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Analysis of Y-Chromosomal STR Population DataUsing the Discrete Laplace Model
9th ICFISLeiden University, August 2014
Mikkel Meyer Andersen, Poul Svante Eriksen and Niels Morling
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Themes
I The discrete Laplace method and its applicationsI Comparing methods for calculating LR for Y-STR data
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Introduction
28
The Discrete LaplaceMethod
2 Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Evidential weight
Hp (prosecutor’s hypothesis): ’The suspect left theY-chromosome DNA in the crime stain.’
Hd (defence attorney’s hypothesis): ’A random man left theY-chromosome DNA in the crime stain.’
E : Evidence (e.g. DNA profile from crime scene)
Match:
LR =1
P (E | Hd )
Non-match:
LR =0
P (E | Hd )
LR =P (E | Hp)
P (E | Hd )
(Ideal situation, no errors, etc.)
Y-STR: Loci are not independent⇒ No product rule
28
The Discrete LaplaceMethod
3 Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Sparsity of Y-STRs
19,630 samples Forensic marker set
MHT SWGDAM PPY12 Yfiler PPY239 loci 11 loci 12 loci 17 loci 23 loci
n = 1 (singletons) 6,083 8,495 9,092 15,263 18,237(31.0%) (43.3%) (46.3%) (77.8%) (92.9%)
n = 2 (doubletons) 1,131 1,227 1,260 1,064 531n = 3 435 436 416 256 64n = 4 226 199 196 94 16n = 5 114 101 106 63 6n = 6 86 85 85 21 2n = 7 63 51 50 12 2n = 8 43 50 41 12 1n = 9 29 29 34 9n = 10 31 21 24 4n = 11 22 24 28 5 1. . .n ∈ (30, 40] 13 11 7 1. . .n ∈ (100, 515] 8 4 4
Purps J, Siegert S, et al. (2014). A global analysis of Y-chromosomal haplotype diversity
for 23 STR loci. Forensic Science International: Genetics, Volume 12, 2014, p. 12-23.
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Estimators
28
The Discrete LaplaceMethod
Introduction
4 Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Estimators
I Forensic ’conservatism’ (innocent suspect): For whom –what about paternity, immigration, etc.?
I Precise (low prediction error) – how do we measure this(more later)?
I Does it work for all datasets, also for those only consistingof singletons?
I Statistical model: Guaranteed behaviour (e.g. probabilitiessum to 1)
I Assign probability to all possible haplotypes (e.g. for mixtureLR)
I Probability mass 1 to be distributed among possiblehaplotypes
28
The Discrete LaplaceMethod
Introduction
5 Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Estimators
I Match probability ≈ DNA profile population frequencyI Count method (works for any trait, e.g. blood type)
I n: Dataset sizeI nx : Number of times x is observed in the datasetI P(X = x) = nx/n
28
The Discrete LaplaceMethod
Introduction
6 Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Estimators
I Include in dataset (new observation)I Additional information: Under Hd , suspect considered as a
random (wrongly accused) individual from the population;the haplotype is just another random sample
I Old dataset: D− of size nI New dataset: D of size n + 1I P(X = x) = (nx + 1)/(n + 1)
I nx = 0: P(X = x) = 1n+1
I∑
x∈Dnx
n+1 = 1n+1
∑x∈D nx = n+1
n+1 = 1, hence P(X = x) = 0for x 6∈ D
I Corrected count estimators:I Brenner’s κ (CH Brenner (2010) / HE Robbins (1968))I Generalised Good (IJ Good (1953), G Cereda/R Gill)
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
The Discrete Laplacemethod
28
The Discrete LaplaceMethod
Introduction
Estimators
7 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Motivation
I Haplotype probability distribution (statistical model)I Enables a wide range of inferences using one model:
I Haplotype frequency estimation (observed and unobserved)I Mixtures (e.g. separation and LR)I Cluster analysisI . . .
I Not a new ad-hoc tool for each taskI A statistical model gives desirable properties:
I P(x): Probability mass functionI Consistent: ∑
x∈H
P(x) = 1
I P(x) > 0 for all x ∈ H
28
The Discrete LaplaceMethod
Introduction
Estimators
8 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Model
I Y-STR: Loci not statistically independentI Our approach: Condition on [something] to obtain
independency between loci
28
The Discrete LaplaceMethod
Introduction
Estimators
9 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Discrete Laplace distribution
Discrete Laplace distributed X ∼ DL(p, µ):I Dispersion parameter 0 < p < 1 andI Location parameter µ ∈ Z = {. . . ,−2,−1,0,1,2, . . .}
Probability mass function:
f (X = x ;p, µ) =1− p1 + p
· p|x−µ| for x ∈ Z
Perfectly homogeneous population with 1-locus haplotypes:
P(X = x) = f (X = x ;p, µ)
8 9 10 11 12 13 14 15 16 17 18
x, e.g. Y−STR allele
f(X
= x
; p =
0.3
, µ =
13)
0.0
0.2
0.4
28
The Discrete LaplaceMethod
Introduction
Estimators
10 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Statistical model for Y-STR haplotypes
Perfectly homogeneous population with r -locus haplotypes:
P(X = (x1, x2, . . . , xr )) =r∏
k=1
f (xk ;pk , µk )
I ~µ = (µ1, µ2, . . . , µr ): Central haplotypeI ~p = (p1,p2, . . . ,pr ): Discrete Laplace parameters (one for
each locus)I Mutations happen independently across loci (relative to ~µ)
28
The Discrete LaplaceMethod
Introduction
Estimators
11 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Statistical model for Y-STR haplotypes
Non-homogeneous population with c subpopulations andr -locus haplotypes:
P(X = (x1, x2, . . . , xr )) =c∑
j=1
τj
r∏k=1
f (xk ;pjk , µjk )
I τj : A priori probability for originating from the j ’thsubpopulation (
∑cj=1 τj = 1)
I ~µj = (µj1, µj2, . . . , µjr ): Central haplotype for the j ’thsubpopulation
I ~pj = (pj1,pj2, . . . ,pjr ): Parameters for all loci at the j ’thsubpopulation
I Parameter estimation from observations using R librarydisclapmix
28
The Discrete LaplaceMethod
Introduction
Estimators
12 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Data and fit
● ● ● ● ●
●
●
●
●
● ●0.0
0.1
0.2
0.3
0.4
0.5
DYS392
Pro
babi
lity
6 7 8 9 10 11 12 13 14 15 16
c: Number of subpopulationsP(X = x) =
∑cj=1 τj f (x ;pj , µj)
I 3 subpopulations: µ̂j 11 13 14τ̂j 52% 46% 2%
I Observed vs expected:Allele 11 12 13 14 15Observed 0.5248 0.0567 0.3322 0.0714 0.0083Expected 0.5248 0.0567 0.3315 0.0715 0.0089
28
The Discrete LaplaceMethod
Introduction
Estimators
12 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Data and fit
● ● ● ● ●
●
●
●
●
● ●0.0
0.1
0.2
0.3
0.4
0.5
DYS392
Pro
babi
lity
6 7 8 9 10 11 12 13 14 15 16
● ObservationsEstimated (c = 1)
c: Number of subpopulationsP(X = x) =
∑cj=1 τj f (x ;pj , µj)
P(DYS392 = x) = 1 · f (x ;p = 0.41, µ = 11)
I 3 subpopulations: µ̂j 11 13 14τ̂j 52% 46% 2%
I Observed vs expected:Allele 11 12 13 14 15Observed 0.5248 0.0567 0.3322 0.0714 0.0083Expected 0.5248 0.0567 0.3315 0.0715 0.0089
28
The Discrete LaplaceMethod
Introduction
Estimators
12 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Data and fit
● ● ● ● ●
●
●
●
●
● ●0.0
0.1
0.2
0.3
0.4
0.5
DYS392
Pro
babi
lity
6 7 8 9 10 11 12 13 14 15 16
● ObservationsEstimated (c = 2)
c: Number of subpopulationsP(X = x) =
∑cj=1 τj f (x ;pj , µj)
P(DYS392 = x) =0.519 · f (x ;p = 0.004, µ = 11)+0.481 · f (x ;p = 0.179, µ = 13)
I 3 subpopulations: µ̂j 11 13 14τ̂j 52% 46% 2%
I Observed vs expected:Allele 11 12 13 14 15Observed 0.5248 0.0567 0.3322 0.0714 0.0083Expected 0.5248 0.0567 0.3315 0.0715 0.0089
28
The Discrete LaplaceMethod
Introduction
Estimators
12 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Data and fit
● ● ● ● ●
●
●
●
●
● ●0.0
0.1
0.2
0.3
0.4
0.5
DYS392
Pro
babi
lity
6 7 8 9 10 11 12 13 14 15 16
● ObservationsEstimated (c = 3)
c: Number of subpopulationsP(X = x) =
∑cj=1 τj f (x ;pj , µj)
I 3 subpopulations: µ̂j 11 13 14τ̂j 52% 46% 2%
I Observed vs expected:Allele 11 12 13 14 15Observed 0.5248 0.0567 0.3322 0.0714 0.0083Expected 0.5248 0.0567 0.3315 0.0715 0.0089
28
The Discrete LaplaceMethod
Introduction
Estimators
12 Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Data and fit
● ● ● ● ●
●
●
●
●
● ●0.0
0.1
0.2
0.3
0.4
0.5
DYS392
Pro
babi
lity
6 7 8 9 10 11 12 13 14 15 16
● ObservationsEstimated (c = 3)
c: Number of subpopulationsP(X = x) =
∑cj=1 τj f (x ;pj , µj)
I 3 subpopulations: µ̂j 11 13 14τ̂j 52% 46% 2%
I Observed vs expected:Allele 11 12 13 14 15Observed 0.5248 0.0567 0.3322 0.0714 0.0083Expected 0.5248 0.0567 0.3315 0.0715 0.0089
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Match probability
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete Laplace13 Match probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Simulation study
Simulation study:I Simulate populations (each 7 loci and 20 mio individuals)I Draw random datasetsI Estimate haplotype frequencies of all singletons and
compare with the true valuesI Result: Smaller prediction error than those with count
estimator and Brenner’s κ method
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete Laplace14 Match probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Estimate match probabilityReal data (Y23 dataset)
Singleton proportions
Population Size, n 5 loci 7 loci
World 18,925 0.026 0.108Europe 11,664 0.029 0.101
I Dataset sizes: 200 and 500I Sampling cases with singleton haplotype:
1. Draw dataset, D, from population2. Draw an extra observation, h3. If h ∈ D, skip and go to next sample4. If h 6∈ D: Estimate frequency and compare to nh/n (’true’)
I 100 cases for each dataset size, population and locuscount
I Compare to Brenner’s κ method and Generalised Good
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete Laplace15 Match probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Estimate match probability
5 loci 7 loci
●
●●
●●
●
●
●
●
●
●
●
●
●
●●
102
104
106
108
Bre
nner
Goo
d
Dis
clap
Bre
nner
Goo
d
Dis
clap
Estimator
LR
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete Laplace16 Match probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Prediction error
Probability LR LR inflation
Case 1 p1 = 0.01 LR = 100.0L̂R/LR = 1.005
p̂1 = 0.00995 L̂R = 100.5
Case 2 p2 = 0.0001 LR = 10,000L̂R/LR = 0.667
p̂2 = 0.00015 L̂R = 6,667
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete Laplace16 Match probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Prediction error
Probability LR LR inflation
Case 1 p1 = 0.01 LR = 100.0L̂R/LR = 1.005
p̂1 = 0.00995 L̂R = 100.5
Case 2 p2 = 0.0001 LR = 10,000L̂R/LR = 0.667
p̂2 = 0.00015 L̂R = 6,667
Error type
p̂i − pip̂i−pi
pi(p̂i − pi )
2 (p̂i−pi )2
pilog10
(p̂ipi
)Case 1 −0.00005 −0.005 2.5 · 10−9 2.5 · 10−7 −0.002Case 2 0.00005 0.5 2.5 · 10−9 2.5 · 10−5 0.176
Taking summary (sum, mean, median, ...): What is 0?
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete Laplace17 Match probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Estimate match probability
5 loci 7 loci
0.0000.0030.0060.009
0.00000.00050.00100.00150.00200.0025
0.00
0.05
0.10
0.000.050.100.150.20
Mean
Median
99th QM
axB
renn
er
Goo
d
Dis
clap
Bre
nner
Goo
d
Dis
clap
Estimator
Sum
mar
y of
(p̂h
−p h
)2
p h
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete Laplace18 Match probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
YHRD
I The discrete Laplace method and Brenner’s κ isimplemented in upcoming version ofhttp://www.yhrd.org
I The discrete Laplace method helped finding haplotypeswith wrong metapopulation assignments
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Mixture separation
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
19 Mixture analysis
Cluster analysis
Conclusion
Denmark
Mixture separation
Yfiler trace, 15 loci (DYS385a/b removed):
Locus Alleles
DYS19 14, 15DYS389I 13, 14DYS389II’ 16, 17DYS390 24, 26DYS391 10, 11DYS392 11, 13DYS393 13DYS438 11, 12DYS439 10, 11DYS437 14, 15DYS448 19, 20DYS456 15, 16DYS458 14, 18DYS635 23Y GATA H4 12, 13
213−1 = 4,096 possible contributor pairs
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
20 Mixture analysis
Cluster analysis
Conclusion
Denmark
Mixture separation
Danish Somali German
DEN (21) DEN (15) DEN (10) SOM (10) GER (7)
Loci 21 15 10 10 7n 181 181 181 201 3,443Singletons 181 164 112 56 662
(100%) (90.6%) (61.9%) (27.9%) (19.2%)
I For each dataset, 550 mixtures were simulatedI i ’th contributor pair ci = {hi,1,hi,2}, find p̂i = P̂(hi,1)P̂(hi,2)
I Order all pairs according to the p̂i values (highest tolowest)
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
21 Mixture analysis
Cluster analysis
Conclusion
Denmark
Mixture separation
Probabiliy DEN (21) DEN (15) DEN (10) SOM (10) GER (7)
Rank ≤ 1 13% 26% 45% 72% 53%Rank ≤ 5 33% 55% 84% 94% 89%Rank ≤ 10 42% 69% 93% 98% 97%
Random ≤ 10 0.03% 0.78% 12.15% 26.79 % 53.93%
DEN (21) DEN (15) DEN (10)
SOM (10) GER (7)0.00.20.40.60.81.0
0.00.20.40.60.81.0
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10x
P(T
rue
rank
≤x)
Ranking Discrete Laplace Random
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
22 Mixture analysis
Cluster analysis
Conclusion
Denmark
Mixture LR
Hp : S + UHd : U1 + U2
LR =P(HU)∑
(HU1 ,HU2 )P(HU1)P(HU2)
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DEN (21) DEN (15) DEN (10)
SOM (10) GER (7)
5
10
15
0
5
10
0.0
2.5
5.0
7.5
0.0
2.5
5.0
7.5
10.0
0
2
4
6
5 10 15 20 5 10 15 2.5
5.0
7.5
10.0
2.5
5.0
7.5
10.0 2 4 6
log10(1 P(Hs))
log1
0(LR
)
Predicted rank
●
●
●
●
●
●
1
2−5
6−10
11−50
51−100
>= 101
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Cluster analysis
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
23 Cluster analysis
Conclusion
Denmark
Cluster analysis
I τj = P(From subpopulation j)I Haplotype frequency by summing the contributions from
each subpopulation:
P(Haplotype = x)
=c∑
j=1
τj · P(Haplotype = x | From subpopulation j)︸ ︷︷ ︸Discrete Laplace model
.
I Bayes theorem:
P(From subpopulation j | Haplotype = x)
=τj · P(Haplotype = x | From subpopulation j)
P(Haplotype = x)
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
24 Cluster analysis
Conclusion
Denmark
Cluster analysis of European data7 loci
14,13,29,25,11,13,13 (R1b1b2a2g)
14,13,29,25,10,13,13 (R1b1b2a1)
14,13,29,24,10,13,13 (R1b1b2a2c)
14,13,30,24,10,13,13 (R1b1b2a2g)
14,13,29,23,10,13,13 (R1b1b2a2c)
14,13,30,23,11,13,12 (R1b1b)
14,13,29,23,11,13,13 (R1b1b2a1)
15,13,29,24,11,13,13 (R1b1b2a2g)
14,13,30,24,11,13,13 (R1b1b2a2c)
14,13,29,24,11,13,13 (J1a)
14,14,30,24,11,13,13 (R1b1b2a2c)
14,14,30,24,11,14,14 (N1c)
14,14,30,23,11,14,14 (N1c1)
15,13,29,23,10,14,14 (N1c)
15,14,31,23,10,12,14 (I2b)
15,13,30,23,10,12,14 (I2b)
15,12,29,22,10,11,14 (G2a3)
15,12,29,22,10,11,13 (G2a3b)
15,12,28,22,10,11,13 (G2a3)
14,12,29,22,10,11,13 (I1)
14,12,28,22,10,11,13 (I1)
14,12,28,23,10,11,13 (I1)
15,12,28,24,10,11,12 (J2b2)
15,13,29,23,10,11,12 (J1)
14,13,30,23,10,11,12 (J1e)
14,13,29,23,10,11,12 (J2a8)
13,14,30,24, 9,11,13 (E1b1b1b)
13,13,30,24,10,11,13 (E1b1b1a2)
13,13,31,24,10,11,13 (E1b1b1a)
14,13,29,24,11,11,13 (R1b1b2a2g)
16,13,31,24,11,11,13 (I2a)
16,13,31,24,10,11,13 (I2a)
16,13,29,24,10,11,13 (R1a1a)
16,13,29,25,10,11,13 (R1a1a7)
16,13,30,25,10,11,13 (R1a1a)
15,13,30,25,10,11,13 (D2)
15,13,30,25,11,11,13 (R1a)
16,13,30,25,11,11,13 (R1a)
17,13,30,25,11,11,13 (R1a)
17,13,30,25,10,11,13 (R1a1a7)
First analysed in ’Signature of recent historical events in the EuropeanY-chromosomal STR haplotype distribution’ by Roewer et al. in 2005
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
25 Cluster analysis
Conclusion
Denmark
Cluster analysis of Y2321 loci (from Purps J, Siegert S, et al. (2014))
029:
14,
13,1
6,24
,10,
13,1
3,12
,12,
15,1
9,16
,11,
23,1
2,18
,22,
13,1
2,17
,11
043:
14,
13,1
7,24
,11,
12,1
3,11
,11,
14,1
9,16
,16,
23,1
3,19
,22,
12,1
2,17
,10
041:
14,
13,1
7,23
,11,
13,1
3,12
,12,
15,1
9,17
,16,
23,1
1,18
,22,
12,1
2,17
,08
011:
14,
12,1
5,24
,11,
13,1
3,12
,12,
15,2
1,16
,16,
23,1
1,19
,22,
12,1
4,17
,10
016:
14,
12,1
6,23
,11,
13,1
3,12
,11,
15,1
9,15
,18,
23,1
2,17
,21,
12,1
2,19
,10
040:
14,
13,1
7,23
,11,
13,1
3,12
,11,
15,1
9,16
,20,
23,1
1,17
,21,
12,1
2,17
,10
025:
14,
13,1
6,23
,11,
13,1
3,12
,13,
15,1
8,19
,17,
23,1
1,17
,22,
13,1
2,17
,10
048:
14,
14,1
6,23
,11,
13,1
3,12
,12,
15,1
8,16
,17,
25,1
2,18
,22,
13,1
2,17
,09
052:
14,
14,1
6,24
,11,
13,1
3,12
,11,
15,1
8,16
,17,
23,1
2,18
,22,
13,1
3,20
,10
024:
14,
13,1
6,24
,11,
13,1
3,12
,12,
14,1
8,16
,17,
23,1
1,18
,22,
12,1
2,17
,10
044:
14,
13,1
6,24
,11,
13,1
3,12
,12,
14,1
8,16
,17,
23,1
1,18
,22,
13,1
2,17
,10
071:
14,
13,1
6,24
,11,
13,1
2,12
,12,
14,1
8,16
,18,
23,1
1,18
,22,
14,1
2,17
,10
042:
14,
13,1
6,23
,11,
13,1
3,12
,12,
15,1
9,17
,17,
23,1
1,17
,22,
13,1
2,17
,10
026:
14,
13,1
6,23
,11,
13,1
3,12
,12,
15,1
9,15
,18,
23,1
2,17
,22,
13,1
2,17
,10
037:
14,
13,1
6,24
,11,
13,1
4,12
,11,
15,1
9,15
,19,
23,1
1,17
,22,
12,1
2,17
,10
045:
14,
13,1
7,24
,11,
13,1
2,12
,12,
15,1
9,15
,16,
23,1
2,18
,22,
12,1
2,18
,10
035:
14,
13,1
6,24
,11,
13,1
3,12
,12,
15,1
9,15
,17,
24,1
2,18
,22,
12,1
3,17
,10
019:
14,
13,1
6,24
,11,
13,1
3,12
,12,
15,1
9,15
,17,
24,1
2,18
,22,
13,1
2,16
,11
034:
14,
13,1
6,24
,11,
13,1
3,12
,12,
15,1
9,15
,16,
23,1
2,19
,22,
13,1
2,17
,10
036:
14,
13,1
6,24
,11,
13,1
3,12
,12,
15,1
9,16
,17,
23,1
2,18
,22,
13,1
2,17
,10
033:
14,
13,1
6,24
,11,
13,1
3,12
,12,
15,1
9,15
,18,
23,1
2,18
,23,
13,1
2,17
,10
021:
14,
13,1
6,24
,10,
13,1
3,12
,12,
15,1
9,15
,17,
23,1
2,18
,22,
12,1
2,17
,10
023:
14,
13,1
6,24
,10,
13,1
3,12
,12,
15,2
0,15
,17,
24,1
1,18
,22,
13,1
2,18
,10
060:
14,
12,1
6,24
,10,
13,1
3,12
,13,
15,1
9,16
,16,
23,1
2,17
,22,
13,1
2,16
,10
032:
14,
13,1
6,24
,11,
13,1
3,11
,11,
15,1
9,16
,17,
23,1
2,18
,22,
13,1
2,17
,10
028:
14,
13,1
6,24
,10,
13,1
3,11
,11,
15,1
9,16
,17,
23,1
2,18
,22,
12,1
2,17
,10
050:
14,
14,1
6,24
,10,
13,1
3,12
,11,
15,1
9,16
,16,
23,1
2,17
,22,
13,1
3,18
,10
049:
14,
14,1
6,24
,10,
11,1
2,09
,12,
15,2
0,15
,14,
21,1
1,15
,22,
14,1
3,16
,10
074:
15,
13,1
7,23
,10,
13,1
3,09
,11,
14,1
9,15
,17,
21,1
1,16
,23,
13,1
2,17
,10
046:
15,
14,1
6,23
,10,
13,1
4,10
,11,
14,1
8,15
,16,
21,1
2,17
,24,
13,1
1,18
,11
001:
14,
13,1
6,23
,10,
15,1
3,10
,11,
14,1
9,15
,17,
21,1
2,18
,23,
12,1
2,17
,11
009:
13,
14,1
7,24
,10,
14,1
3,11
,11,
14,1
9,15
,16,
22,1
2,19
,24,
12,1
2,18
,10
004:
13,
13,1
7,24
,10,
14,1
3,11
,12,
14,2
0,15
,16,
22,1
1,18
,25,
12,1
2,18
,10
002:
13,
13,1
7,23
,10,
14,1
3,11
,12,
14,1
9,15
,17,
22,1
2,18
,24,
13,1
1,17
,10
092:
16,
12,1
6,25
,10,
13,1
2,10
,12,
14,1
9,14
,18,
22,1
2,18
,25,
13,1
1,18
,10
061:
15,
12,1
6,24
,10,
13,1
3,10
,12,
15,1
9,15
,16,
22,1
2,18
,24,
11,1
2,15
,11
082:
15,
13,1
6,25
,11,
13,1
4,10
,12,
14,1
8,15
,17,
22,1
1,19
,23,
12,1
0,17
,12
065:
15,
13,1
7,24
,11,
13,1
4,10
,11,
14,1
8,15
,17,
21,1
0,18
,23,
12,1
0,16
,12
063:
15,
12,1
6,23
,10,
14,1
3,10
,12,
14,1
8,15
,16,
21,1
2,17
,24,
12,1
1,19
,11
058:
15,
12,1
6,23
,10,
14,1
3,10
,12,
14,1
8,17
,15,
21,1
2,16
,24,
12,1
1,19
,11
059:
15,
12,1
7,23
,11,
14,1
3,10
,11,
14,1
8,17
,15,
19,1
2,18
,25,
12,1
1,18
,11
091:
16,
12,1
6,23
,10,
14,1
3,10
,11,
14,1
6,17
,18,
21,1
2,16
,27,
13,1
1,18
,11
066:
16,
12,1
8,24
,11,
13,1
3,10
,11,
14,2
1,14
,18,
21,1
2,17
,25,
12,1
1,19
,11
070:
15,
13,1
6,23
,11,
14,1
4,10
,10,
14,1
9,14
,17,
22,1
2,16
,20,
11,1
1,19
,11
051:
14,
14,1
6,23
,10,
14,1
4,10
,10,
14,1
9,14
,17,
22,1
2,17
,20,
11,1
1,19
,11
053:
14,
14,1
6,24
,11,
14,1
4,10
,10,
14,1
9,14
,17,
21,1
2,18
,20,
11,1
1,19
,11
022:
15,
14,1
6,22
,10,
13,1
3,13
,12,
14,1
8,15
,18,
20,1
2,19
,21,
12,1
1,20
,12
031:
16,
14,1
5,23
,10,
13,1
3,13
,12,
14,1
8,15
,17,
21,1
1,17
,23,
12,1
1,18
,13
057:
14,
13,1
6,23
,10,
14,1
3,10
,12,
15,1
8,15
,15,
20,1
2,17
,24,
12,1
1,20
,11
006:
13,
14,1
6,24
,10,
15,1
4,11
,11,
15,1
8,15
,14,
22,1
0,18
,24,
14,1
2,19
,12
010:
13,
14,1
7,24
,10,
14,1
4,10
,13,
15,2
1,16
,17,
22,1
1,20
,23,
12,1
0,18
,10
030:
14,
12,1
6,24
,10,
14,1
2,11
,12,
15,2
0,15
,18,
20,1
2,18
,23,
12,1
1,18
,11
055:
15,
12,1
6,23
,10,
12,1
2,10
,12,
15,1
9,15
,18,
20,1
2,19
,22,
12,1
1,17
,11
093:
15,
12,1
6,24
,10,
13,1
2,10
,12,
15,2
0,15
,17,
21,1
2,18
,24,
13,1
1,18
,11
027:
14,
12,1
6,22
,10,
14,1
1,10
,12,
15,1
9,15
,15,
23,1
2,17
,23,
13,1
2,15
,10
064:
15,
12,1
6,24
,10,
11,1
2,09
,12,
16,1
9,13
,16,
21,1
1,16
,23,
13,1
2,17
,09
089:
15,
12,1
6,24
,10,
11,1
2,09
,12,
16,1
9,13
,17,
21,1
1,17
,23,
13,1
2,17
,09
067:
15,
14,1
6,22
,10,
11,1
2,09
,11,
14,1
9,15
,18,
20,1
2,18
,23,
13,1
2,16
,09
039:
15,
13,1
6,23
,10,
11,1
2,09
,11,
14,2
0,15
,17,
21,1
1,17
,23,
13,1
1,18
,09
020:
14,
13,1
6,23
,10,
11,1
2,09
,11,
15,2
0,15
,16,
22,1
1,17
,23,
12,1
2,17
,10
054:
14,
14,1
7,23
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11,1
2,09
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15,2
0,15
,16,
21,1
1,17
,22,
12,1
1,17
,10
038:
14,
13,1
7,22
,10,
11,1
2,09
,11,
14,2
1,15
,15,
21,1
1,17
,23,
12,1
1,17
,08
068:
15,
13,1
6,23
,09,
11,1
2,09
,12,
14,2
1,16
,14,
22,1
2,16
,22,
12,1
2,18
,10
007:
13,
14,1
6,24
,09,
11,1
3,10
,10,
14,2
0,16
,18,
21,1
2,18
,27,
11,1
1,22
,12
077:
16,
13,1
8,24
,11,
11,1
3,10
,13,
15,2
0,15
,17,
23,1
1,18
,30,
11,1
3,18
,10
076:
15,
13,1
8,25
,11,
11,1
3,11
,10,
14,2
0,15
,15,
23,1
2,18
,23,
12,1
2,19
,10
084:
16,
13,1
7,25
,11,
11,1
3,11
,10,
14,2
0,15
,15,
23,1
3,18
,23,
12,1
2,19
,10
088:
16,
13,1
7,25
,11,
11,1
3,11
,11,
14,2
0,17
,15,
23,1
3,17
,23,
12,1
2,19
,10
087:
16,
13,1
7,25
,10,
11,1
3,11
,10,
14,2
0,16
,16,
23,1
2,18
,23,
12,1
2,18
,10
090:
16,
13,1
7,25
,10,
11,1
3,11
,11,
14,2
0,16
,16,
23,1
2,18
,25,
12,1
2,19
,10
072:
14,
12,1
6,25
,10,
11,1
3,11
,11,
14,1
9,15
,18,
24,1
1,15
,25,
13,1
2,18
,11
047:
14,
14,1
6,23
,10,
10,1
4,11
,11,
16,1
9,15
,17,
24,1
2,17
,25,
12,1
1,18
,10
056:
15,
12,1
7,22
,10,
11,1
4,10
,12,
16,2
1,15
,17,
21,1
2,17
,21,
12,0
9,18
,11
018:
15,
12,1
7,22
,10,
11,1
4,10
,11,
16,2
1,15
,16,
20,1
2,15
,21,
12,0
9,18
,11
080:
15,
12,1
7,21
,10,
11,1
4,10
,11,
16,2
2,15
,16,
21,1
1,15
,21,
12,1
0,18
,12
012:
14,
12,1
6,22
,10,
11,1
3,10
,11,
16,2
0,14
,15,
22,1
1,17
,25,
13,1
1,19
,12
013:
14,
12,1
6,22
,10,
11,1
3,10
,11,
16,2
0,14
,15,
22,1
1,16
,26,
12,1
1,19
,12
015:
14,
12,1
6,23
,10,
11,1
3,10
,11,
16,2
0,14
,15,
21,1
1,16
,25,
12,1
1,20
,12
083:
16,
12,1
6,25
,11,
11,1
3,10
,11,
15,2
1,14
,16,
21,1
0,16
,23,
12,1
1,18
,13
017:
17,
13,1
5,23
,10,
11,1
3,10
,12,
15,2
1,14
,17,
22,1
2,17
,22,
10,1
2,19
,12
003:
13,
13,1
7,24
,10,
11,1
3,10
,12,
14,2
0,16
,16,
22,1
2,17
,22,
12,1
2,20
,12
008:
13,
13,1
7,24
,10,
11,1
3,10
,12,
14,2
0,16
,15,
22,1
2,18
,22,
12,1
2,19
,12
005:
13,
13,1
7,24
,10,
11,1
3,10
,12,
14,2
0,15
,17,
21,1
1,18
,25,
12,1
1,19
,12
098:
16,
14,1
7,25
,10,
11,1
3,10
,12,
14,1
8,15
,15,
21,1
1,18
,26,
12,1
2,17
,12
078:
15,
13,1
6,23
,10,
11,1
4,10
,12,
14,2
1,15
,16,
21,1
1,18
,24,
12,1
2,16
,09
014:
15,
12,1
7,23
,10,
11,1
3,10
,11,
16,2
0,15
,17,
21,1
1,17
,25,
12,1
1,17
,12
069:
15,
13,1
6,23
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12,1
4,10
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15,2
0,14
,16,
21,1
1,18
,25,
12,1
2,19
,12
073:
15,
14,1
8,23
,10,
12,1
5,10
,11,
14,2
0,14
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21,1
0,17
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12,1
2,19
,12
094:
15,
14,1
7,23
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12,1
4,10
,11,
14,2
0,15
,15,
21,1
1,17
,27,
12,1
2,18
,12
095:
15,
13,1
7,21
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11,1
4,11
,12,
14,2
0,15
,17,
22,1
1,16
,25,
11,1
2,19
,13
086:
16,
13,1
7,21
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11,1
5,11
,12,
14,2
1,15
,16,
21,1
1,17
,25,
11,1
1,18
,13
099:
16,
13,1
8,21
,10,
11,1
5,11
,12,
14,2
1,15
,16,
20,1
1,17
,24,
11,1
1,18
,13
100:
17,
13,1
7,21
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11,1
4,11
,12,
14,2
1,16
,16,
21,1
1,16
,25,
11,1
1,17
,13
096:
17,
13,1
7,21
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11,1
4,11
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14,2
1,17
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21,1
1,18
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11,1
1,17
,13
085:
16,
13,1
7,21
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11,1
3,11
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13,2
1,16
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21,1
1,17
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12,1
1,17
,13
097:
16,
14,1
7,22
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11,1
3,11
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14,2
1,15
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21,1
0,16
,27,
12,1
1,14
,12
079:
15,
13,1
8,21
,10,
11,1
3,11
,12,
14,2
1,15
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21,1
2,15
,26,
12,1
1,19
,15
075:
15,
13,1
8,21
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11,1
3,11
,12,
14,2
1,15
,17,
21,1
2,15
,28,
11,1
1,19
,13
081:
15,
14,1
8,21
,10,
11,1
3,11
,12,
14,2
1,15
,15,
22,1
3,16
,28,
11,1
2,19
,13
062:
15,
12,1
7,23
,10,
10,1
4,09
,12,
16,2
1,15
,19,
21,1
2,20
,24,
12,1
0,18
,11
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
26 Cluster analysis
Conclusion
Denmark
Pairwise population distances
Pairwise population distances:I 7-locus, 12,727 European males (91 locations):
Correlation(AMOVA, discrete Laplace) = 0.90I 10-locus, 2,736 African males (26 locations):
Correlation(AMOVA, discrete Laplace) = 0.82I 21-locus (Y23), 18,925 males (129 locations):
Correlation(AMOVA, discrete Laplace) = 0.78
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Concluding remarks
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
27 Conclusion
Denmark
The discrete Laplace method
I Sound statistical propertiesI Applications
I Estimation of Y-STR haplotype population frequenciesI Mixture analysisI Cluster analysis
I Computationally feasibleI Open source software: R libraries disclap and
disclapmix (and fwsim for simulating populations)I Criticism
I Intermediate alleles (e.g. 10.2)I Duplications (e.g. DYS385a/b)I Copy number variation (e.g. Yfiler Plus)I µ̂’s difficult to estimate (curse of dimensionality)
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
28 Conclusion
Denmark
Conclusion
I Match probability is of great interest and is difficultI Validation of methods
I Open source software (e.g. R library, C++ program):Compare to your own method
I Availability of real data (PPY23)I Purps J, Siegert S, et al. (2014). A global analysis of
Y-chromosomal haplotype diversity for 23 STR loci. ForensicScience International: Genetics, Volume 12, 2014, p. 12-23.http://dx.doi.org/10.1016/j.fsigen.2014.04.008
I R-object: http://people.math.aau.dk/~mikl/?p=y23I Battery of simulated populationsI Measure of prediction error
I For a matching profile (e.g. Y23 or Yfiler Plus), use onlysubset (e.g. 7 or 10 loci) for LR calculations?
I Easier to validate
28
The Discrete LaplaceMethod
Introduction
Estimators
Discrete LaplaceMatch probability
Mixture analysis
Cluster analysis
Conclusion
Denmark
Thank you foryour attention