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# The distribution of common-variant effect sizes

## Abstract

The genetic effect-size distribution of a disease describes the number of risk variants, the range of their effect sizes and sample sizes that will be required to discover them. Accurate estimation has been a challenge. Here I propose Fourier Mixture Regression (FMR), validating that it accurately estimates real and simulated effect-size distributions. Applied to summary statistics for ten diseases (average $$N_{\textrm{eff}} = 169,000$$), FMR estimates that 100,000–1,000,000 cases will be required for genome-wide significant SNPs to explain 50% of SNP heritability. In such large studies, genome-wide significance becomes increasingly conservative, and less stringent thresholds achieve high true positive rates if confounding is controlled. Across traits, polygenicity varies, but the range of their effect sizes is similar. Compared with effect sizes in the top 10% of heritability, including most discovered thus far, those in the bottom 10–50% are orders of magnitude smaller and more numerous, spanning a large fraction of the genome.

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## Data availability

GWAS summary statistics are available at https://alkesgroup.broadinstitute.org/. Numerical results for Figs. 25 are reported in the Supplementary Tables.

## Code availability

Open-source software is available at https://github.com/lukejoconnor63. GENESIS14 software is available at https://github.com/yandorazhang/GENESIS.

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## Acknowledgements

I am grateful to A. Price, J. Ballard, A. Nadig, D. Weiner, O. Weissbrod, G. Getz, B. Neale and E. Lander for suggestions and discussions.

## Author information

Authors

### Contributions

L.J.O. wrote the manuscript.

### Corresponding author

Correspondence to Luke J. O’Connor.

## Ethics declarations

### Competing interests

The author declares no competing interests.

Peer review information Nature Genetics thanks Yan Zhang for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Performance of FMR in simulations at different sample sizes.

I show the true HDM (yellow), estimates for 10 individual simulation replicates (grey), the mean estimate across 20 replicates (blue), and the mean uncorrected estimate. The uncorrected estimate is obtained by running FMR without any correction for sampling variation in the GWAS summary statistics (see Supplementary Note). Data were simulated under a point-normal model with either 1% or 10% of SNPs having nonzero causal effect sizes.

### Extended Data Fig. 2 Calibration of FMR jackknife standard errors.

Simulations were performed under a normal mixture model with small-, medium- and large-effect SNPs (similar to Fig. 1d), at sample size N=460k, N=145k or N=50k. For different effect-size thresholds, I calculated the standard error of the proportion of random-effect heritability explained by SNPs with effect sizes less than that threshold. Bar plots show root-mean-squared jackknife standard errors (blue) and empirical standard errors (orange) based on 25 replicates. At large sample size (N=460k), standard errors were sometimes underestimated, probably due to the nonnegativity constraints in the regression. Caution is needed when making comparisons between the genetic architecture of different traits, as underestimated standard errors could lead to false-positive differences.

### Extended Data Fig. 3 Effect of changing the FMR sampling times and mixture components in simulations.

Simulations were performed under a normal mixture model with small-, medium- and large-effect SNPs (similar to Fig. 1d), at sample size N=460k. I specified a set of 17 mixture components ($$\sigma ^2 = [2^{ - 9},2^{ - 8}, \ldots 2^7]$$) and 17 sampling times ($$t_k = 1/\sigma _k$$), and performed simulations with various subsets of the respective values. In panels a-d, I use the same values of $$\sigma ^2$$ ($$\sigma _3^2,\sigma _4^2, \ldots ,\sigma _{15}^2$$, which correspond to the default FMR model) and various values of $$t$$. In panels e-f, I vary the values of $$\sigma ^2$$. In most cases, very similar results are obtained, except when too few sampling times are used (panel d). 25 replicates are performed (identical between the figure panels), the first 10 of which are plotted in grey. The mean and standard deviation across replicates are shown in blue. (a) $$\sigma ^2 = [\sigma _3^2,\sigma _4^2, \ldots ,\sigma _{15}^2]$$, $${\boldsymbol{t}} = [t_3,t_4, \ldots ,t_{15}]$$; (b) $$\sigma ^2 = [\sigma _3^2,\sigma _4^2, \ldots ,\sigma _{15}^2]$$, $${\boldsymbol{t}} = [t_1,t_2, \ldots ,t_{17}]$$; (c) $$\sigma ^2 = [\sigma _3^2,\sigma _4^2, \ldots ,\sigma _{15}^2]$$, $${\boldsymbol{t}} = [t_5,t_6, \ldots ,t_{11}]$$; (d) $$\sigma ^2 = [\sigma _3^2,\sigma _4^2, \ldots ,\sigma _{15}^2]$$, $${\boldsymbol{t}} = [t_5,t_7, \ldots ,t_{15}]$$; (e) $$\sigma _1^2,\sigma _2^2, \ldots ,\sigma _{17}^2$$, $$t_1,t_2, \ldots ,t_{17}$$; (f) $$\sigma ^2 = \left[ {\sigma _5^2,\sigma _6^2, \ldots ,\sigma _{13}^2} \right]$$, $${\boldsymbol{t}} = [t_5,t_6, \ldots ,t_{13}]$$. I recommend using 13 mixture components and 13 sampling times, even though a smaller number may suffice (Extended Data Fig. 3f).

### Extended Data Fig. 4 Observed number of genome-wide significant SNPs and proportion of heritability explained at N=145k vs 460k.

For numerical results, see Supplementary Table 2.

### Extended Data Fig. 5 Predicted vs. observed heritability explained by genome-wide significant SNPs at different significance thresholds.

%h2GWAS was predicted using interim-release UK Biobank summary statistics (maximum N=145k) and evaluated in the full release (maximum N=460k). Squared correlations between predicted and observed values were 0.94, 0.95, 0.93, and 0.88 in panels a-d respectively. Lower r2 at $$\chi ^2 > 1000$$(panel d) could result from the small number of loci with large effect sizes, which may increase the sampling variance of both the FMR predictions and the observed values. In panel d, the data points for several traits are superimposed near the origin. For numerical results, see Supplementary Table 2.

### Extended Data Fig. 6 Predicted vs. observed number of genome-wide significant SNPs at different significance thresholds.

MGWAS was predicted using interim-release UK Biobank summary statistics (maximum N=145k) and evaluated in the full release (maximum N=460k). Squared correlations between predicted and observed values were 0.92, 0.97, 0.92 and 0.91 in panels a-d respectively. In panel d, the data points for several points are superimposed near the origin. For numerical results, see Supplementary Table 2.

### Extended Data Fig. 7 Consistency of FMR predictions at different sample sizes.

%h2GWAS and MGWAS were predicted for 22 traits based on N=145k vs. N=460k summary statistics, with target sample size equal to 460k, 2M or 10M. Predictions assume that the LD score regression intercept will be equal to what was observed at N=145k for both sets of estimates. Numerical results are presented in Supplementary Table 2.

### Extended Data Fig. 8 Performance of GENESIS vs. FMR predictions in UK Biobank.

FMR and GENESIS were applied to interim-release UK Biobank summary statistics (maximum N=145k) for 22 traits in order to predict the results of the full release (maximum N=460k). Numerical results are presented in Supplementary Table 1.

### Extended Data Fig. 9 Consistency of GENESIS predictions at different sample sizes.

%h2GWAS and MGWAS were predicted for 19 traits (Supplementary Table 3) based on N=145k vs. N=460k summary statistics, with target sample size equal to 460k, 2M or 10M. At N=460k, predictions of large-N %h2GWAS were slightly smaller (panels b-c), while predictions of $$M_{{\mathrm{GWAS}}}$$ were slightly larger (panel f). This difference could result from a less severe form of the power-dependent bias that is known to affect the point-normal (2-component) model when it is misspecified: as sample size increases, SNPs with smaller effect sizes become detectable, and estimates shift toward a larger number of causal SNPs with smaller effect sizes. (This only occurs when the model is misspecified, with a larger-than-expected number of small-effect SNPs). The 3-component model ameliorates this bias by including a small-effect heritability component even at small sample sizes. However, if this model too is misspecified (for example when there is a mixture of small-, medium- and large-effect SNPs), then it would be affected in the same way as the point-normal model, to a lesser degree. Numerical results are presented in Supplementary Table 3. The same analysis using FMR is presented in Extended Data Fig. 7.

### Extended Data Fig. 10 Estimated HDM of height using summary statistics from GIANT vs. UK Biobank.

If results were biased by population stratification, the bottom-left portion of the curve (corresponding to small-effect SNPs) would be inflated for estimates based on GIANT.

## Supplementary information

### Supplementary Information

Supplementary Tables 1, 4, 6 and 8, Figs. 1–7 and Note

### Supplementary Tables

Supplementary Tables 2, 3, 5, 7 and 9–12

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O’Connor, L.J. The distribution of common-variant effect sizes. Nat Genet 53, 1243–1249 (2021). https://doi.org/10.1038/s41588-021-00901-3

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