Kinship Analysis inImmigration Cases

Charles H. Brenner, Ph.D.consulting in forensic mathematicsOakland, Californiahttp://dna-view.com

Outline

• I. Principles of analysis– Likelihood ratio comparing hypotheses

– comparing more than 2 hypotheses

• II. Computer demonstration– DNA·VIEW Kinship program

• III. Genetic anomalies• IV. Role of the laboratory

– attain requisite LR?

– answer questions, or ask them?

I. Principles of analysis

• Likelihood ratio comparing hypotheses– Paternity trio is the prototype– gotta be careful about treating it as archetype

Likelihood ratio(paternity trio)

• Two possible explanations of

• genetic Evidence: M=pr AF=qs C=pq

• LR = PI = P(E if H0) / P (E if H1)

– how much more typical E is of H0, than of H1

– (not how much more likely H0 is, than H1)

M

C

AFAFM

C

Explanation H0 (AF=father) Explanation H1 (AF unrelated)

Likelihood ratio(paternity trio)

• genetic Evidence: M=pr AF=qs C=pq

• LR = P(E if H0) / P (E if H1) = (2pr)(2qs)(¼) / (2pr)(2qs)(½q)

• shortcut to avoid today: Evidence: C=pq– LR = ¼ / ½q

M

C

AFAFM

C

Explanation H0 (AF=father) Explanation H1 (AF unrelated)

Daughter or unrelated?Explanation H0 (daughter) Explanation H1 (unrelated)

Resident

ReferenceApplicant

• genetic Evidence: M= pr Ref= ps Applicant= ps• LR = P(E if H0) / P(E if H1)

–P(E if H0) = P(E and F=ss if H0) + P(E and F=sx if H0) = (2pr)(s2)(½)(½) + (2pr)(2s(1-s))(¼)(¼) –P(E if H1) = (2pr)(s2)(2ps)(½) + (2pr)(2s(1-s))(2ps)(¼)

pr

pspsApplicant

Resident mother

Reference child

Man F (not tested)

• LR = (1+s) / 8ps

demo: DNA·VIEW derives LR formula

demo: DNA·VIEW derives LR formula

DNA·VIEW computes LR

KINSHIP

File: W:\meetings\AABB20~1\DAUGHTER.KIN Analysis mode: Co-dominant Autosomal

; AABB demo Nov 4, 2000; Is Applicant the daughter of Mother, or unrelated?Applicant/? : Mother + FredReference : Mother + FredApplicant psReference psMother prLikelihood ratio = 2.71 (1+s) / 8ps -- --Likelihood ratio:(1+s) / 8psLikelihood ratio = 2.71 (using p=0.2 s=0.3)

Daughter — or sister?Explanation H0 (daughter) Explanation H2 (sister)

Resident

Reference

ApplicantResident mother

Reference child

Applicant

• If H2 is true, probably can disprove H0

• If H0 is true, cannot disprove H2

–LR probably strongly against it though.• LR = 59,900. Forget about “sister”

»(or unrelated — LR > million)

demo: DNA·VIEW computes LR

demo: DNA·VIEW computes LR

DNA·VIEW computation of “daughter or sister” LR

Data for case 1204 2000/11/5 12:31 M Mother 2000-00001 c 2000/11/01 C Child #1 2000-00003 - 2000/11/01 D Child #2 2000-00004 - 2000/11/01Computation per race(s): cGenotype patterns are: THO1 TPOX CSF1PO D3S1358 VWA FGA D8S1179 D21S11 D18S51 D5S818 D13S317 D7S820 D16S539 M pq M p M q M qr M pq M pr M qr M pr M q M pq M pq M pq M pq C pr C p C pq C pr C p C qr C pr C qr C q C q C q C pq C qr D pr D p D pq D pq D pq D pq D pq D qr D pq D pq D pq D pr D pr

;D=daughter or sister of M C, D/? : M + Fred M, ?/D : Granny + Gramps(Caucasian frequencies)-- D=daughter or sister of MTHO1 11p15.5 PCR 2.73 (1+r) / (r+2pr) p=0.237 r=0.331TPOX 2p25-p24 PCR 1.31 2 / (1+p) p=0.528CSF1PO 5q33-34 PCR 3.8 (1+p) / (p+pq) p=0.251 q=0.309D3S1358 3p PCR 5.89 (1+p) / (p+2pq) p=0.126 q=0.256VWA 12p13.3 PCR 0.982 (1+p+q) / (1+p+q+2pq) p=0.131 q=0.0869FGA 4q PCR 6.28 (1+q) / (q+2pq) p=0.171 q=0.135D8S1179 PCR 9.81 (1+p) / (p+2pq) p=0.0761 q=0.221D21S11 PCR 4.21 (1+q) / (q+2qr) q=0.252 r=0.0894D18S51 18q21.33 PCR 0.857 1 / (1+q) q=0.167D5S818 PCR 0.869 (1+p+q) / (1+p+q+2pq) p=0.369 q=0.35D13S317 PCR 0.896 (1+p+q) / (1+p+q+2pq) p=0.305 q=0.307D7S820 7q11 PCR 0.764 1 / (1+2p) p=0.155D16S539 16q24 PCR 5.76 (1+r) / (r+2pr) p=0.107 r=0.167

cumulative LR 59900

DNA·VIEW data for LR example computations

Data for case 1204 2000/11/5 12:31 M Mother 2000-00001 c 2000/11/01 C Child #1 2000-00003 - 2000/11/01 D Child #2 2000-00004 - 2000/11/01Genotype patterns are: THO1 TPOX CSF1PO D3S1358 VWA FGA D8S1179 D21S11 D18S51 D5S818 D13S317 D7S820 D16S539 M pq M p M q M qr M pq M pr M qr M pr M q M pq M pq M pq M pq C pr C p C pq C pr C p C qr C pr C qr C q C q C q C pq C qr D pr D p D pq D pq D pq D pq D pq D qr D pq D pq D pq D pr D pr

lane 1 lane 3 lane 4 M C D locus ReadTHO1 2423 6,7 6,9.3 6,9.3 TPOX 2424 8 8 8 CSF1PO 2425 11 10,11 10,11 D3S1358 2426 15,16 14,16 14,15 VWA 2427 14,19 14 14,19 FGA 2428 21,25 24,25 21,24 D8S1179 2429 14,15 10,15 10,14 D21S11 2430 21,31.2 30,31.2 30,31.2 D18S51 2431 14 14 10,14 D5S818 2432 11,12 12 11,12 D13S317 2433 11,12 12 11,12 D7S820 2434 8,11 8,11 8,12 D16S539 2435 9,10 10,13 9,13

Daughter — or superniece?Explanation H0 (daughter) Explanation H3 (superneice)

Resident mother

Reference child

Applicant

• If H3 is true, likely can disprove H0

• If H0 is true, H3 likely also plausible.

Resident

Reference Applicant

• LR = 26.5. Maybe?

demo: DNA·VIEW computes LR

demo: DNA·VIEW computes LR

DNA·VIEW computation of “daughter or superniece” LR

;D=daughter or superniece of M C : M + Fred D : M/Sophie + Fred M,Sophie: Granny + Gramps(Caucasian frequencies)-- D=daughter or superniece of MTHO1 11p15.5 PCR 1.26 (2+2r) / (1+2p+r+4pr) p=0.237 r=0.331TPOX 2p25-p24 PCR 1.31 2 / (1+p) p=0.528CSF1PO 5q33-34 PCR 1.46 (2+2p) / (1+p+q+2pq) p=0.251 q=0.309D3S1358 3p PCR 1.27 (2+2p) / (1+p+2q+4pq) p=0.126 q=0.256VWA 12p13.3 PCR 1.69 (2+2p+2q) / (1+p+3q+4pq) p=0.131 q=0.0869FGA 4q PCR 1.45 (2+2q) / (1+2p+q+4pq) p=0.171 q=0.135D8S1179 PCR 1.36 (2+2p) / (1+p+2q+4pq) p=0.0761 q=0.221D21S11 PCR 1.65 (2+2q) / (1+q+2r+4qr) q=0.252 r=0.0894D18S51 18q21.33 PCR 0.857 1 / (1+q) q=0.167D5S818 PCR 1.16 (2+2p+2q) / (1+3p+q+4pq) p=0.369 q=0.35D13S317 PCR 1.24 (2+2p+2q) / (1+3p+q+4pq) p=0.305 q=0.307D7S820 7q11 PCR 0.799 (2p+2q) / (3p+q+4pp+4pq) p=0.155 q=0.195D16S539 16q24 PCR 1.61 (2+2r) / (1+2p+r+4pr) p=0.107 r=0.167

cumulative LR 26.5

Daughter — or sister-stepdaughter?Explanation H0 (daughter) Explanation H4 (incest)

Resident mother

Reference child

Applicant

• If H0 is true, H4 is plausible.

Resident

Reference

Applicant

• LR = 3.14 !?

demo: DNA·VIEW computes LR

demo: DNA·VIEW computes LR

DNA·VIEW computation of “daughter or incest” LR

;D=daughter or sister/stepdaughter of M? C : M + Fred M : Sophie + Fred D : M/Sophie + Fred(Caucasian frequencies)-- D=daughter or sister/stepdaughter of M?THO1 11p15.5 PCR 1.11 2 / (1+2p+r) p=0.237 r=0.331TPOX 2p25-p24 PCR 1.31 2 / (1+p) p=0.528CSF1PO 5q33-34 PCR 1.28 2 / (1+p+q) p=0.251 q=0.309D3S1358 3p PCR 1.22 2 / (1+p+2q) p=0.126 q=0.256VWA 12p13.3 PCR 1.03 (1+5p+q) / (1+4p+q+pp+5pq) p=0.131 q=0.0869FGA 4q PCR 1.35 2 / (1+2p+q) p=0.171 q=0.135D8S1179 PCR 1.32 2 / (1+p+2q) p=0.0761 q=0.221D21S11 PCR 1.4 2 / (1+q+2r) q=0.252 r=0.0894D18S51 18q21.33 PCR 0.75 1 / (1+2q) q=0.167D5S818 PCR 0.882 (1+p+5q) / (1+p+4q+5pq+qq) p=0.369 q=0.35D13S317 PCR 0.918 (1+p+5q) / (1+p+4q+5pq+qq) p=0.305 q=0.307D7S820 7q11 PCR 0.61 2 / (2+7p+q) p=0.155 q=0.195D16S539 16q24 PCR 1.45 2 / (1+2p+r) p=0.107 r=0.167

cumulative LR 3.14

I. Principles of analysis• Comparing more than 2 hypotheses

– LR=10,000,000 favoring “daughter” (H0) over “unrelated” (H1)

– LR=60,000 favoring “daughter” over “sister” (H2)

– LR=26 favoring “daughter” over “superniece” (H3)

– LR=3.14 favoring “daughter” over “incest” (H4)

• Hence comparing e.g. H4 vs H1 LR = 10,000,000/3.14

posteriorprobability

Hypoth daughter incest superniece sister unrelated

L(E if H) 10,000,000 3,000,000 400,000 170 1

prior 40% 4% 10% 20% 26%

L·p 4,000,000 120,000 40,000 34 <1

L·p/sum(L·p)) 96% 3% 1%

relativeprobability

Mother or Aunt?Ma vs Unrelated Aunt vs. Unrelated Ma vs. Aunt

general L (=X/Y) (L+1)/2 2 / (1+1/L)

locus 1 5 3 1.67

locus 2 100 50.5 1.98

locus 3 1 1 1

locus 4 0 0.5 0

locus 4m 0.001 0.5 0.002

overall 0.5 75 1/150

Likelihood ratios

III. Genetic anomalies

• Not co-dominant Mendelian inheritance

• Incidence per meiosis (paternal)– 1/500 for VNTR (excluding pH30, PAC425)– 1/400 for CODIS STR loci.

• Incidence per case – 1/100 for VNTR (5 locus assay)– 1/30 for STR (13 locus assay)

STR mutation rates

0

0.001

0.002

0.003

0.004

0.005

0.006V

WA

FG

A

CSF

1P0

TP

OX

TH

O1

CO

DIS

ave

Pat'l (CHB/Fairfax))Pat'l (Brinkmann)Mat'l (Brinkmann)

THO1: 1/316 maternal

2 anomalous loci

• 2 “exclusions” so to speak– RFLP-VNTR

• 1 true paternity case per 25000

– STR • 1 true paternity case per 2500

• Immigration cases– may be more meioses/case

STR paternal inconsistencies

• Inconsistency is like “exclusion” except maybe we don’t exclude

• 1/400 inconsistent loci / true father – 1/500 one-step mutations– 1/20000 two-step mutations– 1/3000 null alleles (e.g. primer dropout)

Reasonable mutation model, STR’s

0%

25%

50%

-3 r

epea

ts

-2 r

epea

ts

-1 r

epea

t

prog

enit

or

+1

repe

at

+2

repe

ats

+3

repe

ats

s=steps from progenitor

Relative mutation probability

Suggested model:

s = /2 if s = 1

s = /20 if s = 2

etc.

where =total mutation rate.

Mutation LR (new, for STR’s)(revised from http://dna-view/gomera.htm#Mutation)

• Data: Mother=PS, Child=PN, Man=RÑ– Explanation #1: Paternity; RÑ N

• 50% chance transmit Ñ = chance Ñ mutates

• m = chance Ñ ends up as N, assuming mutation– m =0.5 if N, Ñ are 1 step apart

– m=0.05 if 2 steps, etc

– Explanation #2: Real father N

• LR= /(4q) (assuming single step) » q=allele frequency of the paternal allele N

IV. Role of the laboratory

• attain requisite LR?– immigration rules

• 99.5%+ (but “contact lab if inconclusive”)» at 50% prior? LR>200

» at case-specific prior? (“possible fraud patterns”) LR> ?

• “distinguish between family members”» need relationship-specific LR

» cannot substitute larger LR threshold for wrong question

• answer questions, or ask them?– advisory

• Comparing more than 2 hypotheses– LR=10,000,000 favoring “daughter” (H0) over “unrelated” (H1)

– LR=60,000 favoring “daughter” over “sister” (H2)

– LR=26 favoring “daughter” over “superniece” (H3)

– LR=3.14 favoring “daughter” over “incest” (H4)

• Hence comparing e.g. H4 vs H1 LR = 10,000,000/3.14

posteriorprobability

Hypoth daughter incest superniece sister unrelated

L(E if H) 10,000,000 3,000,000 400,000 170 1

prior 40% 4% 10% 20% 26%

L·p 4,000,000 120,000 40,000 34 <1

L·p/sum(L·p)) 96% 3% 1%

relativeprobability

LR target not a good criterion

Summary

• Principles of analysis– Likelihoods

– calculate % if priors are given

• Kinship program• Mutation up to 1/30 cases• Advisory role of the laboratory

The end

• Thanks to Fairfax Identity Laboratory for the data from which STR mutation estimates were derived.

• References:– Brenner CH (1997) Symbolic Kinship Program,

Genetics 145:535-542– Forensic mathematics — http://dna-view.com