are
ascribed to *x*≥0
unknown contributors. The probability to see such evidence
under such circumstances is

(*)

where = the total of the frequencies of the alleles that
are in *E* and are not among the *m* alleles *j, k,
... .*

The likelihood ratio for the strength of the mixed stain evidence is therefore the ratio of two probabilities of the form (*).

The Mixed Stain Calculator discussed here lets the user describe a mixed stain scenario into a spreadsheet-like screen shown as Figure 1 below [or, a better image of the screen]. The computer then uses (*) to create a symbolic likelihood ratio expression – 1/(2pq) would be the simplest example – and to evaluate it using numeric frequencies either known to the program from its allele frequency databases, or typed in by the user.

The line with code letters – mostly U, S, and V – gives the
*interpretation* of each allele.

V means an allele that is satisfactorily explained by its presence in the victim or other source (such as a confessed accomplice or a boyfriend) that would be stipulated by both sides (i.e. prosecution and defense).

U codes an allele that is assumed (by both sides) to
have been contributed by one of the *x* unknown persons. The
number *x* may be, usually is, different for prosecution and
defense, and is specified by the boxes after Ho and H1 at the bottom
of the screen.

S codes the alleles that are in dispute – typically that the prosecution ascribes to the suspect, and the defense disagrees. In other words, from the defense point of view S is the same as U.

The algebraic (symbolic) version of the likelihood ratio is completely determined by the interpretation letters and the values in the boxes after Ho and H1.

*Conservative frequencies?*

The mixed stain calculator is a useful tool in part because it can be used to gain insight. For example, it is traditional in stain work to assume that "conservative," i.e. overly large, allele frequencies are generous to the suspect. In the case of a one-person, non-mixed stain, the "conservative" theory makes sense. The matching suspect who wished to allege coincidence certainly would like to claim that his alleles are common.

But what of more complex cases, where some of the alleles present
are not ascribed to the suspect? Might he not then prefer that
*those* alleles are particularly rare? As a first example, try
a PQRS stain and a PQ suspect. The likelihood ratio is 1/(12pq), so
the frequencies of R and S are irrelevant. This example doesn't work.

Pursuing the idea further turns up a similar example that does work. Suppose a PQ stain and homozygous P suspect. The prosecution suggests that there is a Q or a PQ accomplice; the suspect says no, the perpetrator is an unknown PQ person. The likelihood ratio turns out to be 1+q/(2p), so the more common Q is considered to be, the more plausible is the defense. A conservative frequency for Q is no favor to the accused.

*RFLP*

The preceding problem would be particularly annoying if it occurred in a RFLP context, because normally nothing but conservative frequencies are available. Fortunately the scenario is improbable because of the almost invariable rareness of RFLP bands – at least by the prosecution theory, three of four assailant alleles are the same, an unlikely occurrence.

Still it is worth considering cases where the mixed stain evidence shows two bands that are very close to one another. A two-band near-coincidence could happen.

The issue then is, is the formula (*) still valid? A close study
of the role that the definition of plays in the derivation of
(*) shows that it is not. When there are two close-by bands, say
*j* and *k*,ß then among the measurements in the
fragment frequency database there will be some that can alternatively
be mistaken for either of the two bands, and this seems a great
difficulty.

Nonetheless, the right tool might give a helpful result in a particular case. Suppose for example that the RFLP bands are P, Q in the suspect, and the stain shows P, Q, R, and possibly P', where P' is close to P and might even be indistinct from P. The idea is to make various "test" computations.

Assume first that P' is completely distinct from P. With four
completely separate bands the likelihood ratio comes to 1/(12pq)
favoring (H_{0}) Suspect and untyped accomplice over
(H_{1}) two unknown assailants.

Next compute on the assumption that P' and P are the same. The
likelihood ratio favoring H_{0} over H_{1} then would
be (2p+2q+r) / ((p+q+r)(12pq)), which is a larger number.

Even though it is hard to say what the exact likelihood ratio is, it must be somewhere in between the two extremes just described. Therefore in this case the four-band computation can be cited without prejudice to the defense.

Brenner CH (1997) Proof of a mixed stain
formula of Weir,
(Appendix)* J For Sci *42(2)221-222

Weir BS,
Triggs CM, Starling L, Stowell LI, Walsh KAJ, Buckleton J
(1997) Interpreting DNA mixtures. *J For Sci *42(2)213-222

Go to top