# LIKELIHOOD RATIOS

19 Apr

**LIKELIHOOD RATIOS**

**Match probability statement**

DNA analysis of the blood stain from the crime scene gave a full DNA profile that matched that of the suspect. If this blood did not come from the suspect then the STR profile must match by chance. It is estimated that the chance of obtaining these matching profiles if the blood came from a random person unrelated to the suspect is in the order of 1 in a billion (a billion is a thousand million).

Many countries do not use the ceiling value of 1 in a billion and quote the calculated random match probability. In some countries, when the value gets to a certain point the evidence is no longer presented as a random match probability, rather it is presented as coming from the suspect. For example, in the US, random match probabilities of greater than 1 in 260 billion may be considered as identity by the US reporting scientist [6].

**Statement of uniqueness**

Based on the results of the DNA profiling, Mr J. Doe is the source of the DNA obtained from the ev identiary material,to a reasonable degree of scientific accuracy.

The frequent ist approach has the advantage that when dealing with small numbers the phraseology can be understood. It should be noted that part of the duty of the forensic scientist is to make the strength of the evidence understood by the jury or a judge and, therefore, to that extent the frequency approach succeeds. Thus, quoting a random match probability figure of 1 in 1 million, for instance, is relatively simple for the jury to understand and picture. Simplistically, if there are 1 million other people it would be expected that one person in that population would have the same DNA profile. There are however a number of disadvantages with the approach. Assume a full AmpFISTR® SGM Plus® DNA profile where the match probability is greater than 1 in 1 billion of the population. This leads to the observation that there are not 1 billion people in the UK so what population is being described? Indeed there are only 6.5 billion people on the planet at present. Such logical problems with the frequentist approach can be addressed by using a likelihood ratio.

**Likelihood ratios**

A likelihood ratio is the ratio of two competing hypotheses. In terms of a criminal case, it is the ratio of the prosecution hypothesis (Hp) and the defence hypothesis (Hd). The likelihood approach is a more logical way to interpret and present the profile frequency information as it considers an alternative scenario.

**THE EVALUATION AND PRESENTATION OF DNA EVIDENCE**

Three logical principles for the interpretation of DNA evidence and its quantification that have been suggested [7] are as follows:

(1) When evaluating DNA evidence two assumptions should be considered.

(2) Probability of occurrence of the evidence under each of the two assumptions should be quantified.

(3) The ratio of the probabilities under two assumptions should be quantified and considered.

By considering the two propositions where one is the alternative of the other (the prosecution proposition compared to the defence proposition), then the probability of the evidence if the prosecution case is true can be determined. If the DNA profile of a crime scene sample matches a suspect’s DNA profile then there can be two explanations:

(1) Prosecution hypothesis (Hp): the DNA profile originated from the suspect.

(2) Defence hypothesis (Hd): the DNA profile did not originate from the defendant but originated from another person.

The likelihood ratio is described in equation (9.2). It is the probability (Pr) of the DNA evidence (E) given the hypothesis put forward by either the prosecution (Hp) or the defence (Hd):

The prosecution hypothesis is that the defendant left the biological material at the crime scene. The DNA profiles of the defendant and the evidence from the crime scene match, and therefore it is certain under the prosecution’s hypothesis that the defendant left the material, hence Pr(E|Hp) = 1. The probability of occurrence under the defence hypothesis is equal to the probability of observing the profile if its source was somebody other than the suspect in the population to which the defendant belongs. The ratio of the two probabilities is given by:

Let us assume that a certain crime scene stain yielded a DNA profile that has matched that of a suspect who is being prosecuted. The frequency of the matching DNA profile was 1 in a million. The likelihood ratio in this case would be:

Because with DNA evidence the probability under the prosecution’s hypothesis is equal to 1, the value is the same as when calculating a random match probability;

however, the way that the value is expressed differs. The likelihood ratio would nor-mally be expressed in a report with a statement, for example, as in the likelihood ratio statement.

**Likelihood ratio statement**

DNA analysis of the blood stain from the crime scene gave a full DNA profile that matched that of the suspect. If this blood did not come from the suspect then the STR profile must match by chance. The results of the DNA analysis are approximately 1 million times more likely if the DNA came from the suspect than if the DNA came from a random unrelated male in the population.

The figure being quoted in a likelihood ratio is the odds in favour of the proposition putforwardbytheprosecution.Whenthestatementreadsthat‘itis1milliontimesmore likely that the DNA came from the accused than if it came from any unrelated male’, the figure of a million is not a probability but is an odds value, i.e. how many times more likely it is that the DNA matched the crime scene stain if it originates from the suspect, compared to coming from any other unrelated male. The frequency approach has a problem when the chance of a match, or the match probability, exceeds the total sample size. This is not the case when quoting odds in favour as odds can reach near infinity. In a horse race with only five horses, one horse may have odds against of 10 to 1 and here the odds outweigh the number of possibilities. The disadvantage with this statement is that it can seem to be cumbersome when presented to a jury. It is easy to make an error and state the probability that the evidential material came from the suspect instead of the probability that the DNA profile obtained from the evidential material matches that of the suspect. Using the current multiplex STR kits that analyse between 10 and 15 STR loci, the match probabilities and therefore likelihood ratios are extremely high. In order to avoid some of these complications in presenting huge numbers, and also because it is easier for the scientist to express verbally the weight of the evidence, it has been suggested [3, 7] that verbal scales might be used for likelihood ratios as presented in the table below:

Table 9.1 Verbal scales for likelihood ratios