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- 01: Evidence & DNA
- 02: Forensic Biology
- 03: DNA Extraction & Quantitation
- 04: DNA Amplification
- 05: Amplified DNA Product Separation
- 06: STR Data Analysis & Interpretation
- 07: Population Genetics & Statistics
- 08: Communicating Results
- 09: Other DNA Markers & Technologies
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- Laboratory Training Manual
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Advancing Justice Through DNA Technology
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Advancing Justice Through DNA Technology
DNA Analyst Training
Likelihood Ratio - Mixtures
Home > Population Genetics & Statistics > Statistics > Logical Approach > Likelihood Ratio - Mixtures
Although likelihood ratios can be used for determining the significance of single source crime stains, they are more commonly used in mixture interpretation. The following example show the likelihood without a theta correction. A theta correction can be applied to the likelihood ratio calculation. Refer to NRCII formulas 4.10a and 4.10b.
Example of Two Person Mixture |
|||||
Source |
D3S1358 |
vWA |
FGA |
D8S1179 |
D21S11 |
Evidence |
15 |
16,17 |
19,23 |
12,16 |
30,31.2,32.2 |
Victim | 15 |
16,17 |
19 |
12,16 |
30,32.2 |
Suspect | 15 |
16 |
19,23 |
16 |
31.2 |
Two explanations are possible for the above mixture:
- H1 (also stated as
) — Contributors were the victim and the suspect - H0 (also stated as
) — Contributors were the victim and an unknown individual
The evidence is certain under H1. Under H1 the probability of the evidence depends on the chance of obtaining the evidence alleles (and no other alleles) from an unknown individual.
The table below shows all possible genotypes for an unknown individual, given the genotypes of the evidence and the victim.
Possible Genotypes |
|||
Locus |
Evidence |
Victim |
Unknown Individual |
D3S1358 |
15,15 |
15,15 |
15,15 |
vWA | 16,17 |
16,17 |
16,16 or 16,17 or 17,17 |
FGA | 19,23 |
19,19 |
19,23 or 23,23 |
D8S1179 | 12,16 |
12,16 |
12,12 or 12,16 or 16,16 |
D21S11 | 30,31.2,32.2 |
30,32.2 |
31.2,31.2; 30,31.2; or 31.2,32.2 |
The table below shows the equations used to determine the P (E/H1) and P (E/H0) assuming Hardy-Weinberg Equilibrium, where p=allele frequency.
Equations |
|||
Locus |
P(E/H1) |
P(E/H0) |
|
D3S1358 |
1 |
P215 |
|
vWA | 1 |
P216 + P217 + 2p16p17 |
|
FGA | 1 |
p223 + 2p19p23 |
|
D8S1179 | 1 |
p212+p216+ 2p12p16 |
|
D21S11 | 1 |
p231.2 + 2p30p31.2+2p31.2p32.2 |
|
TOTAL (Product) |
1 |
Product of above |
|
For the above example, the following frequencies were used to determine
P(E/H1).
Frequencies |
||||||
Locus |
Allele |
Frequency |
Allele |
Frequency |
Allele |
Frequency |
D3S1358 |
15 |
0.2463 |
||||
vWA |
16 |
0.2015 |
17 |
0.2627 |
||
FGA |
19 |
0.0561 |
23 |
0.1581 |
||
D8S1179 |
12 |
0.1454 |
16 |
0.0138 |
||
D21S11 |
30 |
0.2321 |
31.2 |
0.0994 |
32.2 |
0.1122 |
The following table shows the calculations for P(E/H1) given the above allele frequencies.
Calculations |
|||
Locus |
P(E/H1) |
P(E/H0) |
|
D3S1358 |
1 |
(0.2463)2 = 0.0607 |
|
vWA |
1 |
(0.2015)2 + (0.2627)2 + 2(0.2015)(0.2627) = 0.2154 |
|
FGA |
1 |
(0.1581)2 + 2(0.0561)(0.1581) = 0.0427 |
|
D8S1179 |
1 |
(0.1454)2 + (0.0138)2 + 2(0.1454)(0.0138) = 0.0253 |
|
D21S11 |
1 |
(0.0994)2 + 2(0.2321)(0.0994) + 2(0.0994)(0.1122) = 0.0783 |
|
TOTAL (Product) |
1 |
0.0000011 |
|
To determine the likelihood ratio, the above numbers are inserted into the previous formula as follows:
LR = P(E/H1) / P(E/H0)
LR= 1/0.0000011
LR= 909,091
The results are 909,091 times more likely if the victim and the suspect are the contributors of the mixture rather than the victim and a random individual in the population.18
NOTE: For mixtures with more than one unknown, review Interpreting DNA Evidence: Statistical Genetics for Forensic Scientists, Evett, I.W. and Weir, B.S., Sinauer Associates, Inc., 1998.
The use of any formula for mixture interpretation should only be applied to cases in which the analyst can reasonably assume “that all contributors to the mixed profile are unrelated to each other, and that allelic dropout has no practical impact.”01
