HLA AND DISEASE
ASSOCIATION STUDIES
M.Tevfik Dorak
See also 'Common Concepts in Statistics' and ‘Pitfalls
in Genetic Association Studies {PPT}’
Genetic association studies: Design, analysis & interpretation
by C Lewis (Brieg Bioinform
2002)
Case–Control Genetic Association Analysis by W Li
(Brief Bioinform 2008)
The Lancet Septet on Genetic Epidemiology: Editorial, ( I ), ( II ), ( III )*, ( IV ), ( V )**, ( VI ), ( VII )
* Genetic Association Studies (Cordell &
Clayton, 2005)
** What Makes a Good
Association Study (Hattersley & McCarthy, 2005)
Nature Reviews Genetics Web Focus: Statistical
Analysis: Editorial, ( I )***, ( II ), ( III ), ( IV )
*** Tutorial on Statistical Analysis of Population
Association Studies (DJ Balding, 2006) (PDF)
PopulationBased Association Studies Lecture by David
Clayton
'Human Genome Epidemiology Online Book (2004)'
What is
an association study?
Linkage
and association studies are the two major types of investigations to determine
the contribution of genes to disease susceptibility (or any other phenotype).
While linkage studies can only use family data, association studies can be
family or population based. In addition to having wider applications,
association studies are considered^{ }to be more sensitive (having
greater power) than^{ }linkage methods in a comparably sized study.
Allelic association is useful to refine the location of major genes prior to
positional cloning. This has been the main use of association studies: to map
the unknown disease gene, which is presumed to be in LD with the associated
gene, and to proceed to positional cloning of the disease gene (see Genetic Epidemiology). With
the information obtained from the Human Genome Project, the focus of
association studies is changing to examining the role played by recently
discovered genes in disease development (Burke,
2003).
HLA genes have been known for a long time and the
whole of the HLA complex has been sequenced (The MHC Sequencing
Consortium, 1999; Allcock, 2002; Shiina, 2004; Horton, 2004), so why are we still doing HLA association studies? One of the many
reasons is to identify diseasespecific susceptibility (risk) and protective markers that can be used
in immunogenetic profiling, risk assessment and therapeutic decisions. Perhaps
a more important reason is to refine already known or constantly emerging
associations in the light of expanded knowledge of the HLA genetic map. In the
beginning, only 'classic' HLA genes and few class III genes were known and
could be included even in the most comprehensive association studies. The
latest and perhaps ultimate map of the HLA complex contains more than 100
functional genes and the task is now to figure out the real association behind
HLA associations. Several examples have appeared recently that however strong,
some HLA associations are due to nonHLA genes. One has to remember that
hemochromatosis and congenital adrenal hyperplasia originally showed
associations with HLA antigens but not with HFE or CYP21A2. Therefore, from
basic science point of view, the ultimate goal of HLA association studies is to
find out how genes cause the disease or modify susceptibility or course of it.
In the modern era of HLA association studies, the conceptualization, design,
execution and interpretation have gained new dimensions. Hypothesisdriven
studies are replacing fishing expeditions, extreme polymorphism of HLA genes forces
researchers to take statistical power into consideration in sample size
determination more frequently, and to consider replication more than ever to
avoid wellrecognized complications of multiple comparisons. Ever increasing
global ethnic diversity, on the other hand, requires extraordinary care to
prevent spurious associations resulting from population stratification. A large
number of sophisticated methods have been developed by biostatisticians and
statistical geneticists to handle the data generated from comprehensive
association studies especially in diseases with complex genetic basis (Lewis, 2002; Forabosco, 2005; Cordell, 2005). This
review is on the basic principles of traditional HLA association studies in
terms of statistical interpretation with the ultimate aim of providing an
insight to the modern methods of association data analysis.
General considerations and the multiple comparisons issue
The most common ways to make an inference for the population from which the analyzed sample has been drawn are significance probability calculation or confidence interval (CI) estimation. A CI contains the result of a significance test, but a significance test cannot provide the confidence limit. The statistical tests are essentially a test of whether the confidence limits include unity.
Each experiment starts with a null hypothesis which simply states that there is no difference between the two samples; that is 'two samples have been drawn from the same population.' A test statistics generally relies on the comparison with the observed distribution of what is expected if the null hypothesis is true. If the null hypothesis is rejected by statistical testing, the alternative hypothesis (the effect is not zero; there is a difference, etc.) is accepted. The general test statistics is as follows:
test statistics = (observed value  hypothesized value) / standard error of the observed value
The choice of the test to do this statistics depends on what is being compared. The usual tests used in HLA association studies and also in population genetic analysis of the HLA system are briefly described later. Test statistics yield a probability of observing the value we found (or an even more extreme value) when the null hypothesis is true. If the statistics gives a P (probability of error) value of <0.05, this means that the observed value would have a probability of occurrence somewhere in the extreme 5% of the relevant distribution curve for the data and imply an unusual finding.
In significance testing, a type I error (a falsepositive finding) is considered to be more serious, and therefore more important to avoid, than a type II error (missing a positive finding). If 0.05 is the predetermined level of statistical significance, this equals to reaching one incorrect conclusion (false positive) with no biological relevance in twenty tests ^{1}. The value 0.05 is the probability of getting an erroneous result every time one compares two groups, which are in fact equivalent. This level of error can be accepted in the context of a single test of a prespecified hypothesis. The probability of getting at least one statistically significant result as a result of type I error in multiple comparisons is, however, much higher than this. This probability can be calculated with the formula: 1(1a )^{n}, where n is the number of comparisons and a is the arbitrarily chosen significance level which corresponds to the accepted failure rate for each comparison ^{1}^{}^{4}. If the probability of getting the right result is 0.95 (P = 0.05) in a single test, it would be (0.95)^{20 }= 0.36 in 20 tests. Thus, the probability of getting it wrong is 1  0.36 = 0.64. This is to say that, for 20 comparisons and the significance level of 0.05, the probability of getting at least one erroneous result is 0.64 (which is 0.92 for 50 comparisons). This probability is smaller when the significance level is lower (0.18 for 20 comparisons and the significance level of 0.01). This argument applies to situations when the multiple comparison tests are independent tests.
To avoid a type I error, when independent multiple comparisons are carried out, and all genotypes examined have the same chance of being increased or decreased, a statistical safeguard should be applied. There is no definite limit for the number of comparisons that makes this necessary but it is a must if it is greater than 20. This is usually done by multiplying the P value by the number of comparisons (Bonferroni inequality method) ^{3;5;6}. This is basically lowering the statistical significance level and making the statistical test more conservative. For small numbers of comparisons (say up to five) its use is considered to be reasonable, but for larger numbers and for the multiple tests that are not independent (highly correlated), it is highly conservative ^{1;6} (see also Bland & Altman, 1995). The precise correction is obtained by the formula:
P_{corrected}_{ }= 1  (1P)^{n}
where P is the uncorrected P value, and n is the number of comparisons ^{7}. The number to use for correction is not (as frequently done) the number of alleles or genotypes detected in the study but the number of comparisons one or more of which shows a significant result ^{8} (see below). For P values of less than 0.01, this formula gives almost the same result as simple multiplication ^{8}. If the 'corrected' P value is still less than the predetermined significance level (such as 0.05), then the result is significant.
There has always been a debate about what number to use in the multiplication. In contrast to the most common practice, this is not supposed to be just the number of alleles in the locus analyzed ^{9}. If the allele frequencies are compared between patients and controls, the number of alleles is important as well as the number of comparisons in terms of age groups, sex groups, clinical subgroups, etc. When two antigens in linkage disequilibrium are investigated in an association study, the correction issue becomes slightly more complicated. Svejgaard & Ryder ^{8} has recently discussed this problem. When two loci without linkage disequilibrium between them are studied, Svejgaard recommends considering them separately in multiplication ^{9}.
Another way to correct for the bias due to multiple comparisons is to do a second study to check the frequency of the same specificity in the same disease ^{5;9;10}. If the uncorrected P value is £ 0.05 as in the first study, then the second study has confirmed the results of the preliminary study. The second study is said to be done with a specific hypothesis, therefore, an expected association would not be regarded as an artifact of multiple comparisons.
It has been argued, however, that by using the multiplication rule to avoid type I errors, the risk for type II errors increases ^{2;11}. This means that some genuine finding may be ruled out as a chance finding where it is worth pursuing. This is a good point since in association studies in a disease with multifactorial etiology, there will rarely be a very significant association when a single individual factor is considered. Any significant association, regardless of the corrected P value, should be critically evaluated and pursued further if biologically plausible. As mentioned above, a second study may offer the best explanation to these kinds of marginal findings.
Murray has proposed another method for the design of the study, which would preserve the sensitivity while avoiding a high, risk of type I error ^{12}. This method may be most useful for HLA association studies. This is to set down a priori a hierarchy of comparisons of interest. There should be a single comparison of primary interest, upon which the sample size (power) calculation would be based, and the analysis of which would be taken at face value. Thus the sensitivity is preserved for the most important question, since one has to achieve P < 0.05 in order to claim significance. There would then be a limited number of prespecified secondary comparisons, which would carry a lesser weight. But if significant after the correction for multiple comparisons, it cannot be too lightly dismissed. After that, any other comparisons of interest would be made in a purely exploratory fashion, in an attempt to generate hypotheses for future studies. Even the most extreme P values emerging in these final comparisons should be given very little weight until there is independent supporting evidence from other studies (a second study based on a specific hypothesis).
In their discussion of the statistical analysis of retrospective studies ^{13}, Mantel and Haenszel state that "if the purpose of the retrospective study is to uncover leads for fuller investigation, it becomes clear there is no real multiple significance testing problem. A single retrospective study does not yield conclusions, only leads." This situation corresponds to a full HLA locus analysis in a disease with no a priori hypothesis for the nature of an association. This effort would then be called a hypothesis generation approach. Mantel and Haenszel go on to say "also, the problem does not exist when several retrospective and other type studies are at hand, since the inferences will be based on a collation of evidence, the degree of agreement and reproducibility among studies, and their consistency with other types of available evidence, and not on the findings of a single study." In the HLA field, this is the case when there are several independent studies on the same association in the same disease or when there is a second study performed by a hypothesis.
A specific example for the application of Murray's proposition in the light of Mantel and Haenszel's view to the HLA field would be something like that: a study can be designed to investigate the relevance of homozygosity for an HLA class II supertype in childhood leukaemias as the same genotype has been found to be increased in two adult leukaemias. Therefore, the specific a priori hypothesis is that homozygosity for this supertype is increased in childhood leukaemias. The comparison of its frequency between patients and controls would not require any statistical safeguard for multiple comparisons. If an increase is shown, then one would like to check the individual members of this supertypic family to see whether this increase is due to an increase in any of them. These comparisons would require the correction for the multiple comparison before any increase is declared to be significant. The third step would be to analyze all HLADR types since the data is available anyway. If any of the other HLADR types is found to be increased or decreased, caution should be taken in making a big event out of this. Such a finding should be subject to confirmation with a second study unless, of course, the P value can already stand multiplication by the number of all these comparisons made.
More recently, Klitz suggested that if a Gtest is applied to the overall distribution of genotype frequencies between patients and controls and if this yields a significant result, the individual associations cannot be taken as the result of multiple comparisons ^{14}. He reminds us that looking for an association using individual 2x2 tables for each antigen is the old method stemmed from the presence of blank alleles at the time of serological typing. This would certainly lead to the multiple comparisons problems. Nowadays, blank alleles are not a problem anymore and a Gtest (or C^{2}test) on a 2xN table, where N is the number of alleles/haplotypes or genotypes, is a sensible approach. A significant deviation in this analysis justifies further scrutiny of the data to single out the main association. Klitz also suggests grouping of very rare alleles in one category to increase the sensitivity of the test.
Confusion may occasionally
arise through wrong usage of the terms allele, gene, or marker in an
association study. Some investigators state that they compare allele or
haplotype frequencies, but only count each individual once. They, therefore,
refer to what used to be phenotype frequencies in serological HLA studies, or
in the case of genotyping studies, to marker frequencies (MF), which correspond
to inferred phenotype frequencies if it is an expressed genotype. Allele (AF)
or haplotype frequency (HF) is analogous to gene frequency (GF) in that they
are always calculated in terms of the total number of chromosomes not
individuals.
For a comprehensive
discussion of the multiple comparisons issue, see Intuitive
Statistics: Multiple
Comparisons.
Casecontrol design and related
issues
Population stratification can be
thought of as confounding^{ }by ethnicity. If ethnicity of cases and
controls are reliably known, a stratified analysis would eliminate this problem.
However, it is the unknown stratification within the population that causes
this undesirable effect. To overcome this problem, software designed by Pritchard,
Structure
and Strat can be used with
genomic controls (Devlin, 1999; Pritchard, 1999).
In
casecontrol studies, ideally there should be at least one control per case. If
the number of cases is limited and cannot be increased easily, it may be an
idea to increase the number of control to increase statistical confidence. The
law of diminishing returns, however, dictates that a maximum of five controls
per one case is the limit (Epidemiology for the Uninitiated by
Coggon, Rose and Barker. BMJ Publishing Group, 1997). A higher control to case
ratio will not provide further benefit and may even result in type I
errors.
The
following quality control questions for clinical casecontrol studies also
apply to genetic association studies:
1. Were the
groups similar apart from the exposure under question?
2. Were the
data on outcomes collected identically in both groups?
3. Is there
a doseresponse effect?
4. Does the
relationship make biological and chronological (temporal) sense?
5. How
strong is the association and how precise is the estimate?
Power
calculations and expectation should be realistic in a genetic association study
with a complex (multifactorial) disease. Genetic susceptibility to such
diseases involves a large number of alleles, each conferring only a small
genotypic risk (like OR = 1.2 to 2.0), that combine additively or
multiplicatively to confer a range of susceptibilities in interaction with
environmental factors. Apart from increasing the numbers of cases and controls
to have greater power and validity, hypernormal
controls or genetically severe cases (like cases selected for a family history
of disease) can be used. Other strategies to increase statistical power in an
association study are the use of haplotypes rather than alleles and even an
ancestral haplotype approach in a cladistic haplotype
analysis. Underpowered studies are one of the main reasons for failure to
replicate an initial association study.
Constructing
a 2x2 contingency table
When two
groups are to be compared in a casecontrol study, it is necessary to have a
2x2 contingency table crosstabulating the frequencies. This table is required
for significance testing, relative risk (RR) or odds ratio (OR) estimation and
CI calculation. A contingency table may have more than two rows and columns but
the 2x2 table approach stems from classic casecontrol studies in epidemiology
as the most elemental data structure leading to ideas of association ^{15}.
The current trend in genetic association analysis is to consider the genotypes
as the genetic factor or alleles if a multiplicative risk model is appropriate
(Sasieni, 1997; Lewis, 2002; Cordell, 2005), see Genetic Models. For a multiallelic locus like the HLA loci, however, the number
of genotype categories is large. This is why originally the comparisons based
on the presence or absence of an allele were thought
to be more parsimonious and was established as the standard approach for HLA
association studies ^{5}. Another reason for using individual 2x2
tables for each antigen in HLA association studies was the presence of blank
alleles at the time of serological typing (which is no longer the case) ^{14}.
This contingency table is one of several ways of analysis genetic association
study data and does not take into account underlying genetic model {recessive,
dominant, codominant, additive, multiplicative)
(Lewis, 2002; Sellers, 2004; Cordell,
2005). HLA association tests relying on the presence of an
allele in cases vs controls implicitly use dominant genetic model (carrying at
least a copy of the allele is all that matters) but in compliance with the
current trends, all genetic models
should be explored if the genetic model for susceptibility is unknown (Lewis, 2002; Sellers, 2004; for online genetic modelbased analysis: DeFinetti; MODEL).
A general layout of a
contingency table for a conventional HLAdisease association study is as
follows:



allele i 




Present 

Absent 
Row totals 

Patients 
a 

b 
a+b 
Outcome 






Controls 
c 

d 
c+d 

Column totals 
a+c 

b+d 
N=a+b+c+d 
The number in
each cell is a count, i.e., a nonnegative integer. Each subject must appear in
one, and only in one, cell. In the table, switching rows and columns will not
alter the result. When a contingency table is subject to a hypothesis testing,
the null hypothesis is that there is no association between the two variables,
the alternative being that there is an association of any kind (twosided
test). To do the hypothesis test, it is generally necessary to calculate the
expected frequencies for each cell (see below). If the Chisquared test is to
be used, all expected frequencies should be at least 1 and at least 80% of them
should be more than 5 (Cochran rule) ^{16}. This means that for a 2x2
table, all expected frequencies should be more than 5. As will be discussed
below, the best approach is to use an exact test which is not based on
approximation or assumptions and does not require any kind of correction. Such
tests used to computationally highly demanding but this is no longer the case.
The usual choice for testing the significance of an HLAdisease association studies is the Fisher’s exact test ^{9}. This test is described below. Several other tests based on approximations can also be applied for the same analysis ^{15;17}. These are the ordinary Chisquared test (with or without Yates's correction) ^{18}, twosample Ztest ^{6;18}, Woolf's method ^{19}, and the MantelHaenszel C^{2} test ^{13;20;21}. When one or more of the expected numbers in the 2x2 table is less than five, and when the overall sample size is small, the Fisher’s test is commonly believed to be the only reliable one ^{9;17;22}, although there are also strong arguments against this ^{15}.
Yates's correction for discontinuity
If the C^{2} test is used in the statistical analysis of the results, it may be necessary to use the Yates's continuity correction for small samples ^{15;17;23}. The Chisquared distribution is continuous. That means that the curve of the C^{2} distribution model is continuous without any breaks. The values we calculate in C^{2} are, however, discrete values. This is because observed frequencies vary in discrete units (the number of occurrences of a gene the entries in the 2x2 table may be 4 or 5 but not 4.6). With degrees of freedom greater than 1 and with expected frequencies of at least 5 in each cell, this is not a problem as the difference between the statistics and the true sampling distribution is so small. For example, the difference between 100 and 101 is negligible (1%) compared to the difference between 4 and 5 (25%). Any difference between observed and expected frequencies will appear large when cell frequencies are small and may result in a type I error.
The Yates's correction helps make the discrete data generated by the test statistics [å (OE)^{2}/E] more closely approximate to the continuous Chisquared distribution. This is achieved by changing the above formula to [å (½OE½ 0.5)^{2}/E]. By doing so, the discrete data distribution and continuous data distribution are approximated better. This will result in a smaller calculated value of C^{2} and will reduce the risk of a type I error. In relatively large samples, this would not make an important difference. Most textbooks recommend that the Yates's continuity correction should be applied when the sample size is small or the contingency table contains any number less than 10 ^{24;25}. This correction, however, tends to overcompensate for discontinuity and may result in a more conservative decision than necessary ^{15;26}. As a simple rule, if a result is still significant after correction, or already nonsignificant without it, there is no problem. When a significant result becomes nonsignificant with the correction, this might be due to overcompensation. It is argued that the Yates’s correction produces severe conservative bias in the C^{2} test for association (C^{2} test for independence or homogeneity) where the observed marginal frequencies in the contingency table are subject to sampling variability ^{15;26;27}. Some statisticians strongly argue that the Yates’s correction is only necessary when the experimenter fixed both sets of marginal totals and when the cell frequencies are small ^{15;26}. For other types of contingency tables, Yates’s correction may indeed be too conservative but the current tendency is stronger towards its usage ^{1}.
In the study of biological variations, a P value of less than 0.05 is generally considered to be statistically significant. In HLA association studies, because of the nature of the HLA system and random variations in gene frequencies, it is not infrequent that a P value of this magnitude is obtained. When the significance level is chosen as P = 0.05, most reported associations cannot be confirmed in subsequent studies ^{8}. It has been suggested that a P value of less than 0.01 should be used for the significance of an HLA association, and the description 'highly significant' should be reserved for P values less than 0.001 (Ref. 8).
Occasionally, a relative risk (RR) or odds ratio (OR) is quoted as a result of comparison between patients and controls without a P value or CI. A RR may be more than 1.0 or even greater, especially in small groups, but without statistical significance it is meaningless. If CIs are calculated for such RRs (i.e., not associated with a P £ 0.05), it would be a very wide one extending to both sides of 1.0. In one report, a RR similar to the one reported for an HLA association in Hodgkin's disease can be found even for comparisons between two control groups of the same study ^{28}. A RR/OR should always be reported together with its statistical interpretation and CIs.
The obtained P value (or significance probability) should be interpreted properly. Statistics does not prove the truth of anything, it just provides more or less evidence for its validity. The significance probability describes the extent to which the data support the null hypothesis: if the statistical experiment were to be repeated on many subsequent occasions and if the null hypothesis were true, the P value represents the proportion of further experiments that would support the null hypothesis ^{29}. More technically, it is the probability of obtaining a value of the test statistics as large as or larger than the one computed from the data when in reality there is no difference between the proportions. Basically, a P value is the probability of being wrong when asserting that a true difference exists. If this probability is less than 5%, we usually reject the null hypothesis. Just as P £ 0.05 does not prove anything, P > 0.05 does not necessarily mean the null hypothesis is correct. These may be due to type I and type II errors, respectively. In any case, a significant result only adds weight to the alternative hypothesis. A negative result should be evaluated taking into account the power of the test. If a genotype occurs in two patients (n = 100), its total absence in 300 controls would yield a P value of 0.06 (Fisher’s exact test) with a RR of 15.3 (WoolfHaldane analysis). This result cannot be easily dismissed as nonsignificant.
Some comparisons, like those between two means or two proportions, can be evaluated by onesided or twosided P values (all comparisons of three or more groups are twosided) ^{30}. When a P value is chosen, say P = 0.05, we assume that if the difference between the two values (p_{1} and_{ }p_{2}) is so unusual that if we repeat the sampling from the population 100 times, in 95 of these, the difference will be bigger than the observed, we will consider this difference significant. But we do not state the direction of the difference (both p_{2}p_{1} and p_{1}p_{2} are of interest). If we are evaluating a difference in either direction, a twosided P value is required. If, however, the direction of the difference is stated in the hypothesis; for example, if the alternative hypothesis is that there is a difference and p_{1} is greater than p_{2}, the two proportions can be evaluated by a onesided P value. However, Bland & Altman ^{30} state that "in general, a onesided test is appropriate when a large difference in one direction would lead to the same action as no difference at all. Expectation of a difference in a particular direction is not adequate justification". Inappropriate use of the onesided test would double the risk of a spurious significant difference. Twosided tests should always be preferable unless there is a very good reason for doing otherwise.
Technically, the twosided P value corresponds to the total area in both ends (tails) of the distribution curve for symmetrical distributions. If a onesided P value is going to be used, it will correspond to the area at one tail of the curve and it will be half the value of the twosided P value. A C^{2}test with 1 d.f. is essentially a twosided test, whereas Fisher's test usually gives a onesided P value and there are several ways to calculate the twosided P value.
The number of degrees of freedom is a way of referring to the number of independent variables involved. This is usually one less than the total number of variables. In a 2x2 table containing the number of patients and controls for the presence or absence of an allele or genotype, the degrees of freedom is one. This is because the number of independent variables is one. If anybody has an allele, that person cannot also be negative for it. Similarly, a person is either a patient or a control. The general rule to calculate the degrees of freedom is to multiply the degrees of freedom for the rows and columns [i.e., d.f. = (r1)(c1)] ^{1;25}.
In the study of a biallelic locus, despite having three possible genotypes (say, AA, Aa, aa) the degrees of freedom is still one. These three genotypes are not independent of one another. Given the gene frequency of either A or a, the other one's frequency, and subsequently all genotype frequencies can be calculated. Thus, the number of independent variables is one. If a study is investigating the association of five different independent phenotypic markers in patients and controls (two groups), the degrees of freedom will be (51)(21) = 4. The 'degrees of freedom' is taken into account in the translation of the C^{2} value to a P value. This is because the C^{2} distribution is different for each degrees of freedom.
For the most common situation in the HLA field the comparison of two independent proportions (p_{1}, p_{2}) equivalent hypothesis tests, the Chisquared test, the twosample Ztest, the Gtest, or the Fisher’s exact test can be used ^{6;17} (for a link to Online Analysis of a 2x2 Table, see the end of this document).
To calculate the C^{2} for the same data, a 2x2 table is first constructed, actual numbers of occurrences are placed, and the expected frequencies in each cell are calculated. The expected frequency in a cell is the product of the relevant row and column totals divided by the sample size (grand total; N = a+b+c+d). For the cell with observed frequency 'a', for example, the expected value is (a+b)(a+c)/N. The difference between observed (O) and expected (E) values (residual) is the same for each cell but with different signs ( or +). This means that there is only one independent observation rather than four, so just one degree of freedom. The difference between O and E for each cell (OE) is then calculated. The C^{2} is calculated as the sum of (OE)^{2}/E for all four cells:
C^{2 }= å [(OE)^{2}/E]
Intuitively, as the expected frequencies are calculated for each cell, our entries should be observed frequencies (there are other tests to compare an observed proportion with an expected one). The Chisquared test relies on an normal approximation to the distribution of the cell counts, and this approximation may be poor for small sample sizes.
Alternatively, a different version of the C^{2} formula can be used ^{6;24}. This version uses the observed frequencies in cells and avoids the need to calculate the expected values explicitly:
C^{2}^{ }= [N (adbc)^{2}] / [(a+b) (a+c) (b+d) (c+d)]
This formula for C^{2} for a 2x2 table is mathematically identical to the general formula for C^{2} but can only be justified for large samples ^{15}. As for the Ztest, the P value for the C^{2} test is obtained from tables.
There is one welljustified argument against the use of the Chisquared test. Williams reminds that Pearson originally developed the goodness of fit test as a computationally convenient approximation of the maximum likelihood statistics or the Gtest ^{31}. It can be shown mathematically that the approximation is valid if the deviation of observed from expected frequencies is smaller than the expected frequency (½OE½ < E ). Thus, if even in one cell, ½OE½ > E, the Chisquared test should not be used. The Chisquared distribution is usually poor for the test statistics G^{2} when N / rc is smaller than five ^{21}. Williams also reminds us that it is now computationally easy to calculate maximum likelihood statistics and there seems to be no reason for the continued use of Pearson's approximation.
A common mistake in the analysis of HLA sharing data is to
use the ordinary C^{2}test when C^{2}trend test is more appropriate ^{32}. When there are more than
two categories (number of antigens shared in two loci is 1 to 4) compared
between two groups (those having recurrent spontaneous miscarriages and fertile
couples), and there is an ordered increasing or decreasing difference along the
N categories, the appropriate test to use is a trend test for 2xN tables ^{17;33;34}.
In this situation, the C^{2}test completely ignores the order
of columns. Alternative tests suitable for similar situations are the WilcoxonMannWhitney test or the ttest with use of
ordered scores ^{33;35}. In addition to the
fetal loss studies, the trend test has also been used in an analysis of
increasing heterozygosity in the HLAA and B loci in the elderly compared to
children ^{36}. (See Analysis of Association,
Confounding and Interaction for an explanation of the trend test.)
Armitage trend test can also be used for casecontrol study association data
when the underlying genetic model is presumed to be additive {where risk
increases by rfold for heterozygotes and 2rfold for homozygotes for the risk
allele} (Lewis, 2002). This gradual increase
is best assessed by the trend test by collapsing all other alleles into one
category so that there will be three genotypes to compare in cases and controls.
For SNP data, usually there are three genotypes (AA, AB, BB) and the trend test
is the best test if the genetic model is additive. One other advantage of trend
test is its robustness against deviations from HWE (Sasieni, 1997; Xu, 2002).
Another misuse of the Chisquared test is to use it when McNemar's test should be used. The Chisquared test of independence of the two variables in a 2x2 table provides a test for the equality of two proportions when these proportions are estimated from two independent samples. If the two variables have derived from the same individuals, the Chisquared test cannot be used as the variables are not from independent samples. The appropriate test for equality of proportions in paired samples is the McNemar's test ^{17;37}. This test finds its use in situations where the same sample is used to find out the agreement (concordance) of two diagnostic tests or difference (discordance) between two treatments. The difference is similar to that of twosample ttest and paired ttest for interval data. Like the trend test, the McNemar's test gives a smaller P value than the ordinary Chisquared test for the same 2x2 table. Thus, it is again for the benefit of the researcher to use it when it is more appropriate to use. The calculation is not much different from what is done to obtain the ordinary C^{2} value, but only the discordant values in the 2x2 table are included (with Yates's correction) ^{37} (Online McNemar's Test).
GTest
The likelihood ratio (Chisquared) test or maximum likelihood statistics are usually known as the Gtest or Gstatistics ^{38}. Whenever a Chisquared test can be employed, it can be replaced by the Gtest. In fact, the Chisquared test is an approximation of the loglikelihood ratio, which is the basis of the Gtest. Pearson originally worked out this approximation because the computation of the loglikelihood was inconvenient (but it no longer is). The Pearson's statistics, C^{2} = å [(OE)^{2}/E] is mathematically an approximation to the loglikelihood ratio or G= 2 å O ln (O/E)
The value called G approximates to the C^{2} distribution. The G value can also be expressed as
G = 2 [å O lnO  å O lnE] = 4.60517 [å O log_{10}O  å O log_{10}E]
The Gtest as calculated above is as applicable as a test for goodness of fit using the same number of degrees of freedom as for Chisquared test. It should be preferred when for any cell ½OE½ > E.
For the analysis of a contingency table for independence, Wilks ^{39} formulated the calculation of the G statistics as follows:
G = 2 [ å å f_{ij} ln f_{ij}  å R_{i} ln R_{i}  å C_{j} ln C_{j} + N ln N ]
where f_{ij} represents entries in each cell, R_{i} represents each row total, C_{j} represents each column total, and N is the sample size. The same formula can be written using logarithm base 10 as follows:
G = 4.60517 [ å å f_{ij} log_{10} f_{ij}  å R_{i} log_{10} R_{i}  å C_{j} log_{10} C_{j} + N log_{10} N ]
The G value approximates to C^{2} with d.f. = (r1)(c1). When necessary, Yates' correction should still be used and the formula needs to be modified accordingly. With the exception of the above mentioned condition that ½ OE½ should be smaller than E for the Chisquared test to be valid, there is not much difference between the two tests and they should result in the same conclusion. When they give different results, the Gtest may be more meaningful. The Gtest has been gaining popularity in HLA and disease association studies ^{14;40} (Online Gtest).
Fisher’s exact test
When the Chisquared test is used for small samples, Yates's correction may need to be used to make adjustments for continuity. This correction, however, does not remove the requirement for the expected frequencies. As mentioned above, the expected frequencies in each cell (of a 2x2 table) should be at least five for the C^{2} test to be reliable (note that the observed frequencies may be less than five). The general belief is that when there is one or more expected frequency of less than five, the alternative approach for significance testing is the Fisher’s exact test ^{1;6;9;38}. There is no equivalent method of estimation for comparing proportions from very small samples and Fisher test is the best choice for such data ^{22}. The result of a Fisher’s test is usually more or less the same as the result of the Chisquared test with Yates's correction. Fisher's exact test does not depend on an approximation to the probability distribution of the cell counts. There are, however, statisticians defending the idea that this is a misconception in statistics and the Fisher’s exact test is not suitable for small sample sizes ^{15}. Yates himself replied to such criticism and defended the suitability of both Fisher's test and Yates's correction where appropriate ^{41}. He also quotes Fisher as stating that the exact test should always be used whether or not the margins are determined in advance.
The Fisher’s exact test
also uses the frequencies in a 2x2 table but the calculations are different
from that of the other significance tests. It is based on the observed row and
column totals. The method consists of constructing all possible 2x2 tables
giving the same row and column totals as the observed data. For each table,
probability for such data to arise if the null hypothesis is true is
calculated. Then, the overall probability of getting the observed data is
calculated. To get the probability, the probability of the observed data and
all other probabilities for alternative 2x2 tables equal or more extreme than
that of the observed data are added up ^{6}. Calculations are
mathematically very complex, include factorial
calculations, which may be very cumbersome for big numbers. The Fisher’s
exact test can now be applied by means of computers (which can cope with the
calculations of factorials for large numbers) even to large samples. This test
is essentially onesided. To get the twosided P value, if required, the
P value may be doubled ^{1} (to perform Fisher's Exact Test
online, see the links at the end). There are exact tests available other than
Fisher's and one of them RxC is designed by Mark Miller and can perform
the exact test for large contingency tables.
It is still common to mix the results of Chisquared and Fisher's exact test in the same study. If some comparisons require the use of Fisher's exact test because of small expected numbers then it is best to use the Fisher's test for all comparisons. It does not make much sense to report some results by one test and others by another when all can be done most reliably by an exact test.
For the Ztest, only the two proportions (p_{1}, p_{2}) together with the sample sizes (n_{1}, n_{2}), and the estimated proportion (p_{e}) in the population are required ^{6;18}. The desired condition for the validity of this test is large enough (³ 30) size of both samples. The basic formula for this test is:
Z = (p_{1}p_{2}) / SE_{diff}
The calculation of the SE of the difference (SE_{diff}) is slightly different for this test. First, the expected (combined) proportion in the population is calculated:
p_{e}_{ }= (p_{1+}p_{2}) / (n_{1}+n_{2})
This is then used to calculate SE_{diff}:
SE_{diff}_{ }= [p_{e}(1p_{e}) (1/n_{1}+1/n_{2})]^{1/2}
In HLA studies, it may be necessary to compare an observed value with an expected value. The expected value may be a homozygosity rate calculated from the gene or haplotype frequency in the same sample of the population assuming HardyWeinberg equilibrium.
The statistical test for the null hypothesis 'population proportion is equal to a specified proportion' is the Ztest for single proportion ^{6;18}. The Z value for this comparison is calculated as follows:
Z = p_{o}p_{e} / [p_{o}(1p_{o})/N]^{1/2}
where p_{o} is the observed and p_{e} is the expected proportion. The corresponding P value for the Z statistics can be found from the standard normal distribution tables. For a twosided test, if Z is greater than 1.96 (or smaller than  1.96), P < 0.05.
Confidence interval estimation
When a value (a proportion or OR obtained from a sample) is the estimate of an unknown "true" value (within the population from where the sample has been drawn), CIs can be applied to them. CI is more informative than the simple results of hypothesis tests, where a null hypothesis is rejected or accepted. One of the traps of the hypothesis testing is that nonsignificance may be equated with accepting the null hypothesis whereas it may just be a failure to reject it (due to a small sample size, for example). In the case of comparing two groups, a CI enables the researcher to see how large the difference between two proportions may be, not simply whether it is different from zero. It also shows whether a nonsignificant result suggests an outright rejection of the alternative hypothesis lack of difference, i.e., CI includes zero and does not reach the critical value for the difference, or an inconclusive result, resulting from lack of evidence (CI includes zero but also exceeds the critical value) ^{42}.
CIs provide a range of plausible values for the unknown 'constant' parameter (such as the mean or expected frequency in the population from which the sample has been drawn). CIs can be calculated for different confidence levels. If a CI is calculated at a 95% level (as usually done), 5% of the time the true population parameter will not be contained within the interval calculated from the sample statistics. More technically, it means that 95% of all samples drawn from the population will have the population parameter within this interval. In a way, this is the acknowledgement of the fact that a different sample from the same population may produce a different result.
The width of the CI also gives us an idea about how uncertain we are about the unknown population parameter (the mean, for example). The most common CIs are calculated for a mean (or a single proportion), for the difference between two means and for a RR or OR. A very wide interval may indicate that the sample size should be increased to be more confident about the parameter. If the hypothesis test yields a significant result at the 5% level, the lower limit of the 95% CI for the difference between two means will be more than 0, and it will be more than 1.0 (unity) for RR or OR.
Calculation of confidence interval
In general, for any statistics that has a normal sampling distribution (such as the difference between means or proportions), a CI is constructed by adding to or subtracting from an estimate, a multiple of its standard error ^{18}. For example, a 95% CI is given by
statistics ± 1.96 SE
where SE refers to the standard error of the statistics, and 1.96 is the critical Z value (Z_{c}) for P = 0.05 (or 2.58 for 99% CI interval). The value of the Z_{c} does not depend on the sample size.
CI for a single proportion can be calculated using this principle:
CI = proportion ± Z_{c} SE
The SE for a single proportion (p) from a sample with size n is:
SE = [p(1p)/n]^{1/2}
Thus, 95% CI of a single proportion becomes ^{43}:
p ± 1.96 [p(1p)/n]^{1/2}
As seen in this formula, sample size has an inverse relationship with the CI. Larger samples yield a narrower CI. It can be shown mathematically that p(1p) can simply be replaced by 0.5. Whatever the value of p may be, p(1p) will never be greater than 0.25. If we use 0.5/Ö n, we will be using a number equal to or larger than the real SE in the CI formula. If anything, this could only result in a wider CI.
A 95% CI of the difference between two proportions is calculated as ^{43}:
P_{diff} ± 1.96 SE_{diff} ........... SE_{diff}_{ }= 1/n [(b+c)((bc)^{2}/N) ]^{1/2}
(b and c are from the 2x2 table; N is the grand total in the 2x2 table)
An alternative formula for 95% CI of the difference between two proportions (p_{1} from n_{1} samples, and p_{2} from n_{2} samples) ^{6;18}:
(p_{1}  p_{2}) ± 1.96 [p_{1}(1p_{1})/n_{1}+p_{2}(1p_{2})/n_{2}]^{1/2}
When a difference in the frequency of a gene or genotype is reported, the 95% CI for the difference should accompany this finding together with a P value rather than CIs of each mean separately ^{4;43}. It is now possible to analyze a 2x2 table and obtain OR together with its CIs online (see the end of the document for the link).
Relative risk / Odds ratio calculation
The calculation of RR conferred by an HLA antigen / haplotype / genotype is usually done by Woolf's method ^{19} which was later modified by Haldane ^{44}. Woolf defined the relative incidence (RI) as follows:
RI = ad / bc (a, b, c, d are the entries in the 2x2 table presented above)
Conventionally, RR is used in HLA and disease studies instead of relative incidence ^{5}. To confuse the terminology further, the crossproduct ratios described above as RI actually gives what is called odds ratio (OR) in epidemiology. This discrimination is not always made properly in publications on HLA associations. In the study of HLA specificities, some of the cell frequencies in the 2x2 table may be very small or even zero. If all the patients are positive for the HLA specificity or if none of the controls has it, the denominator would be zero. In this case, the RR would be undefined. For such situations Haldane modified Woolf's formula for RR:
RR = (2a+1)(2d+1) / (2b+1)(2c+1)
More precisely, the RR in a
prospective epidemiological study is defined differently (this is the real RR
as it is used in epidemiologic studies] ^{6}^{;13;29}. In the
2x2 table, the proportion of persons with the risk factor (allele i in this case) having the disease is a/(a+c), and the corresponding proportion
for those lacking the risk factor is b/(b+d). Because
relative risk is obtained by dividing the risk of development of disease among
subjects with the risk factor by the risk of development of disease among
subjects without the risk factor:
RR = (a/(a+c)) / (b/(b+d)) which equals to
RR = a(b+d) / b(a+c)
Especially for small values of a and b compared to c and d, ad/bc (which is the OR) is a close approximation ^{1;6;13}. If the probability of an event is p, then the odds of that event is p/(1p). The OR is the ratio of the odds of the risk factor in a diseased group and in a nondiseased (control) group (RR is the ratio of proportions in two groups). OR is more appropriate for retrospective casecontrol studies ^{1;6}. In retrospective studies, the selection of the subjects is based on the outcome (the rows) such as patients and healthy controls. In prospective studies, however, it is based on the characteristic defining the groups (the columns, presence or absence of an HLA type). Thus, in a prospective study, it is known how many individuals in the risk group developed the disease. In a retrospective study, RR would not be a precise estimate as the risk of the outcome cannot be evaluated (because of the way the subjects were sampled). In the above RR formula, if a and b are small as usually are (hence the choice of retrospective design because of the rarity of the disease), RR is approximated to OR (ad/bc). In other words, both values of (1p) for exposed and nonexposed groups will be close to 1 and can be ignored. This is why in both retrospective and prospective HLA and disease association studies, ad/bc is calculated and quoted as either OR or RR. In cohort studies, however, and especially when the outcome is frequent enough (>10%), the OR does not approximate to RR ^{45}. Under the null hypothesis, the expected value of RR or OR is 1.0 (unity).
Several tests have been described to compute homogeneity of odds ratios for stratified data (combining estimates of the odds ratio) ^{17;20;21;46}^{}^{48}. The most popular of these is the MantelHaenszel test for the common odds ratio for stratified data ^{17;20;21}. A Chisquared test for homogeneity of ORs over different strata can be performed using Woolf's method ^{17}.
Confidence interval for relative risk or odds ratio
As mentioned above, for variables with a normal sampling distribution, we need the SE of the statistics to calculate the CI. RR or OR itself does not have a normal sampling distribution, therefore, to achieve approximate normality, it has to be transformed. Since the natural logarithm of the RR or OR (lnRR or lnOR) has normal sampling distribution, the same principle can be applied to it. It can be shown mathematically that the standard error of the transformed RR is equal to ^{18}:
SE (lnRR) = (lnRR) / Ö C^{2}
Substituting this into the general equation gives the 95% CI for lnRR as:
lnRR ± 1.96 (lnRR) / C
After some mathematical transformation which simplifies to:
RR^{(}^{1 }^{±}^{ 1.96/}^{C}^{ )} and similarly OR^{(1 }^{±}^{ 1.96/}^{C}^{ )}
A general formula for the calculation of CI of RR / OR at any significance value is:
RR^{(}^{1 }^{±}^{ Zc/}^{C}^{ )} or OR^{(1 }^{±}^{ Zc/}^{C}^{ )}
Confidence limits obtained with this approach are reasonably good approximations to the exact limits, especially when RR / OR is not too far from unity. An alternative approach to calculating the CI for OR is as follows ^{17}:
e ^{ln}^{(OR) }^{±}^{ Zc} ^{[1/a + 1/b + 1/c + 1/d]1/2}
Estimation of haplotype frequencies
A haplotype contains a minimum of two loci on the same chromosome. If these two loci are not in linkage disequilibrium (LD), it can be said that the frequencies of their alleles are independent of each other. In this case, the expected frequency of a twolocus haplotype can be calculated as the probability of the occurrence of two independent (or joint) events simply by multiplying their gene frequencies. The same line can be followed for three or multiplelocus haplotypes.
In the case of LD, however, the gene frequencies are not independent and the presence of one allele may influence the presence of a particular allele at the other locus or loci. It is safer to assume LD and calculate the LD parameter D in each case. For the expected frequency, D is added to the product of gene frequencies. If there is no LD, D will be zero (or not significantly different from zero), if there is positive LD it will be a positive value. It can also be negative if the two alleles tend not to occur together.
Ideally, the haplotype frequencies should be calculated from family typing data. Obviously, this gives the most accurate results. In practice, however, when family data are not available, D and haplotype frequencies are calculated from a sample of the population data by constructing a 2x2 contingency table if no software is accessible ^{49}. A contingency table for this purpose contains the individual (observed) values crossclassified by levels in two different attributes. A common 2x2 table constructed in HLA studies is as follows:


allele i 





Present (+) 

Absent () 
Row totals 

Present (+) 
a (+/+) 

b (+/) 
a+b 
allele j 






Absent () 
c (/+) 

d (/) 
c+d 

Column totals 
a+c 

b+d 
N=a+b+c+d 
Counts for each combination of levels (presence or absence) of the two factors (alleles) are placed in each cell. The corresponding D _{ij} is estimated by the formula:
D_{ij}_{ }= (d/N)^{1/2}
– [((b+d)/N) ((c+d)/N)]^{1/2} (originally described by Bodmer & Bodmer, 1970;
see also Schipper, 1998)
The haplotype frequency (HF_{ij}) equals to GF_{i}_{ }x GF_{j}_{ }+ D _{ij}, where GF is the gene frequency (the proportion of the chromosomes carrying a particular allele). The haplotype frequency calculated with this formula from the population data compares reasonably well with the estimates obtained directly from counting haplotypes constructed from family segregation data ^{49}. A recent study concluded that this method generates a reliable estimate of a haplotype frequency with the exception of very small haplotype frequencies (Pearson's r = 0.9949) (Ref. 50).
Statistical Methods
for LD Estimation
While the Mattiuz
formula can be used to calculate twolocus haplotype frequencies manually, an
alternative method, the maximum likelihood estimation, can only be used if
computing facilities are available. This test was originally described by
Yasuda and Tsuji ^{51}, compared with other methods by Schipper et al ^{50}. It can also be used for
multiplelocus haplotypes ^{52}. Userfriendly software to calculate
linkage disequilibrium values can be downloaded from the internet which will
allow estimation of a variety of LD measures. One of the most
sophisticated population genetic data analysis packages ARLEQUIN
uses EM algorithm to calculate multilocus LD. Some of the other software to
perform LD analysis are: Genetic Data Analysis, EH, MLD, LDA, MIDAS, Haploview
and PopGene. LD analysis can also be performed online (Online LD Analysis, Genotype2LDBlock).
Interpretation of LD
Data
The patterns of LD observed
in natural populations are the result of a complex interplay between genetic
factors and the population's demographic history. LD is usually a function of
distance between the two loci. This is mainly because recombination acts to
break down LD in successive generations. However, physical distance could
account for less than 50% of the observed variation in LD. Other factors that
influence LD include changes in population demographics (such as population
growth, bottlenecks, geographical subdivision, admixture and migration),
selective forces and local variation in recombination rates. An extraordinary
example of the effect of recombination rates on LD is the discrepancy between
genetic distance and physical distance between HFE and HLAA, which generates
strong LD despite 5Mb distance. Regional LD may also be variable according to
haplotype. An example has been presented for HLA haplotypes (Ahmad, 2003). Haplotypespecific
patterns of LD may reflect haplotypespecific recombination hotspots as has
been shown for mouse MHC (Yoshino, 1994). For more on LD, see Basic Population Genetics.
Other statistical tests
Loglinear modeling: A loglinear model can be used in the analysis of contingency tables especially in those larger than 2x2 ^{38;53}. It allows more than two discrete variables to be analyzed and interactions (associations) between any combinations of them to be identified. An association in a contingency table appears as an interaction in a loglinear model. In a multidimensional table, each interaction (second order or higher order) can be analyzed without constructing separate smaller tables. In a loglinear model, the counts in cells are the response (dependent) variable and is modeled using the explanatory (independent) variables (categorical variables defining the rows and columns of a contingency table). The interpretation is made by means of the differences in the regression deviances of alternative models. The deviance has a Chisquared distribution.
Practical uses include comparison of two 2x2 tables. The most frequent use of loglinear models in HLA research is the estimation of linkage disequilibrium values ^{54;55}. To make inferences about linkage disequilibrium with a loglinear approach, a series of multiplicative models is constructed that includes different numbers of disequilibrium terms. The difference in regression deviances for two models that differ only by one term can be used to assess the significance of linkage disequilibrium represented by that term. The need for loglinear model is obvious when two LD values are to be compared. This could be a comparison of LD between the two sexes, or between patients and controls. In this case, a model is set up and the threefactor (two alleles and sex/disease status) interaction is assessed. A similar use may be assessment of the significance of the difference in the association in two subgroups or two sexes. For online loglinear analysis of two 2x2 tables, see the link at the end.
Logistic regression: Simple logistic regression is used when the outcome variable is binomial (such as dead/alive, disease absent or present, metastasis absent or present). It is ideally suited for the analysis of casecontrol studies where the outcome is either being a case or control ^{47;56;57}. The explanatory variables may be binomial, categorical or continuous (such as smoker or not, WBC count, height, age at first menstruation, etc). When the model selection is done and computation is complete, the outcome is a logistic regression equation in the following format:
logarithm of odds = a + b_{1}x_{1} + b_{2}x_{2} + ... + b_{n}x_{n}
Here, logarithm of odds is the natural logarithm of the overall odds ratio for all variables included in the model and by using different values of the explanatory variables in the formula, different odds ratios for any combination of the values the variables can take (for example, being male or female, smoking status, previous history of vaccination, age, etc) can be calculated. Each coefficient (b) provides a measure of the degree of association between each variable and the outcome. This coefficient is the logarithm of the odds ratio for that variable (OR = e^{b}) controlled (adjusted or corrected) for the other variables in the model. It is also possible to calculate confidence intervals for the estimated odds ratio as well as the statistical significance of each coefficient ^{17}. This property of logistic regression, which enables the calculation of controlled odds ratios, makes it unique in the analysis of multivariable data (Katz, 2003). The only concern is that depending on which variables are included in the model, these coefficients will change and the model selection is a relatively subjective procedure. Model selection should not be left to the computer, which would use statistical stepwise methods to choose the best selection, but the hypothesis and biological plausibility should be considered too. Logistic regression can be used for analysis of association with codominant loci where the disease risk associated with heterozygotes lies between that of two homozygotes but not in the specific relationship of a multiplicative or additive model {where risk increases by rfold for heterozygotes and r^{2}fold for homozygotes for the risk allele in a multiplicative model; or rfold for heterozygotes and 2rfold for homozygotes on an additive model} (Lewis, 2002).
It is also possible to
examine interactions between any combinations of variables using logistic
regression. The interaction term will also have a coefficient but this does not correspond to an adjusted odds
ratio. Some examples of the use of logistic regression in HLA research are, HLA
sharing studies (outcome: fetal loss vs live birth) ^{58}, HLA effects
on disease severity in rheumatoid arthritis ^{59}, and the effect of
HLA compatibility on graft failure following bone marrow transplantation ^{60}.
Resampling Statistics: Resampling
procedures (bootstrap, permutation and other simulations) have recently become
more popular as the method of choice for hypothesis testing and estimation of
confidence intervals. With resampling, the data are repeatedly resampled,
if needed according to a model suggested by the data, to assess the variability
of a statistic or estimate calculated from the observed data. The software Haploview
can be used to do permutation
test most easily. See Resampling: The New
Statistics book by JL Simon, 1997.
Association and Causality?
However strong, an association does
not necessarily mean causation. Several criteria have been proposed to assess
the role of an associated marker in causation. Some of those are as follows:
1. Biological plausibility
2. Strength of association (this is not
measured by the P value)
3. Dose response (are the
heterozygotes intermediate between the two homozygotes, or is homozygosity
showing a stronger association than just having the marker?)
4. Time sequence (this is inherent
in the germline nature of HLA genes)
5. Consistency (see below for
reasons for inconsistency in HLA association studies)
6.
Specificity of the association to the disease studied
Some
authors also include consideration of alternative explanations, coherence of
the findings with current knowledge and experimental support into what is
cumulatively called Hills's Criteria of Causality; see also Assessing Disease Associations and Interactions by Julian Little in Human Genome Epidemiology (a CDC book).
Why Are the Inconsistencies?
Large metaanalysis
studies and systematic reviews have shown that at best no more than half of
original associations can be consistently replicated. This lack of consistency
is a widely recognized limitation of association studies, and is often ascribed
to inadequate statistical power, population substructure, and
populationspecific linkage disequilibrium.
1. Mistakes in genotyping (lack of HWE in
controls is usually an indication of problems with typing rather than
selection, admixture, nonrandom mating or other reasons for violation of HWE)
2. Poor control selection (would your
controls be in the case group if they had the disease, and would the cases be
in your control group if they were free of the disease; are they from the same
study base as cases; are they in any way matched to cases at least in age and
sex?). Did you make any effort to establish a comparable control group or did
you settle for a convenience group?
3. Design problems including the statistical
power issue (negative results due to lack of statistical power should be
distinguished from truly negative results)
4. Publication bias (are there many more
studies with negative results but we have never heard about them?)
5. Disease misclassification or
misclassification bias
6. Excessive type I errors (are the positive
results due to using P < 0.05 as the statistical significance?)
7. Posthoc and
subgroup analysis (are positive results are due to fishing?)
8. Unjustified multiple comparisons
9. Failure to consider the mode of inheritance
in a genetic disease
10. Failure
to account for the LD structure of the gene (only haplotypetagging markers
will show the association, other markers within the same gene may fail to show
an association)
11.
Likelihood that the gene studied account for a small proportion of the
variability in risk
12. True
variability among different populations in allele frequencies.
Most of the
possible flaws listed above can be avoided by strict adherence to the standards
of modern epidemiological studies (Potter, 2001 & 2003;
Tabor, 2002; Little, 2002; Ransohoff, 2004; Hattersley, 2005).
Pointbypoint list of evaluation criteria for genetic
casecontrols association studies has been generated (Silverman, 2000; Weiss, 2001; Healy, 2006) which are accessible online. Editorials in Nature Genetics
(1999; PDF); Nature Medicine (2005; PDF) and other reviews (Cardon, 2001; Schork, 2001; Little, 2002; Campbell & Rudan, 2002;
Cooper, 2002; Romero, 2002; Lewis, 2002; Colhoun, 2003; Little, 2004; Freimer, 2005; Hattersley & McCarthy,
2005; Healy,
2006)
highlighted the features of a good association study which provides further
guidelines for study design, execution, analysis and interpretation (see Checklist for Statistical Adequacy).
The readers are strongly recommended to consult these before beginning their
association studies. In general, an association study should aim for large
datasets, small P values and
independent replication if results are to be reliable (Vieland, 2001; Dahlman, 2002). See also STrengthening
the Reporting of OBservational studies in Epidemiology (STROBE) checklists.
Basic statistical
checklist for genetic association studies:
 In a casecontrol study:
Cases and
controls derive from the same study base:
YES or NO
There are
more controls than cases (up to 5to1, for increased statistical power): YES
or NO
There are
at least 100 cases and 100 controls: YES or
NO
 Statistical power calculations are presented (at least
retrospectively): YES or
NO
 HardyWeinberg equilibrium (HWE) is checked and
appropriate tests are used: YES or NO (for HWE in
genetic association studies, see Population
Genetics)
 If HWE is violated, no allelic association test is used
(because independence assumption is not met):
YES or NO
 Possible genotyping errors and countermeasures are
discussed: YES or
NO
 All statistical tests are twotailed: YES
or NO
 Temptation to extract favorable P values from subgroups
(data dredging) was resisted: YES or
NO
 Alternative genetic models of association considered: YES
or NO
 The choice of marker/allele/genotype frequency (for
comparisons) is justified: YES or
NO
 For HLA associations, a global test for association
(Gtest, RxC exact test) for each locus is used (if
necessary, with correction for multiple testing): YES
or NO
 Chisquared and Fisher tests are NOT used
interchangeably: YES or
NO
 P values are
presented without spurious accuracy (with two decimal places): YES
or NO
 Strength of association has been measured (usually odds
ratio and its 95% CI): YES or
NO
 In a retrospective casecontrol study, ORs
are presented (as opposed to RRs): YES
or NO
 Multiple comparisons issue is handled appropriately (this
does not necessarily mean Bonferroni
corrections): YES or
NO
 Alternative explanations for the observed associations
(chance, bias, confounding) are discussed:
YES or NO
If all of the answers to above questions are not YES, think
again before submission of your manuscript (or asking a colleague to read it
for you).
For population genetics concepts concerning the HLA field,
see Basic Population Genetics Notes.
1. Bland M. An Introduction to Medical Statistics. Oxford: Oxford University Press, 1996.
2. Rothman KJ. No adjustments are needed for multiple comparisons. Epidemiology 1990; 1: 4346.
3. Bland JM, Altman DG. Multiple significance tests: the Bonferroni method. British Medical Journal 1995; 310: 170170.
4. Bulpitt CJ. Confidence intervals. Lancet 1987; 1: 494497.
5. Tiwari JL, Terasaki PI. The data and statistical analysis. In: Tiwari JL, Terasaki PI, eds. HLA and Disease Associations, New York: SpringerVerlag, 1985: 1822.
6. Altman DG. Practical Statistics for Medical Research. London: Chapman & Hall, 1992.
7. Sham P. Statistics in Human Genetics. London: Arnold, 1998.
8. Svejgaard A, Ryder LP. HLA and disease associations: detecting the strongest association. Tissue Antigens 1994; 43: 1827.
9. Svejgaard A, Jersild C, Nielsen LS, Bodmer WF. HLA antigens and disease. Statistical and genetical considerations. Tissue Antigens 1974; 4: 95105.
10. Thomson G. A review of theoretical aspects of HLA and disease associations [Review]. Theoretical Population Biology 1981; 20: 168208.
11. Perneger TV. What's wrong with Bonferroni adjustments. BMJ 1998; 316: 12361238.
12. Murray GD. Statistical aspects of research methodology. British Journal of Surgery 1991; 78 : 777781.
13. Mantel N, Haenszel W. Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute 1959; 22: 719748.
14. Klitz W, Aldrich CL, Fildes N, Horning SJ, Begovich AB. Localization of predisposition to Hodgkin disease in the HLA class II region. American Journal of Human Genetics 1994; 54: 497505.
15. Richardson JTE. The analysis of 2x1 and 2x2 contingency tables: an historical review. Statistical Methods in Medical Research 1994; 3: 107133.
16. Cochran WG. Some methods for strengthening the common X^{2} tests. Biometrics 1954; 10: 417451.
17. Rosner B. Fundamentals of Biostatistics. Belmont: Duxbury Press, 1999.
18. Daly LE, Bourke GJ. Interpretation and Uses of Medical Statistics. Oxford: Blackwell Scientific Publications, 2000.
19. Woolf B. On estimating the relation between blood group and disease. Annals of Human Genetics 1955; 19: 251253.
20. Mantel N. Chisquare test with one degree of freedom: extensions of the Mantel Haenszel procedure. Journal of the American Statistical Association 1963; 58: 690700.
21. Agresti A. Categorical Data Analysis. New York: John Wiley & Sons, 1990.
22. Kangave D. More enlightenment on the essence of applying Fisher's Exact test when testing for statistical significance using small sample data presented in a 2 x 2 table. West Afr J Med 1992; 11: 179184.
23. Yates F. Contingency tables involving small numbers and the X^{2} test. Journal of the Royal Statistical Society 1934; Suppl 1: 217235 (JSTORUK)
24. Dyer P, Middleton D. Histocompatibility Testing: A Practical Approach. Oxford: IRL Press, 1993.
25. Dyer P, Warrens A. Appendix: statistical notes. In: Lechler R, ed. HLA and Disease, London: Academic Press, 1994: 113121.
26. Haviland MG. Yates's correction for continuity and the analysis of 2x2 contingency tables. Statistics in Medicine 1990; 9: 363383.
27. Conover WJ. Some reasons for not using the Yates continuity correction on 2x2 contingency tables. Journal of the American Statistical Association 1974; 69: 374376.
28. Dorak MT, Mills KI, Poynton CH, Burnett AK. HLA and Hodgkin's disease [letter]. Leukemia 1996; 10: 16711672.
29. Armitage P, Colton T. Encyclopedia of Biostatistics. Chichester: John Wiley & Sons, 1998.
30. Bland JM, Altman DG. One and two sided tests of significance. British Medical Journal 1994; 309: 248248.
31. Williams K. The failure of Pearson's goodness of fit statistics. Statistician 1976; 25: 4949.
32. Hoff C. Chisquare trend analysis in antigen sharing studies. Tissue Antigens 1985; 26: 212213.
33. Armitage P, Berry G. Statistical Methods in Medical Research. Oxford: Blackwell Scientific Publications, 1994.
34. Campbell MJ, Machin D. Medical Statistics. A Commonsense Approach. Chichester: John Wiley & Sons, Ltd, 1999.
35. Moses LE, Emerson JD, Hosseini H. Analyzing data from ordered categories. New England Journal of Medicine 1984; 311: 442448.
36. Yasuda N, Tsuji K, Itakura K. HLA heterozygosity in children and old people. Tokai Journal of Experimental & Clinical Medicine 1980; 5: 165169.
37. Glantz SA. Primer of Biostatistics. New York: McGraw Hill, 1997.
38. Sokal RR, Rohlf FJ. New York: W.H. Freeman & Company, 1994.
39. Wilks SS. The likelihood test of independence in contingency tables. Annals of Mathematical Statistics 1935; 6: 190196.
40. Taylor GM, Gokhale DA, Crowther D, et al. Further investigation of the role of HLADPB1 in adult Hodgkin's disease (HD) suggests an influence on susceptibility to different HD subtypes. British Journal of Cancer 1999; 80: 14051411.
41. Yates F. Tests of significance for 2x2 contingency tables. Journal of the Royal Statistical Society 1984; A147: 426463 (JSTORUK).
42. Berry G. Statistical significance and confidence intervals. Medical Journal of Australia 1986; 144: 618619.
43. Gardner MJ, Altman DG. Confidence intervals rather than P values: estimation rather than hypothesis testing. British Medical Journal 1986; 292: 746750.
44. Haldane JBS. The estimation and significance of the logarithm of a ratio of frequencies. Annals of Human Genetics 1956; 20: 309311.
45. Zhang J, Yu KF. What's the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA 1998; 280: 16901691.
46. Zelen B. The analysis of several 2x2 contingency tables. Biometrica 1971; 58: 129137.
47. Breslow NE, Day NE. Statistical Methods in Cancer Research: The Analysis of CaseControl Studies. Lyon: IARC, 1993.
48. Emerson JD. Combining estimates of the odds ratio: the state of the art. Statistical Methods in Medical Research 1994; 3: 157178.
49. Mattiuz PL, Ihde D, Piazza A, Ceppelini R, Bodmer WF. New approaches to the population genetic and segregation analysis of the HLA system. In: Terasaki P, ed. Histocompatibility Testing 1970, Copenhagen: Munksgaard, 1971: 193205.
50. Schipper RF, D'Amaro J, de Lange P, Schreuder GM, van Rood JJ, Oudshoorn M. Validation of haplotype frequency estimation methods. Human Immunology 1998; 59: 518523.
51. Yasuda N, Tsuji K. A counting method of maximum likelihood for estimating haplotype frequency in the HLA system. Japanese Journal of Human Genetics 1975; 20: 115.
52. Long JC, Williams RC, Urbanek M. An EM algorithm and testing strategy for multiplelocus haplotypes. American Journal of Human Genetics 1995; 56: 799810.
53. Fienberg SE. The analysis of multidimensional contingency tables. Ecology 1970; 51: 419433.
54. Weir BS, Wilson SR. Loglinear models for linked loci. Biometrics 1986; 42: 665670.
55. Weir BS. Genetic Data Analysis II. Sunderland: Sinauer Associates, Inc. Publishers, 1996.
56. Kleinbaum DG, Kupper LL, Morgenstern H. Epidemiologic Research. Principles and Quantitative Methods. Belmont, CA: Lifetime Learning, 1982.
57. Hall GH, Round AP. Logistic regression  explanation and use. Journal of the Royal College of Physicians of London 1994; 28: 242246.
58. Ober C, Hyslop T, Elias S, Weitkamp LR, Hauck WW. Human leukocyte antigen matching and fetal loss: results of a 10year prospective study. Human Reproduction 1998; 13: 3338.
59. Wagner U, Kaltenhauser S, Sauer H, et al. HLA markers and prediction of clinical course and outcome in rheumatoid arthritis. Arthritis & Rheumatism 1997; 40: 341351.
60. Anasetti
C, Amos D, Beatty PG, et al. Effect of HLA compatibility on engraftment of bone
marrow transplants in patients with leukemia or lymphoma. New
England Journal of Medicine 1989; 320: 197204.
Internet Links
Extensive Epidemiology and Biostatistics Links
ASHI 2001 Biostatistics and Population Genetics Workshop
Notes EFI 2006 Teaching Session Notes
Human Genome Epidemiology Online Book
Analysis of
Association, Confounding and Interaction
Experimental Design and Statistical Analysis
in Genetic Association Studies
GENESTAT: Genetic Association Studies Portal
Online Statistics
Fisher's and
Chisquared GStatistics Analysis
of a 2x2 Table: (1) (2) (3)
Loglinear Analysis Logistic
Regression McNemar's Test OneSample
ZTest Bonferroni Correction
GenePop: Online LD
Analysis & Online HWE
Analysis
DeFinetti for Online HWE Analysis for SNP Data
Online Analysis of SNP Associations for Genetic Models
(review the options first)
HLA Data
Analysis (HWE, LD, Association) (AGP,
Universite de Geneve,
Switzerland):
(EFI 2006 Teaching Session & Arlequin Example)
Software for LD Estimations
Arlequin v3.11 for WinXP (2005) by
Laurent Excoffier
PyPop (HLA &
HWE/LD) by Glenys Thomson
PopGene by F.Yeh
EH Program from Rockefeller University
Genetic Data Analysis
Software [multilocus LD] by Paul Lewis
EMLD by Qiqing
Huang
. Genotype2LDBlock by Kun Zhang
LDA (SNP only) (Ding, 2003)
Software for Haplotype Construction
PHASE UNPHASED HAPLOVIEW (Tutorial) MIDAS
Downloads
Purcell
Laboratory (WHAP, PLINK, LPOP and more)
InStat Demo Version for Contingency Table Analysis including Trend Test
CLUMP (Monte Carlo Statistics
for CaseControl Association Studies)
STRUCTURE
& STRAT  LPOP
(Genomic Controls and Test of
Association)
RxC Exact Test for Contingency Table by
Mark Miller
MVSP Demo Version for Correspondence Analysis
Epi Info from CDC
Please update your
bookmark: http://www.dorak.info/hla/stat.html
M.Tevfik Dorak, MD, PhD
Last updated 27 July 2009
HLA MHC Genetics
Population Genetics Genetic
Epidemiology Bias &
Confounding Evolution Infection
& Immunity Biostatistics Homepage