# lattice

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```					KINSHIP ANALYSIS BY DNA
WHEN THERE ARE MANY
POSSIBILITIES

Charles Brenner
– visiting Dept of Genetics, University
of Leicester, UK
– forensic mathematics
Kinship analysis
Q: How are these people related?
• Genetic evidence
• Likelihood ratio
• Kinship program
– ref: Brenner, CH ―Symbolic Kinship Program‖,
Genetics 145:535-542, 1997 Feb
What a likelihood ratio is
• Compares two explanations for data
• Example: man & child both have Q allele
explanations:
– paternity + some coincidence
– non-paternity + lots of coincidence
Likelihood ratio for Paternity (PI)
• Data: Mother=PS, Child=PQ, Man=RQ
– explanation #1: man is father     PS        RQ
• (2ps)(2qs)(1/4) event               PQ

– explanation #2: not father; his Q is coincidence
• (2ps)(2qs)(q/2) event        PS               RQ
PQ
• LR=1/(2q)
– If q=1/20, data 10 times more characteristic of
―father‖ explanation
Paternity Index exegesis
• PI = X/Y, where
– X=P(genetic types | man=father)
– Y=P(genetic types | man not father)
• Interpretations:
– Odds favoring paternity over non-paternity
assuming all other evidence is equally divided
– Evidence is PI times more characteristic of
paternity
Kinship I (basic)
• paternity (Is this man the father?)
• avuncular (Is this man the uncle?)
– (Latin ―avunculus‖ = uncle)
• missing person (Is this corpse the missing
relative?
• More than two scenarios
– Three
– Many
• disaster
• inheritance
• immigration
• Can always compare two at a time.
• The trick is to organize the work.
Three scenarios —

• Father?
• Uncle?
• Unrelated?
Father/Uncle/Unrelated analysis
Likelihoods of data, assuming man
is      Father              Uncle                  Unrelated
X               (X + Y)/2                   Y

Likelihood ratios
Father vs Unrelated Uncle vs Unrelated Unrelated vs Unrelated
X/Y            (X/Y+1)/2                 1

If for example X/Y=5,
5       :             3        :          1

So, LR for tested man being father, vs uncle, is 5:3
Likelihood ratios are
―multiplicative‖
• means that if explanation ―father‖ is 2 times
better than explanation ―uncle‖
• and ―uncle‖ is 10 times better than
―unrelated‖
• then ―father‖ explains data 20 times better
than ―unrelated.‖
Many-scenario kinship cases
• missing person
– disaster
• inheritance
• immigration
Swissair flight 111 crash
Swissair example

• DNA data
– crash victims (unknowns)
– relatives & effects (references)
• Tentative families
– per Benoit Leclair program
• Too many possibilities!
• Bottom-up approach
• Top-down approach
Five of the X— family are lost

XX
Sylvie Jean-L
X
Jöelle

XX
Clelia   Yves   Albon

• Living reference = Albon = E
• Body parts G,F,D,C,M share DNA with Albon
• (of which G,D,M are female, F,C are male)

G      F                               M
D       C       E               ?
Too many possibilities!
GF M         DF M              ?F M           ?? M
DCE          GCE               ?CE            DFE

?F M         ?C G              GF ?
...
DCE          DFE               D?E

?? M
??E         ...

Note: G, D, M are female; F, C are male.
E is living reference.
Bottom-up approach
• M=Jöelle              vs.      M=unknown

X
M
X
M

?? M      Albon                          Albon
?? ?
??E
??E

Biggest objection —
Doesn’t use all the information
(e.g. other people similar to both M and Albon)
Lattice
A diagram showing
that some things are
better than others.

Arrow = “better than”

Dot = hypothesis/explanation
Kinship lattice — principle of design
• heuristic assumption: any consistent
explanation is weakened when a person is
removed
GF M
DC
(Obtained by
?F M            GF ?     MF G exchange, not by
DC              DC       DC  removal)
?? M            ?F ?
DC              DC

?? ?
DC

?? ?
?C
Top-down approach
6
Goal is LR>10
GF M
LR=300      DC                       10
8              10
10
8    ?F M          9                 GF M             GF M
10                 10
DC                              D?               ?C
?? M                         ?F ?
DC                           DC           9
10
?? D                           ?? ?
6                   DC
(<1)     ?C       10           >1

?? ?
?C
“Lattice”
X— family conclusion
• GF(DC)M explains the data at least ten
million times better than any other
arrangement of some or all of the DNA
profiles G,F,D,C,M
– except ?F(DC)M is only 300-fold inferior
• Practically speaking, the identifications are
proven.
Summary
• Likelihood ratios are the way to quantify evidence
• Kinship with multiple scenarios:
• Individual likelihoods for several scenarios
• Lattice approach for the most complicated situations
Acknowledgements
• Ron Fourney, George Carmody, Benoit
Leclair, Chantal Frégeau

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