# Polygenic risk score prediction accuracy convergence

Léo Henches, Jihye Kim, Zhiyu Yang, Simone Rubinacci, Gabriel Pires, Clara Albiñana, Christophe Boetto, Hanna Julienne, Arthur Frouin, Antoine Auvergne, et al.

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## To cite this version:

Léo Henches, Jihye Kim, Zhiyu Yang, Simone Rubinacci, Gabriel Pires, et al., Polygenic risk score prediction accuracy convergence. 2024. pasteur-04368572

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# Polygenic risk score prediction accuracy convergence

Léo Henches, Jihye Kim, Zhiyu Yang, Simone Rubinacci, Gabriel Pires, Clara Albiñana, Christophe Boetto, Hanna Julienne, Arthur Frouin, Antoine Auvergne, et al.

<figure><img src="image_1.png" /></figure>

## To cite this version:

Léo Henches, Jihye Kim, Zhiyu Yang, Simone Rubinacci, Gabriel Pires, et al.. Polygenic risk score prediction accuracy convergence. Human Genetics and Genomics Advances, 2023, 6 (3), pp.100457. 10.1016/j.xhgg.2025.100457. pasteur-04368572v2

HAL Id: pasteur-04368572

https://pasteur.hal.science/pasteur-04368572v2

Submitted on 13 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
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abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
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<figure><img src="image_2.png" /></figure>
# Polygenic risk score prediction accuracy convergence

Léo Henches<fnref n="1" />, Jihye Kim<fnref n="2" />, Zhiyu Yang<fnref n="3" />, Simone Rubinacci<fnref n="4" />, Gabriel Pires<fnref n="1" />, Clara Albiñana<fnref n="5" />, Christophe Boetto<fnref n="1" />, Hanna Julienne<fnref n="1" />, Arthur Frouin<fnref n="1" />, Antoine Auvergne<fnref n="1" />, Yuka Suzuki<fnref n="1" />, Sarah Djebali<fnref n="6" />, Olivier Delaneau<fnref n="4" />, Andrea Ganna<fnref n="3" />, Bjarni Vilhjálmsson<fnref n="5" />,<fnref n="7" />, Florian Privé<fnref n="5" />, Hugues Aschard<fnref n="1" />,<fnref n="2" />*

## Abstract

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<fnref n="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.

<footnote n="1">¹ Institut Pasteur, Université de Paris, Department of Computational Biology, F-75015 Paris, France</footnote>

<footnote n="2">² Department of Epidemiology, Harvard T.H. Chan School of Public Health, Boston, USA</footnote>

<footnote n="3">³ University of Helsinki, Helsinki, Finland</footnote>

<footnote n="4">⁴ Department of Computational Biology, University of Lausanne, Lausanne, Switzerland</footnote>

<footnote n="5">⁵ National Centre for Register-Based Research, Aarhus University, Aarhus 8210, Denmark</footnote>

<footnote n="6">⁶ IRSD, Université de Toulouse, INSERM, INRAE, ENVT, Univ Toulouse III - Paul Sabatier (UPS), Toulouse, France</footnote>

<footnote n="7">⁷ Bioinformatics Research Centre, Aarhus University, Aarhus 8000, Denmark</footnote>

<footnote>* Corresponding authors</footnote>
# Polygenic risk score prediction accuracy convergence

Léo Henches,<fnref n="1" /> Jihye Kim,² Zhiyu Yang,³ Simone Rubinacci,³ Gabriel Pires,<fnref n="1" /> Clara Albiñana,⁴ Christophe Boetto,<fnref n="1" /> Hanna Julienne,<fnref n="1" /> Arthur Frouin,<fnref n="1" /> Antoine Auvergne,<fnref n="1" /> Yuka Suzuki,<fnref n="1" /> Sarah Djebali,⁵ Olivier Delaneau,⁶ Andrea Ganna,³ Bjarni Vilhjálmsson,⁴,⁷ Florian Privé,⁴ and Hugues Aschard<fnref n="1" />,²,<fnref n="8" />,*

## Summary

Polygenic risk scores (PRSs) models 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 among experts about their strengths and limitations. The performance of existing polygenic scores is still limited but is expected to improve with increasing GWAS sample sizes and the development of new, more powerful methods. Theoretically, the variance explained by PRS can be as high as the total additive genetic variance, but it is unclear how much of that variance has already been captured by PRS. Here, we conducted a retrospective analysis to assess progress in PRS prediction accuracy since the publication of the first large-scale GWASs, using data from six common human diseases with sufficient GWAS information. We show that although PRS accuracy has grown rapidly over the years, the pace of improvement from recent GWAS has decreased substantially, suggesting that merely increasing GWAS sample sizes may lead to only modest improvements in risk discrimination. We next investigated the factors influencing the maximum achievable prediction using whole-genome sequencing data from 125,000 UK Biobank participants and state-of-the-art modeling of polygenic outcomes. Our analyses suggest that increasing the variant coverage of PRS, using either more imputed variants or sequencing data, is a key component for future improvements in prediction accuracy.

## Introduction

Most common human diseases exhibit strong polygenic inheritance, characterized by a very large number of genetic variants with small effects. This scattered distribution of risk has severely hampered the initial goal of using genetic association studies for personalized medicine through individualized disease risk predictions, prevention strategies, and treatments.<fnref n="1" />⁻<fnref n="8" /> This issue was recognized early in the genome-wide association study (GWAS) era, and the community developed a strong case for genetic risk profiling based on polygenic risk scores (PRSs) derived from GWAS results.⁹ In its simplest form, a PRS for an individual is the summation of multiple single-nucleotide polymorphisms (SNPs) weighted by their effect sizes estimated from independent GWAS data. Initially, PRSs were constructed from a small number of independent genome-wide significant variants, but they have evolved to include thousands to millions of variants selected from full GWAS results using optimized selection criteria and weighting schemes.¹⁰,¹¹ As with the prediction of any highly multifactorial outcome, PRS accuracy largely depends on the sample size of the dataset used to estimate individual predictor effects. GWAS sample sizes—and therefore the predictive performance of

PRSs—have increased substantially over time. However, it remains unclear how much predictive accuracy has already been achieved for common human diseases and how much further improvement can be attained through future, larger GWAS.

Characterizing the PRS accuracy-sample size relationship in real data is challenging for two reasons. First, this relationship is expected to vary across disease parameters, including prevalence, heritability,¹²⁻¹⁴ and the distribution of effects at causal variants,¹⁵,¹⁶ which can be difficult to estimate in practice. Second, there is notable heterogeneity in the implementation of PRSs.¹⁷⁻¹⁹ Existing studies have used a variety of methods, models, data sources, and populations, including multi-ancestry²⁰⁻²⁴ and multi-phenotype approaches.²⁵⁻²⁷ This heterogeneity limits our ability to determine the key drivers of PRS improvement. A formal evaluation of PRS performance as a function of sample size requires GWASs conducted using populations of similar genetic ancestry, similar statistical tests, comparable numbers of input variants, and consistent modeling (e.g., including the same covariates). Furthermore, the PRS must be derived using the same approach and applied to a single test dataset that is not included in the GWAS used to build the PRS to avoid overfitting. This point is particularly challenging, as most existing disease GWASs

<footnote n="1">¹Institut Pasteur, Université de Paris, Department of Computational Biology, 75015 Paris, France; ²Department of Epidemiology, Harvard T.H. Chan School of Public Health, Boston, MA, USA; ³Institute for Molecular Medicine Finland (FIMM), University of Helsinki, Helsinki, Finland; ⁴National Centre for Register-Based Research, Aarhus University, 8210 Aarhus, Denmark; ⁵IRSD, Université de Toulouse, INSERM, INRAE, ENVT, University Toulouse III - Paul Sabatier (UPS), Toulouse, France; ⁶Department of Computational Biology, University of Lausanne, Lausanne, Switzerland; ⁷Bioinformatics Research Centre, Aarhus University, 8000 Aarhus, Denmark</footnote>

<footnote n="8">⁸Lead contact</footnote>

<footnote>*Correspondence: hugues.aschard@pasteur.fr</footnote>

<footnote>https://doi.org/10.1016/j.xhgg.2025.100457.</footnote>

<footnote>© 2025 The Authors. Published by Elsevier Inc. on behalf of American Society of Human Genetics.</footnote>

<footnote>This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</footnote>
## Main

Most common human diseases harbor a strong polygenic inheritance, characterized by a very large number of genetic variants of small effects. This scattered distribution of risk has severely hampered the initial goal of using genetic association studies for personalized medicine through individualized disease risk predictions, prevention strategies and treatments²,⁴. This issue has been recognized early on in the GWAS era, and the community has developed a strong case for genetic risk profiling on the basis of polygenic risk scores (PRSs), derived from genome-wide association studies (GWAS) results⁵. In its simplest form, a PRS for an individual is a summation of multiple single nucleotide polymorphisms (SNPs) weighted by their effect sizes estimated from independent GWAS data. Initially PRSs were constructed from a small number of independent genome-wide significant variants, but they have evolved to include thousands to millions of variants selected from the complete GWAS results, using optimized selection criteria and weighting schemes⁶,⁷. As for predicting any highly multifactorial outcome, the accuracy of PRS predictions mostly depends on the sample size of the dataset used to estimate individual variant effects. However, despite a substantial increase in GWAS sample size over time, the predictive performance of existing PRSs remains low, raising concerns about their value for clinical purposes⁸⁻¹⁰.

Several studies have estimated the potential for genetic risk prediction in polygenic diseases describing links between prediction accuracy, typically defined by the area under the receiver operating characteristic curve (AUC), disease prevalence and heritability¹¹⁻¹³. More recent studies also proposed theoretical frameworks to estimate future improvement in PRS-based prediction based on the distribution of effects at causal variants¹⁴,¹⁵. However, none of those examined performances retrospectively. This is partly explained by a lack of standard in the implementation of PRS, and the impact of various factors inducing heterogeneity in the PRS performances¹⁶,¹⁷. Here we examine how PRS prediction accuracy of six diseases evolved during the past 15 years and how existing models, state-of-the-art sequencing data, and functional annotation data can inform potential future improvements. We use only PRS derived from population of European ancestry. Using PRS derived in non-European population would be challenging because of the current lack of large-scale GWAS data. However, the linear relationship observed in cross-ancestry PRS transferability¹⁸,¹⁹ suggests the trends should remain consistent across populations.

## PRS prediction accuracy and sample size

We first pulled from the literature and curated previous reports of genetic risk score prediction accuracy, expressed as the AUC, for coronary artery disease, breast cancer, type 2 diabetes, Alzheimer disease, asthma, and obesity (Supplementary Note and Table S1). The studies spanned from 2006 to 2020 and the effective sample size $N_{eff}$ ranged from 981 to 339,224. The reported predictive power showed a very modest association trend with sample size and was instead characterized by substantial heterogeneity (Figure 1a). AUC increase was nominally significant for breast cancer ($\beta$ = 3.3 x10⁻⁴ per 1K increase in sample size, P=0.018) and almost nominally significance for type 2 diabetes ($\beta$ = 3.9x10⁻⁴, P=0.091) and obesity ($\beta$ =
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Study flowchart</figcaption></figure>

The primary analyses conducted in the study for the six diseases included assessing the accuracy of polygenic risk scores (PRSs) using the area under the receiver operating characteristic curve (AUC) as reported in the literature; estimating PRS performance using genome-wide association study (GWAS) summary statistics and a harmonized pipeline; evaluating the maximum AUC based on estimated disease parameters and UKB sequencing data; and analyzing the enrichment between genomic coverage and functional annotations.

are meta-analyses that include all available data. These constraints substantially limit the number of GWAS datasets that can be used. After an extensive literature review, we identified six diseases for which sufficient data meeting these criteria were available: coronary artery disease (CAD), breast cancer, type 2 diabetes, Alzheimer disease (AD), asthma, and obesity. We conducted a retrospective study to examine how PRS prediction accuracy for these six diseases has evolved with increasing GWAS sample sizes over the past 15 years. This evaluation was carried out using two independent datasets: the FinnGen cohort and non-European ancestry participants from the UK Biobank (UKB). We further investigated how existing models, state-of-the-art sequencing data, and functional annotation data may inform potential future improvements. An overview of the study design is presented in Figure 1.

# Material and methods

## GWAS data assembly

We collected publicly available GWAS summary statistics for six outcomes: type 2 diabetes, coronary heart disease, breast cancer, AD, asthma, and body mass index (BMI), for which we used to study obesity. The design of each study was carefully assessed to ensure it met stringent inclusion criteria for our analysis: (1) all studies had to include a majority of individuals of European ancestry; (2) studies with limited genetic coverage, such as exome-wide screening or genotyping chips without imputation (e.g., MetaboChip, ImmunoChip), were excluded; and (3) pheno-

type definitions had to be relatively homogeneous, although we retained some studies with heterogeneity for comparative purposes (e.g., early-onset asthma, AD proxy defined from parental status). In some meta-analyses, we also accessed GWAS results from individual cohorts, which were used as additional data points. After quality control filtering, 30 GWASs summary statistics with the required baseline data (coded allele, signed statistics, and *p* value) remained for analysis. We completed our panel by conducting five additional GWASs of modest sample size in the UKB using unrelated participants of European ancestry, with cases sampled from the entire cohort for four outcomes: breast cancer, CAD, asthma, and BMI. All 35 GWAS were harmonized and converted to hg38 using the *liftover* package.²⁸ As one of the primary goals was to compare the predictive accuracy of PRS scores in the FinnGen cohort, we retained only variants available in that dataset. Unless otherwise specified, we defined the effective sample size of binary GWAS outcomes as $N_{eff} = 4/[1/N_{case} + 1/N_{controls}]$, which re-scales cohorts with unequal case-control numbers to a common unit.²⁹ The list of GWAS used is presented in Table 1.

## Estimation of PRS prediction performances in real data

For each GWAS, we derived a vector of variant-specific PRS weights $\gamma = (\gamma_1, \gamma_2, \dots, \gamma_M)$, where $\gamma_i$ is the weight for variant *i* and *M* is the number of variants available in the GWAS. We used the LDpred2 approach¹¹ with the “auto” option, which automatically estimates hyperparameters, including PRS sparsity and SNP heritability. We employed 30 Gibbs sampling chains. Because the GWASs included predominantly individuals of European ancestry and in-sample linkage disequilibrium (LD) was not
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 Nₐff=4,522 to Nₐff=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 weights¹⁶,²⁰.

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 cohorts²¹ (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 cohort²² (Table S3). Difference in genetic ancestry induces lower absolute predictive performance, however, because the PRS portability across population is expected to be linear¹⁸,¹⁹, 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 AD²³. This likely induces reduced prediction in FinnGEN in our analysis.
<table><caption>Table 1. GWASs used to derive polygenic risk scores</caption><thead><tr><th>Phenotype</th><th>Reference</th><th>N<sub>case</sub></th><th>N<sub>cont</sub></th><th>N<sub>eff</sub></th><th>No. of variants<sup>a</sup></th></tr></thead><tbody><tr><td rowspan="7">CAD</td><td>UK Biobank (5K)</td><td>5,000</td><td>50,000</td><td>18,182</td><td>1,041,959</td></tr><tr><td>C4D et al.<sup>30</sup></td><td>15,420</td><td>15,062</td><td>30,478</td><td>540,233</td></tr><tr><td>Schunkert et al.<sup>31</sup></td><td>22,233</td><td>64,762</td><td>66,204</td><td>2,420,308</td></tr><tr><td>Nikpay et al.<sup>32</sup></td><td>60,801</td><td>123,504</td><td>162,973</td><td>9,455,779</td></tr><tr><td>Nelson et al.<sup>33</sup></td><td>71,602</td><td>260,875</td><td>224,727</td><td>9,020,475</td></tr><tr><td>Harst et al.<sup>34</sup></td><td>34,541</td><td>261,984</td><td>122,070</td><td>7,947,838</td></tr><tr><td>Harst et al.<sup>34</sup></td><td>122,733</td><td>424,528</td><td>380,832</td><td>7,947,838</td></tr><tr><td rowspan="6">Type 2 diabetes</td><td>Aragam et al.<sup>23</sup></td><td>181,522</td><td>984,168</td><td>613,021</td><td>20,073,070</td></tr><tr><td>Cai et al.<sup>35</sup></td><td>9,978</td><td>13,348</td><td>22,839</td><td>8,924,493</td></tr><tr><td>Morris et al.<sup>36</sup></td><td>12,171</td><td>56,862</td><td>40,101</td><td>2,473,442</td></tr><tr><td>Scott et al.<sup>37</sup></td><td>26,676</td><td>132,532</td><td>88,825</td><td>12,056,347</td></tr><tr><td>Xue et al.<sup>38</sup></td><td>62,892</td><td>596,424</td><td>227,571</td><td>5,053,016</td></tr><tr><td>Mahajan et al.<sup>39</sup></td><td>74,124</td><td>824,006</td><td>272,026</td><td>23,465,133</td></tr><tr><td rowspan="5">Breast cancer</td><td>UK Biobank (5K)</td><td>5,000</td><td>50,000</td><td>18,182</td><td>1,041,959</td></tr><tr><td>Michailidou et al.<sup>40</sup> (GWAS meta-analysis)</td><td>14,910</td><td>17,588</td><td>32,277</td><td>11,792,543</td></tr><tr><td>Michailidou et al.<sup>41</sup> (iCOGS GWAS)</td><td>46,785</td><td>42,892</td><td>89,508</td><td>11,792,543</td></tr><tr><td>Michailidou et al.<sup>42</sup> (OncoArray GWAS)</td><td>61,282</td><td>45,494</td><td>104,442</td><td>11,792,543</td></tr><tr><td>Zhang et al.<sup>43</sup></td><td>133,384</td><td>113,789</td><td>245,620</td><td>11,792,543</td></tr><tr><td rowspan="4">Alzheimer disease</td><td>Li et al.<sup>44</sup></td><td>753</td><td>736</td><td>1,489</td><td>391,067</td></tr><tr><td>Lambert et al.<sup>45</sup></td><td>17,008</td><td>37,154</td><td>46,669</td><td>7,053,170</td></tr><tr><td>Kunkle et al.<sup>46</sup></td><td>21,982</td><td>41,944</td><td>57,693</td><td>11,480,633</td></tr><tr><td>Jansen et al.<sup>47</sup></td><td>71,880</td><td>383,378</td><td>242,124</td><td>13,367,300</td></tr><tr><td rowspan="6">Asthma</td><td>UK Biobank (1K)</td><td>1,000</td><td>10,000</td><td>3,636</td><td>1,041,959</td></tr><tr><td>Shrine et al.<sup>48</sup></td><td>5,135</td><td>25,675</td><td>17,117</td><td>33,771,859</td></tr><tr><td>Moffatt et al.<sup>49</sup></td><td>10,365</td><td>16,110</td><td>25,228</td><td>567,590</td></tr><tr><td>Zhu et al.<sup>50</sup></td><td>14,085</td><td>76,768</td><td>47,606</td><td>7,488,536</td></tr><tr><td>Demenais et al.<sup>51</sup></td><td>23,948</td><td>118,538</td><td>79,692</td><td>2,001,257</td></tr><tr><td>Han et al.<sup>52</sup></td><td>64,538</td><td>329,321</td><td>215,851</td><td>9,572,557</td></tr><tr><td rowspan="6">BMI</td><td>UK Biobank (20K)</td><td>20,000</td><td>20,000</td><td>20,000</td><td>1,041,959</td></tr><tr><td>UK Biobank (80K)</td><td>80,000</td><td>80,000</td><td>80,000</td><td>1,041,959</td></tr><tr><td>Speliotes et al.<sup>53</sup></td><td>125,865</td><td>125,865</td><td>125,865</td><td>2,471,517</td></tr><tr><td>Locke et al.<sup>54</sup></td><td>322,206</td><td>322,206</td><td>322,206</td><td>2,554,638</td></tr><tr><td>Yengo et al.<sup>55</sup></td><td>795,640</td><td>795,640</td><td>795,640</td><td>2,336,270</td></tr><tr><td>Elsworth et al.<sup>56</sup></td><td>454,884</td><td>454,884</td><td>454,884</td><td>9,851,867</td></tr></tbody></table>

available, we used European descent participants from the UKB as the reference panel for LD derivation. The use of external LD data matched to the GWAS ancestry is expected to have a negligible impact on PRS accuracy.⁵⁷,⁵⁸ We restricted the analysis to GWAS variants overlapping with 1,054,330 HapMap3 variants.⁵⁹ This restriction was due to the computational cost and memory demands of LDpred2 (also existing for other PRS software), which scale quadratically with the number of variants. For each GWAS,

we also estimated per-variant effective sample sizes ($N_{eff,SNP}$) and filtered out variants with values lower than 50% or higher than 110% of the expected maximum to avoid miscalibration of regression coefficients.⁵⁷,⁶⁰ We then constructed PRSs in independent test datasets using the derived weights. The score for an individual was computed as $PRS = \gamma^t X$, where $X$ is a genotype matrix with alleles coded to match the original GWAS. Because most European ancestry cohorts were used in the original GWASs, we

<footnote><sup>a</sup>Total number of variants available in the original GWAS.</footnote>
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 models¹³ (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 approach¹⁴. 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 three-component 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 SBayesS²⁴, 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 GENESIS¹⁴ and SBayesS²⁴ 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, sumHer²⁵, LDSC²⁶, and MiXeR²⁷, to
focused our application on evaluating relative performance and trends in area under the receiver operating characteristic curve (AUC) using two datasets.

The first dataset included 16,000 unrelated UKB participants of non-European ancestry, categorized into 6 groups (Ashkenazi, Iranian, Indian, Chinese, Caribbean, and Nigerian) as defined by Privé et al.⁶¹ These data were also used for PRS fine-tuning and averaging weights across Gibbs sampling chains. The second dataset comprised 392,423 Finnish ancestry participants from the FinnGen cohort (see supplementary methods). Although genetic differences between training and test populations reduce absolute prediction performance, PRS portability across populations is expected to be approximately linear. Therefore, GWAS sample size vs. PRS performance trends should remain informative. For instance, the top GWAS hits between FinnGen and European samples showed high concordance (mean squared correlation of effect estimates = 0.56; Figure S1).

We evaluated PRS performance using the AUC. In the non-European UKB samples, AUCs were first computed separately for each ancestry group. Due to small case numbers per group, we meta-analyzed the results across groups using standard inverse-variance meta-analysis: $AUC_{combined} = \frac{\sum_{i=1,...,s}[AUC_i/\sigma_i^2]}{\sum_{i=1,...,s}[1/\sigma_i^2]}$, and $SE(AUC_{combined}) = \frac{1}{\sum_{i=1,...,s}[1/\sigma_i^2]}$, where $AUC_i$ and $\sigma_i^2$ represent the AUC and variance for population $i = 1...5$. Importantly, no additional conventional risk factors (e.g., clinical, demographic, environmental) were included in the models, so the reported AUCs reflect the marginal predictive power of polygenic models alone.

## Variance captured by genotyped and imputed variants

We examined the variance captured by genotyped and imputed variants using real genetic data and simulated genetic effects. Consider a standardized phenotype $Y$ drawn from a polygenic model and defined as the linear additive effect of $M$ standardized causal variants. Its total variance equals $V_Y = h^2 + V_e$, where $h^2$ is the genetic variance (heritability) and $V_e$ is the residual environmental variance. Assuming that genetic effects at causal variants are independent of variant correlations, heritability can be approximated as $h^2 = \sum_M \beta_i^2$, where $\beta_i$ is the effect of a standardized variant $i$. We estimated the proportion of $h^2$ that could be recovered under an infinite GWAS sample size when only a subset $S$ of variants has been genotyped, while the remaining $M \notin S$ are either imputed or missing. Two key metrics were considered: $h_I^2$, the additive genetic variance captured by all $M$ variants, both genotyped and imputed using advanced methods,⁶² and $h_G^2$, the genetic variance captured using only the subset $S$ of genotyped variants but accounting for the contribution of the $M \notin S$ untyped variants through LD. The first metric is defined as $h_I^2 = \sum_M \beta_I^2 h_{imput}^2$, where $h_{imput}^2$ is the squared correlation between the sequenced variant $i$ and its imputed value. The second metric is defined as $h_G^2 = \sum_S \beta_I^2 + \sum_M \beta_I^2 \rho_I^2$, where $\rho_I^2$ represents the squared correlation between the untyped variants $j$ and the set of genotyped variants. In theory, $h_G^2$ should closely approximate GWAS-based heritability ($h_{GWAS}^2$) as estimated by existing software.⁶³ The former ($h_I^2$) is expected to capture additional variance from untyped variants poorly tagged by the genotyped ones.

Both metrics were estimated using a subset of 125,152 UKB participants of European ancestry with both genome-wide genotyping and whole-genome sequencing data. For $h_I^2$, we used imputed SNP-array data from the UKB. Imputed variants were lifted over to

GRCh38, retaining 99.5% of sites. We then computed $r_{imput}^2$ between each sequenced variant and its imputed dosage using standard univariate linear regression. For $h_G^2$, we used genotyped variants available on the UKB Axiom array⁶⁴ as the baseline. The $\rho^2$ values were computed using adjusted squared correlation obtained from standard multiple linear regressions, where each non-genotyped variant was predicted from a set $\Omega$ of nearby genotyped variants within a $\pm 1.5$-Mb window. For simplicity, both $r_{imput}^2$ and $\rho^2$ were calculated using only chromosome 22, assuming that it is representative of the entire genome. This chromosome includes 12,968 genotyped variants. Of the 659,092 sequenced variants, 503,195 remained after filtering out those with minor allele frequency (MAF) < 0.001%. As expected, the two metrics were highly correlated ($cor(\rho^2, r_{imput}^2) = 0.71$), but the average $r_{imput}^2$ was substantially higher than $\rho^2$ ($r_{imput}^2 = 0.50, \rho^2 = 0.25$).

In our simulations, the proportion of heritability captured was derived using both $r_{imput}^2$ and $\rho^2$ and effect size $\beta = (\beta_1...\beta_M)$ drawn from a normal distribution under the alpha model, where the expected effect of a variant is proportional to its variance raised to a power $\alpha$.⁶⁵⁻⁶⁷ This model posits that rare variants have larger per-allele effects than common variants when $\alpha < 0$. The genetic effect for variant $i$ was drawn from $\beta_i|\rho_i \sim N(0, \sigma_{\alpha,i}^2 [2\rho_i(1 - \rho_i)]^\alpha)$, where $\rho_i$ is the MAF and $\sigma_{\alpha,i}^2$ is scaled to match a predefined heritability. In our analysis, we considered $\alpha$ values ranging from $-1.5$ to 0 and sampled $\sigma_{\alpha,i}^2$ uniformly over $[0, 1]$. Additionally, we explored an attenuated alpha model that reduces the influence of rare variants using an ad hoc iterative weighting function (see supplementary methods).

## Maximum achievable AUC

The expected maximum achievable AUC from a polygenic model is determined primarily by the proportion of genetic variance captured by the variants used to construct the PRS. This maximum can be derived using the approximation proposed by Wray et al.:ⁱ⁴ $AUC_{max} \approx \Phi((i - v)h^2) / \sqrt{h^2[(1 - h^2i(i - T)) + (1 - h^2v(v - T))]}$, where $h^2$ is the heritability on the liability scale, $\Phi$ is the cumulative density function of the normal distribution, $v$ is the height of the standard normal density at the threshold $T = \Phi^{-1}(1 - K)$ and with $i = z/K$ and $v = -z/(1 - K)$, where $K$ is the disease prevalence. We confirmed the validity of this approximation through simulation models involving independent causal variants with linear additive effect $h^2$ values in [0.2, 0.7] and prevalence $K$ in [0.01, 0.25] (supplementary methods). Conditional estimates of AUC were derived by substituting $h^2$ with either $h_G^2$ or $h_I^2$, themselves derived based on alpha drawn in $[-1.5, 0]$ and $\beta = (\beta_1...\beta_M)$ coefficients sampled from a normal distribution. For real-data analysis, disease prevalence values were sourced from the Centers for Disease Control and Prevention (CDC) website. We derived three estimates of heritability on the liability scale: (1) the total heritability derived from twins studies for CAD (0.55),⁶⁸ type 2 diabetes (0.72),⁶⁹ breast cancer (0.27),⁷⁰ AD after excluding the effect of APOE (0.49),⁷¹,⁷² asthma (0.70),⁷³ and BMI (0.75)⁷⁴; (2) $h_{GWAS}^2$, the heritability captured by GWAS variants and derived using five alternative approaches: SBayesS,⁷⁵ sumHer,⁷⁶ LD score regression (LDSC) regression,⁷⁷ GENESIS,¹⁵ and MiXeR⁷⁸ (supplementary methods); and (3) $h_I^2$, the heritability captured by genotyped and imputed variants, which requires an estimate of the total heritability, the proportion of heritability captured by genotyped and imputed variants given $\alpha$, and a value of $\alpha$.
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 cost-genomic coverage ratio²⁸ 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 variance²⁹) 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 studies³⁰⁻³⁵. 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 so-called alpha model—where the expected effect of a variant is proportional to its variance power $\alpha$ ³³,³⁶ (see Methods)— and derived two metrics: i) $h_I^2$, the additive genetic variance captured by both genotyped variants and variants imputed using advanced methods based on haplotype inference³⁷, 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 software³⁸. 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 $\alpha$=-1.5 to 96% for $\alpha$=0. The heritability captured by genotyped variants only is substantially smaller, with $h_G^2$ varying from 5% for $\alpha$=-1.5 to 90% for $\alpha$=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
For the total heritability, we used the twins studies estimates. For the proportion of heritability captured, we used $r_{input}^2$, the estimate derived using the UKB sequencing data. The choice of $\alpha$ was more challenging, and we ultimately used an *ad hoc* approach. Further details are provided below.

Most real-data estimates of $h_{GWAS}^2$ are derived from genotyped variants and a modest subset of imputed variants filtered for high quality (typical info score $\ge$0.8). As a result, assuming our estimation of the heritability captured by genotyped variants is valid, the previously described $h_G^2$ is expected to approximate $h_{GWAS}^2$ if provided a relevant value of $\alpha$. We assessed this equality using $\alpha$ derived from various approaches (SBayesS, sumHer, and individual-level data from the UKB), but we found large discrepancies. Ultimately, we adopted an *ad hoc* "best-fit" $\alpha$ approach: for each disease, we selected $\alpha$ so that $h_G^2$ equals the median of the five GWAS-based heritability estimates. This $\alpha$ was then used to derive $h_I^2$ and compute AUC predictions accordingly.

## Genetic coverage and functional annotations

We evaluated the association between a range of functional annotations and the imputation quality ($r_{input}^2$, the squared correlation between sequenced and imputed genotypes in the UKB). We used a total of 1,099 functional annotations pulled from multiple sources: baseline GENCODE annotations, including intron, gene, exon, coding DNA sequence (CDS), transcription start site (TSS), transcription termination site (TTS), and untranslated regions (UTRs); epigenetic features across tissues and cell types: transcription factor binding sites (TFBSs) and functional annotation of the mammalian genome version 5 (FANTOM5); and regulatory elements: promoters, enhancers, dyadic annotations from Roadmap, DNase I hypersensitive sites (DHSs) from two sources, and super-enhancers. The analysis was performed on the same dataset used to derive $h_I^2$ and $h_G^2$ (503,195 sequenced chromosome 22 variants from 125,000 UKB participants of European ancestry). Associations between each functional annotation $A_i$ and imputation quality were estimated using a standard univariate linear model: $r_{input}^2 \sim \delta_i A_i$. A sensitivity analysis was also conducted using a model adjusted for GENCODE annotations, $r_{input}^2 \sim \delta_i A_i + \sum_{k \in GENCODE} \delta_k A_k$, where $A_k$ denotes GENCODE category indicators.

## Results

### PRS prediction accuracy and sample size

To illustrate the challenge of characterizing the sample size-PRS accuracy relationship in real data, we reviewed the literature and curated previous reports of genetic risk score prediction accuracy, expressed as the AUC, for CAD, breast cancer, type 2 diabetes, AD, asthma, and obesity (Table S1; supplementary methods). The studies spanned from 2006 to 2023 and the effective sample size ($N_{eff}$) ranged from 981 to 453,912. The reported predictive power showed a very modest linear trend with sample size and was instead characterized by substantial heterogeneity (Figure 2A). AUC increases were nominally significant for breast cancer ($\beta = 2.5 \times 10^{-4}/1,000$ increase in sample size, $p = 0.0096$) and obesity ($\beta = 3.1 \times 10^{-4}, p = 0.0038$), but were not significant for the other outcomes. For example, there was no clear trend

for CAD ($\beta = -1.8 \times 10^{-5}, p = 0.90$) despite an effective sample size ranging from $N_{eff} = 4,522$ to $N_{eff} = 184,305$. This is likely due to several factors already discussed in the literature and complex to disentangle, including heterogeneity in the population characteristics (age, sex, fine-scale genetic ancestry within European population), disease definition, and the method used to derive the PRS weights.<sup>17,19</sup>

Prediction accuracy was then assessed using a harmonized pipeline, in which PRSs were derived from 35 curated GWASs and applied to two independent cohorts: FinnGen and non-European UKB participants. It shows a clear, non-linear increasing trend as a function of sample size, starting with a sharp rise for the first few studies, followed by a gradual decline in improvement (Figures 2B and S2; Tables S2–S4). The patterns are highly concordant across the two test datasets, displaying only the expected offset due to genetic ancestry differences between the training and test data.<sup>61,79</sup> The flattening of AUC improvement is especially striking for type 2 diabetes, obesity, breast cancer, and CAD. Asthma displays a noisier trend, potentially reflecting challenges in disease definition and diagnosis. AD shows a continuous increase in AUC for the non-European UKB samples, but a negligible increase for the most recent and largest GWAS in FinnGen, despite a 5-fold increase in effective sample size. This may be explained by the specificity of the AD PRS (Figure S3) and, in particular, the use of a proxy for disease status—a score based on the AD status of participants' parents—in this GWAS, which could introduce variability in heritability estimates of AD<sup>80</sup> and likely affect disease risk prediction.

Sensitivity analyses yield comparable results. Using alternative approaches to derive the PRS from the same curated GWAS data produces qualitatively similar prediction accuracy, but with higher heterogeneity (Figure S4; Table S5). Trends in prediction accuracy, measured by the coefficient of determination ($R^2$) and the odds ratios of the disease comparing the top 5% and top 1% PRS strata to the remaining population, are fully consistent with those observed for the AUC (Figure S5; Table S5), with additional variability only in the top 1% PRS, likely due to the modest sample size when using this stringent threshold. We also confirm the relevance of our stringent GWAS curation, in which we removed all GWASs displaying potential overlap with our testing samples. As shown in the experiment from Figure S6, even a small overlap between the training and test samples can artificially inflate prediction accuracy.

### Modeling and robustness of AUC convergence

The convergence of prediction accuracy toward its maximum can be demonstrated using simple theoretical models<sup>14</sup> (Figure S7). Predicting the convergence rate in real data is considerably more challenging and requires estimating multiple parameters of the disease genetic model, including heritability, polygenicity, the distribution of
variants³⁶. 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 (AUCtwin), GWAS-based heritability derived using five approaches (AUCGWAS, Table S5-S6), and heritability captured by all imputed variants (AUCImputed). 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 AUCGWAS across the five approaches (see Methods). The comparison of AUC is summarized in Figure 2d. First, the AUCGWAS are relatively close to the AUCs reported in recent studies (AUCcurr, 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 AUCcurr for T2D and asthma display sizeable gaps with AUCGWAS, 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 AUCtwin and AUCGWAS 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, AUCImputed is substantially higher than AUCGWAS 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 regions³⁹⁻⁴². 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 (r², the squared-correlation between sequenced and imputed variants used in the derivation of h²). 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).
<figure><img src="image_4.png" /><figcaption>**Figure 2.** PRSs predictive accuracy as a function of sample size</figcaption></figure>

(A) AUC reported in the literature for PRS as a function of the effective sample size across six diseases: coronary artery disease, type 2 diabetes, breast cancer, Alzheimer disease (AD), asthma, and obesity. The color gradient represents the number of variants used, ranging from a few top-associated variants (light green) to millions (dark blue), and the size is proportional to the logarithm of the effective sample size.

(B) AUCs for the 6 outcomes derived using a harmonized pipeline. PRSs were trained from 35 GWASs using the LDpred2 approach and tested with individual-level data from the FinnGen cohort (black) and in 6 non-European ancestry UKB populations (red, meta-analysis over 6 populations). The AUCs are plotted against the effective sample size of the corresponding GWAS. Missing values indicate instances where there was a sample overlap between the test and train sets. The 95% confidence intervals are shown as shaded red and gray for the UKB and FinnGen analyses, respectively.

causal genetic effects, and the dependence of those effects
on LD, functional annotations, and MAF. We assessed the
ability of GENESIS,¹⁵ an approach that utilizes some of
the aforementioned disease parameters, to predict the
observed AUC trend using each of the 35 GWASs summary
statistics. As shown in Figure S8A, the predictions diverge
substantially from the trend observed in FinnGen. This is
likely partly due to uncertainty in the estimated disease pa-
rameters (Figures S8B and S8C). Estimates of these parame-

ters, obtained from alternative tools,¹⁵,⁷⁵⁻⁷⁷,⁸¹ show similar
variability depending on the input GWAS, with confidence
intervals often not overlapping across GWASs for the same
outcome (Figures S8 and S9; Table S6). The reasons for this
variability are unclear, but until progress is made in
deriving these estimates, our ability to model the conver-
gence rate of PRS predictive power will likely remain
limited, highlighting the importance of conducting retro-
spective studies.
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 $r^2$ ($P = 1.8 \times 10^{-307}$), 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 ($P = 2.0 \times 10^{-131}$) with an average $r^2$ of 0.37 and 0.40 for exonic and non-exonic 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 ($P = 2.1 \times 10^{-14}$). 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 ($P = 2.5 \times 10^{-270}$, Fig. 3c and Table S9). However, other annotations positively associated with $r^2$ also display a significant enrichment for rare variants (e.g. intron, $P = 1.0 \times 10^{-8}$), 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$^{34,41}$, an assumption that is now commonly incorporated in disease parameter estimation tools$^{43}$. 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$^{16}$, 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 pre-processing and analysis pipelines. These analyses also show that the prediction accuracy of PRS
We next investigated the extent to which the observed decrease in AUC improvement with sample size might be confounded by increasing heterogeneity (e.g., variability in disease definition, clinical and environmental characteristics of the participants) arising from meta-analyzing an increasing number of studies. We first compared the observed AUC trend with the trend derived using GWAS training and test data sampled exclusively from UKB participants of European ancestry, thereby reducing potential heterogeneity. We focused on obesity, one of the few scenarios that allowed for achieving a reasonably large sample size within a single homogeneous cohort. Overall, the AUC from those experiments followed the same trend as those derived in FinnGen (Figure S10) and did not suggest any effect of heterogeneity on the AUC trend in these data. We also assessed the performance of PRSs derived from the CAD GWAS in predicting seven intermediate CAD phenotypes in FinnGen. Although the absolute accuracy increased almost linearly with stricter outcome definitions in the test set, the trends remained highly consistent across all intermediate phenotypes (Figure S11).

Further direct assessment of the impact of heterogeneity is challenging, as it would require repeating the analysis from Figure 2B across homogeneous population strata and refined disease definitions. Such data are typically not available; however, some complementary indirect evaluations are possible. Heterogeneity in GWAS design and population is expected to produce heterogeneity in variant-outcome associations, such that increasing heterogeneity with sample size should lead to reduced heritability and increased polygenicity. As shown in Figures S8 and S9, we did not observe any clear evidence of such effects across the GWAS used, nor any marked difference in the trends of genetic parameter estimates when compared to the aforementioned homogeneous UKB experiment (Figures S10A–S10C).

## Maximal achievable prediction

The maximum achievable AUC from a polygenic model relies primarily on variant coverage—that is, the proportion of genetic variability—and concurrently, the proportion of heritability, captured by the variants used to derive the PRS. Existing PRSs are built entirely from genome-wide genotyping arrays complemented by imputation, which offer cost-effective coverage of common genetic variants⁸² in very large cohorts. This implies that even with very large GWAS sample sizes, maximum prediction accuracy is still bounded by this sub-sampling of existing variants, as compared to using whole-genome sequencing data.⁶⁵,⁸³⁻⁸⁷ As shown in the theoretical models from Figure 3A, the proportion of total heritability captured by genotyped variants ($h_G^2$) can vary substantially depending on the relationship between effect sizes and MAFs, as parametrized by the so-called alpha model⁶⁵,⁶⁶ (Figures S12–S14). Using all imputed variants, including those with modest or poor imputation quality—typically

filtered out in GWAS studies—can recover a substantial share of total heritability. For example, assuming a random distribution of genetic effects across the genome, $h_I^2$ varies from 29% for $\alpha = -1.5$ to 96% for $\alpha = 0$. In comparison, $h_G^2$ varies from 5% for $\alpha = -1.5$ to 90% for $\alpha = 0$. Notably, previous studies have argued that the alpha model might overestimate the effect of rare variants.⁶⁶ To address this potential limitation, we devised an attenuated alpha model that implies a reduced contribution of rare variants (Figures S12B–S12D; supplementary methods). However, when comparing these attenuated models to the baseline alpha model using real data, we found no evidence for improved fit (Figures S15 and S16).

The proportion of heritability captured can be translated into maximum achievable AUC. As expected, this maximum increases with higher heritability and lower disease prevalence¹⁴ (Figure 3B). We compared the expected maximum AUC in real data for each of the six diseases using three heritability estimates: twins study heritability (AUCmax-twin), GWAS-based heritability derived from five approaches (AUCmax-GWAS; Table S7), and heritability captured by all imputed variants, regardless of their imputation quality (AUCmax-Imputed). Figure 3C presents all estimates in a single panel. First, the AUCmax-GWAS estimates are relatively close to the AUCs reported in recent studies (AUCcurr; Table S7) for CAD, breast cancer, AD, and obesity, confirming that increasing GWAS sample size for these outcomes is unlikely to dramatically improve prediction performance. Conversely, AUCcurr for T2D and asthma display sizable gaps compared to AUCmax-GWAS, suggesting that despite the trend observed in Figure 2B, future GWASs with larger sample sizes may still provide a slow but continuous improvement in prediction. Second, the gap between AUCmax-twin and AUCmax-GWAS is large for all outcomes except for breast cancer, suggesting that increasing variant coverage in future studies could dramatically improve the predictive power of PRSs for the former outcomes. Third, AUCmax-Imputed is substantially higher than AUCmax-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 predictive power without requiring costly sequencing data. As a sensitivity analysis, we also re-estimated all maximum achievable AUC values using prevalence data from other sources and found no qualitative impact on the results (Figure S17).

## Effect distributions and predicted performances

In previous analyses, we assumed that causal variants were randomly distributed across the genome and modeled only the relationship between effect sizes and MAFs. However, there is strong evidence that causal variants are highly enriched in certain functionally annotated regions.⁸⁸⁻⁹¹ Modeling this conditional distribution may affect both the convergence of the AUC and the expected maximum achievable AUC. Nonetheless, the precise relationship
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 size–
might 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 ancestry¹⁸,¹⁹ and phenotype definition²³,⁴⁴ 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 works³², 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<sup>14,15,27,33,45,46</sup>. 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
<figure><img src="image_5.png" /><figcaption>Figure 3. Heritability captured and maximum achievable AUC</figcaption></figure>

(A) The expected proportion of heritability captured by genotyped variants only (dashed line) and by both genotyped and all imputed variants, regardless of their imputation quality (solid line), conditional on the minor allele frequency (MAF)-effect size relationship (alpha). Individual-level data from the UKB were used to estimate the squared correlation between sequenced variants and either genotyped variants ($p^2$) or imputed variants ($r_{imput}^2$). Genetic effects were assumed to be distributed following an alpha model.

(B) The corresponding maximum achievable AUC for heritability of 0.1 and 0.5 and disease prevalences of 0.01 and 0.2.

(C) Estimates of the maximum achievable AUC for six outcomes: coronary artery disease, type 2 diabetes, breast cancer, AD, asthma, and obesity (using body mass index GWAS). These AUCs were derived based on US disease prevalence and various heritability estimates: twins studies (AUC<sub>Twin</sub>), heritability derived using five competitive approaches (LDSC regression, sumHer, SBayesS, GENESIS, and MiXer, labeled AUC<sub>GWAS</sub>) applied to the largest GWAS available for each disease, and twins study heritability captured by imputed variants (AUC<sub>Imputed</sub>). The black dashed lines indicate the most recent AUC estimates from real data in the literature, derived from approximately 1 million HapMap3 variants. For AD, heritability estimates and current AUC estimates exclude the APOE region.

between functional annotations and causal variants is not yet fully understood and likely depends on the specific outcome studied. Instead, we estimated the association between the quality of imputation and 1,099 annotations (Figure S18; Tables S8 and S9), including gene elements, DHSs, enhancers, and promoters. These annotations cover 0.001%-64.7% of the whole genome (Table S9).

The strongest associations with imputation quality were observed for gene elements (Figures 4A and S19A). CDSs were significantly negatively associated with $r_{imput}^2$ ($p = 1.8 \times 10^{-307}$), with average $r_{imput}^2$ values of 0.27 and 0.40 for CDSs and non-CDS variants, respectively. Exonic variants had lower imputation quality ($p = 2.0 \times 10^{-131}$), with average $r_{imput}^2$ values of 0.37 and 0.40 for exonic and non-exonic variants, respectively. Other annotation categories exhibited weaker but statistically significant negative and positive associations (Figure 4B). Enhancers, super-en-

hancers, DHSs, and TFBSs generally showed reduced imputation quality compared to the average of the genome, while variants within promoters tended to have slightly higher imputation accuracy. No clear enrichment was observed for specific cell types or tissues among other top significant annotations (Figure S20). The observed reduction in imputation quality within certain annotations may partly reflect an enrichment for rare variants. For example, coding regions displayed a significantly higher proportion of rare variants ($p = 2.5 \times 10^{-270}$; Figure 4C; Table S10). As discussed in prior research, causal variants may have lower LD with neighboring variants because of selective pressure,<sup>86,90</sup> an assumption that is now commonly incorporated in disease parameters estimation tools.<sup>92</sup> Lower LD leads to lower $r_{imput}^2$, meaning the observed lower coverage in CDSs, enhancers, DHSs, and TFBSs may indicate an
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)⁴⁷. 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 lifestyle⁴,⁴⁸,⁴⁹. For example, there are increasing evidence that heritability might vary with age and sex⁵⁰,⁵¹. 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 approaches⁵²⁻⁵⁴, where the predictive power can be boosted by partly circumventing the challenge of assembling very large sample size.
<figure><img src="image_6.png" /><figcaption>C</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>B</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>D</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 4.** Imputation quality varies across functional annotations</figcaption></figure>

(A–D) We computed the relationship between functional annotations and imputation quality, measured as the squared correlation ($r_{input}^2$) between true and imputed genotypes.

(A) The difference in the average $r_{input}^2$ for chromosome 22 across GENCODE annotations: introns, genes, exons, CDS (coding DNA sequence), tss (transcription start site), tts (transcription termination site), and UTR (untranslated region). Red bars indicate $\pm$2 SDs from the mean, encompassing 95% of the annotated regions.

(B) Standardized regression coefficients from the univariate association between the measured $r_{input}^2$ and each of the 1,092 functional annotations across 8 categories: TFBS (transcription factor binding site), FANTOM5 (functional annotation of the mammalian genome version 5) regulatory regions, promoters, enhancers, and dyadic regions from Roadmap, DHS (DNase I hypersensitive sites) derived from two studies, and super-enhancers. Horizontal black lines represent the average per category, and the red dashed lines indicate the significance threshold after correction for multiple testing.

(C) The cumulative distribution of variants for each GENCODE category as a function of the MAF.

(D) The distribution of annotation frequencies, grouped by category.

enrichment for rare, poorly tagged causal variants in these regions. In such scenarios, our estimates of the maximum achievable AUC from imputed variants (Figure 3) could be slightly overestimated. These findings further suggest that improving variant coverage, especially in underrepresented functional regions, will be crucial for advancing PRS predictive accuracy in future studies (Figure S20).

## Discussion

Polygenic risk prediction holds significant potential to transform the diagnosis, treatment, and prevention of many common human diseases. However, the timescale and magnitude of this transformation remain uncertain,

and concerns persist within the scientific community regarding the practical relevance of the PRS. These concerns are partly due to inconsistencies in reported performances metrics. Many existing studies lack complete documentation, and while guidelines have been proposed,¹⁷⁻¹⁹ adherence to these guidelines remains limited. Heterogeneity in methodology, study populations, and included covariates often impedes replication and complicates formal comparisons of PRS performance. As demonstrated in this study using six outcomes, even rigorous curation of published results may not be sufficient to clarify the relationship between prediction performance and study parameters.

Here, we demonstrate that the relationship between PRS prediction accuracy and GWAS sample size is unequivocal
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and highly replicable when using harmonized data pre-
processing and analysis pipelines. Our results show that
the accuracy of PRSs derived from existing GWASs has
begun to plateau for most studied diseases. As a conse-
quence, expanding current efforts (i.e., solely increasing
GWAS sample sizes) may result in only modest improve-
ments in predictive power. For some outcomes, including
CAD, breast cancer, and obesity, GWAS-based PRS perfor-
mance appears to be nearing the expected maximum
based on GWAS-derived heritability. For type 2 diabetes
and asthma, a larger gap remains between current and
theoretical maximum AUC values, suggesting that further
gains are still achievable, albeit requiring very large GWAS
sample sizes. Our analyses also suggest that without
large-scale whole-genome sequencing data, the ceiling
for prediction accuracy in GWAS-based PRS will remain
constrained by the variant coverage of GWAS array.
Importantly, this study supports previous findings<sup>85</sup> advo-
cating for the inclusion of both rare and common
imputed variants with modest imputation quality. These
can serve as an effective intermediate strategy to enhance
PRS performance, although this will require substantial
methodological development.

Several other factors could further improve PRS predic-
tive power and raise the ceiling on maximum achievable
AUC. First, the observed plateau in AUC may partly reflect
growing heterogeneity in large meta-analyses (e.g., looser
disease definitions or more genetically diverse popula-
tions), undertaken to allow for a broader inclusion of par-
ticipants. Although our results did not show strong evi-
dence of such an effect across the studied outcomes, the
influence of heterogeneity in genetic ancestry<sup>61,79</sup> and
phenotype definition<sup>80,93</sup> is well documented. Second,
we assumed homogeneity of genetic effects across individ-
uals. Future PRSs may benefit from accounting for poten-
tial effect heterogeneity linked to demographic character-
istics, basic health parameters, and lifestyle.<sup>7,94,95</sup> For
instance, heritability has been shown to vary with age
and sex.<sup>96,97</sup> Third, both the PRSs and their maximum ex-
pected AUCs depend on estimates of genetic architecture
parameters. As illustrated in the present study, estimates
of heritability, polygenicity, and the effect size-MAF rela-
tionship (alpha) vary substantially across methods and
input GWASs. Resolving these inconsistencies will be
key to improving both PRS construction and the modeling
of their predictive ceilings. Additionally, we had to
perform extensive preprocessing of GWASs summary sta-
tistics. Improved quality control and variants filtering<sup>60,98</sup>
may also yield more reliable parameter estimates.

This study has several limitations in its design. First,
we assumed that causal variants are randomly distributed
across the genome and only modeled the relationship
between MAF and effect size. This assumption is likely an
oversimplification of the true underlying model, as prior
research has shown enrichment of causal variants in specific
LD patterns and functional annotations.<sup>15,16,65,78,99,100</sup>
Rather than investigating specific models, we examined

the relationship between variant coverage and functional
annotations, observing both positive and negative enrich-
ment across features. However, except for CDSs that show
substantially poorer coverage, these differences did not
qualitatively affect our conclusions. Second, we focused
on PRSs derived from standard univariate GWASs and did
not consider more recent, potentially more powerful
methods, such as multi-ancestry<sup>20-24</sup> or multi-phenotype
approaches.<sup>25-27</sup> These methods may increase power by
improving effect size estimation, potentially mimicking
an effective sample size increase. However, they are not
likely to alter the fundamental convergence pattern toward
the theoretical maximum AUC. Third, our models included
PRSs only and did not incorporate additional non-genetic
risk factors. Thus, the reported AUCs reflect only the mar-
ginal predictive power of polygenic risk models. In practice,
the incremental value of a PRS depends on how much pre-
dictive power is already captured by conventional clinical
risk factors and their correlation with the PRS.<sup>101,102</sup> Fourth,
for real-data heritability estimates, we relied on twins studies
as proxies for total additive genetic variance. These esti-
mates may be biased due to shared environmental effects
and non-additive interactions, such as gene-gene and
gene-environment effects.<sup>103</sup> Fifth, we used prevalent cases
for our evaluations. Recent research suggests that PRS accu-
racy may differ when predicting incident cases across
various follow-up periods, potentially affecting PRS perfor-
mance trends.<sup>104</sup>

This study also has limitations in scope. It includes only
six diseases. Although we considered additional condi-
tions (e.g., chronic obstructive pulmonary disease, Crohn
disease, hypertension), data availability was insufficient to
support rigorous analysis. Future works could investigate
whether the trends observed here generalize to other dis-
eases and quantitative traits. Finally, our analysis focused
on PRSs derived from GWASs that included predomi-
nantly European ancestry participants. This limitation
was dictated by the current imbalance in genetic study
data across populations.<sup>105</sup> The extent to which our re-
sults apply to other ancestries remains uncertain. Howev-
er, our analysis of UKB participants of non-European ge-
netic ancestry (Figure 2B), along with recent work on
PRS portability,<sup>61,79</sup> suggests that similar trends are likely.
Nonetheless, assembling GWASs of large sample sizes in
non-European populations remains essential. PRS perfor-
mance is currently much lower in these groups, and
increasing their representation is crucial for improving
prediction.<sup>105</sup>

Data and code availability

Individual-level data from the UKB were accessed from the UKB Resource under application nos. 42260 and 66995. Individual-level data from FinnGen were conducted by coauthors from the University of Helsinki with privileged access.

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the NHGRI-EBI Catalog of human genome-wide association studies, and the FinnGen GWAS repository.

CDC disease prevalence: https://www.cdc.gov/datastatistics/index.html

FinnGen results: https://risteys.finnngen.fi

Functional annotations: https://github.com/gkichaev/PAINTOR_V3.0/wiki/2b.-Overlapping-annotations

GWAS catalog: https://www.ebi.ac.uk/gwas

HapMap3: https://www.sanger.ac.uk/resources/downloads/human/hapmap3.html

All analyses were conducted using existing open-source software programs, which are freely available from the following URLs:

GCTA: https://yanglab.westlake.edu.cn/software/gcta/

GENESIS: https://github.com/yandorazhang/GENESIS

LDpred2: https://privefl.github.io/bignsnpr/articles/LDpred2.html

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

MiXeR: https://github.com/precimed/mixer

SBayesS: https://cnsgenomics.com/software/gctb/

sumHer: https://dougspeed.com/sumher

During the preparation of this work, the authors used GPT-4o to correct grammatical and syntax errors. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

## Acknowledgments

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

## Author contributions

H.A. and L.H. conceived and supervised the project. L.H., J.K., Z. Y., S.R., and G.P. carried out the primary analyses. C.A., C.B., H.J., A.F., and Y.S. conducted the secondary analyses. Z.Y. and A.G. led the analyses involving individual-level data from FinnGen. S.R., O.D., and L.H. led the analysis involving individual-level sequencing data from the UKB. L.H., F.P., B.V., C.A., and H.A. designed the LDpred2 pipeline for PRS analysis. H.A. and L.H. co-wrote the manuscript, with input from all other authors. All authors contributed to discussions.

## Declaration of interests

B.V. serves on Allelica's international advisory board.

## Supplemental information

Supplemental information can be found online at https://doi.org/10.1016/j.xhgg.2025.100457.

Received: April 3, 2025

Accepted: May 12, 2025

## References

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2. Wald, N.J., and Old, R. (2019). The illusion of polygenic disease risk prediction. Genet. Med. 21, 1705–1707.

3. Polygenic Risk Score Task Force of the International Common Disease Alliance (2021). Responsible use of polygenic risk scores in the clinic: potential benefits, risks and gaps. Nat. Med. 27, 1876–1884.

4. Sud, A., Turnbull, C., and Houlston, R. (2021). Will polygenic risk scores for cancer ever be clinically useful? NPJ Precis. Oncol. 5, 40.

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**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.
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**Figure 2. Heritability captured and maximum achievable AUC**

Panel 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 estimate the squared-correlation between sequenced variants and either genotyped ($\rho^2$) 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.

<figure><img src="image_4.png" /></figure>
**Figure 3. Variability in imputation quality across functional annotations**

Relationship between functional annotations and the quality of imputation measured as the squared-correlation ($r^2$) between true and imputed genotypes. Panel a) displays the change in the average $r^2$ of chromosome 22 for GENCODE annotations: intron, gene, exon, CDS (coding DNA sequence), tss (transcription start site), tts (transcription termination site), and UTR (untranslated regions). Red bars indicate ±2 standard deviation from the mean and include 95% of the annotated regions. Panel b) displays standardized regression coefficients from univariate association between the measured $r^2$ and each of the 1,092 functional annotations from eight categories: TFBS (transcription factor binding site), FANTOM5 (functional annotation of the mammalian genome version 5) regulatory regions, promoters, enhancers, and dyadic from Roadmap, DHS (DNase I hypersensitive sites) derived from two studies, and super enhancers. Horizontal black lines indicate the average per category, and the red dashed lines indicate the significance threshold after correction for multiple testing. Panel c) shows the cumulative distribution of variants for each GENCODE category as a function of the minor allele frequency. Panel d) shows the distribution of frequencies of each annotation, grouped by category.

<figure><img src="image_5.png" /></figure>
# 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<sub>cases</sub>=5,000; N<sub>controls</sub>=50,000); coronary artery disease (N<sub>cases</sub>=5,000; N<sub>controls</sub>=50,000); asthma (N<sub>cases</sub>=1,000; N<sub>controls</sub>=10,000); and BMI (N=20,000 and N=80,000). All 34 GWAS were harmonized and converted to hg38 using the liftover package<sup>55</sup>. 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 $\gamma=(\gamma_1, \gamma_2 \dots \gamma_M)$, where $\gamma_i$ is the weight of variant *i* and *M* is the number of variants available in the GWAS, using the LDpred2 approach<sup>7</sup>, 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<sup>42</sup>. 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<sup>56</sup>.

We use those weights to construct genetic risk scores in two test datasets (Table S3). The scores were derived as $PRS = \gamma' X$, 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<sup>18</sup>. 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 unit⁵⁷. 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:
$$AUC_{combined} = \sum_{i=1...5} [AUC_i/\sigma_i^2] / \sum_{i=1...5} [1/\sigma_i^2], \text{ and } SE(AUC_{combined}) = 1/ \sum_{i=1...5} [1/\sigma_i^2], \text{ where } AUC_i \text{ and } \sigma_i^2 \text{ are the AUC and variance derived for population } i=1...5.$$

## 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 $h^2=\sum_M \beta_i^2$, where $\beta_i$ 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 $M \notin S$ 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 $M \notin S$ variants captured through linkage disequilibrium. The first metric $h_I^2$ is defined as $h_I^2=\sum_M \beta_i^2 r_i^2$, 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 $h_G^2=\sum_S \beta_i^2 + \sum_{M \notin S} \beta_j^2 \rho_j^2$, where $\rho_j^2$ 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 & Howie⁵⁸, which is expected to be an unbiased estimator of $r_I^2$. 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 $r_I^2$ 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 $r_I^2$ 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 array²¹ as our baseline. The $\rho_j^2$ were derived as the adjusted
squared-correlation obtained from a standard multiple linear regression with the *Im*() 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 $\rho^2$ 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 ($\rho^2$, $r^2$)=0.71) but, the average $r^2$ was substantially larger than $\rho^2$ ($r^2$=0.50, $\rho^2$=0.25, **Fig. S9b**).

In our simulations, the proportion of heritability captured was derived using both parameters ($r^2$ and $\rho^2$), and $\beta$=(β₁...β_M) coefficients drawn from a normal distribution using the alpha model³⁶. 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. ⁵⁹). In practice, the genetic effect of variant *i* is drawn from the following: $\beta_i | p_i N(0, \sigma_{g,\alpha}^2 [2p_i(1-p_i)]^\alpha)$, where $p_i$ is the minor allele frequency, and $\sigma_{g,\alpha}^2$ is a constant constraining the outcome heritability (**Fig. S10a**). In our analysis we considered $\alpha$ values in the range [-1.5 ; 0.0], and draw random $\sigma_{g,\alpha}^2$ 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 al³: $AUC_{max} \approx \Phi((|i-v|h^2)/\sqrt{h^2[(1-h^2i|i-T|)+(1-h^2v|v-T|)]})$, where $h^2$ is the heritability on the liability scale, $\Phi$ is the cumulative density function of the normal distribution, $z$ is the height of the standard normal density at the threshold $T=\Phi^{-1}(1-K)$, 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 $\beta=(\beta_1...\beta_M)$ 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)⁶⁰, type 2 diabetes (0.72)⁶¹, breast cancer (0.27)⁶², Alzheimer disease after excluding the effect of APOE (0.49)⁶³,⁶⁴, asthma (0.70)⁶⁵, and body mass index (0.75)⁶⁶; ii) $h_{GWAS}^2$, the heritability captured by GWAS variants derived using five alternative approaches: SBayesS²⁴, sumHer²⁵, LDSC regression²⁶, GENESIS¹⁴, and MiXeR²⁷ (**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
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 result, 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 $h_G^2$ equals the median of $h_{GWAS}^2$ derived over the five aforementioned approaches (Table S6). This “Best fit” alpha was used to derive $h_I^2$ 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 $h_I^2$ and $h_G^2$ (503,195 variants from chromosome 22 from 125K UK Biobank participants of European ancestry).

## 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. Individual-level data from FinnGEN were conducted by co-authors from the University of Helsinki with privileged access.

## URL resources

- GWAS catalog: https://www.ebi.ac.uk/gwas/

- GCTA: https://yanglab.westlake.edu.cn/software/gcta/

- SBayesS: https://cnsgenomics.com/software/gctb/

- MiXeR: https://github.com/precimed/mixer

- GENESIS: https://github.com/yandorazhang/GENESIS

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

- sumHer: https://dougspeed.com/sumher/
FinnGen results: https://risteys.finnngen.fi/

HapMap3: https://www.sanger.ac.uk/resources/downloads/human/hapmap3.html

CDC disease prevalence: https://www.cdc.gov/datastatistics/index.html

Functional annotations: https://github.com/gkichaev/PAINTOR_V3.0/wiki/2b.-Overlapping-annotations

## 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 thank 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).

## 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.