Download Genetic biases in using `Mendelian randomization` to compare

LETTERS TO THE EDITOR
1167
Genetic biases in using ‘Mendelian randomization’ to compare transplantation with
chemotherapy
From ROBERT N CURNOW
Sirs—The recent paper by Wheatley and Gray1 pointed out that
Mendelian randomization may not entirely remove biases in
the comparison of stem cell transplantation (SCT) with
chemotherapy. These biases arise from the probable delays in
SCT treatment and because of prognostic factors that may be
correlated with the number of siblings in the family.
There may also be biases because patients with an
HLAcompatible sibling will have HLA genotypes in differing
proportions to those without a compatible sibling. This will only
be important if the relative effectiveness of the treatments is
related directly to the patient’s HLA genotype or to any
genotypes at other loci that are associated with the HLA
genotypes.
Consider a single locus with n alleles, A1, A2, . . . , An, that
have frequencies in the population pi, i = 1, 2, . . . , n, and
assume that the there is random mating in the parental
generation; that a match with a sibling means a complete
genotypic match at the locus and that the patient who is a
candidate for a transplant has only one sibling available for
testing as a donor. Table 1 shows the parental matings that may
produce siblings that are identical at the locus, either both
homozygous, A1A1, or both heterozygous, A1Ax. In Table 1 X, Y,
and Z represent all possible distinct alleles that are not A1. The
probability of each parental mating is obtained by multiplying
the product of the frequencies of the four alleles involved by a
coefficient obtained by multiplying factors of 2 for each
heterozygous parent as in the Hardy–Weinberg formula and
further by a factor of 2 if the parents differ in genotype. The
latter allows for the two genotypes being either male/female or
female/male. Also shown for each parental mating are the
probabilities of the matched pairs of siblings, A1A1 or A1Ax. By
summing the product of the probabilities of the parental mating
type and the probabilities of the matching pairs over all possible
pairs of distinct alleles Ax and Ay, the probability of a matching
pair of homozygous A1A1 siblings is
(1)
P(A1A1, A1A1) p21(1 p1)2 4.
In this summation the frequency of parents A1Ax, A1Ay needs to
be halved since otherwise the summation over both x and y
counts each parental mating type twice.
The probability of a matching pair of heterozygous siblings,
for example A1A2, is
/
/
p(A1A2, A1A2) p1p2(1 p1 p2 2p1p2) 2.
(2)
Equations (1) and (2) compare with the corresponding
probabilities for a random patient with no sibling available for
testing, p12 and 2p1p2.
Summing (1) and (2) over the n alleles and n(n 1)/2 pairs
of alleles gives, respectively, the probability of a homozygous
and a heterozygous match.
School of Applied Statistics, The University of Reading, PO Box 240, Reading
RG6 6FN, UK. E-mail: [email protected]
Table 1 Parental frequencies and probabilities of matched sibling pairs,
homozygous A1A1 and heterozygous A1Ax, for a single locus
Parental
genotypes
Frequency
Probability
A1A1 sibling
pair
Probability
A1Ax sibling
pair
1
0
A1A1
A1A1
p41
A1A1
A1Ax
4p13px
1/4
1/4
A1Ax
A1Ax
4p12p2x
1/16
1/4
A1Ax
A1Ay
8p12pxpy
1/16
1/16
A1A1
AxAx
2p12p2x
0
1
A1A1
AxAy
4p12pxpy
0
1/4;
A1Ax
AxAx
4p1p3x
0
1/4;
A1Ax
AxAy
8p1p2x py
0
1/16
A1Ay
A1Ax
4p1p2x py
0
1/4
A1Ay
AxAy
8p1pxpy2
0
1/16
A1Ay
AxAz
8p1pxpypz
0
1/16
x, y, z and 1 indicate different alleles.
Matching of siblings is by phenotype not genotype with
some alleles at each of the three HLA loci, A, B, and DR,
merged into broad antigenic equivalents. The chromosome
sections with the merged alleles will be referred to as
haplotypes. Providing there are no recombinations within the
HLA locus, haplotypes can be treated for our purposes as alleles
at a single locus with one exception. Two siblings can be
phenotypically identical although not sharing identical
haplotypes. As an example, among the 15 haplotypes
commonest in Caucasian American populations (Mori et al.2)
there is one set of four haplotypes that can in two pairs achieve
the same phenotype. The pairs are A1B8DR3/A2B57DR7 and
A1B57DR7/A2B8DR3. The parental haplotypes must have been
A1B8DR3/A1B57DR7 and A2B8DR3/A2B57DR7 with probability
8p1p2p3p4 where p1, p2, p3 and p4 are the frequencies of the
four haplotypes. Patients with matching siblings will, therefore,
be equally A1B8DR3/A2B57DR7 and A1B57DR7/A2B8DR3 each
with probability p1p2p3p4 = 0.000000032. The corresponding
probabilities of a child with a matching sibling through identical
haplotypes are 0.000196 and 0.000044. Clearly the matches
with non-identical haplotypes can safely be ignored.
1168
INTERNATIONAL JOURNAL OF EPIDEMIOLOGY
Table 2 Probability of a matched sibling pair and comparisons
between a random individual and a patient with a matched sibling of
the probabilities of a homozygous individual and the probabilities of
haplotypes and phenotypes associated with the three most frequent
haplotypes H1, H2, and H3 in a North American Caucasian population
Random
individual
P (Matched pair)
Patient with a
matching sibling
0.2531
P (Homozygous)
0.0062
0.0065
H1
0.0518
0.0542
H2
0.0263
0.0268
H3
0.0215
0.0218
H1H1
0.00268
0.00293
H1H2
0.00272
0.00291
H1H
0.00223
0.00237
H2H2
0.00069
0.00072
H2H3
0.00113
0.00117
H3H3
0.00046
0.00048
The National Marrow Donor Program Donor Register
(www.ashi-hla.org/resourcesfiles/resources-frequencies.html)
provides estimates of the ABDR haplotype frequencies for three
North American populations. Using the data on the Caucasian
American population in which 9642 haplotypes were estimated
to have non-zero frequencies, only 184 had estimated
frequencies 0.1%. The three highest estimated frequencies
were 0.0518, 0.0263, and 0.0215 (Mori et al.2).
Table 2 shows that the probability of a matched sibling
pair in this population is 0.2531. With the large number of
haplotypes and the small frequencies associated with each,
most parental pairs of identical siblings will be heterozygous
with different haplotypes in the two parents. All such parental
pairs would give a probability of 0.25 of a matched sibling
pair and phenotypic frequencies the same as in a random
population. The additional probability of 0.0033 arises from
other types of parental pairs. The frequencies of the three
commonest haplotypes and the six corresponding phenotypes
shown in Table 2 demonstrate that this small set of additional
parental pairs results in very small differences between the
frequencies in the general population and in patients with a
matching sibling. Clearly the genetic bias is very small and
can safely be ignored particularly remembering that, to be
important, the difference in phenotype has to affect
differentially the consequences of the two treatments.
Corresponding results for the African American and Asian
American populations were even closer to 0.25, 0.2508 and
0.2516, respectively.
The differences will be doubled if the patient has a sibling
who does not match, but the difference will still be negligible.
The effect on the comparison when the patient has more than
one sibling with varying numbers of them matching their
phenotype is more difficult to quantify but again unlikely to be
important.
Clearly, the number of possible HLA haplotypes at the A, B,
and DR loci and their low frequencies results in the majority of
parents of siblings with identical HLA phenotypes being
heterozygous and sharing no haplotypes in common. This
means that differences in the distribution of phenotypes of
siblings with and without an HLA identical sibling are very
small. They should be ignored and concentration should be
continued on the non-genetic biases that may occur between
the types of patient being used to compare SCT with
chemotherapy.
Acknowledgement
I am grateful to Dr Linda Shelper of UK Transplant for
providing access to the data used in this paper.
References
1 Wheatley K, Gray R. Commentary: Mendelian randomization—an
update on its use to evaluate allogenic stem cell transplantation in
leukaemia. Int J Epidemiol 2004;33:15–17.
2 Mori M, Beatty PG, Graves M, Boucher KM, Milford EL. HLA gene and
haplotype frequencies in the North American population: The National
Marrow Donor Program Donor Registry. Transplantation
1997;64:1017–27.
doi:10.1093/ije/dyi161
Advance Access publication 8 August 2005
Origins of the mutational origin of cancer
From LUTZ EDLER* and ANNETTE KOPP-SCHNEIDER
The authors would like to congratulate the editors of the
International Journal of Epidemiology for their insight into the
important role of theoretical concepts of carcinogenesis for
the understanding, prevention, and treatment of cancer, and for
Department of Biostatistics–C060, German Cancer Research Center,
Im Neuenheimer Feld 280, D-69120 Heidelberg, Germany.
* Corresponding author. E-mail: [email protected]
their initiative to refresh the concept of multi-stage theory
of cancer through reprinting of the pioneering paper of
P Armitage and R Doll1 from 1954. The commentaries of Steven
A Frank,2 Suresh H Moolgavkar,3 and Sir Richard Doll4 provided
further insight into the importance of this theory for examining
and explaining the change in cancer mortality with age. Since
1954 the multi-stage theory has become a necessary prerequisite
for understanding cancer data and developing advanced concepts.