See also:
CH Brenner
The correct method of derivation is indicated, and formulae are given for the various possibilities of gene sharing patterns between child and alleged father. In most cases PI = 1 / 4Pr{shared allele} is an adequate formula.
Key words: Parentage testing, motherless paternity case, deficiency case, paternity index, probability of paternity, DNA typing.
H_{1}. The alleged father is in fact the father.
H_{0}. The father is some unknown, unrelated man.
Blood types (e.g. DNA types) are determined for child, alleged father, and mother if possible, and the ratio X/Y is calculated, where X and Y are the probabilities:
X = Pr{observing child types given adult type and assuming H_{1}}, (1)
Y = Pr{observing child types given adult type and assuming H_{0}}. (2)
X/Y, known as the paternity index or PI, therefore tells how many times more easily the observed results are explained by relationship rather than by coincidence. It can be shown ^{(2), (3), (4)} that PI defined in this way completely summarizes the laboratory evidence. If we suppose, for simplicity and just for illustration, that prior to evaluation of the DNA evidence the probability of paternity is ½, the posterior probability of paternity is^{(5)}
So long as the various systems tested are statistically independent, PI for a battery of systems is simply the product of the PI's for the constituent systems. Therefore it will be sufficient to consider the PI calculation for a single system at a time, such as a single DNA locus.
When there is no mother available for typing a conclusion can still be derived based on the types of the child and alleged father. As an example, consider the diagram
Child | P | Q | |
Alleged father | Q | R |
meaning that the heterozygous child and heterozygous alleged father have matching alleles or DNA RFLP fragments of size Q. In this context "match" means that the fragments cannot be clearly distinguished, either as matter of experimental fact or of laboratory protocol. The "matching probability" Pr{Q} is the probability that a fragment selected at random – e.g. from the sperm of some random man – would match Q.
There is a notion whose origin is apparently lost in antiquity that a motherless PI is the same as the PI for the trio case obtained by introducing a fictitious mother identical to the child. That would mean
Fictitious mother | P | Q | |
Child | P | Q | |
Alleged father | Q | R |
The PI corresponding to this diagram is easily calculated. Let p=Pr{P} and q=Pr{Q}. Then from (1) and (2) X and Y are the probabilities^{(6)}
and
hence
PI_{hoary} = X/Y = ½ / (p+q). (3)
The fallacy in this approach can be seen immediately by considering the following related scenario:
Fictitious mother | P | Q |
Child | P | Q |
Alleged father #2 | P | Q |
As full (trio) cases, the paternity index is the same in both cases – X=¼ both times since X depends only on the pattern of matching, and Y is clearly the same both times because Y only depends on the first two rows of the tableaux. Consequently the same PI would be obtained for the two corresponding motherless cases (obtained by looking at just the "child" and "alleged father" rows), even if P were a common fragment and Q were a rare one.
Imagine that the picture represents something like hemoglobin types, with, say, P=normal gene, p>96%, and Q=sickle cell, q<3%. Alleged father #2 has in common with the child only the gene that every man has. There is no evidence against him. But the first alleged father shares a rare trait with the child. Surely that must mean something.
(4)
(5)
where AF=alleged father, M=biological mother, and RM=random man.
This formulation holds whether the mother is typed or not. The motherless case is distinguished from the trio case only in the evaluation of probabilities of maternal contributions like Pr{M passes P}.
Probabilities always only make sense relative to some presumed state of knowledge. From the point of view of a laboratory that knows the mother's phenotype, Pr{M passes P} is 0, ½, or 1. The motherless situation, on the other hand, presumes total ignorance about the mother. Thus we must take the point of view that the maternal gametetic allele is simply a random representative from all alleles in the population, which is to say that it is a P with probability p. This is of course the same as the reasoning appropriate to transmission probabilities for an unknown father ("random man").
Applying these observations to equations (4) and (5) listed above, in the motherless case where the child is PQ we have
and
Y_{m} = 2pq.
Depending on the exact pattern of comparison between child and alleged father, there are several cases to consider.
Child P Q
Alleged father Q R
we have
PI_{(a)} = X/Y = ½p / 2pq = 1 / 4q.
Comparing this result with the incorrect formula (3) shows that (3) works when P and Q are
equally common, and when it is unfair to the man it is unfair by at most a factor of two. As a
possible explanation of the origin of the "fictitious mother" scheme, it may relevant to note that it
is appropriate for calculating the probability of
exclusion^{(7)}.
It is interesting to consider the benefit of typing the mother. For the common trio situation wherein the mother shares only P with the child, PI=1/2q, so when the man is in fact the father the likely benefit of typing the mother is seen to be a factor of 2 in each system. Of course, when the man is not the father typing the mother may also have the beneficial effect of excluding him.
Child P Q
Alleged father Q
PI_{(b)} = p / 2pq = 1/2q. The benefit of typing the mother is again a factor of 2.
Child P Q
Alleged father P Q PI_{(c)} = (p/2 + q/2) / 2pq
= ( 1/4q ) + ( 1/4p ). Typing the mother in this case may have a paradoxical effect of decreasing the paternity index. This will happen when the mother's type proves that the commoner of P and Q is the paternal fragment, and also when the mother matches both of the child's sizes.
Child Q
Alleged father Q R
PI_{(d)} = ½q / q^{2} = 1 / 2q so the same as case (b).
In this and the next case typing the mother is inconsequential.
Child Q
Alleged father Q
PI_{(e)} = 1q / q^{2} = 1 / q.
probe | FRAGMENT SIZES (KB) | Frequencies for child's fragments | PATERNITY INDEX | |||||
Alleged father | Child | Wrong formula | Right formula | |||||
Case A | ||||||||
YNH24 | 3.49 | 4.56 | 4.56 | 4.85 | 0.048 | 0.098 | 3.4 | (a) 5.2 |
TBQ7 | 3.63 | 4.00 | 3.63 | 4.00 | 0.089 | 0.063 | 6.6 | (c) 6.8 |
Product | 23 | 35 | ||||||
Case B | ||||||||
3'HVR | 3.68 | 2.34 | 2.34 | 0.218 | 2.3 | (d) 2.3 | ||
YNH24 | 3.98 | 3.7 | 3.7 | 0.136 | 3.7 | (d) 3.7 | ||
TBQ7 | 3.51 | 3.51 | 5.65 | 0.082 | 0.032 | 8.8 | (b)6.1 | |
Product | 74 | 50 | ||||||
Case C | ||||||||
3'HVR | 2.65 | 3.37 | 3.37 | 3.05 | 0.061 | 0.023 | 6 | (a) 4.1 |
YNH24 | 3.61 | 4.83 | 4.83 | 9.85 | 0.097 | 0.01 | 4.7 | (a) 2.6 |
TBQ7 | 4.13 | 6.02 | 6.02 | 3.63 | 0.05 | 0.09 | 3.6 | (a) 5 |
Product | 99 | 53 | ||||||
Illustration | ||||||||
Q | R | Q | P | 0.01 | 0.1 | 4.55 | (a) 25 | |
R | P | Q | P | 0.01 | 0.1 | 4.55 | (a) 2.5 | |
(a) etc. refer to the various subcases discussed in the text. |
1. This work was supported in part by the DNAVIEW Users' Group.
2. Brenner, C. Calculation of Paternity Index. In: Walker R, ed. Inclusion Probabilities in Parentage Testing. Arlington, VA: American Association of Blood Banks, 1983:633-638.
3. Nijenhuis, LE. A Critical Evaluation of Various Methods of Approaching Probability of Paternity. In: Walker R, ed. Inclusion Probabilities in Parentage Testing. Arlington, VA: American Association of Blood Banks, 1983:103-112.
4. Morris, JW. Relationships Between Power of Exclusion and Probability of Paternity. In: Walker R, ed. Inclusion Probabilities in Parentage Testing. Arlington, VA: American Association of Blood Banks, 1983:103-112.
5. Essen-Möller E, Die Beweiskraft der Ähnlichkeit im Vaterschaftsnachweis – Theoretische Grundlagen. Mitt Anthrop Ges (Wein) 1938;68;9-53.
6. Some people prefer to use double these values for each of X and Y. Of course the ratio remains the same.
7. Mayr WR. Paternity Testing with Unavailable Putative Father or Mother. In: Walker R, ed. Inclusion Probabilities in Parentage Testing. Arlington, VA: American Association of Blood Banks, 1983:377.