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Genetic correlations of polygenic disease traits: from theory to practice

Abstract

The genetic correlation describes the genetic relationship between two traits and can contribute to a better understanding of the shared biological pathways and/or the causality relationships between them. The rarity of large family cohorts with recorded instances of two traits, particularly disease traits, has made it difficult to estimate genetic correlations using traditional epidemiological approaches. However, advances in genomic methodologies, such as genome-wide association studies, and widespread sharing of data now allow genetic correlations to be estimated for virtually any trait pair. Here, we review the definition, estimation, interpretation and uses of genetic correlations, with a focus on applications to human disease.

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Acknowledgements

W.v.R. was funded by the ALS Foundation Netherlands. W.J.P. was funded by an NWO Veni grant (91619152). S.H.L. is an ARC Future Fellow (FT160100229). N.R.W. acknowledges funding from the Australian National Health and Medical Research Council (1078901, 1087889 and 1113400). W.v.R. and N.R.W. acknowledge funding from the EU Joint Programme – Neurodegenerative Disease Research (JPND) project (Australia, NHMRC 1151854; The Netherlands, ZonMW project number 733051071). The authors thank K. Tilling, G. Davey Smith and the members of the University of Queensland Program in Complex Trait Genomics for their insightful discussions.

Competing interests

The authors declare no competing interests.

Reviewer information

Nature Reviews Genetics thanks D. Balding, B. Pasaniuc and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Author information

All authors researched data for the article, made substantial contributions to discussions of the content and reviewed and/or edited the manuscript before submission. W.v.R. and N.R.W. wrote the article.

Correspondence to Wouter van Rheenen or Naomi R. Wray.

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Related links

BUHMBOX: http://software.broadinstitute.org/mpg/buhmbox/

fastPAINTOR: https://github.com/gkichaev/PAINTOR_V3.0

GCTA: http://cnsgenomics.com/software/gcta/

GNOVA: https://github.com/xtonyjiang/GNOVA

GWIS: https://sites.google.com/site/mgnivard/gwis

JPND: www.jpnd.eu

LCV: https://github.com/lukejoconnor/LCV

LDAK: http://dougspeed.com/ldak

LD Hub: http://ldsc.broadinstitute.org/ldhub/

LDSC: https://github.com/bulik/ldsc

MR Steiger: https://github.com/explodecomputer/causal-directions

MTAG: https://github.com/omeed-maghzian/mtag

PCGC: https://data.broadinstitute.org/alkesgroup/PCGC/

ρHESS: https://github.com/huwenboshi/hess

PleioPred: https://github.com/yiminghu/PleioPred

Popcorn: https://github.com/brielin/Popcorn

RiVIERA: https://github.com/yueli-compbio/RiVIERA-beta

SMTpred: https://github.com/uqrmaie1/smtpred

SumHer: http://dougspeed.com/sumher/

Supplementary information

Supplementary Information

Glossary

Parameter

A numerical value that summarizes a characteristic of a population, such as the mean height of men, the lifetime risk of schizophrenia or the heritability of a specific trait.

Traits

Measurements or phenotypes that are usually studied as the outcome of statistical analyses. They can be quantitative (for example, height) or dichotomous (for example, schizophrenia).

Estimates

Approximations of a parameter based on a sample of observed data drawn from a population.

Ascertainment biases

Types of bias that occur when the studied trait or disease affects how data were ascertained. For example, patients with a family history of diabetes may have more frequent examinations for cardiovascular diseases.

Genome-wide association studies

Studies in which up to millions of mostly common single-nucleotide polymorphisms from across the genome are each tested for association with a trait.

GWAS summary statistics

The output of statistical tests of association of a trait with each single-nucleotide polymorphism generated by a genome-wide association study (GWAS), typically including the effect allele, signed effect estimate, standard error, test statistic (for example, a z-score) and/or p-value.

Power

The probability that a study correctly rejects the null hypothesis of no association or correlation, also described as 1– type II error.

Bias

Phenomenon where statistical analyses produce estimates in observed data that systematically overestimate or underestimate the population parameter. Bias can arise from the ascertainment of the observed data or the statistical procedures used to generate the estimates.

Linkage disequilibrium

(LD). The non-random segregation of alleles at two distinct loci. LD induces a correlation between two single-nucleotide polymorphism (SNP) genotypes in the population and is caused by the fact that alleles of neighbouring SNPs are transmitted together until broken down by recombination events.

Genetic value

(g). The sum of the total effects of all genetic loci on the trait in an individual, that is g =  where X is a vector of genotypes for all loci and ß is a vector with additive allelic effects on the trait. It is also called the genotypic value, true polygenic (risk) score or breeding value.

Covariance

(\({\sigma }_{x,y}\)). The expected product of the deviation of two random variables from their mean (\({\sigma }_{x,y}=E[(X-{\mu }_{x})(Y-{\mu }_{y})]\)).

Genetic variance

(\({\sigma }_{g}^{2}\)). The expected squared deviation of genetic values from the mean genetic value (\({\sigma }_{g}^{2}=E[{(G-{\mu }_{g})}^{2}]\)), and can also be considered the covariance of a genetic value with itself.

Heritability

(h2). The proportion of phenotypic variance (parameter \({\sigma }_{P}^{\,2}\), estimate VP) attributable to variance in genetic factors. In the context of human traits, most often only additive genetic factors are considered for the genetic variance (parameter \({\sigma }_{A}^{2}\), estimate VA) and the ratio of variances is the narrow-sense heritability.

Latent model

A collection of formalized assumptions to describe a data-generating process through which observed variables (such as disease occurrence) can be used to identify unobserved (latent) variables (for example, genetic parameters: heritability and genetic correlation).

Phenotypic variance

(\({\sigma }_{P}^{2}\)). Variance of phenotypic values (for example, height or disease liability) after accounting for the variance attributable to fixed effects (for example, sex). When phenotypes are standardized, these phenotypic values are scaled such that µP = 0 and \({\sigma }_{P}^{2}\) = 1.

Coheritability

(hxy). The genetic covariance of standardized traits. This is a useful measure for comparisons of coheritabilities and heritabilities on the same scale.

Linear mixed model

(LMM). A linear model that includes both fixed and random effects to describe phenotypic values and that allows a correlation structure between the random effect levels.

Restricted maximum likelihood

(REML). A method for maximum likelihood estimation of variance–covariance components of the parameters in linear mixed models.

Liability threshold model

A model that describes a dichotomous trait (disease) as a threshold partitioning of ‘liability’, which is a latent variable assumed to follow a standard normal distribution in the population. The liability threshold (T) defines lifetime risk (K) of disease as the proportion of individuals exceeding this threshold.

Risk ratio

Ratio between the risk of disease in a specific group (for example, relatives of affected individuals) and the risk of disease in the general population.

Tetrachoric correlation

The correlation between two latent normally distributed liability phenotypes assumed to underlie dichotomous population data and estimated from an observed 2 × 2 frequency table.

Genomic relationship matrix

(GRM). A matrix whose off-diagonal elements represent a coefficient of genetic sharing between individuals to describe the variance–covariance structure between their genetic values calculated from observed single-nucleotide polymorphism (SNP) data. GRM coefficients can be calculated based on different assumptions of the expected distribution of per-SNP heritability.

SNP-based heritability

An estimate of the proportion of the total phenotypic variance attributable to the additive effects of the class of variants (that is, common single-nucleotide polymorphisms (SNPs)) that are typically genotyped and imputed in pursuit of a genome-wide association study. It is often shortened to SNP heritability, but this should be avoided.

Genotype by environment (G × E) interaction

Differences in size and/or direction of the effect of genotype on disease risk in two different environments.

Sample heterogeneity

Differences in the effects of genotype on disease risk in two different cohorts. Potential causes include differences in phenotype criteria, ascertainment methods and unknown environmental differences with genotype by environment interaction.

Infinitesimal model

This model assumes that a trait is shaped by a very large number of variants with small (infinitesimal) effects resulting in a normally distributed phenotype. A polygenic architecture of >~10 causal variants is approximated well by normal distribution infinitesimal model theory.

Haseman–Elston regression

Regression of the product of the standardized phenotypes of pairs of individuals on their coefficient of genetic sharing as defined in the genomic relationship matrix.

Confounding bias

A type of bias that emerges when a covariate, a ‘confounder’, causally influences the predictor variable and outcome variable. When the confounder is not accounted for, the relationship between predictor and outcome may be biased (confounded).

Assortative mating

Mating selection on a trait where the phenotypes of mates are positively correlated. Examples of assortative mating in humans include height or educational attainment.

Collider bias

A type of bias that emerges when estimates are conditioned on a covariate, a ‘collider,’ that is causally influenced by both the predictor variable and outcome variable.

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Fig. 1: Different mechanisms of pleiotropy between two diseases.
Fig. 2: Genome-wide genetic correlation versus regional genetic correlation.
Fig. 3: Relation between cross-disorder relative risk (CDRR) and genetic correlation.
Fig. 4: Precision of genetic correlation estimates compared to heritability estimates and power for GREML and LDSC.
Fig. 5: Bias in estimated genetic parameters.
Fig. 6: Prediction accuracy increases when correlated traits are combined.