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# Likelihood Ratio

Conditional probabilities can assist when estimating the probability that evidence came from an identified source. The probability estimate is based on calculation of a likelihood ratio (LR).03 The likelihood ratio is the ratio of two probabilities of the same event under different hypotheses. Thus for events A and B, the probability of A given that B is true (hypothesis #1), divided by the probability of event A given that B is false (hypothesis #2) gives a likelihood ratio. The likelihood ratio is a ratio of probabilities, and can take a value between zero and infinity.02 The higher the ratio, the more likely it is that the first hypothesis is true.

In forensic biology, likelihood ratios are usually constructed with the numerator being the probability of the evidence if the identified person is the source of the evidence, and the denominator being probability of the evidence if an unidentified person is the source of the evidence.

The results can be interpreted as follows:

• LR < 1 — the genetic evidence has more support from the denominator hypothesis
• LR = 1 — the genetic evidence has equal support from both numerator and denominator hypotheses
• LR > 1 — the genetic evidence has more support from the numerator hypothesis16

These likelihood ratios can be translated into verbal equivalents that depict, in a relative way, the strength of the particular likelihood ratio in consideration.16,03 These verbal equivalents, however, are only a guide.17

Table of Verbal Equivalents

Limited evidence to supportLR <1-10
Moderate evidence to supportLR 10-100
Moderately strong evidence to supportLR 100-1000
Strong evidence to supportLR 1000-10000
Very strong evidence to supportLR >10000

The following equation can be used to determine the probability of the evidence given that a presumed individual is the contributor rather than a random individual in the population.

LR  =  P(E/H1) / P(E/H0)

P(E/H1) is the probability of the evidence given a presumed individual is the contributor.

P(E/H0) is the probability of the evidence given the presumed individual is not the contributor of the evidence.

In the case of a single source sample, the hypothesis for the numerator (the suspect is the source of the DNA) is a given, and thus reduces to 1. This reduces to:

LR  =  1/ P(E/H0) which is simply 1/P, where P is the genotype frequency.

The use of the likelihood ratio for single source samples is simply another way of stating the probability of the genotype and, while stated differently, is the same as the random match probability approach.