relationship testing based on dna mixtures · mortera (roma tre&bristol) relationships &...

Relationship testing based on DNA mixtures

Julia Mortera

Università Roma Tre & University of Bristol

Workshop on Advanced Statistical Methods for Complex Data, University of Cape TownBased on joint work with Peter Green

Mortera (Roma Tre&Bristol) Relationships & DNA mixtures Cape Town, 28-31 January 2020 1 / 40

Outline

1 Motivating example

2 Bayesian Networks

3 DNA mixtures

4 Methods for inference about relationships

5 Results from real casework examples

6 Relationships among contributorsPopulations with high relatednessSpecific close relationships

Motivating example

The Murder of Yara Gambirasio

On 26/11/2010 a 13-year-old YG left home to go to the gym. An hour and ahalf later she left the gym never to return home. Three months later her bodywas found. DNA was extracted from the victim’s clothes. This appeared as amixtureM of the victim’s profile and one or two male individuals.Familial search showed that two brothers, unrelated to the crime, sharedmany alleles with the mixture and could therefore potentially be related to themurderer. A DNA sample from their mother revealed that she shared noalleles withM.

Motivating example

The Murder of YG

The brothers’ father, GG, was a bus driver who had died 11 years before thecrime. DNA was extracted from his exhumed body.

GG must have had an illegitimate child. The investigators then decided toscreen women who potentially could have borne him a child decades earlier.A woman, EA was found whose DNA was compatible with that of the motherof U. Thus EA’s son, MGB, became the chief suspect.

In 2016, MGB was sentenced to life imprisonment.

Motivating example

The Murder of YG

Motivating example

The Murder of YG

Motivating example

Electropherogram (EPG)

Peak height is roughly proportional to the amount of the allele pre-PCR.

Motivating example

Extract of DNA Data

The full data consists 20 DNA mixture samples having 17 markers each.

Marker Alleles Peak height victim GG EAD2S1338 16 77

17 1379 17 (24) 1720 75 2021 713 2122 4223 497 23

D8S1179 11 9312 1265 12 1213 2724 13 13 1314 858 14

Bayesian Networks

Bayesian Networks (BNs) and conditional probabilitytables (CPTs)

In a BN model, the joint distribution of all variables factorises as a product of‘parent to child’ conditional distributions

p(x) =∏i∈V

p(xi |xpa(i))

Bayesian Networks

p(x) =∏i∈V

p(xi |xpa(i))

When variables are discrete, the conditional probabilities p(xi |xpa(i)) can berepresented as conditional probability tables.

Note that the size of each table is proportional to the numbers of values of thevariables concerned.

Bayesian Networks

p(x) =∏i∈V

p(xi |xpa(i))

When variables are discrete, the conditional probabilities p(xi |xpa(i)) can berepresented as conditional probability tables.

Note that the size of each table is proportional to the numbers of values of thevariables concerned.

Bayesian Networks

Representing genotypes

Some markers can have many alleles, so CPTs for genotypes become large.This can be avoided by representing genotypes by vectors of allele counts:nia = 0,1 or 2, for alleles a = 1,2, · · · ,A, for individual i . For an individualdrawn from the gene pool,

(nia)Aa ∼ Multinomial(2, (qa)

where qa is the frequency of allele a for this marker, in the relevant population.

Bayesian Networks

Markov genotype representation

Imagine the 2 alleles of a genotype being allocated sequentially. Let Sia bethe partial sums Sia =

∑b≤a nib. Then ni1 ∼ Bin(2,q1) and

ni,a+1 | (ni1,ni2, · · · ,nia) ∼ Bin

(2− Sia,qa+1/

∑b>a

Hence a genotype is modelled by a Markov structure and computations canbe done linearly in number of alleles.

DNA mixtures

Statistical model for DNA mixtures

Joint model for EPG, genotypes and relationships.

Combine1 model for genotypes;2 model for peak heights for fixed genotypes;3 models for inference about relationships

DNA mixtures

We need a joint model for electropherogram (peak heights z), genotypes(allele counts n) and (later) relationship data and hypotheses (R and H).

DNA mixtures

The gamma model for peak heights CGLM (2016)

We built a statistical model for p(z|n) – i.e. for peak heights z given counts ofalleles n for the contributors to the mixture – representing their genotypes).It allows for differing proportions φ of DNA for the different contributors, andthe two most important artefacts prevalent in EPG data, arising from the PCRamplification: stutter and dropout.

DNA mixtures

Likelihood function

For given genotypes n, given φ and fixed values of parameters (ρ, ξ, η) allobserved peaks are independent. Thus the conditional likelihood functionbased on observations z = {zma}m∈M,a∈Am is

p(z|n) = p(z|n; ρ, ξ, φ, η) =∏m

Lma(zma)

Lma(zma) =

{g{zma; ρDa(φ, ξ,n), η} if zma > CG{C; ρDa(φ, ξ,n), η} otherwise.

with g and G denoting the gamma density and cdf respectively, and whereDa(φ, ξ,n) =

∑i∈I φi [(1− ξ)nia + ξni,a+1] is the effective number of alleles of

type a after stutter, where ξ denotes the mean stutter proportion.

DNA mixtures

Full likelihood

For a given hypothesis H on the number of contributors, the full likelihood isobtained by summing over all possible combinations of genotypes

L(H) = Pr(E |H) =∑

p(z|n; ρ, ξ, φ, η)p(n |H).

with probabilities associated with the hypothesis, p(n |H). This sum isastronomical in size for any hypothesis which potentially involves unknowncontributors to the mixture.

However, the sum can be calculated efficiently by appropriate use of Bayesiannetwork (BN) techniques.

DNA mixtures

Full likelihood

For a given hypothesis H on the number of contributors, the full likelihood isobtained by summing over all possible combinations of genotypes

L(H) = Pr(E |H) =∑

p(z|n; ρ, ξ, φ, η)p(n |H).

with probabilities associated with the hypothesis, p(n |H). This sum isastronomical in size for any hypothesis which potentially involves unknowncontributors to the mixture.

However, the sum can be calculated efficiently by appropriate use of Bayesiannetwork (BN) techniques.

DNA mixtures

Computational aspects

The main bottleneck is the need to sum over possible genotypes (many timesin numerical likelihood maximisation), especially when there are many allelesfor each marker.We use two tricks:

1 Representing genotypes by allele counts, and using a Markov genotyperepresentation.

2 Computation by auxiliary nodes in a Bayesian network.

DNA mixtures

Benefits

Exact evaluation of likelihood function and other quantities by probabilitypropagation.We can introduce auxiliary likelihood nodes to also represent familyrelationships.No approximations are introduced to be able to perform computations(apart from numeric maximisation and differentiation for computing MLEs)We can currently handle up to 6 unknown contributors.

DNA mixtures

Bayesian network for two person mixture and peakheight observations

Methods for inference about relationships

Relationship Identification from DNA mixtures

Examples are:Is a contributor to the mixture the son of individual GG?Is a contributor to the mixture the son of GG and EA?Is GG a family relative of one or both contributors to a mixture?

The evidence is E={DNA mixture, genotype of measured individuals}

General Setup

Let Ui = U be a specified contributor to the mixture, and let Ugt denote thegenotype of U.

We are interested in assessing a potential relationship between U and one ormore other individuals who have a known relationship to each other. Let Rdenote the genotypes of these individuals.

Hp : U has the specified relationship with individuals whose genotypes are inRH0 : U is unrelated with individuals whose genotypes are in R

General Setup

Let Ui = U be a specified contributor to the mixture, and let Ugt denote thegenotype of U.

We are interested in assessing a potential relationship between U and one ormore other individuals who have a known relationship to each other. Let Rdenote the genotypes of these individuals.

Hp : U has the specified relationship with individuals whose genotypes are inRH0 : U is unrelated with individuals whose genotypes are in R

Note R ⊥⊥ z | Ugt. We assume Ugt ⊥⊥ R | H0.

Under Hp

Under H0

Likelihood Ratio

The LR for Hp : U has the specified relationship with individuals whosegenotypes are in R; against the contrary hypothesis H0 is

LR =P(R, z|Hp)

P(R, z|H0)=

P(R, z|Hp)

P(R|H0)p(z|H0)

∑Ugt P(R|Hp,Ugt)p(z|Ugt)P(Ugt)

P(R|H0)p(z)

=∑Ugt

LRUgt × p(Ugt|z)

(this assumes we use the same parameter values for both Hp and H0.)

Examples from the murder case

R is GG’s genotype fgtR is genotypes of GG and EA, fgt and mgt

and we are testing whether U is the son of fgt.

Only father genotyped

LRUgt =P(fgt|Ugt,Hp)

P(fgt|H0)=

{nia/2qa if fgt = {a,a},nia/4qa + nib/4qb if fgt = {a,b}

The only allele counts for Ui needed as graphical parents for defining the CPTfor the likelihood node are those (a and/or b) in fgt.Similarly for mgt.

Examples from the murder case

R is GG’s genotype fgtR is genotypes of GG and EA, fgt and mgt

and we are testing whether U is the son of fgt.

Only father genotyped

LRUgt =P(fgt|Ugt,Hp)

P(fgt|H0)=

{nia/2qa if fgt = {a,a},nia/4qa + nib/4qb if fgt = {a,b}

The only allele counts for Ui needed as graphical parents for defining the CPTfor the likelihood node are those (a and/or b) in fgt.Similarly for mgt.

Methods

We have devised various explicit methods for implementing theLR =

∑Ugt LRUgt × p(Ugt|z) calculation, exploiting or adapting the BN in

different ways:ALN Additional likelihood nodes: add more likelihood nodes to

encode P(R|Ugt,Hp)

RPT Replacing probability tables: replace allele count CPTsencoding P(Ugt) by those for P(Ugt|R,Hp)

MBN Meiosis Bayes net: extend BN to include mother & father(Mendel’s law not well suited to Markov genotyperepresentation)

WLR Weighted likelihood ratio: extract p(Ugt|z) and compute LRexplicitly

Methods

encode P(R|Ugt,Hp)

Methods

encode P(R|Ugt,Hp)

Methods

encode P(R|Ugt,Hp)

Methods

encode P(R|Ugt,Hp)

Methods

encode P(R|Ugt,Hp)

Methods

encode P(R|Ugt,Hp)

Methods

encode P(R|Ugt,Hp)

Bayesian network for ALN (homozygous fgt = (2,2))

Results from real casework examples

Italian Criminal Case

A murder case where with 3 mixed crime traces T1,T2 and T3. We also havethe genotypes of victim V and alleged father fgt and/or mother mgt of acontributor to the mixture were also available. We assume that there are atmost 3 contributors to each mixture, the victim V and two unknowns U1 andU2.

The extra contributor U2 is included to account for dropin.

Italian Criminal Case

A murder case where with 3 mixed crime traces T1,T2 and T3. We also havethe genotypes of victim V and alleged father fgt and/or mother mgt of acontributor to the mixture were also available. We assume that there are atmost 3 contributors to each mixture, the victim V and two unknowns U1 andU2.

The extra contributor U2 is included to account for dropin.

MLEs based on combined information from T1,T2,T3

Parameter T1 T2 T3

µ 3857 1289 1836σ 0.408 0.671 0.562ξ 0.127 0.048 0φV 0.22 0.53 0.63φU1 0.71 0.45 0.37φU2 0.067 0.026 0

In T1 the major contributes φU1 = 0.71 > φV = 0.22, whereas they eachcontribute about the same proportion to T2, and in T3 φV = 0.63 > φU1 = 0.37.U2 contributes a negligble amount in all traces.

Have we found the culprit?

In this case we want to compare the hypotheses:

Hp: U1 is the child of fgt (or of mgt) vs.H0: no unknown contributors are related to fgt ( mgt ),

Using the exact methods the likelihood ratio infavour of Hp for fgt is log10 LR = 4.26favour of Hp for mgt is log10 LR = 4.23

Comparison with single trace analysis

separate traces combined tracesT1 T2 T3 T1&T2&T3

LR 17156 103.7 238.64 18046log10 LR 4.23 2.02 2.38 4.26

Using solely T1, (φU1=70%) the LR is reduced only by a factor of 1.05 the LRbased on T1&T2&T3.Using trace T2 (T3) alone yield LR about 174 (76) times smaller than the LRbased on T1&T2&T3.

Computational times

...for the mother-typed-only case

Method Time (seconds)ALN 1.32RPT 1.66MBN 2.82WLR 46.90

For other relationship inferences, which method is better suited will depend onthe complexity of the relationship. For example, ALN in paternity case islinked to 1 or 2 allele counts.

Based on recent experience, it seems RPT is more readily extended to morecomplex problems.

Computational times

...for the mother-typed-only case

Method Time (seconds)ALN 1.32RPT 1.66MBN 2.82WLR 46.90

For other relationship inferences, which method is better suited will depend onthe complexity of the relationship. For example, ALN in paternity case islinked to 1 or 2 allele counts.

Based on recent experience, it seems RPT is more readily extended to morecomplex problems.

Questioning only some of the relationships

Example - paternity testing in the Italian criminal case, including the Victimand 2 unknown contributors in the mixture, one of whom may be the child ofone or both typed parents.

Parenthood tests log10 LRFather 4.26Father, after mother 10.37Mother 4.23Mother, after father 10.35Both 14.60

(using 17 markers)

Relationships among contributors

In criminal cases, we sometimes need to allow for dependence betweengenotypes of contributors. This possibility will obviously affect the evidentialvalue of the mixture.

We distinguish two casespopulations with high relatedness, due to inbreeding, etcspecific close relationships, e.g. father and son, or two brothers engagedin a joint criminal activity

In both cases, the genotypes of two or more actors will be positivelyassociated through identity by descent (IBD), the phenomenon that two genesmay be identical because they are copies of the same ancestor gene, ratherthan being independent draws from the ‘gene pool’. The phenomenon is thesame in both cases, but they require different modelling approaches.

Identity by descent

Relationships among contributors Populations with high relatedness

Population with high relatedness – Ambient IBD

When genotypes are represented by allele counts arrays nia, the number ofalleles a of individual i , the Pólya urn scheme can be expressed through therecursion

n1. ∼DM(2, (αa)Aa=1)

ni.|(nj.)i−1j=1 ∼DM(2, (αa + n<i,a)

where n<i,a =∑i−1

j=1 nja (etc.), and DM denotes the Dirichlet–Multinomialdistribution.

Factorising these conditional distributions over alleles, we find that individualallele counts have Beta–Binomial conditional distributions:

nia|(nj.)i−1j=1 , {nib,b < a} ∼ BB((2− ni,<a), (αa + n<i,a), (α>a + n<i,>a)).

Population with high relatedness – Ambient IBD

When genotypes are represented by allele counts arrays nia, the number ofalleles a of individual i , the Pólya urn scheme can be expressed through therecursion

n1. ∼DM(2, (αa)Aa=1)

ni.|(nj.)i−1j=1 ∼DM(2, (αa + n<i,a)

where n<i,a =∑i−1

j=1 nja (etc.), and DM denotes the Dirichlet–Multinomialdistribution.

Factorising these conditional distributions over alleles, we find that individualallele counts have Beta–Binomial conditional distributions:

nia|(nj.)i−1j=1 , {nib,b < a} ∼ BB((2− ni,<a), (αa + n<i,a), (α>a + n<i,>a)).

Ambient IBD for allele count arrays

Code DNAmixturesUAF is available to augment DNAmixtures to implementthis model for mixture analysis (it just uses the above Beta–binomialconditional probabilities explicitly in replacement CPTs for DNAmixtures).

Relationships among contributors Specific close relationships

Specific close relationships

The general setting is to encode the joint distribution of the genotypes ofspecified individuals in a pedigree, in a form amenable to computationalinference based on peak heights in a DNA mixture with these individuals ascontributors.

For two non-inbred actors, there are only three distinct ibd states at anautosomal locus. The actors can either share none, one or both of their genesibd with probabilities κ = (κ0, κ1, κ2),

∑2i=0 κi = 1 determined by the

pedigree (E. Thompson 2000).

Values of κ for various relationships between two non-inbred individuals.

Pairwise relationship κ0 κ1 κ2

unrelated 1 0 0parent-child 0 1 0sibs 0.25 0.5 0.25half sibs, aunt, grandmother 0.5 0.5 0first cousins, great-grandmother 0.75 0.25 0quadruple half first cousins 0.5312 0.4375 0.0312

Example: Testing whether two contributors have a specific relationship,against all contributors unrelated.

For a simple case, 12 markers:

log10 LRRelationship of U2 to U1 under Hp Case 1 Case 2parent 3.4227 −0.4035sib 3.4125 −0.3018half-sister, aunt, grandmother 2.2301 0.2464cousin, great-grandmother 1.3525 0.2533half-cousin, great2-grandmother 0.7859 0.1884

Easily extends to tests for relatedness with typed individuals, underassumptions of relatedness of contributors.

Example: Testing whether two contributors have a specific relationship,against all contributors unrelated.

For a simple case, 12 markers:

log10 LRRelationship of U2 to U1 under Hp Case 1 Case 2parent 3.4227 −0.4035sib 3.4125 −0.3018half-sister, aunt, grandmother 2.2301 0.2464cousin, great-grandmother 1.3525 0.2533half-cousin, great2-grandmother 0.7859 0.1884

Easily extends to tests for relatedness with typed individuals, underassumptions of relatedness of contributors.

Final RemarksWe have shown that a wide range of relationship inference problems, whereone or more actors appear only as contributors to a DNA mixture, can behandled coherently.

We handle relationships among contributors, and between contributorsand typed individualsOur modular approach provides a toolkit for new problems of this kind

Calculations are done in R using a suite of functions KinMix, freelyavailable. This calls functions in the RHugin package to augment thecapabilities of the DNAmixtures package.

Based on Cowell, Graversen, Lauritzen, Mortera (2015) “Analysis of DNAmixtures with artefacts”. Journal of the Royal Statistical Society C, (withdiscussion); Green and Mortera (2017) “Paternity testing and otherinference about relationships from DNA mixtures”, Forensic ScienceInternational: Genetics; and Graversen, Mortera, Lago (2019) “The YaraGambirasio case: Combining evidence in a complex DNA mixture”Forensic Science International: GeneticsMortera (Roma Tre&Bristol) Relationships & DNA mixtures Cape Town, 28-31 January 2020 39 / 40

The 11th International Conference on ForensicInference and Statistics

ICFIS11 will be held on 15–18 June 2020 at Lund University,Sweden.

relationship testing based on dna mixtures · mortera (roma tre&bristol) relationships &...

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