The statistical weight of mixed samples with allelic

drop outFirst serious attempt by Gill et al. 2006, Forensic Science International 160:90

An important general paper about mixtures: Curran et al. 1999, J. Forensic Science 44:987

Mixed sample with drop out

Standard Mixture Analysis

• Assume there are 2 people and 3 alleles: A1, A2, A3

• There must be a total of 4 alleles allowing for the following possible combinations: (A1,A1,A2,A3) and (A1,A2,A2,A3) and (A1,A2,A3,A3).

• Let the frequency of the 3 alleles bep1

p2

p3

Details of (A1,A1,A2,A3)

• Possible pairs of sampled genotypes are:[A1/A1 and A2/A3] or [A2/A3 and A1/A1][A1/A2 and A1/A3] or [A1/A3 and A1/A2]

• These pairs are chosen with frequencies2[p1

22 p2 p3]2[2 p1 p22 p1 p3]

• The sum of these is 12p12 p2 p3

• Repeating this for the other two orderings and adding them all up gives 12p1p2 p3(p1+p2+p3)

General formula

• let c=number of distinct alleles• x= number of people in the mixture• ui= number of copies of allele i• the frequency of any particular

combination iui

c

ic

ii

pu

x1

1

!

!2

Mixtures with drop out

• Let Q be the dropped out allele

• The frequency of Q is 1-sum(distinct alleles)

• Suppose evidence is A1,A2,Q

• Possible orderings are (A1,A1,A2,Q) and (A1,A2,A2,Q) but not (A1,A2,Q,Q) since we have assumed only one allele dropped out

• frequency is 12p1p2 pQ(p1+p2)

Two people mixtures

Number of alleles out

evidence frequency

1 A1,A2, A3,Q 24p1p2p3 pQ

1 A1,A2,Q 12p1p2 pQ(p1+p2)

1 A1,Q 4p13

pQ

2 A1,A2,Q 12p1p2 pQ2

2 A1,Q 6p12

pQ2

3 A1,Q 4p1 pQ3

Likelihood Ratios

• Compare the probability of two hypotheses, the prosecution and the defense

• Each hypothesis must compute the probability of the observed genetic evidence

• Let L = Prob[evidence|prosecution] / Prob[evidence|defense]

Example

• Three person mixture• Evidence: 9• Suspect: 11, 14• Two alleles dropped out• Let D be the probability that one allele will drop

out.• In this sample the State assumes at least two

alleles dropped out, and four alleles did not:• This probability is: (1-D)4D2

Example: state hypothesis

• (1-D)4D2 {prob[two people with only the 9 allele]}

• (1-D)4D2 p94

Example: defense hypothesis

• There are several possibilities

• No drop out: (1-D)6 p96

• One allele dropped out, five did not: (1-D)5D prob[three people with only the 9 allele and one allele dropped out] = (1-D)5D 6p9

5pQ

• Two alleles dropped out, four did not: (1-D)4D2prob[three people with only the 9 allele and two alleles dropped out] =(1-D)4D215p9

4pQ2

Example Results

• 13 loci with a total of 5 alleles dropped out and a minimum of three people in the mixture, 1 known, 2 unknown

• The lab CPI for Caucasians was 1 in 42 million

D L

0.001 0.0008

0.01 8

0.05 1500

0.1 2800

0.2 3000

0.5 2100

0.9 2900