# Genome-wide complex trait analysis

Genome-wide complex trait analysis (GCTA) Genome-based restricted maximum likelihood (GREML) is a statistical method for variance component estimation in genetics which quantifies the total narrow-sense (additive) contribution to a trait's heritability of a particular subset of genetic variants (typically limited to SNPs with MAF >1%, hence terms such as "chip heritability"/"SNP heritability"). This is done by directly quantifying the chance genetic similarity of unrelated strangers and comparing it to their measured similarity on a trait; if two strangers are relatively similar genetically and also have similar trait measurements, then this indicates that the measured genetics causally influence that trait, and how much. This can be seen as plotting prediction error against relatedness.[1] The GCTA framework extends to bivariate genetic correlations between traits.[2] It can be done on a per-chromosome basis, comparing against chromosome length, and can examine changes in heritability over aging and development.[3] There is an ongoing debate about whether GCTA generates reliable or stable estimates of heritability when used on current SNP data.[4] The method is based on the outdated and false dichotomy of genes versus the environment. It also suffers from serious methodological weaknesses, such as susceptibility to population stratification.[5]

GCTA heritability estimates are useful because they can lower bound[6] the genetic contributions to traits such as intelligence without relying on the assumptions used in twin studies and other family studies, and pedigree analyses, thereby corroborating them,[7][8][9] and enabling the design of well-powered Genome-wide association study (GWAS) designs to find the specific genetic variants. For example, a GCTA estimate of 30% SNP heritability is consistent with a larger total genetic heritability of 70%. However, if the GCTA estimate was ~0%, then that would imply one of three things: a) there is no genetic contribution, b) the genetic contribution is entirely in the form of genetic variants not included, or c) the genetic contribution is entirely in the form of non-additive effects such as epistasis/dominance. The ability to run GCTA on subsets of chromosomes and regress against chromosome length can reveal whether the responsible genetic variants cluster or are distributed evenly across the genome or are sex-linked. Examining genetic correlations can reveal to what extent observed correlations, such as between intelligence and socioeconomic status, are due to the same genetic traits, and in the case of diseases, can indicate shared causal pathways such as the overlap of schizophrenia with other mental diseases and intelligence-reducing variants.

## History

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.

## Benefits

### Robust heritability

Twin and family studies have long been used to estimate variance explained by particular categories of genetic and environmental causes. Across a wide variety of human traits studied, there is typically minimal shared-environment influence, considerable non-shared environment influence, and a large genetic component (mostly additive), which is on average ~50% and sometimes much higher for some traits such as height or intelligence.[17] However, the twin and family studies have been criticized for their reliance on a number of assumptions that are difficult or impossible to verify, such as the equal environments assumption (that the environments of monozygotic and dizygotic twins are equally similar), that there is no misclassification of zygosity (mistaking identical for fraternal & vice versa), that twins are unrepresentative of the general population, and that there is no assortative mating. Violations of these assumptions can result in both upwards and downwards bias of the parameter estimates.[18] (This debate & criticism have particularly focused on the heritability of IQ.)

The use of SNP or whole-genome data from unrelated subject participants (with participants too related, typically >0.025 or ~fourth cousins levels of similarity, being removed, and several principal components included in the regression to avoid & control for population stratification) bypasses many heritability criticisms: twins are often entirely uninvolved, there are no questions of equal treatment, relatedness is estimated precisely, and the samples are drawn from a broad variety of subjects.

In addition to being more robust to violations of the twin study assumptions, SNP data can be easier to collect since it does not require rare twins and thus also heritability for rare traits can be estimated (with due correction for ascertainment bias).

### GWAS power

GCTA estimates can be used to resolve the missing heritability problem and design GWASes which will yield genome-wide statistically-significant hits. This is done by comparing the GCTA estimate with the results of smaller GWASes. If a GWAS of n=10k using SNP data fails to turn up any hits, but the GCTA indicates a high heritability accounted for by SNPs, then that implies that there are a large number of polygenic variants and thus that much larger GWASes will be required to accurately estimate each SNP's effects and directly account for a fraction of the GCTA heritability.

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, meta-analysis must be done.
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.

## Interpretation

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.

## Implementations

Original author(s) Jian Yang 30 August 2010 1.25.2 / 22 December 2015 C++ Linux (Mac/Windows support dropped at v1.02) English genetics GPL v3 cnsgenomics.com/software/gcta/; forums: gcta.freeforums.net 22 May 2016

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

Other implementations and variant algorithms include:

## Traits

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)"):

## References

1. ^ Figure 3 of Yang et al 2010, or Figure 3 of Ritland & Ritland 1996
2. ^
3. ^
4. ^ Krishna Kumar, Siddharth; Feldman, Marcus W.; Rehkopf, David H.; Tuljapurkar, Shripad (2016-01-05). "Limitations of GCTA as a solution to the missing heritability problem". Proceedings of the National Academy of Sciences of the United States of America. 113 (1): E61–70. doi:10.1073/pnas.1520109113. ISSN 1091-6490. PMC 4711841. PMID 26699465.
5. ^ BURT, CALLIE H.; SIMONS, RONALD L. (February 2015). "HERITABILITY STUDIES IN THE POSTGENOMIC ERA: THE FATAL FLAW IS CONCEPTUAL". Criminology. 53 (1): 103–112. doi:10.1111/1745-9125.12060.
6. ^ "A common misconception about SNP-chip heritability estimates calculated with GCTA and LDSC is that they should be similar to twin study estimates, when in reality twin studies have the advantage of capturing all genetic effects—common, rare and those not genotyped by available methods. Thus, the assumption should be that h2SNP < h2TWIN when using GCTA and LDSC, and this is what we observe for PTSD, as has been observed for many other phenotypes.54" --Duncan et al 2017
7. ^ Eric Turkheimer ("Still Missing", Turkheimer 2011) discusses the GCTA results in the context of the twin study debate: "Of the three reservations about quantitative genetic heritability that were outlined at the outset—the assumptions of twin and family studies, the universality of heritability, and the absence of mechanism—the new paradigm has put the first to rest, and before continuing to explain my skepticism about whether the most important problems have been solved, it is worth appreciating what a significant accomplishment this is. Thanks to the Visscher program of research, it should now be impossible to argue that the whole body of quantitative genetic research showing the universal importance of genes for human development was somehow based on a sanguine view of the equal environments assumption in twin studies, putting an end to an entire misguided school of thought among traditional opponents of classical quantitative (and by association behavioral) genetics (e.g., Joseph, 2010; Kamin & Goldberger, 2002)"; see also Turkheimer, Harden, & Nisbett: "These methods have given scientists a new way to compute heritability: Studies that measure DNA sequence variation directly have shown that pairs of people who are not relatives, but who are slightly more similar genetically, also have more similar IQs than other pairs of people who happen to be more different genetically. These “DNA-based” heritability studies don’t tell you much more than the classical twin studies did, but they put to bed many of the lingering suspicions that twin studies were fundamentally flawed in some way. Like the validity of intelligence testing, the heritability of intelligence is no longer scientifically contentious."
8. ^ "This finding of strong genome-wide pleiotropy across diverse cognitive and learning abilities, indexed by general intelligence, is a major finding about the origins of individual differences in intelligence. Nonetheless, this finding seems to have had little impact in related fields such as cognitive neuroscience or experimental cognitive psychology. We suggest that part of the reason for this neglect is that these fields generally ignore individual differences.65,66 Another reason might be that the evidence for this finding rested largely on the twin design, for which there have always been concerns about some of its assumptions;6 we judge that this will change now that GCTA is beginning to confirm the twin results." --"Genetics and intelligence differences: five special findings", Plomin & Deary 2015
9. ^ "Top 10 Replicated Findings From Behavioral Genetics", Plomin et al 2016: "This research has primarily relied on the twin design in which the resemblance of identical and fraternal twins is compared and the adoption design in which the resemblance of relatives separated by adoption is compared. Although the twin and adoption designs have been criticized separately (Plomin et al., 2013), these two designs generally converge on the same conclusion despite being based on very different assumptions, which adds strength to these conclusions...GCTA underestimates genetic influence for several reasons and requires samples of several thousand individuals to reveal the tiny signal of chance genetic similarity from the noise of DNA differences across the genome (Vinkhuyzen, Wray, Yang, Goddard, & Visscher, 2013). Nonetheless, GCTA has consistently yielded evidence for significant genetic influence for cognitive abilities (Benyamin et al., 2014; Davies et al., 2015; St. Pourcain et al., 2014), psychopathology (L. K. Davis et al., 2013; Gaugler et al., 2014; Klei et al., 2012; Lubke et al., 2012, 2014; McGue et al., 2013; Ripke et al., 2013; Wray et al., 2014), personality (C. A. Rietveld, Cesarini, et al., 2013; Verweij et al., 2012; Vinkhuyzen et al., 2012), and substance use or drug dependence (Palmer et al., 2015; Vrieze, McGue, Miller, Hicks, & Iacono, 2013), thus supporting the results of twin and adoption studies."
10. ^
11. ^
12. ^ Visscher et al 2007, "Genome partitioning of genetic variation for height from 11,214 sibling pairs"
13. ^
14. ^
15. ^
16. ^ "GCTA: A Tool for Genome-wide Complex Trait Analysis", Yang et al 2011
17. ^ "Meta-analysis of the heritability of human traits based on fifty years of twin studies", Polderman et al 2015
18. ^ Barnes, J. C.; Wright, John Paul; Boutwell, Brian B.; Schwartz, Joseph A.; Connolly, Eric J.; Nedelec, Joseph L.; Beaver, Kevin M. (2014-11-01). "Demonstrating the Validity of Twin Research in Criminology". Criminology. 52 (4): 588–626. doi:10.1111/1745-9125.12049. ISSN 1745-9125.
19. ^ "GCTA will eventually provide direct DNA tests of quantitative genetic results based on twin and adoption studies. One problem is that many thousands of individuals are required to provide reliable estimates. Another problem is that more SNPs are needed than even the million SNPs genotyped on current SNP microarrays because there is much DNA variation not captured by these SNPs. As a result, GCTA cannot estimate all heritability, perhaps only about half of the heritability. The first reports of GCTA analyses estimate heritability to be about half the heritability estimates from twin and adoption studies for height (Lee, Wray, Goddard, & Visscher, 2011; Yang et al., 2010; Yang, Manolio, et al" 2011), and intelligence (Davies et al., 2011)." pg110, Behavioral Genetics, Plomin et al 2012
20. ^
21. ^ "Phenome-wide Heritability Analysis of the UK Biobank", Ge et al 2016
22. ^ Pasaniuc & Price 2016, "Dissecting the genetics of complex traits using summary association statistics"
23. ^ "LD Score Regression Distinguishes Confounding from Polygenicity in Genome-Wide Association Studies", Bulik-Sullivan et al 2015
24. ^
25. ^
26. ^ "Fast and Accurate Construction of Confidence Intervals for Heritability", Schweiger et al 2016
27. ^
28. ^
29. ^ "Deep Sequencing of 10,000 Human Genomes", Telenti 2015
30. ^
31. ^
32. ^
33. ^
34. ^ Methods of Meta-Analysis: Correcting Error and Bias in Research Findings, Hunter & Schmidt 2004
35. ^
36. ^ "Improved linear mixed models for genome-wide association studies", Listgarten et al 2012
37. ^
38. ^
39. ^ "Genome-wide efficient mixed-model analysis for association studies", Zhou & Stephens 2012
40. ^
41. ^ "Advanced Complex Trait Analysis", Gray et al 2012
42. ^
43. ^
44. ^
45. ^ "Mixed Models for Meta-Analysis and Sequencing", Bulik-Sullivan 2015
46. ^
47. ^ Speed et al 2016, "Re-evaluation of SNP heritability in complex human traits"
48. ^
49. ^
50. ^ "The Evolutionary Genetics of Personality", Penke et al 2007; "The Evolutionary Genetics of Personality Revisited", Penke & Jokela 2016