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  • Review Article
  • Published:

Bayesian statistical methods for genetic association studies

Nature Reviews Genetics volume 10, pages 681–690 (2009)Cite this article

Key Points

  • p-values are commonly used as summaries of evidence for association between a genetic variant and phenotype, but they have an important limitation in that they are unable to quantify how confident one should be that a given SNP is truly associated with a phenotype.

  • Bayesian methods provide an alternative approach to assessing associations. We show that Bayesian analyses are not too difficult and can be rewarding — for example, unlike p-values, a Bayesian probability of association is comparable across SNPs and across studies.

  • For a Bayesian analysis of single-SNP association in a case–control study, we discuss genetic models that can form an alternative to the null hypothesis of no association, in addition to effect-size distributions for the parameters of these models. An alternative Bayesian analysis derives a posterior distribution for effect size, without reference to a null hypothesis.

  • We give an example of a multi-SNP Bayesian analysis for fine-scale mapping and discuss Bayesian approaches to multiple testing and meta-analysis.

  • Broad guidelines are suggested for editors and reviewers of Bayesian analyses.

Abstract

Bayesian statistical methods have recently made great inroads into many areas of science, and this advance is now extending to the assessment of association between genetic variants and disease or other phenotypes. We review these methods, focusing on single-SNP tests in genome-wide association studies. We discuss the advantages of the Bayesian approach over classical (frequentist) approaches in this setting and provide a tutorial on basic analysis steps, including practical guidelines for appropriate prior specification. We demonstrate the use of Bayesian methods for fine mapping in candidate regions, discuss meta-analyses and provide guidance for refereeing manuscripts that contain Bayesian analyses.

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Figure 1: Dependence of the Bayes factor on minor allele count and on the prior standard deviation of the effect size.

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Acknowledgements

We thank C. Hoggart for providing R code to compute the normal-exponential-gamma probability density function and J. Wakefield for helpful discussions and critical reading of an early draft. We thank R. Krauss for access to the CRP genotype and phenotype data that we analysed here. We are also grateful to W. Astle, A. Ramasamy, L. Bottolo, L. Coin, P. O'Reilly and H. Eleftherohorinou for discussions. The authors' work is supported in part by National Institutes of Health grants HL084689 (to M.S.) and EP/C533542 (to D.J.B.).

Author information

Author notes

  1. David J. Balding

    Present address: Current address: Institute of Genetics, University College London, London WC1E 6BT, UK. mstephens@uchicago.edu; d.balding@ic.ac.uk,

Authors and Affiliations

  1. Departments of Statistics and Human Genetics, University of Chicago, Chicago, 60637, Illinois, USA

    Matthew Stephens

  2. Department of Epidemiology and Public Health, Imperial College, St Mary's Campus, Norfolk Place, W2 1PG, London, UK

    David J. Balding

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FURTHER INFORMATION

Matthew Stephens' homepage

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BIMBAM

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WinBUGS

Nature Reviews Genetics series on Modelling

Nature Reviews Genetics series on Genome-wide Association Studies

Glossary

Frequentist

A statistical school of thought in which inferences about unknowns are justified not with reference to probabilities for the inferred value, but on the basis of measures of performance under imaginary repetitions of the procedure that was used to make the inference.

Population association

Also known as true association. An association between a SNP and a phenotype that is present in the population from which a sample is taken. A population association can arise owing to population structure, but for simplicity we assume here that this possibility has been eliminated (for example, by covariate adjustment) and hence that population associations are caused by a functional SNP, either directly or through linkage disequilibrium.

p-value

The probability, if the null hypothesis were true, that an imaginary future repetition of the study would generate stronger evidence for association than that actually observed. A p-value is conventionally interpreted as measuring the strength of evidence for association, but there is no simple relationship between a p-value and the probability that the association is genuine.

Power

For a given population association, the power of a statistical test is the probability that the null hypothesis is rejected under imaginary repetitions of the study.

Bayesian

A statistical school of thought that holds that inferences about any unknown parameter or hypothesis should be encapsulated in a probability distribution, given the observed data. Computing this posterior probability distribution usually proceeds by specifying a prior distribution that summarizes knowledge about the unknown before the observed data are considered, and then using Bayes' theorem to transform the prior distribution into a posterior distribution.

Meta-analysis

The combination of the results of multiple scientific studies that address the same, or similar, hypotheses.

Posterior probability of association

The probability that a SNP is truly associated with a phenotype. The posterior probability of association depends on modelling assumptions that should be made explicit in a careful analysis.

Likelihood ratio

The ratio of the probabilities of the observed data for two different values of the unknown parameter(s) under a given statistical model.

Odds

The probability of the occurrence of a particular event (for example, the onset of disease) divided by the probability of the event not occurring. It is often mathematically convenient to transform a probability, which must lie between zero and one, to odds, which can take any positive value.

Bonferroni correction

When multiple hypotheses are tested, the Bonferroni correction to the overall desired significance level (α) is obtained by dividing it by the number of tests (k), so that each hypothesis is rejected if p-value < α/k.

False discovery rate

For a sequence of hypothesis tests, the false discovery rate is the proportion of times H0 is true among those tests for which H0 is rejected.

Odds ratio

The odds ratio comparing, for example, two genotypes is the odds for individuals with the first genotype divided by the odds for individuals with the second genotype.

Logistic regression

A regression model for binary outcomes (such as case and control) in which the logarithm of the odds is related linearly to one or more predictors, such as SNP minor allele count(s).

Laplace approximation

A method for approximating the integral of a (possibly multidimensional) probability density based on replacing that density by a Gaussian probability density with the same mean and variance–covariance matrix.

Maximum-likelihood estimate

The maximum-likelihood estimate of an unknown parameter in a statistical model is the value of the parameter that maximizes the probability under the model of the observed data.

Statin

A class of drugs that is used to lower cholesterol levels in people with, or at risk of, cardiovascular disease.

Genotype imputation method

A method for estimating ('imputing') the unobserved genotypes of study subjects, both for individuals with missing or unreliable genotypes at a genotyped SNP and for all individuals at an ungenotyped SNP.

Hardy–Weinberg equilibrium

This holds at a given locus in a given population when the two alleles of individuals in the population are mutually independent.

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Stephens, M., Balding, D. Bayesian statistical methods for genetic association studies. Nat Rev Genet 10, 681–690 (2009). https://doi.org/10.1038/nrg2615

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