Genome-wide complex trait analysis

Estimation in biology/animal breeding using standard ANOVA/REML methods of variance components such as heritability, shared-environment, maternal effects etc. typically requires individuals of known relatedness such as parent/child; this is often unavailable or the pedigree data unreliable, leading to inability to apply the methods or requiring strict laboratory control of all breeding (which threatens the external validity of all estimates), and several authors have noted that relatedness could be measured directly from genetic markers (and if individuals were reasonably related, economically few markers would have to be obtained for statistical power), leading Kermit Ritland to propose in 1996 that directly measured pairwise relatedness could be compared to pairwise phenotype measurements (Ritland 1996, "A Marker-based Method for Inferences About Quantitative Inheritance in Natural Populations"[10]).

As genome sequencing costs dropped steeply over the 2000s, acquiring enough markers on enough subjects for reliable estimates using very distantly related individuals became possible. An early application of the method to humans came with Visscher et al. 2006[11]/2007,[12] which used SNP markers to estimate the actual relatedness of siblings and estimate heritability from the direct genetics. In humans, unlike the original animal/plant applications, relatedness is usually known with high confidence in the 'wild population', and the benefit of GCTA is connected more to avoiding assumptions of classic behavioral genetics designs and verifying their results, and partitioning heritability by SNP class and chromosomes. The first use of GCTA proper in humans was published in 2010, finding 45% of variance in human height can be explained by the included SNPs.[13][14] (Large GWASes on height have since confirmed the estimate.[15]) The GCTA algorithm was then described and a software implementation published in 2011.[16] It has since been used to study a wide variety of biological, medical, psychiatric, and psychological traits in humans, and inspired many variant approaches.

1. Limited inference: GCTA estimates are inherently limited in that they cannot estimate broadsense heritability like twin/family studies as they only estimate the heritability due to SNPs. Hence, while they serve as a critical check on the unbiasedness of the twin/family studies, GCTAs cannot replace them for estimating total genetic contributions to a trait.
2. Substantial data requirements: the number of SNPs genotyped per person should be in the thousands and ideally the hundreds of thousands for reasonable estimates of genetic similarity (although this is no longer such an issue for current commercial chips which default to hundreds of thousands or millions of markers); and the number of persons, for somewhat stable estimates of plausible SNP heritability, should be at least n>1000 and ideally n>10000.[19] In contrast, twin studies can offer precise estimates with a fraction of the sample size.
3. Computational inefficiency: The original GCTA implementation scales poorly with increasing data size (${\displaystyle {\mathcal {O}}({\text{SNPs}}\cdot n^{2})}$), so even if enough data is available for precise GCTA estimates, the computational burden may be unfeasible. GCTA can be meta-analyzed as a standard precision-weighted fixed-effect meta-analysis,[20] so research groups sometimes estimate cohorts or subsets and then pool them meta-analytically (at the cost of additional complexity and some loss of precision). This has motivated the creation of faster implementations and variant algorithms which make different assumptions, such as using moment matching.[21]
4. Need for raw data: GCTA requires genetic similarity of all subjects and thus their raw genetic information; due to privacy concerns, individual patient data is rarely shared. GCTA cannot be run on the summary statistics reported publicly by many GWAS projects, and if pooling multiple GCTA estimates, a meta-analysis must be performed.
   In contrast, there are alternative techniques which operate on summaries reported by GWASes without requiring the raw data[22] e.g. "LD score regression"[23] contrasts linkage disequilibrium statistics (available from public datasets like 1000 Genomes) with the public summary effect-sizes to infer heritability and estimate genetic correlations/overlaps of multiple traits. The Broad Institute runs LD Hub which provides a public web interface to >=177 traits with LD score regression.[24] Another method using summary data is HESS.[25]
5. Confidence intervals may be incorrect, or outside the 0-1 range of heritability, and highly imprecise due to asymptotics.[26]
6. Underestimation of SNP heritability: GCTA implicitly assumes all classes of SNPs, rarer or commoner, newer or older, more or less in linkage disequilibrium, have the same effects on average; in humans, rarer and newer variants tend to have larger and more negative effects[27] as they represent mutation load being purged by negative selection. As with measurement error, this will bias GCTA estimates towards underestimating heritability.

GCTA estimates are often misinterpreted as "the total genetic contribution", and since they are often much less than the twin study estimates, the twin studies are presumed to be biased and the genetic contribution to a particular trait is minor.[28] This is incorrect, as GCTA estimates are lower bounds.

A more correct interpretation would be that: GCTA estimates are the expected amount of variance that could be predicted by an indefinitely large GWAS using a simple additive linear model (without any interactions or higher-order effects) in a particular population at a particular time given the limited selection of SNPs and a trait measured with a particular amount of precision. Hence, there are many ways to exceed GCTA estimates:

1. SNP genotyping data is typically limited to 200k-1m of the most common or scientifically interesting SNPs, though 150 million+ have been documented by genome sequencing;[29] as SNP prices drop and arrays become more comprehensive or whole-genome sequencing replaces SNP genotyping entirely, the expected narrowsense heritability will increase as more genetic variants are included in the analysis. The selection can also be expanded considerably using haplotypes[30] and imputation (SNPs can proxy for unobserved genetic variants which they tend to be inherited with); e.g. Yang et al. 2015[31] finds that with more aggressive use of imputation to infer unobserved variants, the height GCTA estimate expands to 56% from 45%, and Hill et al. 2017 finds that expanding GCTA to cover rarer variants raises the intelligence estimates from ~30% to ~53% and explains all the heritability in their sample;[32] for 4 traits in the UK Biobank, imputing raised the SNP heritability estimates.[33] Additional genetic variants include de novo mutations/mutation load & structural variations such as copy-number variations.
2. narrowsense heritability estimates assume simple additivity of effects, ignoring interactions. As some trait values will be due to these more complicated effects, the total genetic effect will exceed that of the subset measured by GCTA, and as the additive SNPs are found and measured, it will become possible to find interactions as well using more sophisticated statistical models.
3. all correlation & heritability estimates are biased downwards to zero by the presence of measurement error; the need for adjusting this leads to techniques such as Spearman's correction for measurement error, as the underestimate can be quite severe for traits where large-scale and accurate measurement is difficult and expensive,[34] such as intelligence. For example, an intelligence GCTA estimate of 0.31, based on an intelligence measurement with test-retest reliability ${\displaystyle r=0.65}$, would after correction (${\displaystyle {\frac {0.31}{0.65}}}$), be a true estimate of ~0.48, indicating that common SNPs alone explain half of variance. Hence, a GWAS with a better measurement of intelligence can expect to find more intelligence hits than indicated by a GCTA based on a noisier measurement.

The original "GCTA" software package is the most widely used; its primary functionality covers the GREML estimation of SNP heritability, but includes other functionality:

• Estimate the genetic relationship from genome-wide SNPs;
• Estimate the inbreeding coefficient from genome-wide SNPs;
• Estimate the variance explained by all the autosomal SNPs;
• Partition the genetic variance onto individual chromosomes;
• Estimate the genetic variance associated with the X-chromosome;
• Test the effect of dosage compensation on genetic variance on the X-chromosome;
• Predict the genome-wide additive genetic effects for individual subjects and for individual SNPs;
• Estimate the LD structure encompassing a list of target SNPs;
• Simulate GWAS data based upon the observed genotype data;
• Convert Illumina raw genotype data into PLINK format;
• Conditional & joint analysis of GWAS summary statistics without individual level genotype data
• Estimating the genetic correlation between two traits (diseases) using SNP data
• Mixed linear model association analysis
  — GCTA, cnsgenomics.com/software/gcta/

Other implementations and variant algorithms include:

• FAST-LMM[35]
• FAST-LMM-Select:[36] like GCTA in using ridge regression[37] but including feature selection to try to exclude irrelevant SNPs which only add noise to the relatedness estimates
• LMM-Lasso[38]
• GEMMA[39]
• EMMAX[40]
• REACTA (formerly ACTA) claims order of magnitude runtime reductions[41][42]
• BOLT-REML/BOLT-LMM[43] (manual), faster & better scaling;[44] with potentially better efficiency in the meta-analysis scenario[45]
• MEGHA[46]
• PLINK >1.9 (December 2013) supports "the use of genetic relationship matrices in mixed model association analysis and other calculations"
• LDAK:[47] loosens the GCTA assumption that all SNPs, regardless of genotyping quality or frequency, have same averaged expected effect, allowing for potentially finding much more SNP heritability
• GREML-IBD:[48] GCTA for identity by descent, attempting to estimate heritability based on shared genome segments in distant otherwise-unrelated relatives, in order to capture the effect of rarer variants which are not measured by SNP panels or otherwise imputed

GCTA estimates frequently find estimates 0.1-0.5, consistent with broadsense heritability estimates (with the exception of personality traits, for which theory & current GWAS results suggest non-additive genetics driven by frequency-dependent selection[49][50]). Traits univariate GCTA has been used on (excluding SNP heritability estimates computed using other algorithms such as LD score regression, and bivariate GCTAs which are listed in genetic correlation) include (point-estimate format: "${\displaystyle h_{SNP}^{2}}$(standard error)"):

• Animal breeding
• Best linear unbiased prediction
• Mendelian randomization
• Pleiotropy
• "The Correlation between Relatives on the Supposition of Mendelian Inheritance", Fisher 1918

• "Research review: Polygenic methods and their application to psychiatric traits", Wray et al. 2014
• "Heritability in the genomics era — concepts and misconceptions", Visscher et al. 2008
• "Uncovering the Genetic Architectures of Quantitative Traits", Lee et al. 2016
• "Estimating heritability using genomic data", Stanton-Geddes et al. 2013
• "MultiBLUP: improved SNP-based prediction for complex traits", Speed & Balding 2012
• "Advantages and pitfalls in the application of mixed-model association methods", Yang et al. 2013
• "Prediction of Total Genetic Value Using Genome-Wide Dense Marker Maps", Meuwissen et al. 2001
• "Understanding and using quantitative genetic variation", Hill 2009
• "One Hundred Years of Statistical Developments in Animal Breeding", Gianola & Rosa 2015
• "Conditions for the validity of SNP-based heritability estimation", Lee & Chow 2014
• "Measuring missing heritability: Inferring the contribution of common variants", Golan et al. 2014
• "Concepts, estimation and interpretation of SNP-based heritability", Yang et al. 2017
• Maier et al. 2017, "Embracing polygenicity: a review of methods and tools for psychiatric genetics research"
• Ronald & Pain 2018, "A systematic review of genome-wide research on psychotic experiences and negative symptom traits: new revelations and implications for psychiatry"

• GCTA-GREML Power Calculator ("Statistical Power to Detect Genetic (Co)Variance of Complex Traits Using SNP Data in Unrelated Samples", Visscher et al. 2014)
• "Genomics, Big Data, Medicine, and Complex Traits" (Peter Visscher talk)
• "The Genetic Architectures of Psychological Traits", Lee 2014 slides
• "Heritability-based models for prediction of complex traits", David Balding 2015