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5391
CORRESPONDENCE
Probability Calculations for a Matched Donor in the Extended Family
To the Editor:
Recently, Schipper et al’ described a computer program that calculates a patient’s probability of finding a matched bone marrow donor
in the extended family. The formulas they use for the probabilities of
at least one match among auntshncles, cousins, and both combined
(equations 1, 4, and 6, respectively) are only valid for events that
are statistically independent. Because they potentially involve the
unknown haplotypes of the patient’s grandparents, the genotypes of
blood-related aunts/uncles and cousins on the same side of the family
arenot statistically independent. Using formulas for independent
events will give overestimates of the true probability. The magnitude
of the error grows with the number of family members.
For example, the output of their Fig2 for a particular patient
shows the calculated probability of a match among N = 2 aunts/
uncles on the mother’s side as 0.036378. A calculation taking into
account the dependence of these two people and using the given
haplotype frequencies [h(A) = 0.07198, h(C) = 0.010071 puts the
true probability at 0.032035. The formula for independent events
gives a 13% relative error in this case. When N = 3 auntduncles,
we would generally have close to 30%error, and when N = 4 aunts/
uncles, the error would generally be about 45%. Thus, in certain
situations, this computer program will significantly overstate the true
probability.
The investigators correctly note the dependence between cousins
of the same parents, but ignore the dependencies among bloodrelated aunts/uncles; between auntshncles and cousins; and among
cousins of different parents. To calculate the overall probability of
a match on, eg, the father’s side, we need to condition on the full
genotypes of the paternal grandparents.
Let Oi&) = 2*HFc*(1 - HFc)*(.75)” + (HFC)**(.5)”+ (1 HFC)’ denote the probability that a paternal sibling with the genotype
A N and n offspring does not have a match for the patient among
hisher offspring (the investigators use haplotype N to denote anything that is neither A nor C). Similarly, we have O&t) = 2*HFA*(1
- HF,4)*(.75)”+ (ffFA)’*(.s)” + (1 - HFA)*; Oh(n) = 2*HFc*(1
- HFC)*(S)” + (HFC)**(O)” + (1 - HFC)’; and O&n) = 2*HFA*
(1 - HFA)*(.5)”+ (HFA)’*(0)” + (1 - HFA)2,where 0’ = 1 . Let
I be the number of paternal siblings and n,be the number of offspring
of the if* aunt/uncle i = 1,2, . . . , I . Given that the patient’s father
inherited haplotype A from one grandparent (labeled no. 1) and
haplotype B from the other (labeled no. 2), we have Table 1 of
possible genotypes for the paternal grandparents.
The total probability of no matches among blood-related aunts/
uncles or cousins on the paternal side can be calculated by summing
the product of the second and third columns of each row. A similar
calculation can be performed for the maternal side. The overall
probability of no match from either the paternal or maternal side is
the product of these two quantities: Pr(no march) = Pr(no march on
paternal side)*Pr{nomatch on maternal side). Finally, the overall
probability of a match is this quantity subtracted from 1: Pr(march)
= 1 - Pr(no match]. Admiaedly, this formula is more involved
Table 1. Nine Possible Genotype Combinations forthe Patarnal Grandparents GivenThat the Father Is AB
Grandparent
No. 1
Grandparent
No. 2
AN
BN
Conditional Probability of No Matches Among BloodRelated AuntsIUncles or Cousins on Paternal Side
Probability of
Genotypes
I HFN)’
I
R {O0.5+02w(nj)
+ 0.5)
1
I=
AN
BA
H F g HFA
AN
BC
HF,+ HF,
l
i= 1
AA
6N
AA
BA
AA
BC
HF,,* HF,
AC
6N
HF&iFN
AC
BA
HFp HF.,
AC
BC
( HFc)’
HFA*HFN
I
I
n
,=1
(0.25*0&InJ + 0.25*0&(ni) + 0.25*O&(n,)}
From www.bloodjournal.org by guest on July 28, 2017. For personal use only.
5392
CORRESPONDENCE
than that used by Schipper et al,’ but it is easily programmed into
a computer with little additional effort. An analogous formula can
be used when either the mother or father is homozygous.
When feasible, it might also be interesting to consider grandparents as potential donors. The probability that grandparent no. 1 would
be a match for the patient is HF,, which is greater than that of any
aunt/uncle or cousin on the paternal side of the family in virtually
every case. Similarly, the probability that whichever maternal grandparent gave the mother the C haplotype would match the patient is
HF,, which is greater than that of any aunduncle or cousin on the
maternal side in virtually every case.
Craig Kollman
National Marrow Donor Program
Minneapolis, MN
REFERENCE
1. Schipper RF, D’Amaro J, Oudshoorn M: The probability of
finding a suitable donor for bone marrow transplantation in extended
families. Blood 87:SoO. 1996
Response
Dr Kollman discusses certain parts of our model, stating that
three of our equations (1, 4, and 6) are only valid for statistically
independent events. He further states that these events are dependent,
because they concern related individuals that inherited part of their
genome from the same parents and grandparents. We fully agree
with Dr Kollman and will modify the computer program accordingly.
R.F. Schipper
J. D’Amaro
M. Oudshoorn
Immunohematology
Leiden University Hospital
Lriden, The Netherlands
From www.bloodjournal.org by guest on July 28, 2017. For personal use only.
1996 87: 5391
Probability calculations for a matched donor in the extended family
[letter; comment]
C Kollman
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