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Population Genetics and Statistics for Forensic Analysts

Likelihood Ratio - Mixtures

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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

SourceD3S1358vWAFGAD8S1179D21S11
Evidence1516,1719,2312,1630,31.2,32.2
Victim1516,171912,1630,32.2
Suspect151619,231631.2

Two explanations are possible for the above mixture:

  • H1 — Contributors were the victim and the suspect
  • H0 — 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

LocusEvidenceVictimUnknown Individual
D3S135815,1515,1515,15
vWA16,1716,1716,16 or 16,17 or 17,17
FGA19,2319,1919,23 or 23,23
D8S117912,1612,1612,12 or 12,16 or 16,16
D21S1130,31.2,32.230,32.231.2,31.2; 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

LocusP(E/H1)P(E/H0)
D3S13581P215
vWA1P216 + P217 + 2p16 p17
FGA1p223 + 2p19 p23
D8S11791p212 +p216 + 2p12 p16
D21S111p231.2 + 2p30 p31.2 +2p31.2 p32.2
TOTAL (Product)1
 
Product of above
 

 

For the above example, the following frequencies were used to determine
P(E/H1).

Frequencies

LocusAlleleFrequencyAlleleFrequencyAlleleFrequency
D3S1358150.2463    
vWA160.2015170.2627  
FGA190.0561230.1581  
D8S1179120.1454160.0138  
D21S11300.232131.20.099432.20.1122

The following table shows the calculations for P(E/H1) given the above allele frequencies.

Calculations

LocusP(E/H1)P(E/H0)
D3S13581(0.2463)2 = 0.0607
vWA1(0.2015)2 + (0.2627)2 + 2(0.2015)(0.2627) = 0.2154
FGA1(0.1581)2 + 2(0.0561)(0.1581) = 0.0427
D8S11791(0.1454)2 + (0.0138)2 + 2(0.1454)(0.0138) = 0.0253
D21S111(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 EvidenceStatistical 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

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