The drop-out/drop-in model

31 May 2012
The Likelihood Ratio model
Hinda Haned
h.haned@nfi.minvenj.nl

Outline
Illustration of the LR principle applied to DNA
mixtures
Two-person mixtures to explain the principle
(but no general formula is given!)
Example with and without allelic dropout
The LR model— Avila June 2013 1

DNA mixtures
Two or more individuals contributing to the sample
More than two peaks per locus

Why are mixtures challenging?
What genotypes created the mixture?
12,12/13,15
12,15/13,15
12,13/13,15
...

ISFG DNA commission recommendations
The likelihood ratio is the preferred approach to mixture
interpretation. DNA commission 2005
Probabilistic approaches and likelihood ratio principles are superior
to classical methods.
DNA commission 2012

The Bayesian framework: likelihood ratios
LR =
Pr(data|Hprosecution)
Pr(data|Hdefence)
data: alleles and their peaks
ratio of two probabilities or,
ratio of two likelihoods

Interpretation
Need for an interpretation framework that applies to all types
of samples:
• High template
• Low template: PCR-related stochastic eﬀects are exacerbated,
creating uncertainty about the composition of the
crime-sample
Reporting oﬃcers make pre-case assessments and formulate the
propositions to be evaluated within the likelihood ratio framework.

Dropout/Drop-in deﬁnitions
Allele or locus dropout is deﬁned as a signal that is below the limit
of detection threshold, it occurs when one or both alleles of a
heterozygote fail to PCR-amplify.
Allele drop-in is an allele that is not associated with the
crime-sample and remains unexplained by the contributors under
either Hp or Hd.

Low/High template DNA
High template DNA
The epg reﬂects the composition of the sample:
• no dropout
• no drop-in
Low level DNA
The epg does not reﬂect the composition of the sample:
• allele dropout
• allele drop-in
• stutters
• ...

Part 1: High template DNA, the epg reﬂects the
composition of the sample.

Two-person mixture example
Two-person mixture

Two-person mixture example
Locus1
Evidence 9,11,12
Suspect 9,11
Victim 11,12
Hp: Suspect + Victim contributed to the sample
Hd : Victim + Unknown person (unrelated to the suspect)
contributed to the sample

Two-person mixture: Under Hp
Locus1
Evidence 9,11,12
Suspect 9,11
Victim 11,12
Hp: Suspect + Victim contributed to
the sample
Pr(Evidence|Hp) = 1

Two-person mixture: Under Hd
Locus1
Evidence 9,11,12
Victim 11,12
Unknown ?
Hd : Unknown + Victim contributed
to the sample

Two-person mixture: Under Hd
The victim’s proﬁle explains 11 and 12
The unknown has to have allele 9: allele 9 is constrained
Locus1
Evidence 9,11,12
Victim 11,12
Unknown 9,11
9,12
9,9
Pr(evidence|Hd ) =
2p9p11 + 2p9p12 + p2
9

Two-person mixure: LR
Hd : Victim + Unknown person (unrelated to the suspect)
Pr(Evidence|Hp) = 1
Pr(evidence|Hd ) = 2p9p11 + 2p9p12 + p2
9
LR =
1
2p9p11 + 2p9p12 + p2
9

What is the underlying model?
LR is a function of the genotypic frequencies
Assumes independent association of the alleles within loci:
Hardy Weinberg equilibrium
Multiply between loci: Linkage equilibrium
The product rule

Summary
Derive the possible genotypes for the unknowns
Determine the genotypic probabilities
Sum up the probabilities for all plausible genotypes
Calculate the ratio of the probabilities under Hp and under Hd
You should not do this by hand!
usually, analysis of 15 or more loci simultaneously
calculations get complicated with two or more unknowns

What happens if there are two unknowns under Hd?
Hd : Two Unknown individuals (unrelated to the suspect)
Locus1
Evidence 9,11,12
Unknown 1 ?
Unknown 2 ?
Have to consider all the
plausible genotypic
combinations for the unknown
that explain alleles 9,11,12
observed in the crime-sample.

Under Hd: two unknowns
Unknown 1 Unknown 2
9,9 11,12
11,11 9,12
12,12 9,11
9,11 9,12
9,11 11,12
9,12 11,12
Pr(Evidence|Hd ) = 2(p2
92p11p12 + p2
112p9p12 + p2
122p9p11+
2p9p112p9p12 + 2p9p112p11p12 + 2p9p122p11p12)

LR: two unknowns
LR =
1
2(p2
92p11p12 + p2
112p9p12 + p2
122p9p11 + 2p9p112p9p12 + 2p9p112p11p12 + 2p9p122p11p12)
Increasing the number of unknowns increases the number of
terms under Hd

Part 2: Low template DNA, the epg does not reﬂect
the composition of the sample.

Likelihood ratios vs. Low template DNA
Classical approach of the LR: the product rule
Main source of uncertainty in previous examples: Genotypes
of unknown contributors
We will now see how we can modify the classical LR approach to
account for uncertainty in the data, due to low template DNA
conditions

Uncertainty in the data: single-source example
Locus1
Evidence 11
Suspect 9,11
Hp: Suspect contributed to the sample
Hd: Unknown person (unrelated to
the suspect) contributed to the sample
Classical LR: Pr(Evidence|Hp) = 0
LR with dropout and drop-in: Pr(Evidence|Hp) = 0

LR with dropout and drop-in
Main theory described by:
• Haned et al, FSIG, 2012
• DNA commission ISFG, FSIG 2012
• Gill et al, FSI 2007
• Curran et al, FSI, 2005
Two key parameters in the model
• dropout: Heterozygote, Homozygote
• drop-in: not treated here
Basic model: qualitative data only, also called the drop-model.

LR with dropout and drop-in
An allele drops out with a probability of d
An allele does not drop out with a probability of 1 − d
Allele dropout from a heterozygote: d
Allele dropout from a homozygote: d

Single-source example: Under Hp
Hp: Suspect contributed to the sample
dropout
Allele 9 yes
Allele 11 no
Pr(evidence|Hp) = Pr(dropout of 9) × Pr(non-dropout of 11)
= d × (1 − d)

Single-source example: Under Hd
Unknown contributed to the sample
Locus1
Evidence 11
Unknown ?

The Q alleles
What are the possible genotypes for the unknown?
• The dropped out alleles are gathered under a virtual alleles Q
• Q is a ‘place-holder’ to all possible genotypes!
• The Unknown’s genotype has to explain allele 11 (no drop-in)

Under Hd
Locus1
Evidence 11
Unknown 11,11
11,Q
Q can be anything except 11
Unknown genotype must explain 11
This leaves us with two possibilities:
• Homozygote: 11, 11
• Heterozygote 11, Q

Q allele
• Locus L has ﬁve alleles: {9, 10, 11, 12}
• p9 + p10 + p11 + p12 = 1
• pQ = 1 − p11
• pQ = p9 + p10 + p12
11,Q can be:
• 9,11
• 10,11
• 11,12
No need to worry about deriving all the
genotypes!
All thee genotypes are regrouped
under 11Q with frequency: 2p11pQ

Summary
Two possible genotypes: 11,11 and 11Q
Dropout Genotype probability
11,11 (1 − d ) p2
11
11Q (1 − d)d 2p11pQ
LR =
d(1 − d)
(1 − d )p2
11 + (1 − d)d2p11pQ

LR vs. probability of dropout

Low-template mixture
Low-template DNA mixture

Two-person mixture example: one dropout, no drop-in
Locus1
Evidence 9,10,12
Suspect 9,11
Victim 10,12
Hp: Suspect + Victim
Hd: Two unknowns (unrelated to
suspect/victim)

Under Hp: Dropout from the suspect
Suspect 9,11 d(1-d)
Victim 10,12 (1-d)2
Pr(Evidence|Hp) = d(1 − d)3

Under Hd: dropout is possible
Hd: two unknowns
Dropout is possible: Q allele, can be anything except 9, 10, 12
9,9 10,12
No-dropout
10,10 9,12
12,12 9,10
9,12 9,10
9,12 10,12
10,12 9,10
9Q 10,12
One dropout10Q 9,12
12Q 9,10

Under Hd: dropout is possible
Hd: two unknowns
Dropout is possible: Q allele, can be anything execept 9, 10,
12
Dropout Genotype Prob.
9,9 10,12
(1 − d )(1 − d)2
p2
9 × 2p10p12
10,10 9,12 p2
10 × 2p9p12
12,12 9,10 p2
12 × 2p9p10
9,12 9,10
(1 − d)4
2p9p12 × 2p9p10
9,12 10,12 2p9p12 × 2p10p12
10,12 9,10 2p10p12 × 2p9p10
9Q 10,12
d(1-d)3
2p9pQ × 2p10p12
10Q 9,12 2p10pQ × 2p9p12
12Q 9,10 2p12pQ × 2p9p10

Likelihood ratio

LR vs. dropout probability
0.0 0.2 0.4 0.6 0.8 1.0
51015202530
d
LR
LR vs. Drop−out

How about drop-in probability?
Under Hp: Dropout from the suspect
Suspect 9,11 d(1-d)
Victim 10,12 (1-d)2
If drop-in=0 Pr(Evidence|Hp) = d(1 − d)3
If drop-in = 0: Pr(Evidence|Hp) = d(1 − d)3 × (1 − c)
c is the probability of drop-in

Under Hd: two unknowns
Dropout is possible, no drop-in: Q allele, can be anything
except 9, 10, 12
If drop-in is possible: Q allele can be anything!
So the genotypes of the unknown have no longer to explain
alleles 9, 10, 12.
This increases the number of terms under Hd

Think of drop-in as a scaling factor
If an allele is a drop-in: multiply by c× frequency of allele i.
If an allele is not a drop-in, multiply by (1 − c)

LR vs. dropout and drop-in probability
0.0 0.2 0.4 0.6 0.8 1.0
51015202530
d
LR
LR vs. Drop−out
drop−in=0
drop−in=0.01
drop−in=0.05

Summary
Derive the possible genotypes for the unknowns
Determine the genotypic probabilities
Sum up the probabilities for all plausible genotypes
Determine the corresponding dropout probabilities
Calculate the ratio of the probabilities under Hp and under Hd

Software
Derive genotypes of the unknowns is the key issue
Assign genotype probability to each genotype
The number of possibilities increases with the number of
contributors, deriving LRs for mixtures by hand is not realistic!

Casework example 1: A 3-person mixture
Victim is major contributor
At least two minor contributors

Casework example 1: A 3-person mixture
Hp: Victim + Suspect + Unknown
Hd: Victim + two unknowns

Sensitivity analysis: Overall LR
Same dropout probability for all
contributors
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
d
log10LR
0.01 0.20 0.40 0.60 0.80 0.99
Overall LR for the 10 SGM+ loci

Sensitivity analysis: Overall LR
Average probability vs. Splitting
dropout/contributor =⇒ No
signiﬁcant diﬀerences between the
models!
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
d
log10LR
0.01 0.20 0.40 0.60 0.80 0.99
Basic model
SplitDrop model

Plausible ranges for PrD?
LR dropout
≤ 1010
0.01 ≤ D ≤ 0.50
[109
, 108
] 0.50 < D ≤ 0.99
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
d
log10LR
0.01 0.20 0.40 0.60 0.80 0.99

Casework example 2: two-person mixture
LR dropout
(1) [1010
, 109
] 0 ≤ D ≤ 0.50
(2) [109
, 106
] 0.50 < D ≤ 0.76
(3) [106
, 104
] 0.76 < D ≤ 0.84
(4) [104
, 1] D > 0.84 0
5
10
Probability of dropout
log10LR
0.01 0.50 0.76 0.93
(1) (2) (3) (4)

Casework example 3: three-person mixture
LR dropout
(1) [1014
, 109
] 0 ≤ D ≤ 0.08
(2) [109
, 106
] 0.08 < D ≤ 0.53
(3) [106
, 104
] 0.53 < D ≤ 0.75
(4) [104
, 100] 0.75 < D ≤ 0.86
(5) [100, 1] 0.86 < D ≤ 0.93
0
5
10
15
Probability of dropout
log10LR
0.08 0.53 0.75 0.86
(1) (2) (3) (4) (5)

All models are wrong...
Continuous models are expected to extract more information
from the data, but their implementation is diﬃcult and
tedious in practice
semi-continuous methods are easier to implement and can
serve as a good approximation

How to inform dropout probabilities?
Estimate dropout probabilities via logistic regression
• diﬃcult to extended to > 2-person mixtures
Deﬁne plausible ranges of dropout
• based on expert belief
• based on maximum likelihood principle
Bayesian approach: combine prior belief and likelihood to
yield a posterior distribution

The drop-out/drop-in model

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The drop-out/drop-in model