Download Allele frequency estimation in the human ABO blood group system

Allele frequency estimation in the human
ABO blood group system
Pedro J.N. Silva
Faculdade de Ciencias da Universidade de Lisboa
Campo Grande, C2, 4o. piso
P-1700 LISBOA PORTUGAL
[email protected]
2002
Table of Contents
OVERVIEW
1
THEORY
2
Population genetics
Genetics nomenclature
The ABO system
Hardy-Weinberg frequencies
2
2
2
3
ABO allele frequency estimators
Bernestein (1925)
Bernestein (1930)
Wiener (1929)
Maximum Likelihood and the EM algorithm
3
3
4
4
5
Statistics
Maximum Likelihood
The EM algorithm
Log-likelihood ratio test
2
Pearson's χ test
5
5
5
6
6
S2 ABOESTIMATOR
7
Description
7
How to get the latest version
7
RECOMMENDED READING
8
Overview
We deal here with the estimation of allele frequencies of the human ABO blood group system.
It is assumed that
• the ABO system is determined by three alleles of a single gene, call them A, B and O
•
•
•
A and B are codominant, and both are dominant over O
this gene is in Hardy-Weinberg frequencies in the population
the data are a random sample from the population
You should be familiar with classical population genetics, maximum likelihood estimation and the EM
algorithm, as well as statistical testing in general and goodness-of-fit tests in particular.
Towards the end, there is a plug for a computer program that you may find useful for the actual calculations.
See some suggested bibliography at the end, and a brief summary of relevant theory follows.
ABO allele frequency estimation
1
Theory
Population genetics
Genetics nomenclature
A gene is a unit of hereditary transmission (or, as some whould say, a gene is whatever geneticists study...).
Different forms of the same gene are known as alleles (e.g., A and a; A, B and O).
Alleles may be combined in genotypes (e.g., AB, or OO), which may or may not have distinct phenotypes
(e.g., white or red flowers; different blood groups), depending on dominance relationships. For example,
since AA and AO have the same phenotype (blood group A), different from that of OO, we say A is dominant
over O; on the other hand, AA, AB and BB all have distinct phenotypes (blood groups A, AB and B,
respectively), so we say A and B are codominant.
The relative proportion of each allele in a population is called its allele frequency; similarly, the relative
proportion of each genotype is its genotypic frequency and, as you can guess, the relative proportion of each
phenotype is the phenotypic frequency. As long as there is no dominance, the frequency of one allele can
be estimated from the genotypic frequencies by adding the homozygote frequencies and half the
heterozygote frequencies (for the respective allele). For example, for two alleles,
pA =
2N AA + N Aa
1
= n AA + n Aa
2N
2
.
However, if there is dominance we cannot distinguish (some of) the homozygotes and (some of) the
heterozygotes, so this simple procedure cannot be used, and we can run into trouble.
The ABO system
The ABO is a blood group system notorious for being responsible for blood transfusion accidents. It was
among the first human traits proven to be mendelian. It was often used in forensic (identitication and
paternity) studies, but has been superceded in this by other genetic markers. It remains clinically important,
and a great system for teaching.
We assume that
·
the ABO system is determined by three alleles of a single gene, call them A, B and O
·
A and B are codominant, and both are dominant over O
·
this gene is in Hardy-Weinberg frequencies in the population
Note that these assumptions are not necessarily true. Why the Hardy-Weinberg assumption? For without it,
estimation of allele frequencies is not possible in this case (because of dominance). Wanna try?… :-)
Because of its importance in the estimation proceedings, this assumption should always be tested.
These assumptions, and some of its consequences, are summarized in the following table, where p, q and r
are the frequencies of alleles A, B and O, respectively:
2
Pedro J.N. Silva
Phenotype
Genotype
Phenotypic
Genotypic
Expected
(Blood group)
frequency
frequency
frequency
--------------------------------------------------------------------------------------------------------------------------------A
AA + AO
nA
nAA+nAO
p 2 + 2 pr
B
AB
BB + BO
AB
nB
nAB
nBB+nBO
nAB
q 2 + 2 pr
2 pq
O
OO
nO
nOO
r2
--------------------------------------------------------------------------------------------------------------------------------Total
n
n
1
It is interesting to note that the genetic basis of the ABO system was not determined by family investigations,
as might be expected, but by testing the predictions of the competing genetic hypotheses (two genes with
two alleles each vs. the above model) against actual population data, using the Hardy-Weinberg law.
Hardy-Weinberg frequencies
While the (complete set of) genotypic frequencies always determine the allelic frequencies, the reverse is not
necessarily true, that is, we cannot always calculate the genotypic frequencies from the allelic.
Given some assumptions -- random union of gametes (with or without random mating), very large population
size (in theory, infinite), absence of selection, migration, etc. --, however, the genotypic frequencies
eventually take a form that depends only on the allele frequencies. For example, for an autosomal gene with
just two alleles (A and a) with respective frequencies p and q, we have three genotypes (AA, Aa and aa),
whose frequencies are
p 2 , 2 pq and q 2 .
These genotypic frequencies can be thought of as the development of the square of the sum of the allele
frequencies:
( p A + q a ) 2 = p 2A + 2 p A q a + q a2 .
This result was published independently by the british mathematician G.H. Hardy and the german physician
W. Weinberg in 1908.
For more than two alleles we have
( p1 + p2 + ... + pn )2 = ( p12 + 2 p1 p 2 + ... + 2 p1 pn + p 22 + ... + p n2 + ... + 2 p n−1 p n ).
ABO allele frequency estimators
Bernestein (1925)
Let us agree to name the three allele frequencies of the ABO system p (of allele A), q (of B) and r (of… you
guessed it!).
The oldest estimator of the ABO allele frequencies is due to Bernstein (1925), who had determined the
genetic basis of this blood group just the year before using Hardy-Weinberg frequencies.
Since the expected (Hardy-Weinberg) frequency of individuals with blood group O is r 2 , a fairly obvious
estimate of r is
ABO allele frequency estimation
3
r' =
nO
n
On the other hand, the expected combined frequency of blood groups A and O is
 n + nO 
2
2
E A
 = ( p + r ) = (1 − q )
n


and therefore q can be estimated by
q' = 1−
n A + nO
n
and in a similar way we obtain
p '= 1 −
n B + nO
n
So, p', q' and r' are Bernstein's 1925 estimators. They (normally) do not add up to one, which of course is
not altogether desirable.
Bernestein (1930)
As noted above, Bernstein's 1925 estimators do not necessarily add up to one. To solve this, we could
simply divide them by their sum, but in 1930 Bernstein suggested a much better procedure.
Let d be the difference between the sum of Bernstein's 1925 estimates and unity:
d = 1 − ( p '+ q '+ r')
The new estimators are then
d

p" = p ' 1 + 
2

 d
q" = q' 1 + 
2

d  d 

r" =  r '+ 1 + 
2 
2

They still don't quite add up to one, but the difference is much smaller ( d 2 4 , as you should check). In fact,
they are usually quite close to the maximum likelihood estimators (especially if the Hardy-Weinberg
assumption holds).
Wiener (1929)
In 1929, Wiener suggested an alternative to Bernstein's 1925 estimators. They are, perhaps, more intuitive,
but seldom work any better (a tribute to Bernestein's insight).
4
Pedro J.N. Silva
The estimator of r is actually the same as Bernstein's 1925. Remembering that the expected (HardyWeinberg) frequency of individuals with blood group O is r 2 , we get
r ' "=
nO
n
The expected combined frequency of blood groups A and O is (still)
 n + nO 
2
E A
 = ( p + r)
n


so p can be estimated by
p' " =
n A + nO
nO
−
n
n
and similarly
q' " =
n B + nO
−
n
nO
n
Like the other heuristic estimators, these do not normally add up to one.
Maximum Likelihood and the EM algorithm
See below, under statistics
Statistics
Maximum Likelihood
Suppose we want to estimate a parameter from given observations. If we have a probabilistic model for the
estimation of the data (such as the binomial model for the tossing of a coin), we can (in principle) calculate
the probability of getting our observations for each value that the parameter can take. The maximum
likelihood method consists in choosing that parameter value that maximizes the probability of the data, also
known as the likelihood of the parameter.
The method can be justified using Bayes theorem (with uniform priors, or large enough sample sizes to
overcome whatever priors we have), or by its results. In fact, maximum likelihood estimators tend to be
consistent and efficient, but are often biased.
Estimation is not the only application of likelihoods. For example, they can also be used for hypothesis
testing.
The EM algorithm
The EM algorithm is a general method to obtain maximum likelihood (ML) estimates, starting from
reasonable guesses. It is not the only method, and cannot always be used, but when applicable tends to
work well in pactice. Here is a brief description of the EM algorithm applied to the ABO case.
ABO allele frequency estimation
5
The general idea is simple. We start from estimates of the allele frequencies, and use them to calculate the
expected frequencies of all genotypes (step E of the EM algorithm), assuming them to be Hardy-Weinberg
frequencies. Then, we use those fake but complete genotypic frequencies to obtain new estimates of the
allele frequencies, using maximum likelihood (the step M). We then use these new allele frequency
estimates in a new E step, and so forth, in an iterative fashion, until the values converge or we get tired.
Log-likelihood ratio test
This test compares the unconstrained likelihood of the data with the (smaller) likelihood imposing the (null)
hypothesis under test, in our case, that the ABO gene is in Hardy-Weinberg frequencies in the population. If
the hypothesis is true, the difference in likelihoods should be small and, conversely, if it is false the difference
should be large.
We can use the fact that the distribution of twice the difference of the logarithms of the likelihoods (or, which
amounts to the same, twice the logarithm of the ratio of the likelihoods) tends asymptotically (as the sample
2
size increases) to the χ distribution to perform an actual statistical test, i.e., to help us decide whether the
difference is large enough to reject the null.
The number of degrees of freedom depends on whether the hypothesis is extrinsic (fully specified in absence
of the sample) or intrinsic (depend on parameters that have to be estimated from the sample). To be
concrete, let us think about goodness-of-fit tests. For extrinsic hypotheses, the number of degrees of
freedom is simply the number of classes minus one; for intrinsic hypotheses it is usually determined as the
number of classes minus one minus the number of independent parameters estimated from the sample.
Let the data be categorized in k classes, each with Oi observations, and let the corresponding expected
(derived from the hypothesis) numbers be Ei . In practice, the goodness-of-fit test statistic can be computed
as
k
O 
G = 2 ∑ Oi ln  i 
 Ei 
i =1
Pearson's χ2 test
This procedure tests the goodness-of-fit of a given hypothesis to the data. Let the data be categorized in k
classes, each with Oi observations, and let the corresponding expected (derived from the hypothesis)
numbers be Ei .
The test statistic is then
k
(Oi − Ei )2
i
Ei
X2 = ∑
and its asymptotic (as the sample size increases) distribution is the
χ2 .
The number of degrees of freedom is calculated as for the log-likelihood ratio test above.
6
Pedro J.N. Silva
S2 ABOestimator
Description
S2 ABOestimator is a program to estimate the allele frequencies of the ABO blood group system, and
perform a couple of statistical tests on the data. It requires MsWindows (95+) or a good emulator.
It is very simple to use, and is meant to be used in teaching, particularly
1. to compare simple heuristic estimates of the allele frequencies
2. to show the EM algorithm in action, to obtain maximum likelihood (ML) estimates of the allele
frequencies
3. to perform goodness-of-fit tests of the Hardy-Weinberg assumption
Some example data sets are provided to get you started. You can also use your own data (or make them
up, and experiment). A rather extensive help file is provided (from which most of this text is excerpted).
Comments (pan or praise) welcome!
N.B. This program is NOT meant to be used in clinical applications, or any life-threatning situations. It is
offered "as is", with no warranties whatsoever.
The program is Copyright
 Pedro J.N. Silva, 2000.
How to get the latest version
S2 ABOestimator is postcardware – meaning, if you like it, you should send me a nice postcard from your
home town, but you should never have to pay for it, or ask anyone to pay you for it.
Even if you are so mean as to actually use the child of my labors and not send me a postcard, please send
me an email, so I know who is using the program and what for, and can tell you about bugs and new
versions.
You can always get the latest version of S2 ABOestimator from its home page, currently
http://alf1.cii.fc.ul.pt/~pedro/Soft/ABOestimator/
N.B. This program is NOT meant to be used in clinical applications, or any life-threatning situations. It is
offered "as is", with no warranties whatsoever.
Have much fun!
Pedro J.N. Silva
Faculdade de Ciencias da Universidade de Lisboa
Campo Grande, C2, 4o. piso
P-1700 LISBOA PORTUGAL
[email protected]
ABO allele frequency estimation
7
Recommended reading
Hartl, D.L. and Clark, A.G. 1989. Principles of Population Genetics, 2nd ed. Sinauer. (Hardy-Weinberg law,
EM algorithm)
Li, C.C. 1978. First course in Population Genetics.
Weinberg law, ML estimation, goodness-of-fit tests)
Boxwood.
(Genetics of the ABO system, Hardy-
Sokal, R.R. and Rohlf, F.J. 1995. Biometry, 3rd ed. Freeman. (ML estimation, goodness-of-fit tests)
Vogel, F. and Motulsky, A.G. 1986. Human Genetics, 2nd ed. Springer-Verlag. (Genetics of the ABO system,
Hardy-Weinberg law, ML estimation, goodness-of-fit tests)
Weir, B. 1998. Genetic data analysis, 2nd ed. Sinauer. (ML estimation, EM algorithm, goodness-of-fit tests)
8
Pedro J.N. Silva