## Mixture likelihood ratio based on exclusion20 June 2000 |
Forensic mathematics index Likelihood ratio exclusion |

The likelihood ratio method – for example for paternity, or for a mixed stain in a criminal situation – consists of formulating two hypotheses and comparing the probability to see the evidence, assuming each hypothesis in turn.

If we take a limited view of the evidence, the same scheme produces an "exclusion likelihood ratio." The limited view consists in ignoring the details of the man's type, and only taking cognizance of the fact that he is "excluded." I put "excluded" in quotes because it is a questionable, if not downright phoney, concept.

In the case of a mixed stain, we probably define exclusion like this:

A man is excluded if he has any allele at any locus not detected in the stain.

Normally we do not take any account of number of contributors. Thus, if the stain is PQRS, a homozygous Q man is not excluded even if the witness swears there were only two assailants.

We define A=probability to exclude a random non-contributor. Given the stain, then, there is a formula to evaluate A. For the likelihood ratio formulation, we let

E=evidence= DNA types of the stain, plus the fact that the suspect is not excluded. |

H_{0} = "suspect contributed" |

H_{1} = "suspect is unrelated to any contributor" |

Then X=P(E|H_{0}) = 1

Y = P(E | H_{1}) = 1-A.

Therefore LR_{exclusion}= 1/(1-A).

The formulas come out naturally in terms of 1-A and
1-A* _{i}*, so we denote these quantities by B and
B

If we believe in the product rule across loci,
B=**PROD** B* _{i}*.

To see how to evaluate the locus-specific inclusion probability
B_{i}_{}, consider an example. Suppose that
*p, q, r, s* are the frequencies of alleles P, Q, R, S
respectively, and that *o* (which may be 0) is the rate of
null or undetected alleles.

Then for the locus *i*
mentioned above, any man both of whose alleles are among P,Q,R,S or
null will be *included*.
If you accept the product rule within a locus,

B* _{i}* =
(

It is easiest to understand this expression by taking the point of
view that P, Q, etc are sub-types of some type Z={P, Q, R, S, null}.
The set of eligible men is then the set of men who are "homozygous
for Z." Since the frequency of the type Z is
*p*+*q*+*r*+*s*+*o*, the
expression for B_{i}_{} is now obvious.

Alternatively, by expanding the square we get

B* _{i}* =

So that B_{i}_{} represents the sum
of the frequencies of all genotypes consistent with the stain.
Therefore, if we want to abandon the assumption of the product rule
for genotype frequencies and instead follow the NRC II recommendation
of compensating for homozygous types based on population
substructure, instead of *p*^{2} we should write
*p*^{2} + *p*(1-*p*)theta. This is what the
DNA•VIEW
**DNA exclusion
** command does.