Lucie Troubat

Deniz Fettahoglu

Léo Henches

Hugues Aschard

Hanna Julienne
email: hanna.julienne@pasteur.fr

Jonas Lefevre-Daumal

Arthur Frouin

Yuka Suzuki

Multi-trait GWAS for diverse ancestries: Mapping the knowledge gap

Keywords: statistical genetics, GWAS, multi-trait GWAS, diverse ancestries

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Background

After 15 years of intensive GWAS analyses, investigators are reaping tangible translational benefits such as drug repurposing opportunity, and actionable polygenic risk scores [START_REF] Abdellaoui | 15 years of GWAS discovery: Realizing the promise[END_REF]. Yet, ~95% of samples analyzed in GWAS are from European ancestry [START_REF] Mills | The GWAS Diversity Monitor tracks diversity by disease in real time[END_REF]. Genetic insights gained on European ancestry are only partially transferrable to other ancestries: polygenic risk scores have lessened accuracy [START_REF] Ding | Polygenic scoring accuracy varies across the genetic ancestry continuum in all human populations[END_REF][START_REF] Privé | Portability of 245 polygenic scores when derived from the UK Biobank and applied to 9 ancestry groups from the same cohort[END_REF][START_REF] Wang | Theoretical and empirical quantification of the accuracy of polygenic scores in ancestry divergent populations[END_REF], and genetic testing yields more often ambiguous results in non-European ancestries [START_REF] Manrai | Genetic Misdiagnoses and the Potential for Health Disparities[END_REF]. Altogether, this gap in data and analyses is a major bias in the existing research that can result in increased health disparities [START_REF] Martin | Clinical use of current polygenic risk scores may exacerbate health disparities[END_REF][START_REF] Pereira | African genetic diversity and adaptation inform a precision medicine agenda[END_REF]. The genetic community is increasingly aware of this issue and strongly recommend to include more individuals from non-European ancestries in GWAS studies [START_REF] Mills | The GWAS Diversity Monitor tracks diversity by disease in real time[END_REF][START_REF] Martin | Clinical use of current polygenic risk scores may exacerbate health disparities[END_REF][START_REF] Atkinson | Cross-ancestry genomic research: time to close the gap[END_REF][START_REF] Manolio | Using the Data We Have: Improving Diversity in Genomic Research[END_REF][START_REF] Peterson | Genome-wide Association Studies in Ancestrally Diverse Populations: Opportunities, Methods, Pitfalls, and Recommendations[END_REF][12]. Increasing the sample size coverage of non-European ancestries has been identified as one of the most important objectives for the genetic community in the coming years [START_REF] Abdellaoui | 15 years of GWAS discovery: Realizing the promise[END_REF].

Increasing the sampling of non-European ancestries is paramount, but computational method development is also an area of improvement that is less often discussed [START_REF] Manolio | Using the Data We Have: Improving Diversity in Genomic Research[END_REF][START_REF] Peterson | Genome-wide Association Studies in Ancestrally Diverse Populations: Opportunities, Methods, Pitfalls, and Recommendations[END_REF]. Many statistical genetics methods are developed and tested primarily on the European population.

To ascertain this bias, we surveyed 25 recent methods in statistical genetics (Table S1).

Amongst these methods 20 conducted an application on human real data, all included European data, and 13 focused on European data exclusively. When developed on European data exclusively, using these computational methods on European ancestry becomes all the more easy: input files (e.g. specific formats of reference panels) are often provided, performances and potential pitfalls are known, and protocols (e.g. parameters) are specific. In comparison, applying these methods to diverse ancestries can require substantial additional work and can be perceived as riskier since potential performance discrepancies are undocumented.

This European by default mode creates a snowballing effect where secondary analysis increases further the genetic knowledge gap between European and non-European ancestries. Furthermore, the analysis of non-European do not only address a public health gap, but also offer opportunities and several recent publications illustrate the benefit of methodological development for the analysis of diverse populations in various applications such as: polygenic risk scores [START_REF] Ruan | Improving polygenic prediction in ancestrally diverse populations[END_REF], association testing [START_REF] Smith | Enrichment analyses identify shared associations for 25 quantitative traits in over 600,000 individuals from seven diverse ancestries[END_REF], and TWAS [START_REF] Li | METRO: Multi-ancestry transcriptome-wide association studies for powerful gene-trait association detection[END_REF].

In this context, we tested if the properties of our previously developed JASS (Joint Analysis of Summary Statistics) pipelineits ability to detect biologically relevant associations missed by univariate GWAS [START_REF] Julienne | Multitrait GWAS to connect disease variants and biological mechanisms[END_REF][START_REF] Julienne | JASS: command line and web interface for the joint analysis of GWAS results[END_REF] would generalize to non-European ancestries. To this end, we conducted the joint GWAS of 19 hematological traits and glycemic traits retrieved from two large trans ancestry studies [START_REF] Chen | The transancestral genomic architecture of glycemic traits[END_REF][START_REF] Chen | Trans-ethnic and Ancestry-Specific Blood-Cell Genetics in 746,667 Individuals from 5 Global Populations[END_REF]. We applied our pipeline to five ancestries (European (EUR), Admixed American (AMR), African (AFR), East Asian (EAS), South-East Asian (SAS)), and investigated the performance and robustness of each step of the pipeline across ancestries. For the given input data, the pipeline generalized properly to three out of the four non-European ancestries: AMR, AFR, and EAS. We detected 367 new genome-wide significant associations in non-European populations (15 in AMR, 72 in AFR and 280 in EAS).

We then report to what extent multi-trait testing increases the replication of European associated loci in non-European ancestry. Finally, we investigate the conservation of pleiotropy across ancestries, and tie the observed differences to known evolutionary divergences. Altogether these analyses suggest that multi-trait GWAS methods can be valuable to take advantage of modest size cohorts and to narrow the genetic knowledge gap between European and non-European populations.

Methods

Survey of ancestries used in recently published methods in statistical genetics

To ascertain a potential bias in ancestries used to develop and test recent statistical genetic tools, we surveyed 25 publications. We looked up ancestries studied in: i) methodology focused publications of the American Journal of Human Genetics published after January 2023, ii) Bioinformatics publications in the "Genetics and Population Analysis" section of the March, April and May issues, and iii) the first page of results of the Google Scholar query "gwas summary statistics methods" ordered by relevance and published after 2021 (query was performed by Dr. Julienne on May 17 th 2023). This survey does not pretend to comprehensively ascertain the ancestry bias in method development in statistical genetics. Its aim is rather to provide a snapshot of common practices of the field at the time of this study.

following inline options: --l2 --ld-wind-kb 1000. LD matrices for imputation were generated using the generate_genome_matrices function from the RAISS python package (see https://statistical-genetics.pages.pasteur.fr/raiss/#precomputation-of-ld-correlation).

RAISS Imputation parameters were chosen to ensure high accuracy for all traits and ancestries as described in the documentation (https://statisticalgenetics.pages.pasteur.fr/raiss/#optimizing-raiss-parameters-for-your-data). We systematically assessed RAISS accuracy for all traits and ancestries on chr22. We set RAISS key parameters to --R2-threshold 0.6, --minimum-ld 5 and --eigen-threshold 0.05 and report corresponding accuracies in Table S3. The correlations between the imputed and initial values range from 0.93 to 0.99.

Links towards the JASS pipeline and associated computational tools are provided in the "Availability of data and material" section.

LD independent regions

For each ancestry, we computed LD independent regions using the R package bigsnpr [START_REF] Privé | Efficient analysis of large-scale genomewide data with two R packages: bigstatsr and bigsnpr[END_REF] designed for massive analysis of SNP arrays. The function snp_ldsplit splits a correlation matrix in quasi-independent blocks using dynamic programming [START_REF] Privé | Optimal linkage disequilibrium splitting[END_REF]. We optimized settings to reach a tradeoff between the sum of squared correlations outside the blocks (cost) and the number of regions found in a chromosome (n_block). On the chromosome 22 of each ancestry, we have varied the minimum number of variants in each block (min_size) between 250 and 10 000, the maximum number of variants in each block (max_size) between 1000 and 50 000, and the threshold under which squared correlations are ignored (thr_r2) between 0.005 and 0.4. By analyzing the results, we selected thr_r2 = 0.05, min_size = 4000, max_size = 5000, max_K = 400 as it provides a balance between the cost and the number of regions for all chromosomes of all ancestry (Figure S1).

The omnibus multi-trait test and contrast with univariate GWAS

To perform the omnibus multi-trait tests, JASS relies on a theoretical framework and a software architecture described in [START_REF] Julienne | Multitrait GWAS to connect disease variants and biological mechanisms[END_REF][START_REF] Julienne | JASS: command line and web interface for the joint analysis of GWAS results[END_REF]. Briefly, assuming an infinitesimal model and under the null hypothesis of no genetic effects, the statistic of the omnibus test (equation 1) follows a distribution where denotes the number of traits considered in the study. The omnibus test statistic is defined as:

(equation 1)
where is the vector of the z-score across traits and is the residual z-score covariance under the null hypothesis. As derived in [START_REF] Bulik-Sullivan | An atlas of genetic correlations across human diseases and traits[END_REF], for a pair of traits i and j the expected covariance under the null hypothesis is equal to , where is the total covariance between traits and , is the number of samples shared between studies and , and and are the sample sizes of studies and .

Validation of the estimation of by the LDscore regression in non-European ancestries

To ensure that the parameter of the omnibus test is correctly estimated in non-European ancestries, we compared the direct estimate of computed in UK Biobank and an indirect estimate derived from the LDscore regression applied to GWAS summary statistics. We used the ancestry segmentation presented in [START_REF] Privé | Portability of 245 polygenic scores when derived from the UK Biobank and applied to 9 ancestry groups from the same cohort[END_REF] and worked with the Indian (equivalent to SAS), Nigerian (equivalent to AFR), and Caribbean (equivalent to AFR with admixture with EUR) individuals. We restricted our analysis to complete cases (Sample sizes of 5565, 3427, and 1959 for Indian, Nigerian, and Caribbean individuals respectively) for hematological traits so the term

𝑛 𝑠 √ 𝑛 1 𝑛 2 equals 1.
For each population, we derived a direct estimate of by computing the Pearson correlation between hematological traits in UK BioBank. We then derived LDscore estimates by: i) computing univariate GWAS on each trait using plink, ii) applying the JASS pipeline on the resulting summary statistics. To compute the LDscore for each variant, we used the SAS 1000G reference panel for the Indian population and the AFR 1000G reference panel for the Caribbean and Nigerian populations (Availability of data and material).

Characterization of new genetic associations

JASS identifies significant genetic variants using a two step procedure. First, large LD independent regions (see paragraph above for region computation) are scanned for new associations. For each LD independent region, the minimum association p-value for the univariate test across traits and for the joint test are retrieved. In regions containing significant association for JASS, we searched for potential additional lead SNPs using the PLINK clump function [START_REF] Purcell | PLINK: A Tool Set for Whole-Genome Association and Population-Based Linkage Analyses[END_REF] with the following parameters: a significance threshold equal or greater than 5.10 - 8 (--clump-p1), a significance threshold for other SNPs in the clump equal or greater than 5.10 - 4 (--clump-p2) and a linkage disequilibrium of =0.2 (--clump-r2). We filtered out isolated lead SNPs (clumps containing only the lead SNPs) to remove poor quality associations. A lead SNPs is considered as a new association if the p-value of the omnibus test is significant while the minimum univariate test p-value across traits is not. Note that we choose to apply no multi testing correction to the minimum univariate p-value as we wish to contrast JASS results with all associations reported previously in the literature.

Trans ancestry meta analysis

We performed a trans ancestry multi-trait GWAS by adapting the omnibus test to the trans ancestry setting. Under the null hypothesis, the omnibus statistic follows a distribution with degree of freedom where denotes the number of traits considered. As a cohort of a given ancestry is independent from cohorts of other ancestries, the sum of the omnibus statistics across all ancestries, , follows a chi-square with degree of freedom.

For a given SNPs:

where is the vector of z-scores across traits, is the ancestry considered and is the residual covariance between Z-score under the null for the ancestry .

To contrast the multi-trait trans ancestry analysis against univariate meta-analysis we performed a meta-analysis on Z-score (assuming fixed genetic effects) and retrieved the minimum p-value across traits.

For a given trait, we computed the trans ancestry meta analysis Z-score as:

As LD independent regions cannot be defined across ancestries, we define regions as 1Mbp blocks and select one lead SNPs per block as the one with a minimum p-value.

Simulating a smaller sample size for European data

To simulate a diminished sample size for the European data, we scaled down Z-scores of lead SNPs with the following formula:

Once scaled down, new p-values for the univariate test and the omnibus test were computed.

We varied the simulated sample size from a fraction of 0 to 100% of the initial sample by increments of 1%.

Functional annotation

We mapped lead SNPs to genes through positional mapping and eQTLs (expression Quantitative Trait Loci) using Functional Mapping and Annotation of Genome-Wide Association Studies (FUMA GWAS) [START_REF] Watanabe | Functional mapping and annotation of genetic associations with FUMA[END_REF]. For position based mapping, SNPs were mapped to their nearest gene if the distance was under 10kb. For eQTL mapping, we selected blood and To compute the genomic distance between novel associations mapped to the same genes in the EUR and EAS ancestries, we retrieved genes linked by significant eQTLs to SNPs discovered by JASS exclusively in each ancestry. Then, using the intersection of EAS and EUR genes, we computed the genomic distance between the linked SNP in EAS and the linked SNP in EUR for each gene. If a gene was linked to multiple SNPs, the pair of SNPs with the smallest distance was retained.

For the investigation of newly detected genes in African ancestry, we focused on genes tied to SNPs discovered by JASS exclusively and belonging to at least one GeneOntology (GO)

set. Then, we queried each of this gene function on The Human Gene Database GeneCards (www.genecards.org, [START_REF] Safran | The GeneCards Suite[END_REF]).

Results

Overview of the study Our analysis focused on 15 hematological and four glycemic quantitative traits in five superpopulations (Figure 1). We retrieve summary statistics from two large trans ancestry meta-analyses each focusing either on hematological traits or on glycemic traits. Using a single source for summary statistics by group of traits allows to minimize potential discrepancies in data preprocessing across ancestries. It also allows for an increased sample overlap between traits, which can be beneficial for the statistical power of the multi-trait test [START_REF] Julienne | Multitrait GWAS to connect disease variants and biological mechanisms[END_REF][START_REF] Suzuki | Trait selection strategy in multi-trait GWAS: Boosting SNPs discoverability[END_REF][START_REF] Porter | Multivariate simulation framework reveals performance of multitrait GWAS methods[END_REF]. Sample size varies widely between populations: GWAS on hematological phenotypes studied 563,946 individuals from EUR ancestry, 151,807 for the EAS ancestry, 9,368 for AMR ancestry, 15,171 for AFR ancestry, and 8189 for SAS ancestry (Table S2).

While the sample size across hematological traits is relatively homogeneous, the sample size in glycemic traits presents large discrepancies. For instance, in the AFR population, ten times more individuals were analyzed for fasting glucose phenotype than for the two-hour glucose tolerance test. We applied a previously developed pipeline [START_REF] Julienne | JASS: command line and web interface for the joint analysis of GWAS results[END_REF] to: i) curate and harmonize GWAS summary statistics, ii) impute missing statistics within each study using the RAISS software (Robust and Accurate Imputation from Summary Statistics, [START_REF] Julienne | RAISS: robust and accurate imputation from summary statistics[END_REF]), and iii) compute multi-trait GWAS. Entry files for the Nextflow pipeline were generated and made publicly available to facilitate the analysis of summary statistics from all ancestries (Supplementary Note 1 and Availability of data and material).

The median increase in the number of variants across traits after imputation is 6%, 13%, 22%, 21% and 27% for AFR, AMR, EUR, SAS and EAS ancestries, respectively (Table S2 and

Figure S2

). The imputation method relies on the linkage disequilibrium (LD) structure; the number of variant increases seems to reflect the differences in LD block length across ancestry [START_REF] Pereira | African genetic diversity and adaptation inform a precision medicine agenda[END_REF][START_REF] Auton | A global reference for human genetic variation[END_REF]. Imputation performances and the impact of imputation on the number of significant loci with the univariate test are congruent with our previous observations [START_REF] Julienne | RAISS: robust and accurate imputation from summary statistics[END_REF] (Table S3). All traits available by ancestry were analyzed jointly using an omnibus test implemented in JASS [START_REF] Julienne | Multitrait GWAS to connect disease variants and biological mechanisms[END_REF][START_REF] Julienne | JASS: command line and web interface for the joint analysis of GWAS results[END_REF] (applied to association z-scores), grouping them by hematological traits and glycemic traits. We used the LDscore regression [START_REF] Bulik-Sullivan | An atlas of genetic correlations across human diseases and traits[END_REF] to estimate the covariance of z-scores under the null hypothesis (Table S4)a required input for the omnibus test (Methods)along with heritabilities, and genetic correlation across all traits (Supplementary note 2, Figure S3 and Table S5). The covariance of z-scores under the null ( ) hypothesis is a critical parameter for the validity of the omnibus test. We previously validated its estimation by the LDscore regression in the European population [START_REF] Julienne | Multitrait GWAS to connect disease variants and biological mechanisms[END_REF]. We reproduce a similar validation in non-European ancestries present in UK biobank (namely the Indian, Nigerian and Caribbean populations [START_REF] Privé | Portability of 245 polygenic scores when derived from the UK Biobank and applied to 9 ancestry groups from the same cohort[END_REF]). Briefly, when all samples are shared between phenotypes, is equal to the Pearson correlation matrix of phenotypes. If one has access to individual level data, the Pearson correlation matrix can be compared to its estimation by the LDscore regression (Methods). Overall, LDscore estimates were accurate (mean absolute error = 0.03) even in the Caribbean population, which is admixed between the European and African ancestries [START_REF] Privé | Portability of 245 polygenic scores when derived from the UK Biobank and applied to 9 ancestry groups from the same cohort[END_REF] (Figure S4).

Overall, all analyses passed standard quality controls except for the SAS ancestry. More precisely, we observed an inflation of the joint test statistics on hematological traits after imputation. This limited robustness seems to arise from the higher collinearity of the residual covariance matrix for this ancestry. We discuss this issue in depth in the supplementary material (Supplementary note 3 and Figure S5). Because we cannot rule out additional sources of biases, we decided to exclude both the multi-trait GWAS on hematological traits and on glycemic traits for the SAS ancestry from subsequent analyses.

Using the intersection of available SNPs across ancestries, we first performed a trans-ancestry multi-trait GWAS by summing the omnibus test statistics across ancestries and by contrasting its results with a univariate trans-ancestry meta analysis (see Methods). For glycemic traits, the set of SNPs available after performing the intersection was too small (#Nsnps = 8727).

Hence, the trans-ancestry GWAS was performed only on hematological traits. Focusing on hematological traits, we compared association positioning and features across ancestries, and systematically studied to what extent loci are shared across ancestries. We assessed the biological relevance of significant associations by annotating them with Functional Mapping and Annotation of Genome-Wide Association Studies (FUMA GWAS, [START_REF] Watanabe | Functional mapping and annotation of genetic associations with FUMA[END_REF]) and The Human Gene Database GeneCards (www.genecards.org, [START_REF] Safran | The GeneCards Suite[END_REF]). Associations that are significant only with the joint test (new associations) are distinguished from associations that are significant with univariate tests.

New genetic associations

We report 2,648 new independent associations (Table 1, Tables S6-S10, Figures S6-S10).

Genomic inflation factors are congruent with an adequate control of the type 1 error ranging from 0.99 for the AMR ancestry to 1.23 for the European ancestry (Figures S11-S13). New associations are LD-independent associations significant exclusively for the joint test (Methods). The univariate p-value is defined as the minimum p-value of univariate tests across traits. Although this choice leads to an inflation of the univariate p-value, it allows for the inclusion of all associations previously reported in the literature and provides a stringent definition of new associations detected by JASS. For the TRANS ancestry analysis, we derived a TRANS ancestry multi-trait association statistic by calculating the sum of the omnibus test statistics across ancestries. Since samples are independent from one ancestry to another, the obtained statistic follows a with degrees of freedom equal to the number of ancestries times the number of traits (Methods). To contrast this TRANS ancestry multi-trait analysis with a univariate approach, we computed a standard meta-analysis for each trait, assuming fixed genetic effects across ancestries (Methods).

New associations represent 10%, 17%, 5%, 13%, and 2.8% of associations detected by the joint test in the EUR, EAS, AFR, AMR and TRANS GWAS, respectively. The vast majority of associations are associated with hematological traits (97%), following the same trend as univariate association (97.5% of univariate associations are also associated with hematological traits). JASS identified new genetic associations in modest sample size cohorts (i.e 9,368 individuals for AMR) on hematological traits emphasizing the interest in using non-European data when conducting secondary analyses even when they are seemingly underpowered [START_REF] Manolio | Using the Data We Have: Improving Diversity in Genomic Research[END_REF]. Concerning glycemic traits, the very modest number of new signals for AFR and AMR might be explained by the low coverage of glycemic summary statistics for these ancestries (Table S2). Indeed, for these two ancestries, the approximate number of SNPs available for glycemic traits is 2 million whereas more than 8 million SNPs were available for hematological traits. Here, we chose to report all associations below the standard p-value threshold of 5e10 -8 for completeness. However, a stricter choice would have been to apply a

Bonferroni correction taking into account the number of multi-trait GWAS performed in the current report [START_REF] Peterson | Genome-wide Association Studies in Ancestrally Diverse Populations: Opportunities, Methods, Pitfalls, and Recommendations[END_REF]. When applying a Bonferroni correction, the number of new associations is 41, 6, 169, 1479 and 36 in the AFR, AMR, EAS, EUR and TRANS ancestry analyses respectively.

Focusing on hematological trait associations, 38% of new associations detected in the African population arise from absent or rare variants (minor allele frequency [MAF] < 1%) in other studied populations (Table S11). In contrast, 4% and none of the new associations detected in the East Asian or European populations were rare or absent in all other populations. The African population's unique allele diversity appears to be a key factor in explaining the number of associations discovered in this population.

Shared loci across ancestries for hematological traits

We compared association loci across populations to assess the extent of their potential overlap across ancestries. To this end, the genome was segmented into non-overlapping regions of 1 Mbp that were considered detected in one ancestry if the region contained at least a significant association (joint or univariate). We represented (Figure 2A) the overlap of loci across populations with an UpSetR plotan improved Venn diagram for the visualization of numerous set intersections. When including several ancestries, we considered the intersection as newly discovered if at least one of the genetic associations was found by the joint test only (Figure 2A). S15). When the sample sizes were similar in the pair of ancestries, the fraction of loci specific to European ancestry ranged from 0 (when compared to AMR) to ~15%

(when compared to EAS). This observation is in line with most genetic associations sharing the same region within different populations as observed in previous studies [START_REF] Carlson | Generalization and Dilution of Association Results from European GWAS in Populations of Non-European Ancestry: The PAGE Study[END_REF][START_REF] Ishigaki | Largescale genome-wide association study in a Japanese population identifies novel susceptibility loci across different diseases[END_REF][START_REF] Wojcik | Genetic analyses of diverse populations improves discovery for complex traits[END_REF].

Performing the multi-trait GWAS increased the proportion of significant associations detected in the trans-ancestry and European GWAS that were also found in non-European ancestries by 15% (Figure 2A

Similarity of the multi-trait signal between ancestry pairs

To quantify whether ancestries had similar multi-trait association patterns in overlapping significant loci, we compared multi-trait vectors of genetic effects in shared loci across all pairs of ancestry, measured as the squared correlation (R²) of genetic effects across traits (Figures 2B-C). Multi-trait effect size vectors were strikingly similar between the EUR and EAS superpopulations with 75% of SNP pairs in shared loci having a correlation across traits higher than 0.8 (Figure 2C, Figure S14). At first glance, the similarity between the EUR and AFR superpopulations seemed lessened. However most of the low correlation SNPs pairs (80% of the shared loci with a rho <0.8) were located on chromosome 1 near the ACKR1 (Figure 3) locus (Chromosome 1: 159,204,875-159,206,500 forward strand), which is known to be adaptive to overcome the p.vivax induced malaria [START_REF] Pereira | African genetic diversity and adaptation inform a precision medicine agenda[END_REF][START_REF] Mcmanus | Population genetic analysis of the DARC locus (Duffy) reveals adaptation from standing variation associated with malaria resistance in humans[END_REF]. After accounting for the ACKR1 locus the median of correlation between EUR and AFR significant loci was 0.87. Overall, except for the ACKR1 locus, multi-trait patterns were shared between ancestries for hematological traits.

Functional annotation for hematological trait GWAS

To validate the biological relevance of associations found by the joint test, and in particular novel associations, we focused our functional analysis on hematological traits. Indeed, for hematological traits, the joint test detected enough associations in all ancestries to allow for a comparison of the relevance of the functional enrichment across ancestries. Briefly, we mapped lead SNPs using a combination of positional and eQTLs mapping (Table S12 reports the number of genes mapped by each method). Lead SNPs were mapped to their nearest genes if their distance was less than 10kb. For eQTLs mapping, we selected eQTLs detected in the immune system and blood cells (Methods), which corresponds to the trait assessed in the GWAS. To characterize genes found by the annotation step, we looked in which tissue the genes were differentially expressed (Figure S17). For each population, we represented the five first tissues with the highest p-value for upregulated genes (Figure 4A). Blood was significantly enriched for the four populations.

To further analyze the annotation of the associations discovered by the joint test, we selected genes linked to a SNP significant for the joint test. We then selected genes annotated in GO (Gene Ontology) term sets [START_REF]The Gene Ontology Resource: 20 years and still GOing strong[END_REF] (Methods, Tables S13-S16). For European ancestry, 3 342 genes were identified, 388 genes for the EAS population and 49 genes for the AFR population.

For the Admixed American population, there were no genes related to the genetic associations found by the joint test only. We then investigated whether newly detected signals corresponded to the same regulatory regions across ancestries by computing the position shift between European and East Asian populations for SNPs regulating the same gene. (i.e., SNPs that are eQTLs for the same gene, Methods, Table S17). Here, we focused on the European and East Asian ancestries because the two ancestries share enough significant loci (433, Figure 1A) to assess whether these loci systematically regulate the same genes. A total of 177 genes were shared in both ancestries (i.e., associated with a lead SNP through a significant eQTL). When there were several SNPs in one ancestry for one gene, we kept those with the closest pair between populations. The median distance was 86 kb and for 90% of the SNPs, the distance was under 481 kbp (see

Figure 4B

). This suggests that the same regulatory regions are involved in gene regulation in both populations.

Because of the relatively high proportion of newly associated variants specific to the African ancestry population, we conducted a targeted functional analysis to validate genes associated with newly identified SNPs in that population. 49 genes were associated with a significant lead SNP in the AFR population. Each of those genes was looked up in The Human Gene Database GeneCards (www.genecards.org, [START_REF] Safran | The GeneCards Suite[END_REF]) (Table S17). Five genes mapped to hemoglobin subunits (HBG1, HBG2, HBE1, HBB and HBD) involved in beta thalassemia and fetal hemoglobin quantitative trait locus 1 diseases. Eight genes mapped to other blood-related pathways like Tubulin Beta 1 Class VI (TUBB1) expressed in platelets and CD36 molecule (CD36), a gene protein located at the platelet surface. Five genes mapped to immune systemrelated pathways. 20 genes mapped to the olfactory receptor family. This overrepresentation of Olfactory receptors reflects probably more the close proximity of these genes along the genome rather than a genuine functional enrichment. Indeed, 15 SNPs mapping to an olfactory receptor family gene are located in a 1 Mbp region (chr11:4.449.477 bp to chr11:5.539.485 bp). For African ancestry, 37% of associations that were discovered by the joint test and annotated with GO term genes mapped to relevant blood pathways.

Discussion

In this study, we applied JASS, a multi-trait analysis pipeline previously developed and tested on European ancestry to four non-European ancestries. On three (AMR, EAS and AFR) out of four ancestries, our pipeline generalized without barriers and allowed for the detection of new associations while controlling for the genomic inflation factor. We also designed a Transancestry extension of the omnibus test. Altogether, single population and trans-ancestry, omnibus tests identified 367 new associations in non-European ancestry. For hematological traits, these new associations were linked with genes upregulated in blood, a relevant tissue, in all ancestries. Notably for AFR ancestry, a fine analysis of new associations pointed toward blood and immune pathways in 37% of annotated genes. Overall, this study highlights that imputation followed by multi-trait testing can be a valuable tool set for non-European ancestry analysis.

Our extension of the omnibus test to perform trans-ancestry GWAS is an elegant and computationally efficient way to perform trans-ancestry GWAS without assuming homogeneous effects across ancestries. Indeed, the trans-ancestry omnibus test relies on the independence of the chi-squares derived in each ancestry to aggregate them in an unique chisquare test. Hence, in each ancestry, the genetic signals can deviate from the null hypothesis in different directions and contribute to the test statistic. However, our trans-ancestry analysis detected only a handful of new signals, suggesting that the additional degree of freedom in the expected distribution under the null hypothesis dilutes the signal and leads to a lack of statistical power. Incorporating LD local structure of each ancestry into the multi-trait test, as has been done for univariate GWAS [START_REF] Xiao | Leveraging the local genetic structure for trans-ancestry association mapping[END_REF], might enable us to further improve our trans-ancestry multi-trait tests.

Further improvements of JASS could include a better adaptation to admixed populations.

Here, we validated the use of the JASS pipeline for admixed ancestries by showing that the omnibus test is not inflated for Admixed Americans (Figures S10 andS11), and by assessing the accuracy of our estimate of the distribution of Z-scores under the null in the Caribbean population (Figure S4), which is admixed between the African and European ancestries [START_REF] Privé | Portability of 245 polygenic scores when derived from the UK Biobank and applied to 9 ancestry groups from the same cohort[END_REF].

Although our current approach appears valid and was able to detect 15 new associations in the Admixed Americans, it may not be optimal in terms of statistical power or may be biased for higher degrees of admixture. We recommend a careful inspection of the genomic inflation factor when using JASS on admixed populations. Further development of the method may allow JASS to leverage local ancestry in admixed populations to boost GWAS statistical power as previously done on individual level data [START_REF] Atkinson | Tractor uses local ancestry to enable the inclusion of admixed individuals in GWAS and to boost power[END_REF].

We encountered an unforeseen pitfall with the SAS ancestry: the imputation led to an implausible increase in the number of associations for the joint test. After an investigation, this inflation was due to collinearity of traits investigated coupled to a high sample overlap. This specific set of conditions leads to an increased condition number for covariance under the null, or more plainly said, to a null hypothesis that lacks robustness (Supplementary Note 3).

Hence, we do not recommend using the omnibus test in this specific setting. We will further investigate the robustness of the omnibus test on SAS data to confirm that this inflation is not related to other specificities of this population.

A large fraction of new genetic associations matched significant eQTL: 80% (1729/2173) for the European, 60% (163/271) for the East Asian and 82% (59/72) for the African population.

However, for the AMR population, functional annotation was less informative suggesting a lower multi-trait or eQTLs annotation signal quality (only 1.9% of sampled individuals in GTEx self-reported as hispanic [START_REF] The | The GTEx Consortium atlas of genetic regulatory effects across human tissues[END_REF]). Only six significant SNPs out of 57 were eQTLs for a gene.

This observation underlines the importance of representing a diverse population not only in GWAS sampling but also in other genomic assays.

Our assessment of multi-trait genetic signal similarity across ancestries demonstrates that the omnibus test allowed for an increased replication of loci detected in European ancestry in non-European ancestries. We mapped associated SNPs to genes through eQTLs and observed that genes share the same regulatory regions in the EUR and EAS ancestries. We reported a systematic comparison of multi-trait genetic signals in shared loci. We observe strikingly similar multi-trait patterns (with the median of the genetic signal squared correlation at variant pairs being above 0.8) except in ACKR1 locus which is tied to an adaptation specific to African ancestries protective for p.vivax parasites.

Our observations suggest an overall stability of the sign of genetic effects except in the ACKR1 loci and are consistent with previous reports that genetic effects are consistent across ancestries [START_REF] Carlson | Generalization and Dilution of Association Results from European GWAS in Populations of Non-European Ancestry: The PAGE Study[END_REF][START_REF] Gharahkhani | Genome-wide meta-analysis identifies 127 open-angle glaucoma loci with consistent effect across ancestries[END_REF][START_REF] Ntzani | Consistency of genome-wide associations across major ancestral groups[END_REF]. Other reports nuance the extent of this similarity by noting that genetic correlations across ancestries are often significantly lower than 1 [12, [START_REF] Brown | Transethnic Genetic-Correlation Estimates from Summary Statistics[END_REF][START_REF] Giannakopoulou | The Genetic Architecture of Depression in Individuals of East Asian Ancestry: A Genome-Wide Association Study[END_REF][START_REF] Matoba | GWAS of smoking behaviour in 165,436 Japanese people reveals seven new loci and shared genetic architecture[END_REF], and loci replication can be lower for specific traits with culturally dependent definition, such as depression (only 11% of European lead SNPs for depression are replicated East Asian ancestry [START_REF] Giannakopoulou | The Genetic Architecture of Depression in Individuals of East Asian Ancestry: A Genome-Wide Association Study[END_REF]). Hematological traitscontinuous traits with an objective definitioncan be a good setting to assess replication with less confounding.

Our observation at the ACKR1 loci suggests that modification of multi-trait genetic effects is associated with evolutionary forces. We hypothesize that divergences in multi-trait effects are more striking and more telling functionally than a difference in mean between univariate effect sizes. A systematic investigation coupling trans-ancestry multi-trait genetics and evolutionary pressure measures could highlight how recent evolutionary events in the human population transformed genetic effects. In short, through our multi-trait and trans ancestry GWAS we detected relevant new associations and highlighted the similarity of multi-trait genetics across ancestry. We argue that a computationally efficient pipeline such as the JASS pipeline could be a tool of choice to investigate a multi-trait genetic pattern across ancestries and their potential coupling with evolutionary forces.

Conclusion

By conducting multi-trait GWAS on 93 summary statistics originating from five ancestries, we detected 367 new genome-wide significant associations in non-European populations (

  immune system tissues (eQTL catalog (BLUEPRINT monocyte, BLUEPRINT neutrophil, BLUEPRINT T-cell, CEDAR platelet, CEDAR T-cell CD4, CEDAR T-cell CD8, GENCORD Tcell, Kasela 2017 T-cell CD4, Kasela 2017 T-cell CD8, Lepik 2017 blood, Naranbhai 2015 neutrophil CD16), van der Wijst et al. scRNA eQTLs (B cells, Dendritic cells, Natural Killer cells, Monocytes, Classical Monocyte, Non-classical Monocytes, CD4 T cells, CD8 T cells, PBMC), DICE (Naive B cells, Naive CD4 T cells, Naive (activated) CD4 T cells, Naive CD8 T cells, Naive (activated) CD8 T cells, Classical Monocytes, Non-classical Monocytes, Natural Killer cells, TFH CD4 T cells, TH1 CD4 T cells, TH17 CD4 T cells, TH1-17 CD4 T cells, TH2 CD4 T cells, Memory TREG CD4 T cells, Naive TREG CD4 T cells), Blood eQTLs (Westra et al. (2013) BIOS QTL Browser), GTEx v8 Blood Vessel (GTEx Cells EBV-transformed lymphocytes, GTEx Whole Blood)).

Figure 1 .

 1 Figure 1. Overview of the study. After harmonization and imputation of GWAS summary

  , 430 regions were replicated with the univariate test, 495 regions were replicated when considering both tests). A substantial number (367) of loci were shared by the European, Asian and trans-ancestry analyses and 10% of those shared associations are detected by the joint test. Multi-trait GWAS can increase the number of replicated associations with no additional samples.

Figure 2 .

 2 Figure 2. Genetic signal similarity across 4 superpopulations. A) Overlapping loci across

Figure 3 .

 3 Figure 3. Examples of a diverging multi-trait signal: the ACKR1 locus. For each ancestry,

Figure 4 .

 4 Figure 4. Shared functional features across ancestries. A) Top five tissues with the highest

Table 1 . Independent genetic associations detected with the JASS omnibus test using the standard p-value threshold of 5.10 -8 .

 1 Number of associations found for 15 blood and four glycemic traits in four different populations with the joint test and the univariate tests.

		#Associations	European East Asian African Admixed	Trans Total by
		found by				American		Test
	Hematolo	joint test only	2,173	271	72	14	50	2,580
	gical
	traits	univariate test and	14,153	981	875	43	1,457 17,509
		joint test
		univariate test only	4,800	389	420	53	293	5,955
		Total by ancestry	21,126	1,641	1367	110	1,800 26,044
	Glycemic	joint test only	58	9	0	1		68
	traits
		univariate test and	371	32	0	8		411
		joint test
		univariate test only	89	6	1	0		96
		Total by ancestry	518	47	1	9		575

  The joint test increased the discovery of non-European ancestry specific loci by over 33% (from 10 regions to 15 regions). Of these specific loci, 12 regions on 15 were detected in the AFR ancestry and 4 of those associations were discovered exclusively by the joint test. This result is in line with previous reports that African cohorts tend to yield more associations by sample than European cohorts[START_REF] Pereira | African genetic diversity and adaptation inform a precision medicine agenda[END_REF]. The large number of specific loci detected in European ancestry samples likely reflects the larger sample size of the European cohorts. Specific loci in the European population tended to have a weaker signal (i.e., higher SNPs detected in the European population to simulate a range of diminished sample sizes (Methods) and computed the fraction of 1 Mbp loci that remained specific to European ancestry (Figure

p-value) than those shared by at least one other ancestry (rank test p-value = 0.008, Figure

S14

). These weaker signals might be detected in other ancestries when reaching an equivalent sample size. To test this hypothesis, we artificially weakened the signal of lead

  [START_REF] Li | METRO: Multi-ancestry transcriptome-wide association studies for powerful gene-trait association detection[END_REF] in AMR, 72 in AFR and 280 in EAS), which represents respectively 7%, 25% and 21% of all associations in the AFR, AMR and EAS populations. Overall, multi-trait testing increases the replication of European associated loci in non-European ancestry by 15%. Pleiotropic effects were highly similar at significant loci across ancestries (e.g. the mean correlation between multi-trait genetic effect of EUR and EAS was 0.88). For hematological traits, strong discrepancies in pleiotropic effects are tied to known evolutionary divergences: the ARKC1 loci which is adaptive to overcome the p.vivax induced malaria. Altogether these analyses suggest that multi-trait GWAS methods can be a valuable tool to narrow the genetic knowledge gap between European and non-European populations.To facilitate multi-trait GWAS on non European ancestries, we distribute publicly (Availability of data and material section) the JASS pipeline, and curated entry files (summary statistics, Reference panel) issued from this study.

	List of abbreviations
	EUR -European
	AMR -Admixed American
	AFR -African
	EAS -East Asian
	SAS -South-East Asian

Declarations

Ethics approval and consent to participate

Not applicable.

Acknowledgements

We would like to thank members of the Statistical Genetic Unit at the Institut Pasteur who contributed to the quality of this study through their insightful feedbacks, namely: Antoine

This research has been conducted using the UK Biobank

Funding

This research was supported by the Agence Nationale pour la Recherche (ANR-20-CE36-0009-02). This work has been conducted as part of the INCEPTION program (Investissement d'Avenir grant ANR-16-CONV-0005)

http://ftp.ensembl.org/pub/release-105/variation/gvf/homo_sapiens). LD score files were computed using the ldsc.py script available at https://github.com/bulik/

Availability of data and material

All computational tools used in this study are publicly available on the gitlab of the Statistical Genetics group of the Institut Pasteur:

-JASS pipeline: https://gitlab.pasteur.fr/statistical-genetics/jass_suite_pipeline -RAISS: https://gitlab.pasteur.fr/statistical-genetics/raiss -JASS: https://gitlab.pasteur.fr/statistical-genetics/jass -JASS preprocessing https://gitlab.pasteur.fr/statistical-genetics/jass_preprocessing

Consent for publication

Not applicable.

Authors' contributions

HJ conceptualized the study, supervised the analysis. LT and LH performed the data curation, the formal analysis. LT, HJ and DF investigated the results. HJ and LT wrote the original manuscript. HJ, LT, DF and HA reviewed the original draft. LT and HJ prepared visualisation of results. HJ and HA acquired funding for this work. All authors read and approved the final manuscript.

Competing interest

The authors declare that they have no competing interests.


Léo Henches

Jihye Kim

Zhiyu Yang

Simone Rubinacci

Gabriel Pires

Clara Albiñana

Christophe Boetto

Hanna Julienne

Arthur Frouin

Antoine Auvergne

Yuka Suzuki

Sarah Djebali

Olivier Delaneau

Andrea Ganna

Bjarni Vilhjálmsson

Florian Privé

Hugues Aschard

Polygenic risk score prediction accuracy convergence

Polygenic risk scores (PRS) trained from genome-wide association study (GWAS) results are set to play a pivotal role in biomedical research addressing multifactorial human diseases. The prospect of using these risk scores in clinical care and public health is generating both enthusiasm and controversy, with varying opinions about strengths and limitations across experts 1 . The performances of existing polygenic scores are still limited, and although it is expected to improve with increasing sample size of GWAS and the development of new powerful methods, it remains unclear how much prediction can be ultimately achieved. Here, we conducted a retrospective analysis to assess the progress in PRS prediction accuracy since the publication of the first large-scale GWASs using six common human diseases with sufficient GWAS data. We show that while PRS accuracy has grown rapidly for years, the improvement pace from recent GWAS has decreased substantially, suggesting that further increasing GWAS sample size may translate into very modest risk discrimination improvement. We next investigated the factors influencing the maximum achievable prediction using recently released whole genome-sequencing data from 125K UK Biobank participants, and state-of-the-art modeling of polygenic outcomes. Our analyses point toward increasing the variant coverage of PRS, using either more imputed variants or sequencing data, as a key component for future improvement in prediction accuracy.

2.2x10 -4 , P=0.078), but not significant for the other outcomes, with for example, a negative trend for coronary artery disease (β = -6.8x10 -5 , P=0.73) despite an effective sample size varying from N eff =4,522 to N eff =184,305. This is likely due to the limited number of studies available, but also to a number of factors already discussed in the literature and complex to decipher, including heterogeneity in the characteristics of the population (age, sex, fine-scale genetic ancestry within European population), disease definition, and the method used to derive the PRS weights [START_REF] Wand | Improving reporting standards for polygenic scores in risk prediction studies[END_REF][START_REF] Janssens | Strengthening the reporting of genetic risk prediction studies (GRIPS): explanation and elaboration[END_REF] .

To quantify more precisely changes in the predictive accuracy as a function of sample size, we implemented a harmonized pipeline to derive PRS and applied it to a set of highly curated GWAS data. We derived PRS weights for the six outcomes using 29 GWAS summary statistics (with different sample sizes) published since 2007 and including mostly participants of European ancestry. To expend the sample size coverage, we also added five GWAS with modest sample size that we conducted using unrelated participants of European ancestry from the UK Biobank cohorts [START_REF] Bycroft | The UK Biobank resource with deep phenotyping and genomic data[END_REF] (Supplementary Note and Table S2). Those 34 PRS were used to derive AUC in two independent cohorts. Because almost all existing genetic cohorts of European ancestry have been used to produce these GWAS, we focused our application on relative power and investigated the trend in AUC in 16,000 participants from six different non-European ancestries from the UK Biobank cohort, and up to 392,423 participants of Finnish ancestry from the FinnGen cohort [START_REF] Kurki | Unique genetic insights from combining isolated population and national health register data[END_REF] (Table S3). Difference in genetic ancestry induces lower absolute predictive performance, however, because the PRS portability across population is expected to be linear [START_REF] Prive | Portability of 245 polygenic scores when derived from the UK Biobank and applied to 9 ancestry groups from the same cohort[END_REF][START_REF] Ding | Polygenic scoring accuracy varies across the genetic ancestry continuum in all human populations[END_REF] , the trend remains highly informative, and for example, top GWAS hits between FinnGen and European ancestry samples were highly concordant (average squared-correlation between effect estimates equals 0.56, Fig. S1).

The prediction accuracy shows a univocal non-linear increasing trend as a function of sample size that starts with a sharp increase for the few first studies followed by a gradual decline in improvement. The convergence patterns are highly concordant between FinnGen and non-European UK Biobank participants, only displaying the expected offset due to genetic ancestry difference (Fig 1b, Fig S2-S3, and Table S4). It suggests that PRSs based on future GWASs of the target outcome with larger sample size are unlikely to dramatically improve prediction accuracy (as measured by the AUC) by themselves. The AUC convergence is especially striking for type 2 diabetes, obesity, breast cancer and coronary artery disease. Asthma displays a noisier trend, potentially reflecting the challenges in the disease definition and diagnosis. Alzheimer disease (AD) shows a continuous increase in AUC for the non-European UKB samples, but no increase for the last and largest GWAS in FinnGEN despite a five-fold increase in the effective sample size. This might be explained by the use of a proxy for disease status (a score determined by the Alzheimer status of the parents' participants) for the derivation of the PRS weights in the largest GWAS study. AD-by-proxy was also used in the non-European UKB participants because of a limited number of cases, while the true AD status was used in FinnGEN ( Supplementary Note). As recently discussed, cohorts with heterogeneous disease definition and study design can induce variability in heritability estimates of AD [START_REF] Escott-Price | Genome-wide association studies for Alzheimer's disease: bigger is not always better[END_REF] . This likely induces reduced prediction in FinnGEN in our analysis.

Modelling AUC convergence

The convergence of the AUC toward its maximum can be investigated using simulation, and one can easily demonstrate the validity of simple theoretical models 13 (Fig. S4). Predicting the convergence rate in real data is much more challenging and requires the estimation of multiple parameters of the disease genetic model. It includes the heritability, the polygenicity, the distribution (or mixture of distributions) of causal genetic effects, and the dependence of those effects on the linkage disequilibrium, functional annotations, and the minor allele frequency (MAF). We first examined to what extent existing tools are able to fit the AUC derived in real data using the GENESIS approach [START_REF] Zhang | Estimation of complex effect-size distributions using summary-level statistics from genome-wide association studies across 32 complex traits[END_REF] . Briefly, GENESIS infers some, but not all, of the aforementioned parameters and uses them to predict the expected AUCs as a function of future increases in sample size. As recommended by the authors, we focused on a threecomponent mixture of effects, where variants are classified as either non-causal, causal with low effect, or causal with large effect. The shape of the predicted AUC as a function of sample size obtained from GENESIS is reasonably close to the ones observed in FinnGen and the UK Biobank (Fig S5a), confirming the observation of a sharp decline in PRS predictive power improvement. However, the increase rate and expected maximum AUC varied substantially conditional on the input GWAS used, being especially variable for Alzheimer disease, asthma, and obesity, and highlighting the importance of conducting real data retrospective analysis.

We next investigated to what extent the observed convergence in AUC might be explained by heterogeneity arising when meta-analyzing an increasing number of studies. Heterogeneity in genetic ancestry, disease definition, and environment across the cohorts meta-analyzed, might reduce heritability and increase polygenicity, and ultimately lessen the benefits of larger sample size GWAS for predictive purposes. To assess these hypotheses, we examined the variability in AUC derived using the same procedure as in Figure 1b, and disease parameters estimated using SBayesS [START_REF] Zeng | Widespread signatures of natural selection across human complex traits and functional genomic categories[END_REF] , while increasing GWAS sample size, but using the same population for the GWAS and parameter estimation -thus providing a benchmark of the behavior when potential heterogeneity is minimized. We used participants of European ancestry from the UK Biobank (referred further as the EUR-UKB experiment), focusing on obesity and using GWAS of body mass index, one of the sole scenarios allowing to achieve reasonably large sample size within a single homogeneous cohort. While the role of genetic heterogeneity cannot be ruled out, we did not find any strong evidence in favor of an effect of heterogeneity. The polygenicity estimate tends to decrease first (potentially due to limited sample size) until it reaches a minimum and then starts to increase for both the EUR-UKB (Fig. S6a) and most of the six outcomes GWASs (Fig. S7a). Except for the few estimates derived using a limited sample size, heritability tends to be fairly consistent, although much less variable in EUR-UKB (Fig. S6b) than across the GWASs (Fig. S7b). Estimates of alpha, the minor allele frequency (MAF)-effect size parameter, also display a similar pattern across analyses (Fig. S6c, Fig. S7c). Overall, the AUC from the EUR-UKB analysis appears to follow the same trend as the ones derived in FinnGen and non-European UKB participants (Fig. S6d).

As a general observation, the disease parameters estimated from both GENESIS [START_REF] Zhang | Estimation of complex effect-size distributions using summary-level statistics from genome-wide association studies across 32 complex traits[END_REF] and SBayesS [START_REF] Zeng | Widespread signatures of natural selection across human complex traits and functional genomic categories[END_REF] vary substantially conditional on the input GWAS used (Figs. S5b-c, S6), with confidence interval often not overlapping across GWASs from the same outcome. For comparison purposes, we applied three additional methods, sumHer [START_REF] Speed | SumHer better estimates the SNP heritability of complex traits from summary statistics[END_REF] , LDSC [START_REF] Bulik-Sullivan | LD Score regression distinguishes confounding from polygenicity in genome-wide association studies[END_REF] , and MiXeR [START_REF] Holland | Beyond SNP heritability: Polygenicity and discoverability of phenotypes estimated with a univariate Gaussian mixture model[END_REF] , to all available GWAS with an effective sample size larger than 5,000 (Table S5). With a few exceptions, cross-method heritability estimates from the same GWAS tend to be less variable than within-method estimates from GWASs of the same outcome. Polygenicity varies extensively across methods, each of them displaying a different trend as a function of sample size. The reasons for this variability are unclear, and until progresses are made in estimating those and other parameters underlying the distribution of genetic effect, our ability to develop an accurate modelling of the convergence rate of PRS predictive power will likely remain limited, again, highlighting the importance of conducting real data retrospective analyses.

Maximal achievable prediction

The maximum achievable AUC relies on less parameters than the convergence rate. A central feature to derive this maximum is the variant coverage, which itself determines the proportion of heritability captured by the genetic variants used in the PRS. Existing polygenic risk prediction models rely heavily on genome-wide genotyping studies, thanks to a high costgenomic coverage ratio [START_REF] Nelson | Imputation-Based Genomic Coverage Assessments of Current Genotyping Arrays: <em>Illumina HumanCore, OmniExpress, Multi-Ethnic global array and sub-arrays[END_REF] allowing to generate genotype data for very large cohorts, as compared to sequencing studies. This implies that even reaching an infinite sample size, the total genetic variance captured by GWAS, and therefore the maximum prediction accuracy, is bounded by the sub-sampling of all existing variants. How much of the narrow-sense heritability (i.e. the linear additive genetic variance [START_REF] Visscher | Heritability in the genomics era--concepts and misconceptions[END_REF] ) can be captured by GWAS and how much more will be captured using sequencing data has been previously explored using both theoretical model and whole-exome sequencing data of modest size as compared to GWAS studies [START_REF] Wainschtein | Assessing the contribution of rare variants to complex trait heritability from whole-genome sequence data[END_REF][START_REF] Yang | Common SNPs explain a large proportion of the heritability for human height[END_REF][START_REF] Yang | Genetic variance estimation with imputed variants finds negligible missing heritability for human height and body mass index[END_REF][START_REF] Speed | Reevaluation of SNP heritability in complex human traits[END_REF][START_REF] Weiner | Polygenic architecture of rare coding variation across 394,783 exomes[END_REF][START_REF] Manolio | Finding the missing heritability of complex diseases[END_REF] . Although the size of existing genome sequencing data remains limited and does not allow to fully address this question, some recent sequencing efforts provide opportunities to revisit this question. Here, we used individual-level whole-genome sequencing data from 503,195 variants on chromosome 22 measured in 125,152 participants of European ancestry from the UK Biobank study, to quantify the proportion of heritability captured by genotyped variants under various alternative genetic models.

We assumed the relationship between effect size and variants characteristics follow the socalled alpha model -where the expected effect of a variant is proportional to its variance power α 33,36 (see Methods)and derived two metrics: i) h I2 , the additive genetic variance captured by both genotyped variants and variants imputed using advanced methods based on haplotype inference [START_REF] Delaneau | Accurate, scalable and integrative haplotype estimation[END_REF] , and ii) h G 2 , the additive genetic variance captured by genotyped variants and the additional variance captured through linkage disequilibrium between genotyped and untyped variants. In theory, the later parameter should be approximately similar to the GWAS-based heritability (h GWAS 2 ) estimated by existing software [START_REF] Zaitlen | Heritability in the genome-wide association era[END_REF] . Both metrics were derived using sequence data, after considering various alternative approaches (Fig. S8-S9). As shown in Figure 2a, assuming a random distribution of genetic effect in the genome, h I 2 varies from 29% for α =-1.5 to 96% for α =0. The heritability captured by genotyped variants only is substantially smaller, with h G variants [START_REF] Schoech | Quantification of frequency-dependent genetic architectures in 25 UK Biobank traits reveals action of negative selection[END_REF] . To address this possible limitation, we also devised an attenuated alpha model implying a reduced contribution of rare variants (Supplementary Note, Fig. S10). However, when comparing those attenuated models against the standard in real data, we did not find any evidence for an increased fit (Fig. S11-12).

Estimates of heritability captured can be translated into maximum achievable AUC given the prevalence and heritability of the target outcome (see Methods). As expected, this maximum increases with the disease heritability captured, and decreases with lower value of alpha ( Fig. 2b). We compared maximum AUC for each of the six diseases using three heritability estimates: twin studies heritability (AUC twin ), GWAS-based heritability derived using five approaches (AUC GWAS , Table S5-S6), and heritability captured by all imputed variants (AUC Imputed ). Note that the latter AUC requires an alpha value. We attempted to use estimates of alpha obtained from various approaches (Table S6), but none of them match the expected based on GWAS-based AUC, and we ultimately set alpha for each disease so that the AUC from the genotype-based alpha model equals the median of AUC GWAS across the five approaches (see Methods). The comparison of AUC is summarized in Figure 2d. First, the AUC GWAS are relatively close to the AUCs reported in recent studies (AUC curr , Table S6) for coronary artery disease, breast cancer, Alzheimer disease and obesity, confirming that increasing sample size of GWAS for those outcomes is unlikely to dramatically impact prediction performances. Conversely AUC curr for T2D and asthma display sizeable gaps with AUC GWAS , so that, despite the convergence observed in Figure 1b, future GWAS with larger sample size might provide a slow but continuous improvement in prediction. Second, the gap between AUC Twin and AUC GWAS is large for all outcomes, except for breast cancer, suggesting that increasing the variant coverage in future studies can dramatically improve the predictive power of polygenic risk scores. Third, AUC Imputed is substantially higher than AUC GWAS for all outcomes except breast cancer. Hence, future PRSs using an increasing number of imputed variants, even those with poor imputation quality, have the potential to boost prediction power without requiring the generation of costly sequencing data.

Effect distributions and predicted performances

In the previous analyses, we assumed that causal variants are randomly distributed across the genome and only modelled the relationship between the effect size and the minor allele frequency. There is strong evidence that causal variants are highly enriched in some functionally annotated regions [START_REF] Finucane | Partitioning heritability by functional annotation using genomewide association summary statistics[END_REF][START_REF] Won | Disproportionate Contributions of Select Genomic Compartments and Cell Types to Genetic Risk for Coronary Artery Disease[END_REF][START_REF] Gazal | Linkage disequilibrium-dependent architecture of human complex traits shows action of negative selection[END_REF][START_REF] Epigenomics | Integrative analysis of 111 reference human epigenomes[END_REF] . However, the exact link between those functional annotations and causal variants is not fully understood and likely depends on the outcome study. Rather than assuming a specific relationship, which will be subjective given the current knowledge, we assessed the impact of unequal distribution of causal variants in the genome by estimating the association between a range of functional annotations and the quality of imputation (r ², the squared-correlation between sequenced and imputed variants used in the derivation of h I

2

). Here we used a total of 1,099 annotations from nine sources (Table S7-8, Fig. S13), including gene elements (coding DNA sequence, exon, intron, transcription start and termination sites), DNase I hypersensitive sites (DHS), enhancers and promoters. Those annotations cover 0.001% to 64.7% of the whole genome (Table S8).

6

The largest associations between annotation and r ² were observed for gene elements (Fig. S14a). Coding DNA sequence (CDS), that cover 1.9% of chromosome 22, show a highly significant decrease in r ² (P = 1.8 x 10 -307 ), with an average r ² of 0.27 and 0.40 for CDS and non-CDS variants, respectively. Exons, that cover 9.8% of the chromosome 22, are also strongly negatively associated (P = 2.0 x 10 -131 ) with an average r ² of 0.37 and 0.40 for exonic and nonexonic variants (Fig. 3a). Variants from other annotation categories show significant negative and positive associations, but to a lower extent. Enhancers, super enhancers, DHS, and transcription factor binding sites tend to display lower r ² than the average of the genome (Fig. 3b). Conversely, genetic variants within promoters appear to be slightly better captured. Excluding gene elements, the strongest association was observed for transcription start site with an average r ² of 0.36 and 0.40 for annotated and non-annotated variants (P = 2.1 x 10 -14 ). We did not observe any enrichment for specific cell types or tissues among other top significant annotations (Fig. S15).

Part of the decrease in variant coverage (Fig. 3a-b) is likely explained by an enrichment for rare variants with, for example, coding regions displaying a significantly higher proportion of rare variants (P = 2.5 x 10 -270 , Fig. 3c and Table S9). However, other annotations positively associated with r ² also display a significant enrichment for rare variants (e.g. intron, P = 1.0 x 10 -8 ), so that minor allele frequency alone is unlikely to fully explain the observed lower variant coverage. As discussed in previous works, causal variants might display decreased linkage disequilibrium with other variants because of selective pressure [START_REF] Weiner | Polygenic architecture of rare coding variation across 394,783 exomes[END_REF][START_REF] Gazal | Linkage disequilibrium-dependent architecture of human complex traits shows action of negative selection[END_REF] , an assumption that is now commonly incorporated in disease parameter estimation tools [START_REF] Speed | Improved heritability estimation from genome-wide SNPs[END_REF] . Lower linkage disequilibrium implies lower r ², and the observed lower variant coverage for CDS, enhancers, DHS and transcription factor binding sites might embody an enrichment for rare causal variants for those annotations. Under such scenario, the estimated maximum achievable AUCs from imputed variants presented in Figure 2 could be slightly overestimated, which suggest again that future improvement in PRS prediction accuracy is likely going to rely on improving the variant coverage of future genetic association study (Fig. S14b).

Discussion

Polygenic risk prediction has the potential to transform the diagnosis, treatment, and prevention of many common human diseases. However, the timescale and extent of this transformation is partly unknown, and some in the community continue to express concerns about the relevance of PRS. These concerns are partly due to inconsistencies in the reported performances of polygenic risk models. Previous reports of PRS prediction accuracy are not always properly documented, and while guidelines have been proposed [START_REF] Wand | Improving reporting standards for polygenic scores in risk prediction studies[END_REF] , adherence to reporting statements remains limited. Heterogeneity in the methods used, the populations studied, and the covariates accounted for, can prevent the replication of results and a formal assessment of PRS performances. As illustrated in this study for six outcomes, even a careful curation of publication results based on the above criteria might not be sufficient, and the potential relationship between performances and study parameters can remain blurry.

Here, we demonstrate that the relationship between prediction accuracy of polygenic models and sample size is unequivocal and highly replicable when using harmonized data preprocessing and analysis pipelines. These analyses also show that the prediction accuracy of PRS 7 derived from existing GWAS have started to converge for most diseases studied. As a result, expanding the current state-of-the-art -i.e. fully relying on an increase in GWAS sample sizemight only lead to a modest increase in predictive power. For some outcomes, such as coronary artery disease, breast cancer, and obesity, our analyses suggest that the predictive power of GWAS-based PRS is nearing the expected maximum derived based on the GWAS heritability. If confirmed, this implies that improvements in methods used to derive PRS will not either dramatically impact prediction accuracy. For type 2 diabetes and asthma, the gap with the expected maximum remains fairly large, and AUC from future larger GWAS might continue to increase, although any gain will likely require extremely large sample size. Importantly, the observed convergence might be partly explained by increasing heterogeneity arising from large meta-analysis, with for example, very large meta-analysis using a loosen disease definition or increasingly genetically diverse populations, in order to allow for a broader inclusion of participants. Although we did not find any strong evidence for such effect across outcomes we studied, the potential impact of genetic ancestry [START_REF] Prive | Portability of 245 polygenic scores when derived from the UK Biobank and applied to 9 ancestry groups from the same cohort[END_REF][START_REF] Ding | Polygenic scoring accuracy varies across the genetic ancestry continuum in all human populations[END_REF] and phenotype definition [START_REF] Escott-Price | Genome-wide association studies for Alzheimer's disease: bigger is not always better[END_REF][START_REF] Cai | Minimal phenotyping yields genome-wide association signals of low specificity for major depression[END_REF] heterogeneity is well established. Note that there are other sources of heterogeneity that we did not investigate including in particular sample ascertainment, which might influence the disease prevalence and heritability, and therefore the prediction accuracy.

Estimating the trend in prediction accuracy as a function of sample size depends on our ability to estimate for each disease, the distribution of causal variants in the genome and its dependence on other factors. However, as illustrated in this study, estimates of the genetic disease parameters using existing methods vary substantially conditional on, not only the approach and model used, but also the GWAS used as input. Developing a reliable predictor of the AUC trend will remain challenging until more robust estimators are developed. Despite these limitations, some of the current estimators can still be used to investigate the maximum achievable AUC. Altogether our analyses suggest that, assuming rare variants do carry a fair share of heritability, the genetic variance carried by these variants might be poorly captured by genotyped variants, so that until more large-scale whole-genome sequencing become available, the maximum AUC will remain bounded by the variant coverage of existing GWASs. In agreement with previous works [START_REF] Yang | Genetic variance estimation with imputed variants finds negligible missing heritability for human height and body mass index[END_REF] , the present study also support the use of imputed rare variants, even those with poor imputation quality, as an effective intermediate step to improve PRS predictive power.

For most analyses we assumed for simplicity that the causal variants are randomly distributed and we only modelled the relationship between minor allele frequency and effect size. This is clearly an over-simplification of the true underlying model, and as stated above, there is now strong evidence that causal variants distribution also varies across the linkage disequilibrium pattern and functional annotations [START_REF] Zhang | Estimation of complex effect-size distributions using summary-level statistics from genome-wide association studies across 32 complex traits[END_REF][START_REF] Zhang | Assessment of polygenic architecture and risk prediction based on common variants across fourteen cancers[END_REF][START_REF] Holland | Beyond SNP heritability: Polygenicity and discoverability of phenotypes estimated with a univariate Gaussian mixture model[END_REF][START_REF] Speed | Reevaluation of SNP heritability in complex human traits[END_REF][START_REF] O'connor | The distribution of common-variant effect sizes[END_REF][START_REF] Orliac | Improving GWAS discovery and genomic prediction accuracy in Biobank data[END_REF] . The relationship between those factors and the causal effect has not been fully specified, and is expected to vary across outcomes. Instead of investigating specific models, we derived the relationship between the variant coverage and a vast set of functional annotations. We observed significant variability across annotations, with both positive and negative enrichments in variant coverage. However, except for CDS that show substantially poorer coverage, those differences did not have any qualitative impact on our results.

Our study has multiple limitations. First, our study only includes six diseases. We initially intended to include more outcomes, but for all other diseases we considered (e.g. chronic 8 obstructive pulmonary function, Crohn's disease, hypertension, etc) there was not enough available data to ensure a fair and objective analysis. Second, for real data heritability, we used twins study heritability as a proxy for the total additive genetic variance. We appreciate that these estimates might be overestimated because of shared environment and non-additive effects (e.g. gene-gene and gene-environment effects) [START_REF] Zaitlen | Using extended genealogy to estimate components of heritability for 23 quantitative and dichotomous traits[END_REF] . Third, we assumed the genetic effects are homogeneous in the populations studied. Future improvement in predictive performances might depend on our ability to integrate potential heterogeneity in genetic effects conditional on demographic characteristics, basic health parameters, and lifestyle [START_REF] Lewis | Polygenic risk scores: from research tools to clinical instruments[END_REF][START_REF] Mostafavi | Variable prediction accuracy of polygenic scores within an ancestry group[END_REF][START_REF] Duncan | Analysis of polygenic risk score usage and performance in diverse human populations[END_REF] . For example, there are increasing evidence that heritability might vary with age and sex [START_REF] Bernabeu | Sex differences in genetic architecture in the UK Biobank[END_REF][START_REF] Ge | Phenome-wide heritability analysis of the UK Biobank[END_REF] . Fourth, in our derivation of the maximum achievable AUC, we assumed a relatively simple model, focusing on the minor allele frequency-effect size relationship. Future works might incorporate more complex modelling including, in particular linkage disequilibrium (LD) dependencies. Note that assuming enrichment for causal variants in low LD regions will only worsen the gap between genotype-based AUC and imputed-based AUC. Fifth, heritability and AUC were derived using disease prevalence reported by the CDC. Prevalence clearly varies across population and using other estimates will impact the absolute values of both parameters, but not their relative values though. Sixth, we focus all our analyses based on the gold standard approach for PRS, that is, using univariate GWAS with the largest possible sample size. There is now increasing literature on multitrait approaches [START_REF] Truong | Integrative polygenic risk score improves the prediction accuracy of complex traits and diseases[END_REF][START_REF] Albiñana | Multi-PGS enhances polygenic prediction: weighting 937 polygenic scores[END_REF][START_REF] Andrew | Phenotype integration improves power and preserves specificity in biobank-based genetic studies of MDD[END_REF] , where the predictive power can be boosted by partly circumventing the challenge of assembling very large sample size. 9

Figures Figure 1. Polygenic risk scores predictive accuracy as a function of sample size

Panel a) presents the area under the receiver operating characteristic curve (AUC) reported in the literature for polygenic risk score as a function of the effective sample size for coronary artery disease, type 2 diabetes, breast cancer, Alzheimer disease, asthma and obesity. The color gradient corresponds to the number of variants used, from a few top associated variants (light green) to millions (dark blue), and the size is proportional to the log of the effective sample size. Panel b) shows AUCs for the six outcomes derived using a harmonized pipeline. Polygenic risk scores were trained from 34 genome-wide association study (GWAS) using the LDpred2 approach, and tested in individual-level data from the FinnGEN cohort (black) and in six non-European ancestry UK Biobank populations (red, meta-analysis over six populations). The AUCs are plotted as a function of the effective sample size of the corresponding GWAS. Missing values correspond to cases where there was a sample overlap between the test and train sets.

Methods

GWAS data assembling

We collected publicly available genome-wide association study (GWAS) summary statistics from six outcomes: type 2 diabetes, coronary heart disease, breast cancer, Alzheimer disease, asthma, and body mass index (BMI) which was used to study obesity. The design of each study was assessed carefully to ensure it matches stringent criteria for inclusion in our analysis: i) all studies had to include a majority of individual of European ancestry; ii) studies with limited genetic coverage, such as exome-wide screening and ad hoc genotyping chips (e.g. Metabochip or Immunochip) without imputation were excluded; iii) phenotype definition had to be relatively homogeneous, although we kept some study with heterogeneity for comparison purposes (early onset for asthma, and Alzheimer disease proxy derived from parent status). Note that for some meta-analysis, we also had access to GWAS results of individual cohorts, which were used as additional data points. After this quality control filtering, a total of 29 GWAS summary statistics with the baseline required data (coded allele, sign statistics and P value) remained for analysis (Table S2). To complete our panel of GWAS we conducted five additional GWAS in the UK Biobank using unrelated participants of European ancestry, and cases sampled from the entire cohort: breast cancer (N cases =5,000; N controls =50,000); coronary artery disease (N cases =5,000; N controls =50,000); asthma (N cases =1,000; N controls =10,000); and BMI (N=20,000 and N=80,000). All 34 GWAS were harmonized and converted to hg38 using the liftover package [START_REF] Gao | segment_liftover : a Python tool to convert segments between genome assemblies[END_REF] . As one of the ultimate goals of these data is to compare the predictive accuracy of PRS scores in the FinnGen cohort, we only kept variants available in that dataset.

Estimation of prediction performances of polygenic risk score in real data

For each GWAS, we derived a vector of individual variants PRS weights γ= ( γ 1 , γ 2 … γ M ), where γ i is the weight of variant i and M is the number of variants available in the GWAS, using the LDpred2 approach 7 , a popular Bayesian method. We used the 'auto' option that automatically tune hyper-parameters including the PRS sparsity and the SNP heritability, and the recommended number (30) of Gibbs sampling chains. European descent participants from the UKB biobank were used as a reference panel for LD derivation, and we restricted the analysis to the GWAS variants overlapping with a set of 1,054,330 HapMap3 variants [START_REF] Epigenomics | Integrative analysis of 111 reference human epigenomes[END_REF] . For each GWAS, variants with an estimated effective sample size smaller than 50% or larger than 110% of the expected maximum were further filtered out to avoid a miscalibration of the estimated regression coefficients [START_REF] Privé | Identifying and correcting for misspecifications in GWAS summary statistics and polygenic scores[END_REF] .

We use those weights to construct genetic risk scores in two test datasets (Table S3). The scores were derived as PRS=γ t X, where X is a matrix of genotypes from the test dataset with coded alleles matching the coded alleles of the original GWAS. The first dataset included 16,000 unrelated participants of non-European ancestry from the UK Biobank, where ancestry (Ashkenazi, Iranian, Indian, Chinese, Caribbean, and Nigerian) was derived following Privé et al [START_REF] Prive | Portability of 245 polygenic scores when derived from the UK Biobank and applied to 9 ancestry groups from the same cohort[END_REF] . This analysis was also used for the fine tuning of the PRS and to derive the average weights over the 30 Gibbs sampling. The second dataset included 392,423 participants of Finns ancestry from the FinnGen cohort (see Supplementary Notes). Except if specified otherwise, we defined the effective sample size of a binary outcome GWAS as N eff =4 / [ 1/ N case +1/ N controls ], which re- scales cohorts sample size with unequal numbers of cases and controls to the same unit [START_REF] Willer | METAL: fast and efficient meta-analysis of genomewide association scans[END_REF] . We estimated the predictive performance of the PRSs using the Area Under the Roc Curve (AUC). The AUC is a commonly used classifier in genetics, which value equals to the probability of a randomly chosen case to be ranked higher than a randomly chosen control. For the non-European UK Biobank samples, we first derived the AUC per ancestry (Table S3, Fig. S2), and combined results using a standard inverse-variance meta-analysis:

AUC combined = ∑ i=1 …5 [ AUC i /σ i 2 ]/ ∑ i=1 …5
[1/σ i 2 ] , and

SE ( AUC combined )=1/ ∑ i =1 … 5 [1/σ i 2 ]
, where AUC i and

σ i 2
are the AUC and variance derived for population i=1 … 5.

Variance captured by genotyped and imputed variants

Consider a standardized phenotype Y drawn from a polygenic model and defined as the linear additive effect of M standardized causal variants. Its variance equals V y =h 2 + V e , where h 2 is the genetic variance, or heritability and V e is the residual environmental variance. Assuming the genetic effects at the causal variants are independent of variants' correlation, the heritability can be approximated as

h 2 = ∑ M β i 2
, where β i is the effect of a standardized variant i.

We derived the proportion of h 2 that can be recovered given an infinite GWAS sample size, when only a subset S of variants have been genotyped, while the remaining M ∉ S are either imputed or missing. We considered two metrics, h I 2 the genetic variance captured when using all M variants whatever their imputation quality, and h G 2 the genetic variance captured when using only the subset S of genotyped variants, but accounting for the effect of the M ∉ S variants captured through linkage disequilibrium. The first metric h I 2 is defined as

h I 2 = ∑ M β i 2 r i 2 ,
where r i 2 is the squared correlation between the sequence variant i and its imputed value. The

second metric h G 2 is defined as h G 2 = ∑ S β i 2 + ∑ M ∉ S β j 2 ρ j 2
, where ρ j 2 is the squared correlation between the untyped variants j and the genotyped ones, obtained from a multiple regression (see below for the derivation). Both features were derived using a subset of 125,152 participants of European ancestry from the UK Biobank with genome-wide genotyping and whole genome sequencing data available. Note that for h I 2 , we first considered using the imputation quality score, commonly referred as the info_score, as defined by Marchini & Howie [START_REF] Marchini | Genotype imputation for genome-wide association studies[END_REF] , which is expected to be an unbiased estimator of r .

2 . However, this metric appears to be inflated in both simulated and real data, especially for variants with low minor frequencies (Fig. S8). We therefore derived r . 2 using the aforementioned individual-level sequencing data. We used the imputed SNP-array data provided by the UK Biobank. We lifted-over the imputed variants for the 125,152 samples, retaining 99.5% of the original imputed sites in GRCh38. We then derived an r2 between the imputed dosages and the whole-genome sequencing data for each of the overlapping sites.

The r . 2 was estimated from a standard univariate linear model using each sequenced variants as the outcome and the imputed ones as predictors. For h G 2 , we used genotyped variants available on the UK Biobank Axiom array 21 as our baseline. The ρ .

2

were derived as the adjusted 19 squared-correlation obtained from a standard multiple linear regression with the lm() function from the R software. Each non-genotyped variant j was predicted from a set Ω of neighboring variants in a window of ±1.5Mb (Fig. S9a). For simplicity we derived r .

2 and ρ . 2 using only variants from chromosome 22, under the assumption that estimates from this chromosome are representative of the entire genome. There are 12,968 variants genotyped on this chromosome. Out of 659,092 variants available from the sequencing data, a total of 503,195 remained for analysis after removing those with a minor allele frequency (MAF)< 0.001%. As expected, the two metrics are highly correlated (cor ( ρ 2 ,r 2 )=0.71) but, the average r 2 was substantially larger than ρ 2 (r 2 =0.50, ρ 2 =0.25, Fig. S9b). In our simulations, the proportion of heritability captured was derived using both parameters (r 2 and ρ

2 ), and β=( β 1 … β M ) coefficients drawn from a normal distribution using the alpha model [START_REF] Schoech | Quantification of frequency-dependent genetic architectures in 25 UK Biobank traits reveals action of negative selection[END_REF] . The alpha model assumes an inverse relationship between the variant frequency and the per-allele effect, with rare causal variants harboring larger per-allele effect than common variants. This model has been empirically confirmed in many studies (e.g. [START_REF] Marouli | Rare and low-frequency coding variants alter human adult height[END_REF] ). In practice, the genetic effect of variant i is drawn from the following:

β i |p i N ( 0 , σ g , α 2 • [ 2 p i (1-p i ) ] α )
, where p i is the minor allele frequency, and σ g , α 2 is a constant constraining the outcome heritability (Fig. S10a). In our analysis we considered α values in the range [-1.5 ; 0.0], and draw random σ g , α 2 in [0,1]. We also investigated an attenuated alpha model where the contribution of the rarest variants was decreased using an ad hoc iterative weight function (Supplementary Notes).

Maximum achievable AUC

The expected maximum achievable AUC is derived using the approximation proposed by Wray et al [START_REF] Torkamani | The personal and clinical utility of polygenic risk scores[END_REF] :

AUC max ≈ Φ( ((i-v)h 2 )/ √ h 2 [(1-h 2 i(i-T ))+(1-h 2 v (v-T ))])
, where h 2 is the heritability on the liability scale, Φ is the cumulative density function of the normal distribution, z is the height of the standard normal density at the threshold T =Φ -1 (1-K ), and with i=z / K and v=-z /(1-K ), where K is the disease prevalence. We confirmed the validity of the approximation using a simple simulation model involving independent causal variants with a linear additive effect on the outcome and using h 2 in [0.2 ; 0.7] and the prevalence K in [0.01 ; 0.25] (Fig. S4 and Supplementary Notes). Estimation of the AUC conditional on alpha (Fig. 2b) were derived by replacing h 2 by either h G 2 or h I 2 , themselves derived based on alpha drawn in [-1.5, 0] and β=(β 1 … β M ) coefficients generated from a normal distribution.

For real data analysis, disease prevalence from the six outcomes were pulled from the CDC website (https://www.cdc.gov/, Table S6). For the heritability, we considered various estimates : i) the total heritability derived from twins studies for coronary artery disease (0.55) [START_REF] Dai | Genetics of coronary artery disease and myocardial infarction[END_REF] , type 2 diabetes (0.72) [START_REF] Willemsen | The Concordance and Heritability of Type 2 Diabetes in 34,166 Twin Pairs From International Twin Registers: The Discordant Twin (DISCOTWIN) Consortium[END_REF] , breast cancer (0.27) [START_REF] Lichtenstein | Environmental and heritable factors in the causation of cancer-analyses of cohorts of twins from Sweden, Denmark, and Finland[END_REF] , Alzheimer disease after excluding the effect of APOE (0.49) [START_REF] Gatz | Heritability for Alzheimer's disease: the study of dementia in Swedish twins[END_REF][START_REF] Karlsson | Measuring heritable contributions to Alzheimer's disease: polygenic risk score analysis with twins[END_REF] , asthma (0.70) [START_REF] Thomsen | The contribution of twin studies to the understanding of the aetiology of asthma and atopic diseases[END_REF] , and body mass index (0.75) [START_REF] Elks | Variability in the heritability of body mass index: a systematic review and meta-regression[END_REF] ; ii) h GWAS 2 , the heritability captured by GWAS variants derived using five alternative approaches: SBayesS 24 , sumHer [START_REF] Speed | SumHer better estimates the SNP heritability of complex traits from summary statistics[END_REF] , LDSC regression [START_REF] Bulik-Sullivan | LD Score regression distinguishes confounding from polygenicity in genome-wide association studies[END_REF] , GENESIS [START_REF] Zhang | Estimation of complex effect-size distributions using summary-level statistics from genome-wide association studies across 32 complex traits[END_REF] , and MiXeR 27 (Supplementary Notes), and iii) h I 2 , the heritability captured by all imputed variants. The later estimate requires three parameters: the total heritability, the proportion of heritability captured given alpha, and a value of alpha. For the 20 total heritability, we used the twin study estimate. For the proportion of heritability captured, we used the estimate derived using the UK Biobank sequencing data. The choice of alpha was more challenging and is described below.

Most real data estimates of the GWAS variants heritability (h GWAS

2

) are derived based on genotyped variants along a modest subset of imputed variants with very high quality (typically an info_score ≥ 0.8). As a results, assuming our estimation of the heritability captured by genotyped variants based on the UK Biobank sequencing data (Fig. 2a) is valid, the previously described h G 2 is expected to approximate h GWAS 2 providing a relevant value of alpha. We assessed the equality between the two metrics using alpha derived from various approaches (SBayesS, sumHer, and individual-level data from the UK Biobank, Table S6), but none of them match the expected based on h GWAS 2 . Therefore, for each disease, we selected a "Best fit" alpha so that h G 2 is equals the median of h GWAS 2 derived over the five aforementioned approaches (Table S6). This "Best fit" alpha was used to derived h I 2 for the six outcomes.

Genetic coverage and functional annotations

We investigated the association between r ², the squared-correlation between the true and the imputed genotypes and a total of 1,099 functional annotation pulled from nine sources: baseline GENCODE annotations: intron, gene, exon, CDS (coding DNA sequence), tss (transcription start site), tts (transcription termination site), and UTR (untranslated regions) ; and epigenetic annotations across a vast range of tissues and cell types: TFBS (transcription factor binding site) ; FANTOM5 (functional annotation of the mammalian genome version 5) ; promoters ; enhancers, and dyadic from Roadmap, DHS (DNase I hypersensitive sites) derived from two studies, and super enhancer. The analysis was conducted using the same data used to derive h I

14

 14

Figure 2 . 16 Figure 3 .

 2163 Figure 2. Heritability captured and maximum achievable AUCPanel a) shows the expected proportion of heritability captured by genotyped (dashed line) and all imputed variants (plain line). Individual-level data from the UK Biobank were use to estimates the squared-correlation between sequenced variants and either genotyped ( ρ 2 ) or imputed variants (r ²), and genetic effects were assumed to be distributed following an alpha model. Panels b) shows the corresponding maximum achievable AUC for heritability of 0.1 and 0.5, and disease prevalence equals to 0.01 and 0.2. Panel c) shows estimates of the maximum achievable AUC for six outcomes: coronary artery disease, type 2 diabetes, breast cancer, Alzheimer disease, asthma, and obesity (using body mass index GWAS). Those AUCs were derived based on US disease prevalence pooled, and various estimates of heritability: twin studies (AUC Twin ), estimates of GWAS heritability derived using five competitive approaches: LDSC regression, sumHer, SBayesS, Genesis, and MiXer (AUC GWAS ), and twin study heritability captured by imputed variants (AUC Imputed ). The black dashed lines correspond to the most recent estimates of AUC in real data from the literature, derived based on approximately 1M hapmap3 variants. For Alzheimer disease, heritability estimates and current AUC estimates correspond to a model excluding the APOE region.

2 and h G 2 (

 2 503,195 variants from chromosome 22 from 125K UK Biobank participants of European ancestry).

varying from 5% for α =-1.5 to 90% for α =0. This suggests that imputed variants with modest and poor imputation quality, often filtered out in GWAS studies, might capture a substantial share of the total narrow-sense heritability missed by genotyped variants. Of note, previous studies argued that the alpha model might overestimate the effect of rare

Acknowledgments

We want to acknowledge the participants and investigators of the FinnGen study and the UK Biobank cohort. This research has been conducted using the UK Biobank Resource under Application Number 42260 and 66995. We also would like to thanks the authors of GENESIS, sumHer and SBayesS for their helpful recommendations. This work has been conducted as part of tshe INCEPTION program (Investissement d'Avenir grant ANR-16-CONV-0005). This research was supported by the Agence Nationale pour la Recherche (ANR-20-CE36-0009-02 and ANR-20-CE15-0012-01).

Data availability

All GWAS summary statistics have been downloaded from publicly available websites including dedicated page from consortia, the NHGRI-EBI Catalog of human genome-wide association studies, and the FinnGEN GWAS repository. Individual-level data from the UK Biobank were accessed from the UK Biobank Resource under Application Number 42260 and 66995.

Authors contributions

HA and LH conceived and supervised the project. LH, JK, ZY, SR, GP carried out the primary analyses. CA, CB, HJ, AF, YS conducted secondary analyses. ZY and AG led the analyses involving individual-level data from FinnGEN. SR, OD and LH led the analysis involving individual-level sequencing data from the UK Biobank. FP, BJ, CA and HA designed the LDpred2 pipeline for PRS analysis. HA and LH co-wrote the manuscript with input from all other authors. All authors contributed to discussions.

Ethics declarations

Competing interests

The authors declare no competing interests.