Realistic matching odds
An interesting article by Balding led
me to the following line of thought.
Suspect related to donor
In a typical criminal DNA case, the matching odds calculation
presented by the laboratory to the court assumes that the suspect is
either the donor or is unrelated to the donor. Consequently the odds
against a match at random are astronomical, such as 1014
for a typical CODIS (i.e. United States) STR profile.
If instead the suspect can make a case that the donor might be his
brother, the matching odds drop considerably – maybe
106. The most authoratitive
recommendation in the US on this point states:
Recommendation 4.4: If the possible contributors of the
evidence sample include relatives of the suspect, DNA profiles of
those relatives should be obtained. If these profiles cannot be
obtained, the probability of finding the evidentiary profile in those
relatives should be calculated with Formulae 4.8 or 4.9
But what are "possible contributors"?
"Possible contributors"
Since the excerpt states that possible contributors should be tested,
apparently they are considered to be rather definite people. But
that's puts the suspect in an unfair either/or situation:
- Either there is no chance at all that the suspect is related
to the donor (unless by identity), or
- there is a very good chance.
- Nothing in between, such as a slight chance.
But common sense says there is always a slight chance. Maybe
the suspect doesn't even know that he has a brother. Nonetheless, from
the court's point of view he might. And even a slight chance changes
the matching odds radically.
Intermediate view
The point is that the fair computation of matching odds would be
some sort of weighted odds between the "unrelated random man"
computation (e.g. 1014) and the "brother or other close
relative" computation (formulas 4.8 and 4.9, e.g. 105).
Even a very slight weighting on the second possibility gives a result
several orders of magnitude smaller than the "unrelated" computation.
To reiterate – the "unrelated" computation is based on
idealized, i.e. unrealistic, assumptions, such as the assumption that
the suspect is either the donor or is completely unrelated to the
donor. At the other unrealistic extreme, we imagine that the suspect
is either the donor or the brother of the donor. A reasonable
intermediate view, in my opinion, is that assuming the suspect is not
the donor then he is probably nearly unrelated to the donor but there
is some small chance that the donor is his close relative, so a more
relevant matching odds calculations is some weighted average between
the two numbers I mentioned (ok, and also of other intermediate
numbers representing other relationships, more distant than brother).
Weighted average
Here's an example.
| Degree of relationship
| Prob thereof
| Prob relative matches
| Chance of such match
|
|---|
| sibling | 1/4096 | 1/150000
| 1/630e6
|
| parent/child
| 1/4096
| 1/160e6
| 1/640e9
|
| uncle/nephew/halfsib
| 1/2048
| 1/19e9
| 1/40e12
|
| 1st cousin
| 1/1024
| 1/620e9
| 1/630e12
|
| 1/32 allele sharing IBD
| 1/512
| 1/5.9e12
| 1/3e15
|
| 1/64 (=second cousin)
| 1/256
| 1/23e12
| 1/5.8e15
|
| 1/128
| 1/128
| 1/48e12
| 1/6.1e15
|
| 1/256 (=third cousin)
| 1/64
| 1/71e12
| 1/4.6e15
|
| 1/512
| 1/32
| 1/88e12
| 1/2.8e15
|
| 1/1024 (=fourth cousin)
| 1/16
| 1/97e12
| 1/1.6e15
|
| 1/2048
| 1/8
| 1/100e12
| 1/820e12
|
| 1/4096 (=fifth cousin)
| 1/4
| 1/110e12
| 1/420e12
|
| unrelated
| 1/2
| 1/110e12
| 1/220e12
|
| Cumulative weighted matching
chance=
| 1/630e6
|
|---|
Matching odds calculated conventionally are 110e12 (1.1 •
1014). Maybe 630 million is a more meaningful number. The
estimate 1/4096 as the relative chance of a brother being the culprit
(assuming the suspect is not), is a completely arbitrary number.
Notice, though, that the weighted average is 100% dominated by the
"brother" term. The "random man" matching number is completely
irrelevant. The final result depends on only:
- the matching odds for siblings
- the conditional probability a sibling is the donor, given that
the suspect is not.
The DNA•VIEWTM profile
computation tool DNA odds offers the above computation as
an option ("Uniqueness estimate").
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