2. Kinship Analysis in Immigration Cases Charles H. Brenner, Ph.D. consulting in forensic mathematics Oakland, California http://dna-view.com

3. 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?

4. I. Principles of analysis Likelihood ratio comparing hypotheses –Paternity trio is the prototype –gotta be careful about treating it as archetype

5. Likelihood ratio (paternity trio) Two possible explanations of genetic Evidence: M=pr AF=qs C=pq LR = PI = P(E if H 0 ) / P (E if H 1 ) –how much more typical E is of H 0, than of H 1 –(not how much more likely H 0 is, than H 1 ) M C AF M C Explanation H 0 (AF=father) Explanation H 1 (AF unrelated)

6. Likelihood ratio (paternity trio) genetic Evidence: M= pr AF= qs C= pq LR = P(E if H 0 ) / P (E if H 1 ) = (2pr)(2qs)(¼) / (2pr)(2qs)(½q) shortcut to avoid today: Evidence: C= pq –LR = ¼ / ½q M C AF M C Explanation H 0 (AF=father) Explanation H 1 (AF unrelated)

7. Daughter or unrelated? Explanation H 0 (daughter) Explanation H 1 (unrelated) Resident Reference Applicant genetic Evidence: M= pr Ref= ps Applicant= ps LR = P(E if H 0 ) / P(E if H 1 ) –P(E if H 0 ) = P(E and F= ss if H 0 ) + P(E and F= sx if H 0 ) = (2pr)(s 2 )(½)(½) + (2pr)(2s(1-s))(¼)(¼) –P(E if H 1 ) = (2pr)(s 2 )(2ps)(½) + (2pr)(2s(1-s))(2ps)(¼) pr ps Applicant Resident mother Reference child Man F (not tested) LR = (1+s) / 8ps demo: DNA·VIEW derives LR formula

8. 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 + Fred Reference : Mother + Fred Applicant ps Reference ps Mother pr Likelihood ratio = 2.71 (1+s) / 8ps -- -- Likelihood ratio: (1+s) / 8ps Likelihood ratio = 2.71 (using p=0.2 s=0.3)

9. Daughter — or sister? Explanation H 0 (daughter) Explanation H 2 (sister) Resident Reference Applicant Resident mother Reference child Applicant If H 2 is true, probably can disprove H 0 If H 0 is true, cannot disprove H 2 –LR probably strongly against it though. LR = 59,900. Forget about “sister” »(or unrelated — LR > million) demo: DNA·VIEW computes LR

10. 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 - 2000/11/01 Computation per race(s): c Genotype 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 M THO1 11p15.5 PCR 2.73 (1+r) / (r+2pr) p=0.237 r=0.331 TPOX 2p25-p24 PCR 1.31 2 / (1+p) p=0.528 CSF1PO 5q33-34 PCR 3.8 (1+p) / (p+pq) p=0.251 q=0.309 D3S1358 3p PCR 5.89 (1+p) / (p+2pq) p=0.126 q=0.256 VWA 12p13.3 PCR 0.982 (1+p+q) / (1+p+q+2pq) p=0.131 q=0.0869 FGA 4q PCR 6.28 (1+q) / (q+2pq) p=0.171 q=0.135 D8S1179 PCR 9.81 (1+p) / (p+2pq) p=0.0761 q=0.221 D21S11 PCR 4.21 (1+q) / (q+2qr) q=0.252 r=0.0894 D18S51 18q21.33 PCR 0.857 1 / (1+q) q=0.167 D5S818 PCR 0.869 (1+p+q) / (1+p+q+2pq) p=0.369 q=0.35 D13S317 PCR 0.896 (1+p+q) / (1+p+q+2pq) p=0.305 q=0.307 D7S820 7q11 PCR 0.764 1 / (1+2p) p=0.155 D16S539 16q24 PCR 5.76 (1+r) / (r+2pr) p=0.107 r=0.167 cumulative LR 59900

11. 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 - 2000/11/01 Genotype 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 Read THO1 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

12. Daughter — or superniece? Explanation H 0 (daughter) Explanation H 3 (superneice) Resident mother Reference child Applicant If H 3 is true, likely can disprove H 0 If H 0 is true, H 3 likely also plausible. Resident ReferenceApplicant LR = 26.5. Maybe? demo: DNA·VIEW computes LR

13. 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 M THO1 11p15.5 PCR 1.26 (2+2r) / (1+2p+r+4pr) p=0.237 r=0.331 TPOX 2p25-p24 PCR 1.31 2 / (1+p) p=0.528 CSF1PO 5q33-34 PCR 1.46 (2+2p) / (1+p+q+2pq) p=0.251 q=0.309 D3S1358 3p PCR 1.27 (2+2p) / (1+p+2q+4pq) p=0.126 q=0.256 VWA 12p13.3 PCR 1.69 (2+2p+2q) / (1+p+3q+4pq) p=0.131 q=0.0869 FGA 4q PCR 1.45 (2+2q) / (1+2p+q+4pq) p=0.171 q=0.135 D8S1179 PCR 1.36 (2+2p) / (1+p+2q+4pq) p=0.0761 q=0.221 D21S11 PCR 1.65 (2+2q) / (1+q+2r+4qr) q=0.252 r=0.0894 D18S51 18q21.33 PCR 0.857 1 / (1+q) q=0.167 D5S818 PCR 1.16 (2+2p+2q) / (1+3p+q+4pq) p=0.369 q=0.35 D13S317 PCR 1.24 (2+2p+2q) / (1+3p+q+4pq) p=0.305 q=0.307 D7S820 7q11 PCR 0.799 (2p+2q) / (3p+q+4pp+4pq) p=0.155 q=0.195 D16S539 16q24 PCR 1.61 (2+2r) / (1+2p+r+4pr) p=0.107 r=0.167 cumulative LR 26.5

14. Daughter — or sister-stepdaughter? Explanation H 0 (daughter) Explanation H 4 (incest) Resident mother Reference child Applicant If H 0 is true, H 4 is plausible. Resident Reference Applicant LR = 3.14 !? demo: DNA·VIEW computes LR

15. 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=0.331 TPOX 2p25-p24 PCR 1.31 2 / (1+p) p=0.528 CSF1PO 5q33-34 PCR 1.28 2 / (1+p+q) p=0.251 q=0.309 D3S1358 3p PCR 1.22 2 / (1+p+2q) p=0.126 q=0.256 VWA 12p13.3 PCR 1.03 (1+5p+q) / (1+4p+q+pp+5pq) p=0.131 q=0.0869 FGA 4q PCR 1.35 2 / (1+2p+q) p=0.171 q=0.135 D8S1179 PCR 1.32 2 / (1+p+2q) p=0.0761 q=0.221 D21S11 PCR 1.4 2 / (1+q+2r) q=0.252 r=0.0894 D18S51 18q21.33 PCR 0.75 1 / (1+2q) q=0.167 D5S818 PCR 0.882 (1+p+5q) / (1+p+4q+5pq+qq) p=0.369 q=0.35 D13S317 PCR 0.918 (1+p+5q) / (1+p+4q+5pq+qq) p=0.305 q=0.307 D7S820 7q11 PCR 0.61 2 / (2+7p+q) p=0.155 q=0.195 D16S539 16q24 PCR 1.45 2 / (1+2p+r) p=0.107 r=0.167 cumulative LR 3.14

16. I. Principles of analysis Comparing more than 2 hypotheses –LR=10,000,000 favoring “daughter” (H 0 ) over “unrelated” (H 1 ) –LR=60,000 favoring “daughter” over “sister” (H 2 ) –LR=26 favoring “daughter” over “superniece” (H 3 ) –LR=3.14 favoring “daughter” over “incest” (H 4 ) Hence comparing e.g. H 4 vs H 1 LR = 10,000,000/3.14 posterior probability 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% relative probability

17. Mother or Aunt? Ma vs UnrelatedAunt vs. UnrelatedMa vs. Aunt generalL (=X/Y)(L+1)/22 / (1+1/L) locus 1531.67 locus 210050.51.98 locus 3111 locus 400.50 locus 4m0.0010.50.002 overall0.5751/150 Likelihood ratios

18. Avuncular index (Is the man an uncle? the woman an aunt?)

19. 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)

20. STR mutation rates THO1: 1/316 maternal

21. 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

22. 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)

23. Reasonable mutation model, STR’s Suggested model:  s =  /2 if s =  1  s =  /20 if s =  2 etc. where  =total mutation rate.

24. 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

25. 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

26. Comparing more than 2 hypotheses –LR=10,000,000 favoring “daughter” (H 0 ) over “unrelated” (H 1 ) –LR=60,000 favoring “daughter” over “sister” (H 2 ) –LR=26 favoring “daughter” over “superniece” (H 3 ) –LR=3.14 favoring “daughter” over “incest” (H 4 ) Hence comparing e.g. H 4 vs H 1 LR = 10,000,000/3.14 posterior probability 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% relative probability LR target not a good criterion

27. Summary Principles of analysis –Likelihoods –calculate % if priors are given Kinship program Mutation up to 1/30 cases Advisory role of the laboratory

28. 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