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# 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 — 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

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 + 2p16 p17

FGA

1

p223 + 2p19 p23

D8S1179

1

p212 +p216 + 2p12 p16

D21S11

1

p231.2 + 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

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 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