analysis of y-chromosomal str population data 0.5em using...

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

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The Discrete LaplaceMethod

MM [email protected]

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

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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Introduction

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2 Introduction

Estimators


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

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3 Introduction

Estimators


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.

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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Estimators

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Introduction

4 Estimators


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

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Introduction

5 Estimators


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

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Introduction

6 Estimators


Mixture analysis

Cluster analysis

Conclusion

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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)

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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

The Discrete Laplacemethod

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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

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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Model

I Y-STR: Loci not statistically independentI Our approach: Condition on [something] to obtain

independency between loci

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Introduction

Estimators


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

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Introduction

Estimators


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)

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Introduction

Estimators


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

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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Data and fit

● ● ● ● ●

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0.1

0.2

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0.5

DYS392

Pro

babi

lity

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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

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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

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Data and fit

● ● ● ● ●

●

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● ●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)



P(DYS392 = x) = 1 · f (x ;p = 0.41, µ = 11)



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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Data and fit

● ● ● ● ●

●

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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




P(DYS392 = x) =0.519 · f (x ;p = 0.004, µ = 11)+0.481 · f (x ;p = 0.179, µ = 13)



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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Data and fit

● ● ● ● ●

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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






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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Match probability

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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

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Introduction

Estimators


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

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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Estimate match probability

5 loci 7 loci

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102

104

106

108

Bre

nner

Goo

d

Dis

clap

Bre

nner

Goo

d

Dis

clap

Estimator

LR

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Introduction

Estimators


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

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Introduction

Estimators


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?

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Introduction

Estimators


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

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Introduction

Estimators


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

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http://www.yhrd.org

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Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Mixture separation

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Introduction

Estimators


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

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Introduction

Estimators


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)

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Introduction

Estimators


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

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Introduction

Estimators


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

[email protected]

28



Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Cluster analysis

[email protected]

28



Introduction

Estimators


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)

[email protected]

28



Introduction

Estimators


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

[email protected]

28



Introduction

Estimators


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

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28



Introduction

Estimators


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

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28



Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Concluding remarks

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28



Introduction

Estimators


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)

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28



Introduction

Estimators


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

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http://dx.doi.org/10.1016/j.fsigen.2014.04.008

http://people.math.aau.dk/~mikl/?p=y23

28



Introduction

Estimators


Mixture analysis

Cluster analysis

Conclusion

Denmark

Thank you foryour attention

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analysis of y-chromosomal str population data 0.5em using...

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