A genetic association case-control study compares the frequency of alleles or genotypes at genetic marker loci, usually single-nucleotide polymorphisms (SNPs) (see

Box 1 for a glossary of terms), in individuals from a given population—with and without a given disease trait—in order to determine whether a statistical association exists between the disease trait and the genetic marker. Although individuals can be sampled from families (‘family-based’ association study), the most common design involves the analysis of unrelated individuals sampled from a particular outbred population (‘population-based association study’). Although disease-related traits are usually the main trait of interest, the methods described here are generally applicable to any binary trait.

BOX 1. GLOSSARYAdmixture

The result of interbreeding between individuals from different populations.

Cochran-Armitage trend test

Statistical test for analysis of categorical data when categories are ordered. It is used to test for association in a 2 × *k* contingency table (*k* > 2). In genetic association studies, because the underlying genetic model is unknown, the additive version of this test is most commonly used.

Confounding

A type of bias in statistical analysis that occurs when a factor exists that is causally associated with the outcome under study (e.g., case-control status) independently of the exposure of primary interest (e.g., the genotype at a given locus) and is associated with the exposure variable but is not a consequence of the exposure variable.

Covariate

Any variable other than the main exposure of interest that is possibly predictive of the outcome under study; covariates include confounding variables that, in addition to predicting the outcome variable, are associated with exposure.

False discovery rate

The proportion of non-causal or false positive significant SNPs in a genetic association study.

False positive

Occurs when the null hypothesis of no effect of exposure on disease is rejected for a given variant when in fact the null hypothesis is true.

Family-wise error rate

The probability of one or more false positives in a set of tests. For genetic association studies, family-wise error rates reflect false positive findings of associations between allele/genotype and disease.

Hardy-Weinberg equilibrium (HWE)

Given a minor allele frequency of *p*, the probabilities of the three possible unordered genotypes (*a/a*, *A/a*, *A/A*) at a biallelic locus with minor allele *A* and major allele a, are (1 – *p*)^{2}, 2*p* (1 – *p*), *p*^{2}. In a large, randomly mating, homogenous population, these probabilities should be stable from generation to generation.

Linkage disequilibrium (LD)

The population correlation between two (usually nearby) allelic variants on the same chromosome; they are in LD if they are inherited together more often than expected by chance.

*r*^{2}

A measure of LD between two markers calculated according to the correlation between marker alleles.

Odds ratio

A measure of association derived from case-control studies; it is the ratio of the odds of disease in the exposed group compared with the non-exposed.

Penetrance

The risk of disease in a given individual. Genotype-specific penetrances reflect the risk of disease with respect to genotype.

Population allele frequency

The frequency of a particular allelic variant in a general population of specified origin.

Population stratification

The presence of two or more groups with distinct genetic ancestry.

Relative risk

The risk of disease or of an event occurring in one group relative to another.

Single-nucleotide polymorphism (SNP)

A genetic variant that consists of a single DNA base-pair change, usually resulting in two possible allelic identities at that position.

Following previous protocols on study design, marker selection and data quality control^{1–3}, this protocol considers basic statistical analysis methods and techniques for the analysis of genetic SNP data from population-based genome-wide and candidate-gene (CG) case-control studies. We describe disease models, measures of association and testing at genotypic (individual) versus allelic (gamete) level, single-locus versus multilocus methods of association testing, methods for controlling for multiple testing and strategies for replication. Statistical methods discussed relate to the analysis of common variants, i.e., alleles with a minor allele frequency (MAF) > 1%; different analytical techniques are required for the analysis of rare variants^{4}. All methods described are proven and used routinely in our research group^{5,6}.

Conceptual basis for statistical analysis

The success of a genetic association study depends on directly or indirectly genotyping a causal polymorphism. Direct genotyping occurs when an actual causal polymorphism is typed. Indirect genotyping occurs when nearby genetic markers that are highly correlated with the causal polymorphism are typed. Correlation, or non-random association, between alleles at two or more genetic loci is referred to as linkage disequilibrium (LD). LD is generated as a consequence of a number of factors and results in the shared ancestry of a population of chromosomes at nearby loci. The shared ancestry means that alleles at flanking loci tend to be inherited together on the same chromosome, with specific combinations of alleles known as haplotypes. In genome-wide association (GWA) studies, common SNPs are typically typed at such high density across the genome that, although any single SNP is unlikely to have direct causal relevance, some are likely to be in LD with any underlying common causative variants. Indeed, most recent GWA arrays containing up to 1 million SNPs use known patterns of genomic LD from sources such as HapMap^{7} to provide the highest possible coverage of common genomic variation^{8}. CG studies usually focus on genotyping a smaller but denser set of SNPs, including functional polymorphisms with a potentially higher previous probability of direct causal relevance^{2}.

A fundamental assumption of the case-control study is that the individuals selected in case and control groups provide unbiased allele frequency estimates of the true underlying distribution in affected and unaffected members of the population of interest. If not, association findings will merely reflect biases resulting from the study design^{1}.

Models and measures of association

Consider a genetic marker consisting of a single biallelic locus with alleles *a* and *A* (i.e., a SNP). Unordered possible genotypes are then *a/a*, *a/A* and *A/A*. The risk factor for case versus control status (disease outcome) is the genotype or allele at a specific marker. The disease penetrance associated with a given genotype is the risk of disease in individuals carrying that genotype. Standard models for disease penetrance that imply a specific relationship between genotype and phenotype include multiplicative, additive, common recessive and common dominant models. Assuming a genetic penetrance parameter γ (γ > 1), a multiplicative model indicates that the risk of disease is increased γ-fold with each additional *A* allele; an additive model indicates that risk of disease is increased γ-fold for genotype *a/A* and by 2γ-fold for genotype *A/A*; a common recessive model indicates that two copies of allele *A* are required for a γ-fold increase in disease risk, and a common dominant model indicates that either one or two copies of allele *A* are required for a γ-fold increase in disease risk. A commonly used and intuitive measure of the strength of an association is the relative risk (RR), which compares the disease penetrances between individuals exposed to different genotypes. Special relationships exist between the RRs for these common models^{9} (see ).

| **TABLE 1**Disease penetrance functions and associated relative risks. |

RR estimates based on penetrances can only be derived directly from prospective cohort studies, in which a group of exposed and unexposed individuals from the same population are followed up to assess who develops disease. In a case-control study, in which the ratio of cases to controls is controlled by the investigator, it is not possible to make direct estimates of disease penetrance, and hence of RRs. In this type of study, the strength of an association is measured by the odds ratio (OR). In a case-control study, the OR of interest is the odds of disease (the probability that the disease is present compared with the probability that it is absent) in exposed versus non-exposed individuals. Because of selected sampling, odds of disease are not directly measurable. However, conveniently, the disease OR is mathematically equivalent to the exposure OR (the odds of exposure in cases versus controls), which we can calculate directly from exposure frequencies^{10}. The allelic OR describes the association between disease and allele by comparing the odds of disease in an individual carrying allele *A* to the odds of disease in an individual carrying allele *a*. The genotypic ORs describe the association between disease and genotype by comparing the odds of disease in an individual carrying one genotype to the odds of disease in an individual carrying another genotype. Hence, there are usually two genotypic ORs, one comparing the odds of disease between individuals carrying genotype A/A and those carrying a/a and the other comparing the odds of disease between individuals carrying genotype a/A and those carrying genotype a/a. Beneficially, when disease penetrance is small, there is little difference between RRs and ORs (i.e., RR ≈ OR). Moreover, the OR is amenable to analysis by multivariate statistical techniques that allow extension to incorporate further SNPs, risk factors and clinical variables. Such techniques include logistic regression and other types of log-linear models^{11}.

To work with observations made at the allelic (gamete) rather than the genotypic (individual) level, it is necessary to assume (i) that there is Hardy-Weinberg equilibrium (HWE) in the population, (ii) that the disease has a low prevalence ( < 10%) and (iii) that the disease risks are multiplicative. Under the null hypothesis of no association with disease, the first condition ensures that there is HWE in both controls and cases. Under the alternative hypothesis, the second condition further ensures that controls will be in HWE and the third condition further ensures that cases will also be in HWE. Under these assumptions, allelic frequencies in affected and unaffected individuals can be estimated from case-control studies. The OR comparing the odds of allele A between cases and controls is called the allelic RR (γ*). It can be shown that the genetic penetrance parameter in a multiplicative model of penetrance is closely approximated by the allelic RR, i.e., γ ≈ γ* (

ref. 10).

Tests for association

Tests of genetic association are usually performed separately for each individual SNP. The data for each SNP with minor allele

*a* and major allele

*A* can be represented as a contingency table of counts of disease status by either genotype count (e.g.,

*a/a*,

*A/a* and

*A/A*) or allele count (e.g.,

*a* and

*A*) (see

Box 2). Under the null hypothesis of no association with the disease, we expect the relative allele or genotype frequencies to be the same in case and control groups. A test of association is thus given by a simple χ

^{2} test for independence of the rows and columns of the contingency table.

Box 2. CONTINGENCY TABLES AND ASSOCIATED TESTSThe risk factor for case versus control status (disease outcome) is the genotype or allele at a specific marker. The data for each SNP with minor allele *a* and major allele *A* in case and control groups comprising *n* individuals can be written as a 2 × *k* contingency table of disease status by either allele (*k* = 2) or genotype (*k* = 3) count.

Allele count

Allele | a | A | Total |
---|

Cases | *m*_{11} | *m*_{12} | *m*_{1}. |

Controls | *m*_{21} | *m*_{22} | *m*_{2}. |

Total | *m*._{1} | *m*._{2} | *2n* |

- The allelic odds ratio is estimated by .
- If the disease prevalence in a control individual carrying an
*a* allele can be estimated and is denoted as *P*_{0}, then the relative risk of disease in individuals with an A allele compared with an a allele is estimated by .

An allelic association test is based on a simple χ

^{2} test for independence of rows and columns

where

has a χ

^{2} distribution with 1 d.f. under the null hypothesis of no association.

Genotype count

Genotype | a/a | A/a | A/A | Total |
---|

Case | *n*_{11} | *n*_{12} | *n*_{13} | *n*_{1}. |

Controls | *n*_{21} | *n*_{22} | *n*_{23} | *n*_{2}. |

Total | *n*._{1} | *n*._{2} | *n*._{3} | *n* |

- The genotypic odds ratio for genotype
*A/A* relative to genotype *a/a* is estimated by . The genotypic odds ratio for genotype *A/a* relative to genotype *a/a* is estimated by . - If the disease prevalence in a control individual carrying an
*a/a* genotype can be estimated and is denoted as *P*_{0}, then the relative risk of disease in individuals with an A/A [A/a] genotype compared with an *a/a* genotype is estimated by
- A genotypic association test is based on a simple χ
^{2} test for independence of rows and columns where has a χ^{2} distribution with 2 d.f. under the null hypothesis of no association. To test for a dominant (recessive) effect of allele A, counts for genotypes *a/A* and *A/A* (*a/a* and *A/a*) can be combined and the usual 1 d.f. χ^{2}-test for independence of rows and columns can be applied to the summarized 2 × 2 table. - A Cochran-Armitage trend test of association between disease and marker is given by
where
*w* = (*w*_{1}, *w*_{2}, *w*_{3}) are weights chosen to detect particular types of association. For example, to test whether allele *A* is dominant over allele *a w* = (0,1,1) is optimal; to test whether allele *A* is recessive to allele *a*, the optimal choice is *w* = (0,0,1). In genetic association studies, *w* = (0,1,2) is most often used to test for an additive effect of allele *A*. *T*^{2} has a χ^{2} distribut ion with 1 d.f. under the null hypothesis of no association.

In a conventional χ

^{2} test for association based on a 2 × 3 contingency table of case-control genotype counts, there is no sense of genotype ordering or trend: each of the genotypes is assumed to have an independent association with disease and the resulting genotypic association test has 2 degrees of freedom (d.f.). Contingency table analysis methods allow alternative models of penetrance by summarizing the counts in different ways. For example, to test for a dominant model of penetrance, in which any number of copies of allele

*A* increase the risk of disease, the contingency table can be summarized as a 2 × 2 table of genotype counts of

*A/A* versus both

*a/A* and

*a/a* combined. To test for a recessive model of penetrance, in which two copies of allele

*A* are required for any increased risk, the contingency table is summarized into genotype counts of

*a/a* versus a combined count of both

*a/A* and

*A/A* genotypes. To test for a multiplicative model of penetrance using contingency table methods, it is necessary to analyze by gamete rather than individual: a χ

^{2} test applied to the 2 × 2 table of case-control allele counts is the widely used allelic association test. The allelic association test with 1 d.f. will be more powerful than the genotypic test with 2 d.f., as long as the penetrance of the heterozygote genotype is between the penetrances of the two homozygote genotypes. Conversely, if there is extreme deviation from the multiplicative model, the genotypic test will be more powerful. In the absence of HWE in controls, the allelic association test is not suitable and alternative methods must be used to test for multiplicative models. See the earlier protocol on data quality assessment and control for a discussion of criteria for retaining SNPs showing deviation from HWE

^{3}. Alternatively, any penetrance model specifying some kind of trend in risk with increasing numbers of

*A* alleles, of which additive, dominant and recessive models are all examples, can be examined using the Cochran-Armitage trend test

^{12,13}. The Cochran-Armitage trend test is a method of directing χ

^{2} tests toward these narrower alternatives. Power is very often improved as long as the disease risks associated with the

*a/A* genotype are intermediate to those associated with the

*a/a* and

*A/A* genotypes. In genetic association studies in which the underlying genetic model is unknown, the additive version of this test is most commonly used. summarizes the various tests of association that use contingency table methods.

Box 2 outlines contingency tables and associated tests in statistical detail.

| **TABLE 2**Tests of association using contingency table methods. |

Tests of association can also be conducted with likelihood ratio (LR) methods in which inference is based on the likelihood of the genotyped data given disease status. The likelihood of the observed data under the proposed model of disease association is compared with the likelihood of the observed data under the null model of no association; a high LR value tends to discredit the null hypothesis. All disease models can be tested using LR methods. In large samples, the χ^{2} and LR methods can be shown to be equivalent under the null hypothesis^{14}.

More complicated logistic regression models of association are used when there is a need to include additional covariates to handle complex traits. Examples of this are situations in which we expect disease risk to be modified by environmental effects such as epidemiological risk factors (e.g., smoking and gender), clinical variables (e.g., disease severity and age at onset) and population stratification (e.g., principal components capturing variation due to differential ancestry^{3}), or by the interactive and joint effects of other marker loci. In logistic regression models, the logarithm of the odds of disease is the response variable, with linear (additive) combinations of the explanatory variables (genotype variables and any covariates) entering into the model as its predictors. For suitable linear predictors, the regression coefficients fitted in the logistic regression represent the log of the ORs for disease gene association described above. Linear predictors for genotype variables in a selection of standard disease models are shown in .

| **TABLE 3**Linear predictors for genotype variables in a selection of standard disease models. |

Multiple testing

Controlling for multiple testing to accurately estimate significance thresholds is a very important aspect of studies involving many genetic markers, particularly GWA studies. The type I error, also called the significance level or false-positive rate, is the probability of rejecting the null hypothesis when it is true. The significance level indicates the proportion of false positives that an investigator is willing to tolerate in his or her study. The family-wise error rate (FWER) is the probability of making one or more type I errors in a set of tests. Lower FWERs restrict the proportion of false positives at the expense of reducing the power to detect association when it truly exists. A suitable FWER should be specified at the design stage of the analysis

^{1}. It is then important to keep track of the number of statistical comparisons performed and correct the individual SNP-based significance thresholds for multiple testing to maintain the overall FWER. For association tests applied at each of

*n* SNPs, per-test significance levels of α* for a given FWER of α can be simply approximated using Bonferroni (α* = α/

*n*) or Sidak

^{15,16} (α* = 1 − (1 – α)

^{1/n}) adjustments. When tests are independent, the Sidak correction is exact; however, in GWA studies comprising dense sets of markers, this is unlikely to be true and both corrections are then very conservative. A similar but slightly less-stringent alternative to the Bonferroni correction is given by Holm

^{17}. Alternatives to the FWER approach include false discovery rate (FDR) procedures

^{18,19}, which control for the expected proportion of false positives among those SNPs declared significant. However, dependence between markers and the small number of expected true positives make FDR procedures problematic for GWA studies. Alternatively, permutation approaches aim to render the null hypothesis correct by randomization: essentially, the original

*P* value is compared with the empirical distribution of

*P* values obtained by repeating the original tests while randomly permuting the case-control labels

^{20}. Although Bonferroni and Sidak corrections provide a simple way to adjust for multiple testing by assuming independence between markers, permutation testing is considered to be the ‘gold standard’ for accurate correction

^{20}. Permutation procedures are computationally intensive in the setting of GWA studies and, moreover, apply only to the current genotyped data set; therefore, unless the entire genome is sequenced, they cannot generate truly genome-wide significance thresholds. Bayes factors have also been proposed for the measurement of significance

^{6}. For GWA studies of dense SNPs and resequence data, a standard genome-wide significance threshold of 7.2 × 10

^{− 8} for the UK Caucasian population has been proposed by Dudbridge and Gusnanto

^{21}. Other thresholds for contemporary populations, based on sample size and proposed FWER, have been proposed by Hoggart

*et al*^{22}. Informally, some journals have accepted a genome-wide significance threshold of 5 × 10

^{− 7} as strong evidence for association

^{6}; however, most recently, the accepted standard is 5 × 10

^{− 8} (

ref. 23). Further, graphical techniques for assessing whether observed

*P* values are consistent with expected values include log quantile-quantile

*P* value plots that highlight loci that deviate from the null hypothesis

^{24}.

Interpretation of results

A significant result in an association test rarely implies that a SNP is directly influencing disease risk; population association can be direct, indirect or spurious. A direct, or causal, association occurs when different alleles at the marker locus are directly involved in the etiology of the disease through a biological pathway. Such associations are typically only found during follow-up genotyping phases of initial GWA studies, or in focused CG studies in which particular functional polymorphisms are targeted. An indirect, or non-causal, association occurs when the alleles at the marker locus are correlated (in LD) with alleles at a nearby causal locus but do not directly influence disease risk. When a significant finding in a genetic association study is true, it is most likely to be indirect. Spurious associations can occur as a consequence of data quality issues or statistical sampling, or because of confounding by population stratification or admixture. Population stratification occurs when cases and controls are sampled disproportionately from different populations with distinct genetic ancestry. Admixture occurs when there has been genetic mixing of two or more groups in the recent past. For example, genetic admixture is seen in Native American populations in which there has been recent genetic mixing of individuals with both American Indian and Caucasian ancestry^{25}. Confounding occurs when a factor exists that is associated with both the exposure (genotype) and the disease but is not a consequence of the exposure. As allele frequencies and disease frequencies are known to vary among populations of different genetic ancestry, population stratification or admixture can confound the association between the disease trait and the genetic marker; it can bias the observed association, or indeed can cause a spurious association. Principal component analyses or multidimensional scaling methods are commonly used to identify and remove individuals exhibiting divergent ancestry before association testing. These techniques are described in detail in an earlier protocol^{3}. To adjust for any residual population structure during association testing, the principal components from principal component analyses or multidimensional scaling methods can be included as covariates in a logistic regression. In addition, the technique of genomic control^{26} can be used to detect and compensate for the presence of fine-scale or within-population stratification during association testing. Under genomic control, population stratification is treated as a random effect that causes the distribution of the χ^{2} association test statistics to have an inflated variance and a higher median than would otherwise be observed. The test statistics are assumed to be uniformly affected by an inflation factor λ, the magnitude of which is estimated from a set of selected markers by comparing the median of their observed test statistics with the median of their expected test statistics under an assumption of no population stratification. Under genomic control, if λ > 1, then population stratification is assumed to exist and a correction is applied by dividing the actual association test χ^{2} statistic values by λ. As λ scales with sample size, λ_{1,000}, the inflation factor for an equivalent study of 1,000 cases and 1,000 controls calculated by rescaling λ, is often reported^{27}. In a CG study, λ can only be determined if an additional set of markers specifically designed to indicate population stratification are genotyped. In a GWA study, an unbiased estimation of λ can be determined using all of the genotyped markers; the effect on the inflation factor of potential causal SNPs in such a large set of genomic control markers is assumed to be negligible.

Replication

Replication occurs when a positive association from an initial study is confirmed in a subsequent study involving an independent sample drawn from the same population as the initial study. It is the process by which genetic association results are validated. In theory, a repeated significant association between the same trait and allele in an independent sample is the benchmark for replication. However, in practice, so-called replication studies often comprise findings of association between the same trait and nearby variants in the same gene as the original SNP, or between the same SNP and different high-risk traits. A precise definition of what constitutes replication for any given study is therefore important and should be clearly stated^{28}.

In practice, replication studies often involve different investigators with different samples and study designs aiming to independently verify reports of positive association and obtain accurate effect-size estimates, regardless of the designs used to detect effects in the primary study. Two commonly used strategies in such cases are an exact strategy, in which only marker loci indicating a positive association are subsequently genotyped in the replicate sample, and a local strategy, in which additional variants are also included, thus combining replication with fine-mapping objectives. In general, the exact strategy is more balanced in power and efficiency; however, depending on local patterns of LD and the strength of primary association signals, a local strategy can be beneficial^{28}.

In the past, multistage designs have been proposed as cost-efficient approaches to allow the possibility of replication within a single overall study. The first stage of a standard two-stage design involves genotyping a large number of markers on a proportion of available samples to identify potential signals of association using a nominal *P* value threshold. In stage two, the top signals are then followed up by genotyping them on the remaining samples while a joint analysis of data from both stages is conducted^{29,30}. Significant signals are subsequently tested for replication in a second data set. With the ever-decreasing costs of GWA genotyping, two-stage studies have become less common.

Software

Standard statistical software (such as R (

ref. 31) or SPSS) can be used to conduct and visualize all the analyses outlined above. However, many researchers choose to use custom-built GWA software. In this protocol we use PLINK

^{32}, Haploview

^{33} and the customized R package

*car*^{34}. PLINK is a popular and computationally efficient software program that offers a comprehensive and well-documented set of automated GWA quality control and analysis tools. It is a freely available open source software written in C++, which can be installed on Windows, Mac and Unix machines (

http://pngu.mgh.harvard.edu/~purcell/plink/index.shtml). Haploview (

http://www.broadinstitute.org/haploview/haploview) is a convenient tool for visualizing LD; it interfaces directly with PLINK to produce a standard visualization of PLINK association results. Haploview is most easily run through a graphical user interface, which offers many advantages in terms of display functions and ease of use.

*car* (

http://socserv.socsci.mcmaster.ca/jfox/) is an R package that contains a variety of functions for graphical diagnostic methods.

The next section describes protocols for the analysis of SNP data and is illustrated by the use of simulated data sets from CG and GWA studies (available as gzipped files from

http://www.well.ox.ac.uk/ggeu/NPanalysis/ or .zip files as

Supplementary Data 1 and

Supplementary Data 2). We assume that SNP data for a CG study, typically comprising on the order of thousands of markers, will be available in a standard PED and MAP file format (for an explanation of these file formats, see

http://pngu.mgh.harvard.edu/~purcell/plink/data.shtml#ped) and that SNP data for a GWA study, typically comprising on the order of hundreds of thousands of markers, will be available in a standard binary file format (for an explanation of the binary file format, see

http://pngu.mgh.harvard.edu/~purcell/plink/data.shtml#bed). In general, SNP data for either type of study may be available in either format. The statistical analysis described here is for the analysis of one SNP at a time; therefore, apart from the requirement to take potentially differing input file formats into account, it does not differ between CG and GWA studies.