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1
Association Studies
Jennifer H. Barrett
1. Introduction
Aclassicalcase-controlstudydesignisfrequentlyusedingeneticepidemiol-
ogytoinvestigatetheassociationbetweengenotypeandthepresenceorabsence
ofdisease.Associationstudiescanalsobeusefulintheinvestigationofquantita-
tive traits. The aim of such studies is to test for association at the population
levelbetweenthequantitativetraitandgenotypeataparticularlocus.Whether
investigating qualitativeor quantitativetraits, such studiesdepend onthe prior
identification of a candidate gene or genes. The genotyped locus could either
be a polymorphism within a potentially trait-affecting gene or a marker in
linkage disequilibrium with such a gene. Currently, screening of the whole
genome is only feasible using linkage analysis, which is discussed elsewhere,
becauselinkageextendsovermuchgreaterdistancesthandoeslinkagedisequi-
librium.
Quantitative trait association studies are based on a sample of unrelated
subjectsfromthepopulation.Varioussamplingdesignsarepossible,including
random sampling and sampling on the basis of an extreme phenotype. The
advantages and disadvantages of these alternative designs are discussed.
Thebasicmethodofanalysisiscalledanalysisofvariance(seeSubheading
2.1.)astandardstatisticaltechniquefortestingfordifferencesinmeanbetween
two or more groups, on the basis of the comparison of between- and within-
groupvariances.Analternativeifsubjectsaresampledonthebasisofextreme
phenotype is to compare genotypes between groups with high and low trait
values (see Subheading 2.2.).
From:MethodsinMolecularBiology:vol.195:QuantitativeTraitLoci:MethodsandProtocols.
Editedby:N.J.CampandA.CoxHumanaPress,Inc.,Totowa,NJ
3
4 Barrett
2. Methods
2.1. Analysis of Variance and Linear Regression
Thestandardapproachtotheanalysisofquantitativetraitassociationstudies
assumes the following model. The phenotype y of individual i with genotype
ij
j at the locus of interest is given by
y = µ + e (1)
jj j i
whereµ isthemeanforthejthgenotypeande representsresidualenvironmental
j i
andpossiblypolygeniceffectsforindividuali,assumedtobeNormallydistrib-
uted with mean 0 and variance σ2. The data required consist of measured
e
phenotypesandgenotypesonasampleofunrelatedindividuals.Theparameters
µ are estimated in the obvious way by the mean values of individuals with
j
genotype j. The F-statistic from analysis of variance (ANOVA), the ratio of
between- and within-genotype variances, is used to test for the association
between genotype and phenotype, because under the null hypothesis that all
genotypes have the same mean and variance, this ratio should be 1. This
approach has been called the measured-genotype test (1), in contrast to earlier
biometrical methods that use information on the distribution of the phenotype
only (i.e., with unmeasured genotype) discussed briefly in Note 1.
Equivalently, a linear regression analysis of phenotype on genotype can be
carried out, possibly including as covariates other factors that may be related
tophenotype.Wherethegenotypeisdeterminedbyonebiallelicpolymorphism
(with possible genotypes AA, AB, and BB), a test for trend is provided by
regressing the phenotype on the number of copies of the A allele.
There are many examples of this type of approach in the literature. For
example,O’Donnelletal.(2)usedmultiplelinearregressiontoinvestigatethe
relationship between diastolic blood pressure and different genotypes of the
angiotensin-converting enzyme (ACE) gene. Hegele et al. (3) use analysis of
variancetodemonstrateassociationbetweenserumconcentrationsofcreatinine
and urea and the gene encoding angiotensinogen (AGT).
2.2 Analysis of Extreme Groups
Analternativeapproachistouseasamplingschemethatselectsindividuals
on the basis of extreme phenotypes (4,5). There is considerable literature on
the useof suchsampling schemes forsibling pairlinkage studies (e.g.,ref. 6).
Extreme sampling is advocated to increase power and efficiency, as extremes
are more informative. The approach is particularly useful when the phenotype
is relatively easy to measure, so that large numbers of individuals can easily
be screened to select extremes for genotyping.
Association Studies 5
Inassociationstudiesadoptingthismethod,individualsarerandomlyselected
conditional on their phenotype being below a specified lower threshold or
exceeding a specified upper threshold. Alternatively, the upper and lower n
percentilesofarandomsamplefromthepopulationmaybeincluded.Across-
tabulation is then formed by classifying subjects by genotype and by high/low
phenotype.Thegenotypefrequenciesarethencomparedbetweensubjectswith
high and low trait values using a chi-squared test. For example, Hegele et al.
(3) compared allele and genotype frequencies at the AGT locus in subjects
with the lowest and highest quartiles of serum creatinine and urea levels.
3. Interpretation
Incommonwithassociationstudiesforqualitativetraits,asignificantassocia-
tion does not demonstrate an effect of the polymorphism considered, because
it may also arise through linkage disequilibrium with another locus. A further
similarity is that population admixture can lead to spurious associations. For
this reason, family-based approaches, such as the transmission-disequilibrium
test for quantitative traits (7), have been developed (see Chapter 5).
3.1. Heterogeneity
Published results of associations with quantitative as with qualitative traits
arenotalwaysinagreement.Becauseformostcomplextraitstheeffectofany
one locus is likely to be small, individual studies are often not sufficiently
powerful to detect association. To address this issue, Juo et al. (8) carried out
a meta-analysis of studies investigating association between apolipoprotein A-
Ilevelsandvariantsoftheapolipoproteingene,whichhadproducedconflicting
results.Thisisapotentiallyusefulapproach,butmaybeflawedbypublication
bias, which is likely to be more of an issue in epidemiological studies than in
clinical trials. There is also an assumption that patients are genetically and
clinically homogeneous, with similar environmental exposures.
3.2. Using Extremes
An important consideration when using extreme sampling strategies (as in
outlinedinSubheading2.2.)isthatextremesmaybeuntypicalofthequantita-
tive trait as a whole in that they may be under the influence of other genes.
A clear example of this, cited in ref. 4, is that studying individuals with
achondroplastic dwarfism would be inappropriate if the primary interest were
in identifying genes controlling height.
3.3. Power of Association Studies
An attractive feature of association studies is that they may require smaller
sample sizes than methods based on linkage (9).
6 Barrett
Schork et al. (5) investigated the power of the extreme sampling method
analytically (Subheading 2.2.) to detect association between the trait and a
single biallelic marker in linkage disequilibrium with a trait-affecting locus.
Power depends on many factors, including locus-specific heritability, degree
of linkage disequilibrium, allele frequencies, mode of inheritance, and choice
of threshold. In some settings, overall sample sizes of less than 500 provided
adequate power to detect association with a locus accounting for 10% of the
trait variance.
The power of several methods of analysis, variants of those described here,
has been compared in a simulation study (10). Under the models considered,
ANOVA/linear regression (see Subheading 2.1.) generally performed better
than a variant of the extremes method (see Subheading 2.2.), based on the
samenumberofgenotypedindividuals,asmostoftheinformationonphenotype
islostbycategorizinginto“high”and“low”values.Aswithanymethodbased
onselectivesampling,anotherdrawbackisthatitisalsonecessarytophenotype
a larger number of subjects to achieve the same sample size for analysis. The
sameauthorssuggestedavariationonANOVA/linearregression,thetruncated
measuredgenotype(TMG)test,whereonlyextremesareincludedintheanalysis
(see Note 4). This TMG test was found to be more powerful than ANOVA/
linear regression for the same sample size of genotyped individuals, although,
again, a larger number of subjects must be phenotyped to achieve this. These
results are, however, dependent on the underlying genetic model. Allison et
al. (4) showed that extreme sampling can actually lead to a decrease in power
in the presence of another gene influencing the trait.
Page and Amos (10) also found that variants of ANOVA/linear regression
and of the TMG test, which are based on alleles, were more powerful than the
genotype-based methods discussed earlier. In these approaches, the phenotype
ofeachindividualcontributestotwogroups,oneforeachalleleor,inthecase
of homozygotes, contributes twice to one group. Allele-based methods, which
“double the sample size,” are generally only valid under the assumption of
Hardy–Weinberg equilibrium (11). Furthermore, the greater power of this
approachistobeexpectedforthemodelsusedinthesesimulations,allofwhich
assumedanadditiveeffectofthetraitallele,andmaynotapplymoregenerally.
Long and Langley (12) investigated the power to detect association using
a number of single nucleotide polymorphisms in the region of a quantitative
trait locus, but excluding the functional locus itself. Their test statistic was
based on ANOVA (see Subheading 2.1); the significance of the largest F-
statisticobtainedfromanymarkerwasestimatedfromitsempiricaldistribution
based on 1000 random permutations of the phenotype/marker data. From their
simulations,theyconcludedthat,usingabout500individuals,therewasgener-
allysufficientpowertodetectassociationif5–10%ofthephenotypicvariation
wasattributabletothelocus.Furthermore,testsusingsinglemarkershadgreater
Association Studies 7
Table 1
Summary Data onACE LevelsAccording to Genotype
ace geno Mean Std. dev. Freq.
II 74.496732 31.729764 153
ID 90.233871 39.484505 124
DD 103.73913 46.564928 23
Total 83.243333 37.475487 300
power than haplotype-based tests. The latter were based on comparing mean
trait values across all distinct haplotypes, and the authors concede that other
haplotype-basedtestsmakinguseofadditionalinformationmayperformbetter.
4. Software
The basic methods described in this chapter can be carried out in standard
statistical software packages such as Stata (13), which is used here, SAS, or
SPSS. The data would generally be expected to consist of one record for each
subject,recordingtheirmeasuredtraitvalue,theirgenotype,andanycovariates
of interest.
5. Worked Example
5.1. Analysis of Variance
An insertion/deletion (I/D) polymorphism of the ACE gene is associated
withplasmaACElevelsinsomepopulations.PlasmaACElevelsweremeasured
andI/D genotypeobtainedfor 300PimaIndians toinvestigate therelationship
in this population (14). The data consist of 300 records, including ACE levels
(ranging from 7 to 238 units) and genotype (II, ID, or DD).
In Stata, ANOVA can be carried out by the command
oneway ace leve ace geno, tabulate
where ace leve and ace geno are the variables for ACE levels and genotype,
respectively.ThisproducesTables1and2.Table1isproducedbyspecifying
thetabulateoptionaftertheonewaycommand(forone-wayanalysisofvariance)
andprovidesusefulsummaryinformation.In additiontothemeanACElevels
within each genotype group (i.e., estimates of µ , µ , and µ ), the standard
1 2 3
deviation and the number of subjects with each genotype are displayed. It can
be seen that individuals with the DD genotype have much higher levels on
average than those with the II genotype, with intermediate levels found in
heterozygotes.
Table 2 is the basic ANOVA table. The total variability of the data is
measured by the total sum of squares (419,919) (i.e. the sum of squares of the
8 Barrett
Table 2
Analysis of Variance Results for the Data in Table 1
Source SS df MS F Prob > F
Between groups 27426.3358 2 13713.1679 10.38 0.0000
Within groups 392492.901 297 1321.52492
Total 419919.237 299 1404.41216
differencesbetweeneachoftheobservationsandtheoverallmean).Thisfigure
canbeseparatedintothebetween-genotypesumofsquares(thesumofsquares
ofthedifferencebetweenthegroupmeanandtheoverallmean)andthewithin-
genotype sum of squares (the sum of squares of the differences between each
observation and the mean for the corresponding genotype). These are used to
estimate the corresponding variance, shown in the mean square (MS) column,
by dividing by the number of degrees of freedom. [The number of degrees of
freedom is one less than the number of groups or observations within groups
(i.e., 3−1 for between genotypes and 152+123+22 within genotypes).] The
F-statistic (10.38) is the ratio of these estimated variances. Under the null
hypothesis of no difference between groups, its expected value is 1 and it
should follow an F-distribution with (2, 297) degrees of freedom. In this case,
thereisoverwhelmingevidenceforadifferenceinlevelaccordingtogenotype.
The differences in the initial table are not the result of random variation.
The analysis of variance table (Table 2) can also be obtained by using the
Stata command
anova ace leve ace geno
This gives the additional information
R-squared = 0.0653
indicating that the I/D genotype explains 6.5% of the variance in plasma ACE
levels in this population.
Slightly different output, but exactly the same F-test and estimate of R-
squared can alternatively be obtained by carrying out a regression analysis:
xi: regress ace leve i.ace geno
The i in front of the ACE genotype variable shows that this is to be treated
as a categorical variable in the analysis. If, instead, interest was in testing for
a trend in ACE levels with the number of D alleles, then genotype could be
Association Studies 9
Table 3
Genotype Frequencies in Two Extreme Groups Defined by the Top and
Bottom Quintiles ofACE Levelsa
ace geno
Five quantiles of
ace leve II ID DD Total
1 39 20 3 62
62.90 32.26 4.84 100.00
5 17 33 10 122
28.33 55.00 16.67 100.00
Total 56 53 13 122
45.90 43.44 10.66 100.00
aPearsonchi2(2)=15.5722,Pr=0.000.
coded as 0, 1, or 2 to indicate the number of D alleles, and the following
regression carried out:
regress ace leve ace geno
This produces an F-statistic of 20.77 on (1, 298) degrees of freedom.
5.2. Analysis of Extremes
Using the same dataset, a new variable is created, recording the appropriate
quantile for each subject’s ACE level. In this example, quintiles are used,
creating 5 groups of approximately 60 subjects. This is easily done in Stata
as follows:
xtile acegp5=ace leve, nq(5)
A chi-squared test is then carried out comparing the top and bottom quintiles:
tab acegp5 ace geno if acegp5==1 | acegp5==5, chi row
producing Table 3.
The chi-squared statistic of 15.57 on 2 degrees of freedom again indicates
very strong evidence of association between ACE levels and genotype, even
though only 40% of the original subjects are used in the analysis. Nearly 63%
of those with low ACE levels had II genotype compared with only 28% of
those with high levels, and the DD genotype was over three times as common
in those with high levels compared with those with low levels.
6. Notes
1. Commingling analysis. The model underlying ANOVA (see Subheading 2.1.)
assumesthatthedataconsistofamixtureofNormaldistributions,onecorresponding
10 Barrett
to each genotype, each with the same variance. Even in the absence of genotype
data,statisticalmethodscanbeusedtotestforevidenceofamixtureofmorethan
oneNormaldistribution.This“unmeasuredgenotype”approachissometimesknown
as commingling analysis. Evidence for a mixture of two or three distributions is
supportive of the hypothesis that a major gene underlies the trait, although, of
course, environmental factors could also give rise to distinct distributions. Model
fitting allows estimates to be made of parameters of interest such as µ and σ2and
j e
the proportion of subjects in each class.
Inthepresenceofgenotypedatainacandidategene,themethodofcommingling
analysiscanbeextendedtoconditiononthemeasuredpolymorphism(s).Inaddition
to testing for evidence of a mixture of distributions, this method also provides
evidenceofwhetherthemeasuredgenotypeitselfgivesrisetothemixtureorwhether
another polymorphism in the gene is a more likely explanation (15,16).
2. Distributional assumptions. In view of the underlying model for ANOVA, a Nor-
malizingtransformation maybeapplied tothe data.Itis importanttonote thatthe
modelassumesaNormaldistributionwithin eachgenotyperatherthanoverall.(In
comminglinganalysis,Normalizingthedataleadstoaconservativetestformixture,
as this may remove skewness in the overall distribution of the data arising from
themixingofdistributions.)Thefurtherassumptionofacommonwithin-genotype
variance can be tested, and homogeneity of variance may sometimes be achieved
by transformation. In the worked example in this chapter, there is some evidence
forheterogeneityinthevariances.Oneadvantageoftheextremesmethodoutlined
in Subheading 2.2. is that it does not rely on these distributional assumptions.
3. Nonparametric alternatives. Another nonparametric alternative to ANOVA is the
Kruskal–Wallis test. In this approach, the complete set of N trait values is ranked
from 1 to N, and the average rank in each genotype group is calculated. The test
statisticisbasedoncomparingthegenotype-specificaveragerankswiththeoverall
average rank of (N+1)/2. Under the null hypothesis of no genotype–phenotype
association,theteststatisticfollowsa chi-squareddistributionwithtwodegreesof
freedom(assumingthreegenotypes),andasignificantlyhighervalueindicatesthat
thedistributionsdiffer.ApplyingthismethodtotheexampleinSubheading5.,the
test statistic takes the value 18.2 (p=0.0001). This method is only slightly less
powerfulthanANOVAwhenthedataareNormallydistributedandhastheadvantage
that distributional assumptions are not made. However, the test alone is not very
informative, and, in general, the estimates provided by ANOVA are also useful.
4. Analysisofextremes.Analternativesuggestionfortheanalysisofextremesamples,
the TMG method mentioned earlier, is to use analysis of variance, ignoring the
sampling scheme. The analysis of variance assumption of random sampling from
a Normal distribution is violated, but it has been argued that, for large enough
sample sizes, the significance level of the test is still correct (10). The analogs of
thistestandofthoseoutlinedinSubheadings2.1.and2.2.basedonallelesrather
thangenotypes,whereeachindividual’sphenotypecontributestwicetotheanalysis,
violate the further assumption of independence of observations.
Slatkin (17) suggested selecting individuals on the basis of unusually high (or
low)traitvaluesandtesting(1)foradifferenceingenotypefrequencybetweenthe
Association Studies 11
selectedsampleandarandomsampleand(2)fordifferencesinphenotypedistribu-
tionaccordingtogenotypewithintheselectedsample.Thesetwotestsareapproxi-
matelyindependentandsocanbecombinedintooneoveralltest.Thisapproachis
particularlypowerfulwhenarareallelehasasubstantialeffectonphenotype,even
thoughtheoverallproportionofphenotypicvarianceattributabletothelocusissmall.
5. Family-basedsamples.Althoughassociationstudiesasdescribedinthischapterare
applicable to unrelated sets of cases and controls, extensions have been suggested
to allow for relatedness between subjects. Tregouet et al. (18) suggested using
estimatingequations,astatisticalmethodforestimatingregressionparametersbased
on correlated data. They found that, for nuclear families of equal size, the power
of this approach was comparable to maximum likelihood and was similar to the
powerexpectedinasampleofthesamenumberofunrelatedindividuals.However,
thetype1errorratecouldbesubstantiallyinflatedinthepresenceofstrongclustering
if the number of families is relatively small (<50).
References
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Schaefer, E. J., et al. (1998) Evidence for association and genetic linkage of the
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but not women in the Framingham Heart Study. Circulation 97, 1766–1772.
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