## Likelihood Ratio Approach

The likelihood ratio (LR) is the ratio of possibilities under alternative propositions and provides a reliable method that is able to make full use of available genetic data (Evett and Weir 1998). Two competing hypotheses are setup, the hypothesis of the prosecution (Hp), which is that the defendant committed the crime, and the hypothesis of the defense (Hd) that some unknown individual committed the crime. Thus, the likelihood ratio involves a ratio describing the probability of the evidence given the prosecution's hypothesis over the probability of the evidence given the defense's hypothesis:

Unfortunately, determination of which hypotheses to consider is not necessarily straightforward (Ladd et al. 2001). Interpretation of a mixture depends on the circumstances of the case and involves assumptions about the identity and number of contributors to the mixture in question (DAB 2000). LR calculations are more widely used in Europe than the United States for forensic applications. Paternity testing on the other hand routinely uses

LR calculations. In the end though, the LR method makes better use of the available genetic data than does the probability of exclusion method discussed previously.

If evidence contains four alleles at a locus (A1, A2, A3, A4) and the victim possesses A3 and A4 while the suspect exhibits A1 and A2, then the prosecution's hypothesis would be that the DNA evidence is from the victim and the suspect. On the other hand, the defense's hypothesis would be that the DNA evidence is from the victim and an unknown person. The probability of the prosecution's hypothesis is one because their position is that they are 100% confident (probability = 1) that the defendant committed the crime, which is why the trial is occurring in the first place. The defense's hypothesis can vary depending on the circumstances of the case, such as the number of other possible contributors under consideration and the alleles present in the evidentiary DNA profile. In the end, the likelihood ratio describes the relative chance of observing a specific mixture and combination of STR alleles. Some LR examples for various scenarios are listed in Table 22.1.

Using the example shown in Figure 22.1 of evidence containing alleles 11, 12 and 14 at the STR locus D13S317 and a suspect with alleles 11 and 14, we can use the second formula in Table 22.1 to deduce the likelihood ratio for the scenario where three peaks are seen in the crime scene evidence and two of those alleles are present in the suspect. In this case the LR is 8.23 with pj = 0.24834 (allele frequency of D13S317 allele 12 from Caucasians in Appendix II), p2 = 0.33940 (allele frequency of D13S317 allele 11 from Caucasians in Appendix II), and p3 = 0.04801 (allele frequency of D13S317 allele 14 from Caucasians in Appendix II).

### Table 22.1

Likelihood ratios for mixed stains using p2 rule for homozygotes (from Table 5.1 of NRCII, p. 163). Allele frequencies (p) for observed (A) can be obtained from population databases, such as Appendix II.

Crime Scene Suspect

Likelihood Ratio

A1A2A3A4 A1A2

A1A2A3

A2A3

A1 A2 A3

A1A2

A1A2

A1A2 