Dr Eric Laffon
email: elaffon@u-bordeaux.fr

Thomas Hummel

R Marthan

Charlotte Sinding

Modelling analysis of centroid curves of olfactory habituation in humans

Keywords: Human olfactory habituation, Modelling analysis, Curve fitting

Olfactory habituation (OH) can be considered an unconscious signal filtering • Three centroid curves reflecting high-middle-low OH have been previously published • Adjusting parameters of a general derived equation allowed us to fit each curve • High correlation coefficients of 0.9997-0.9995-0.9962 were found

Introduction

Olfactory habituation (OH) can be considered an unconscious filter whose role is to avoid brain overstimulation [START_REF] Thompson | Habituation: a model phenomenon for the study of neuronal substrates of behaviour[END_REF][START_REF] Wilson | Habituation of odor responses in the rat anterior piriform cortex[END_REF][START_REF] Poellinger | Activation and habituation in olfaction-an fMRI study[END_REF][START_REF] Thompson | Habituation: a history[END_REF][START_REF] Rankin | Habituation revisited: an updated and revised description of the behavioral characteristics of habituation[END_REF]. New determinants of OH have been reported in humans by Sinding et al., showing that reduced responsiveness to a continuous or repetitive stimulation depends on some properties of the odorant, notably trigeminality and associated physicochemical characteristics [START_REF] Sinding | New determinants of olfactory habituation[END_REF]. In these previous experiments, the course of odor intensity of 32 different odorants, which differed in physicochemical characteristics and were presented at constant flow during 120s, was rated continuously by 58 participants. A k-means clustering analysis of the 1856 recorded curves allowed the authors to partition the data into 3 clusters, thus providing 3 normalized centroid curves representing high, middle and low habituation (HH, MH, LH), respectively. The current study focuses on the HH -MH -LH experimental data that were re-analyzed by means of a mathematical modelling analysis in order to provide further model-based information about the processes and mechanisms underlying OH centroid curves in humans.

In line with Sinding et al. [START_REF] Sinding | New determinants of olfactory habituation[END_REF], we used, in the current study, the term habituation for the perceptual decrement of odor intensity [START_REF] Thompson | Habituation: a history[END_REF][START_REF] Pellegrino | Habituation and adaptation to odors in humans[END_REF], and adaptation when addressing the underlying mechanisms. The term peripheral adaptation encompasses all the adaptation mechanisms that may occur at the receptor level, and the term central adaptation encompasses all the adaptation mechanisms that tune the olfactory signal coming up from the olfactory neurons [START_REF] Pellegrino | Habituation and adaptation to odors in humans[END_REF]. Likewise, in this study, 2 adaptation mechanisms were considered, namely, intensity-limiting and intensity-decreasing mechanisms, which were not assigned to either peripheral or central adaptation, respectively. More specifically, the term of intensity-limiting mechanism referred to the situation where the odor intensity (i.e., the signal) under constant odorant-concentration conditions cannot increase with time above a certain value. The term of intensity-decreasing mechanism referred to the situation where the odor intensity (i.e., the signal) decreases with time even if the odorant concentration is constant. Noteworthy, this definition only considered a time-dependency of the two mechanisms and investigating the role of a dose-dependency was out of the scope of the current study, as well as of the previous one [START_REF] Sinding | New determinants of olfactory habituation[END_REF].

Although the comparison of HH -MH -LH centroid curves emphasized the differential filtering of the olfactory signal, depending on the odorant, we nevertheless suggest that the 3 different centroid curves share a common set of processes, each involving intensitylimiting and intensity-decreasing mechanisms, without assigning, a priori, any process or any mechanism to either peripheral or central adaptation. Therefore, the aim of the current study was to derive a general equation of the OH centroid curves from a mathematical modelling analysis, involving intensity-limiting and intensity-decreasing parameters whose value can be adjusted to fit each experimental OH centroid curve.

Material and methods

Mathematical modelling analysis

We assume that the rate of change in the OH signal, i.e., dS(t)/dt, can be written as:

dS(t)/dt = L(t) × N0 -kdec × S(t) (1)
where (i) L(t) represents an intensity-limiting time function, (ii) kdec represents an intensitydecreasing time constant (in s -1 ) and (iii) N0 is the odorant molecule amount in the nasal cavity that is constant with time since the OH experiments were performed under continuous delivery of odorant at constant concentration [START_REF] Sinding | New determinants of olfactory habituation[END_REF]. Let us further assume that L(t) can be expressed by means of a sum of exponentially time-decreasing functions (Li(t); amplitude: Li, in arbitrary unit; time constant: kLi, in s -1 ):

L(t) = Σ Li (t) = Σ Li × exp(-kLi×t) (2)
As a consequence, the solution of Equation ( 1) is a sum of separate processes, i.e., Si(t), each involving its own intensity-limiting parameters (Ai, kLi), but all having the same intensitydecreasing time constant (kdec):

S(t) = Σ Si(t) = Σ Si ×[exp(-kLi×t) -exp(-kdec×t)] / [kdec -kLi] (3)
where Si = N0 × Li (in arbitrary unit: a.u.).

Fitting of OH centroid curves

We preliminarily verified the Equation-( 1) assumption of the 1 intensity-decreasing time constant (kdec) by fitting the last part of the HH curve with a mono-exponentially decreasing function (GraphPad Prism software, version 5.00).

Then, since intensity ratings previously reported by Sinding et al. always started at level 6 (by using a device that transformed pressure into a signal ranging between 0 and 10), the fitting of each experimental OH curve was performed by using nonlinear regression (GraphPad Prism software, version 5.00) from this level and over the entire stimulation period of 120 s. Different fittings of Equation ( 3) on HH -MH -LH experimental data were performed assuming L(t) involves 1 or a sum of several intensity-limiting functions, in other words, assuming S(t) involves 1 or a sum of several processes.

Statistical analysis

The fittings allowed us to assess parameters involved in Equation ( 3), i.e., Si, kLi and kdec for each HH -MH -LH curve, respectively, along with their standard deviation (SD). In the framework of a two-process OH signal, i.e., involving the 5 parameters S1,2, kL1,2 and kdec, standard deviation of the ratio S1/S2 was computed from the following equation for each HH -MH -LH curve:

(SDS1/S2) 2 = (S1/S2) 2 [(SDS1/S1) 2 + (SDS2/S2) 2 ] (4)
A weighted mean value of kL1 was obtained from the 3 values and their SD assessed from the 3 HH -MH -LH curves, as:

< kL1> = SDkL1 2 [kL1-HH/(SDkL1-HH) 2 + kL1-MH/(SDkL1-MH) 2 + kL1-LH/(SDkL1-LH) 2 ] ( 5
)
with SDkL1 2 computed from the equation:

SDkL1 2 = [ (1/SDkL1-HH) 2 + (1/SDkL1-MH) 2 + (1/SDkL1-LH) 2 ] -1 (6)
For all statistical tests, a P-value below 0.05 was considered to be statistically significant.

Confidence intervals of the parameter estimates were calculated as 1.96×SD, with 95% reliability.

Results

The -466 -465; P < 0.001), respectively (graphs of one-process HH -MH -LH not shown). We therefore did not further probe a three-process fitting, and S(t) can be expressed as:

S(t) = S1(t) + S2(t) = S1×[exp(-kL1×t) -exp(-kdec×t)] / [kdec -kL1] + S2×[exp(-kL2×t) -exp(-kdec ×t)] / [kdec -kL2] (7)
Figure 2 illustrates the respective part of intensity-limiting and intensity-decreasing mechanisms in the HH signal. 3a, b, c separately shows S1(t) and S2(t) expressed in the right hand side of Equation [START_REF] Pellegrino | Habituation and adaptation to odors in humans[END_REF]. In other words, the sum "S1(t) + S2(t)" of the curves shown in Figure 3a, 3b and 3c is the curve shown in Figure 1a, 1b and 1c for HH -MH -LH, respectively. Figure 4 shows the comparison between S1(t) of HH and S1(t) of both MH and LH, after arbitrarily setting the kdec value of the latter to that of the former (Table 1).

Discussion

Sinding et al. previously showed that the unconscious filter of OH, which is designed to avoid brain overstimulation, depends on the odorant [START_REF] Sinding | New determinants of olfactory habituation[END_REF]. The authors illustrated this finding with 3 normalized centroid curves of different shapes, characterizing High Habituation, Medium Habituation and Low Habituation. These OH curves were obtained under constant olfactory stimulation, but all exhibited a similar pattern consisting in a peak followed by a decrease. Actually, this pattern can be interpreted as a combination of intensity-limiting and intensity-decreasing mechanisms, as illustrated in Figure 2 for HH. An intensity-limiting mechanism alone would lead the OH signal to reaching a plateau, even under conditions of a constant stimulation with an odorant, that differs from that of the maximum intensity perceived reported in literature [START_REF] Chastrette | Modelling the human olfactory stimulusresponse function[END_REF]. The intensity-decreasing mechanism, which is effective simultaneously with the former, modifies this plateau by inducing a decrease. As a result, combining intensity-limiting and intensity-decreasing mechanisms generates a peak then followed by a decrease. Therefore, the current study aimed at deriving a general equation from a mathematical modelling analysis involving these 2 mechanisms (Equations (2-3-7)). 7)) data fitting were high: R = 0.9997 -0.9995 -0.9962 (P < 0.001) for HH -MH -LH (Figure 1), respectively. Moreover, for each quantitative parameter and for each OH degree, the SD was small in comparison with its estimate (Table 1). Hence, we believe that these features support the relevance of the proposed mathematical modelling analysis of the OH centroid curves in humans.

Equation [START_REF] Pellegrino | Habituation and adaptation to odors in humans[END_REF] shows that the OH centroid curves result from the sum of 2 processes, i.e., "S1(t) + S2(t)", that are related to 2 intensity-limiting functions, i.e., L1(t) and L2(t) (Equation ( 2)), that exponentially decrease with a different time constant (Table 1).

Comparison of S1(t) and S2(t) between HH, MH and LH in Figure 3 apparently indicates that both are different in relation to the degree of OH. However, after arbitrarily setting the kdec value in both MH and LH to that of HH (Table 1: kdec = 0.02286 s -1 ), "corrected" S1(t) of MH and LH then appears to be close to that of HH, within 95%-CI limit of HH in Figure 4. This finding is consistent with the comparison of S1 and kL1 values found for HH, MH and LH in Table 1. Thus, the difference occurring in the decreasing part of S1(t) between HH, MH and LH (Figure 3) is related to the value of the intensity-decreasing time constant that diminishes from HH to LH: the lower the kdec value, the shallower the decreasing part of S1(t) and, in terms of OH, the more prolonged the perception of odorant intensity. We therefore suggest that, whatever the OH degree, a first, shared, intensity-limiting mechanism occurs that is, nevertheless, tuned by the specific intensity-decreasing time constant of the OH degree. In contrast, S2 and kL2 values were significantly different between HH, MH and LH, respectively, ruling out such a sharing feature for S2(t) (Table 1). Furthermore, whereas S2(t) reinforces the S1(t) process in HH (Figure 3a), it plays a distinct role from S1(t) in LH (Figure 3c). Indeed, the lower the OH degree, the lower the S2 and kL2 value (Table 1) and the weaker the role of the second process, thus allowing prolonged perception of odor intensity, although at a low level. This prolonged perception in LH is strengthened by a low value of the intensity-decreasing time constant, in line with what has been above commented for S1(t).

An interpretation of the 2 processes involved in the 3 OH centroid curves, i.e., S1(t) and S2(t), may be hypothesized. Figure 3 shows that S1(t) plays a major role in the rising part and in the peak of each OH centroid-curve. We suggest that S1(t) is very likely affected by the somewhat problematic onset ratings (even beginning at y = 0.6) and that, since the current modelling analysis established that S1 and kL1 values were found to be shared by the whole OH centroid curves, these imperfect ratings affected in a similar manner both HH, MH and LH centroid curves. As a consequence, we suggest that the first intensity-limiting mechanism, and, hence, S1(t), may be mainly related to the experimental design involving the whole unknown signal-transmission features occurring from the olfactory epithelium to the response given by the subject, who, furthermore, started intensity ratings at level 6 by using a device that transformed pressure into a signal ranging between 0 and 10. In other words, we suggest that S1(t) is unlikely related to the olfactory habituation itself and, rather, may possibly be of artifactual origin. In contrast, we hypothesize that S2(t) plays a major role in the OH degree, since S2 and kL2 values were found to be significantly different between HH, MH and LH, respectively. However, its origin is not clear and, at this stage of our work, S2(t) cannot be specifically assigned to either peripheral or central adaptation, respectively. Due to the limited knowledge on the central processes involved in adaptation [START_REF] Pellegrino | Habituation and adaptation to odors in humans[END_REF], we can assume that both central and peripheral adaptation very likely occur in combination for achieving OH during a 2-minute continuous odorant-stimulation condition. Finally, it should be noted that the mathematical expression of S1(t) and S2(t) is similar to that of the conductance change due to a presynaptic event [START_REF] Linster | Odor-specific habituation arises from interaction of afferent synaptic adaptation and intrinsic synaptic potentiation in olfactory[END_REF], which thereby involves conductance-limiting and conductancedecreasing mechanisms.

The interpretation of the mathematical model analysis is limited regarding the amplitudes of S1(t) and S2(t), that is, about S1 and S2 in Equation [START_REF] Pellegrino | Habituation and adaptation to odors in humans[END_REF]. This is because the HH -MH -LH curves were previously normalized before being fitted. Therefore, the amplitude comparison was limited to the S1/S2 ratio that was significantly different between HH, MH and LH, respectively (95% reliability). This, nevertheless, remains of limited interest in contrast to comparing absolute values of the corresponding time constants kL1 and kL2 (Figure 3; Table 1). It is worth noting that S1,2 were expressed as S1,2 = L1,2 × N0, and then if the S1/S2 ratio is known, the L1/L2 ratio is known too. Actually, L1,2 involves (i) the total number of different receptors recruited by an arbitrary odorant molecule that might depend on the process and on N0 [START_REF] Chastrette | Modelling the human olfactory stimulusresponse function[END_REF], respectively, and (ii) the whole unknown signal-transmission features occurring from the olfactory epithelium to the intensity ratings that might depend on the process.

Since the current modelling analysis allowed us to distinguish between S1(t) and S2(t), and that S2(t) may play a major role in the OH degree, we suggest that a future goal could be to implement the current modelling analysis in OH centroid curves obtained for different odorants, focusing on S2(t), that is, on S2, kL2 and kdec . In such a framework, curve fitting might be simplified in comparison with the current one by assessing 4 parameters (i.e., S1,2, kL2 and kdec), instead of 5 parameters, since the current results allow us to assume that kL1 is identical for any odorant. Determinants of OH that have been previously reported could be reexamined, as well as combinations of factors [START_REF] Sinding | New determinants of olfactory habituation[END_REF]. Furthermore, a dose-dependency of the intensity-limiting and intensity-decreasing mechanisms, and, hence, of S1(t) and S2(t), could be investigated. Dose-dependent changes in the parameter values of the mechanisms may indeed be expected. The efficiency of the clustering analysis, and, hence, the accuracy of the proposed mathematical modelling analysis, will require a great amount of data for such investigations in order to get rid of random and unwanted factors, including technical and emotional factors [START_REF] Lötsch | Machine learning in human olfactory research[END_REF].

To conclude, for the first time, to the best of our knowledge, a mathematical modelling analysis is proposed enabling us to fit different degrees of human OH corresponding to 3 centroid curves extracted from a clustering analysis [START_REF] Sinding | New determinants of olfactory habituation[END_REF]. By simply adjusting 5 parameters involved in the same general equation, fitting correlation coefficients of HH -MH -LH curves were high: 0.9997 -0.9995 -0.9962, respectively. The general equation suggests that each OH centroid curve results from the simultaneous effect of 2 separate processes, i.e. S1(t) and S2(t), each involving its own intensity-limiting mechanism but having the same intensitydecreasing mechanism, respectively. The first process, i.e. S1(t), is of unclear, possibly artifactual origin, and, thereby, is unlikely related to the olfactory habituation itself. In contrast, we suggest that the second process, i.e. S2(t), which appeared to depend on the odorant, plays a major role in the OH degree and cannot be assigned, a priori, to either peripheral or central adaptation, respectively. The proposed mathematical modelling analysis emphasizes that relevant olfactory research data can be obtained in a non-invasive manner, putting forward the usefulness of the clustering-analysis tool.

  preliminary fitting of the last part of the HH curve with a mono-exponentially decreasing function, yielded a correlation coefficient of R = 0.9989 (range 50 -120 s; graph not shown), thus allowing us the fitting of Equation (3) on HH -MH -LH experimental data. The highest correlation coefficient of Equation-(3) fitting on HH -MH -LH experimental data was obtained when S(t) involves two processes: R = 0.9997 -0.9995 -0.9962 for the graph displaying HH -MH -LH experimental versus theoretical data (Figure 1a,b,c; n = 465

Adjusting 4

 4 intensity-limiting and 1 intensity-decreasing parameters of this general equation enabled us to fit each HH -MH -LH centroid curves, respectively. The finding of 1 intensity-decreasing time constant, i.e. kdec, for each HH -MH -LH curve, respectively, confirmed the preliminary fitting of the last part of the HH curve with a mono-exponentially decreasing function and, thereby, confirmed the Equation-(1) assumption of the 1 intensitydecreasing time constant. Correlation coefficients of experimental versus theoretical (Equation (

Figure 1 .

 1 Figure 1. Fitting of intensity-versus-time OH centroid curves. Intensity in arbitrary unit (a.u.) and time in second (s). Experimental data (dotted line) and fitting outcomes of Equation (7) (full line) for a) HH (R = 0.9997), b) MH (R =0.9995) and c) LH (R = 0.9962),respectively (P < 0.001 for all curves). All fittings started at y = 0.6 since subjects were asked to start evaluating intensity at the level 6 (range 0 to10).

Figure 2 .Figure 3 .

 23 Figure 2. Intensity-limiting versus intensity-decreasing mechanisms for HH.Fitting outcomes of Equation (7) already shown in Figure1a(full line), versus outcomes of Equation[START_REF] Pellegrino | Habituation and adaptation to odors in humans[END_REF] after setting kdec to zero (dashed line).

Figure 4 .

 4 Figure 4. Comparison between S1(t) of HH, MH and LH. S1(t) of HH, MH and LH was compared after arbitrarily setting the same kdec value to both (full line); 95% CI of the HH S1(t) curve is shown (dotted lines).

  of HH, MH and LH (a.u.)

Table 1

 1

summarizes the fitting outcomes of the 5 parameters, i.e., S1,2, kL1,2 and kdec, for HH -MH -LH. Confidence intervals (CI; with 95% reliability) calculated from SD in Table

1

indicate that S2 and kL2 were significantly different between HH, MH and LH, respectively. Moreover, the ratio S1/S2 was significantly different between HH, MH and LH: 6.62 ± 0.39, 11.57 ± 1.00 and 18.07 ± 2.48 (± 95% CI), respectively. A weighted mean kL1 value was computed from HH -MH -LH data: <kL1> = 0.35710 ± 0.00451 s -1 , that is, with ± 1.3% relative measurement uncertainty (95% reliability). Figure

Table 1 . Fitting results (±SD) of intensity-versus-time OH curves, by using Equation (7).

 1

		High Habituation	Middle Habituation	Low Habituation
	S1 (a.u.)	0.245 ±0.003	0.266 ±0.002	0.271 ±0.002
	kL1 (s -1 )	0.32463 ±0.00718	0.36156 ±0.00447	0.35525 ±0.00287
	S2 (a.u.)	0.037 ±0.001	0.023 ±0.001	0.015 ±0.001
	kL2 (s -1 )	0.05416 ±0.00201	0.03220 ±0.00099	0.01042 ±0.00007
	kdec (s -1 )	0.02286 ±0.00027	0.01253 ±0.00020	0.01042 ±0.00049

Author Contributions

Mathematical modelling analysis: E.L.; interpretation of the mathematical modelling analysis: E.L., C.S; design of the study, discussion, writing and reviewing of the manuscript: E.L., T.H., R.M., C.S.

Competing interests

The authors declare no competing interests.


M Graber

S Mohr

L Baptiste

G Duloquin

C Blanc-Labarre

A S Mariet

M Giroud

Y Béjot

Air pollution and stroke A new modifiable risk factor is in the air

Keywords: Stroke, ischaemic and hemorrhagic stroke, mortality, air pollution and air pollutants

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

Introduction

Exposure to air pollution is now well recognized by scientists, media and the population as a major public health issue [START_REF] Desikan | Outdoor air pollution as a possible modifiable risk factor to reduce mortality in post-stroke population[END_REF][START_REF] Delpont | Environmental air pollution : an emerging risk factor for stroke[END_REF][START_REF]Air quality deteriorating in many of the world's cities[END_REF][START_REF] Anderson | Clearing the air: a review of the effects of particular matter air pollution on human health[END_REF][START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF][START_REF] Lee | Air pollution and stroke[END_REF]. It ranks within the top five risk factors for mortality in emerging countries [START_REF] Desikan | Outdoor air pollution as a possible modifiable risk factor to reduce mortality in post-stroke population[END_REF][START_REF] Delpont | Environmental air pollution : an emerging risk factor for stroke[END_REF][START_REF]Air quality deteriorating in many of the world's cities[END_REF][START_REF] Anderson | Clearing the air: a review of the effects of particular matter air pollution on human health[END_REF][START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF][START_REF] Lee | Air pollution and stroke[END_REF]. Forty per cent of all deaths could be induced by outdoor air pollution [START_REF] Anderson | Clearing the air: a review of the effects of particular matter air pollution on human health[END_REF][START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF] and 89 % of death attributed to ambient air pollution are observed in low and medium income countries as a result of rapid industrialization [START_REF] Landrigan | The Lancet Commission on pollution and health[END_REF]. Robust data confirm the high risk of morbidity and mortality due to particulate matter measuring less than 2.5 µm (PM2.5) for myocardial infarction and congestive heart failure [START_REF] Anderson | Clearing the air: a review of the effects of particular matter air pollution on human health[END_REF][START_REF] Lee | Air pollution and stroke[END_REF][START_REF] Mustafic | Main air pollutants and myocardial infarction : a systematic review and meta-analysis[END_REF], lung disease and cancer [START_REF] Desikan | Outdoor air pollution as a possible modifiable risk factor to reduce mortality in post-stroke population[END_REF][START_REF] Anderson | Clearing the air: a review of the effects of particular matter air pollution on human health[END_REF][START_REF] Raaschou-Nielson | Air pollution and lung cancer incidence in 17 European cohorts : prospective analysis from the European Study of Cohorts for Air Pollution Effects (ESCAPE)[END_REF], diabetes [START_REF] Desikan | Outdoor air pollution as a possible modifiable risk factor to reduce mortality in post-stroke population[END_REF] and Nitrogen dioxide (NO2) for fetal growth in multiple pregnancy [START_REF] Mariet | Multiple pregnancies and air pollution in moderately polluted cities : is there an association between air pollution and fetal growth[END_REF]. However, the association between air pollution and stroke overall and by subtype, is less clear [START_REF] Lee | Air pollution and stroke[END_REF][START_REF] Maheswaran | Impact of outdoor air pollution on survival after stroke : population-based cohort study[END_REF][START_REF] Wilker | Residential proximity to high-traffic roadways and poststroke mortality[END_REF][START_REF] Desikan | Effect of exhaust-and non-exhaust-related components of particulate matter on long-term survival after stroke[END_REF][START_REF] Maheswaran | Air pollution and stroke. An overview of the evidence base[END_REF]. The recent works of the Global Burden of Disease study [START_REF] Feigin | Global Burden of stroke and risk factors in 188 countries, during 1990-2013 : a systematic analysis for the Global Burden of Disease Study 2013[END_REF] which evaluated data from 1990 to 2013 in 188 developed and developing countries, demonstrated for the first time that air pollution contributed 30 % to the burden of stroke. This new vascular risk factor can be considered as a potentially modifiable risk factor for stroke that is a major public health problem being the first cause of motor handicap and the second cause of cognitive disorder and death worldwide [START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF][START_REF] Lee | Air pollution and stroke[END_REF][START_REF] Guo | Ambient air pollution and risk for ischemic stroke : a short-term exposure assessment in South China[END_REF]. The aim of this review is to clarify biological and clinical features of the association between air pollution and risk and mortality of stroke.

Gaseous pollution is in the air

Air pollution is a complex association of several components produced by different sources [START_REF] Lee | Air pollution and stroke[END_REF]. Air pollution includes small particulate matter (PM) and gaseous pollution consisting of ozone (03), sulfur dioxide (SO2), nitrogen dioxide (NO2) and carbon monoxide (CO). PM2.5 represent 70 % of PM measuring less than 10 µm (PM10) [START_REF] Maheswaran | Air pollution and stroke. An overview of the evidence base[END_REF] and they are able to penetrate into the alveoli of the lungs [START_REF] Maheswaran | Air pollution and stroke. An overview of the evidence base[END_REF] and the brain cells [START_REF] Maher | Magnetite pollution nanoparticles in the human brain[END_REF]. Combustion of fossil energies, including motorized road traffic, power plants, industrial and home heating using coal, oil or wood [START_REF] Lee | Air pollution and stroke[END_REF], storms and volcanic eruptions, produces PM and SO2. Vehicle engine exhaust (mainly diesel engines) presents high levels of nanoparticles able to have a greater reactive area for a given mass [START_REF] Lee | Air pollution and stroke[END_REF]. NO2 and CO are produced by fuel combustion and O3 and is produced by photochemical reactions [START_REF] Cassee | Particulate matter beyond mass : recent health evidence on the role of fractions, chemical constituent and sources of emission[END_REF]. Air pollutant levels have spatial and temporal variations related to climate conditions (wind, temperature, sunlights, and rain) and geographic sources.

Air pollution and stroke : epidemiological results

Methods

Several resources were evaluated (PubMed, Web of Sciences, Embase) for publications in all languages available from 1960 through August 2018, using classical keywords : stroke, ischemic and hemorrhagic stroke, cerebrovascular disease, transient ischemic attack, incidence, mortality air pollution, air pollutants (PM2.5, PM10,O3, NO2, CO, SO2). We included all types of study design, evaluating short-term and long-term exposures to air pollutants, acute and chronic effects.

To be eligible for the analysis of cause-effect association and its strength, studies should provide clear definitions of stroke [START_REF] Adams | Classification of subtype of acute ischemic stroke. Definitions for use in a multicenter clinical trial. TOAST trial of Org 10172 in acute stroke treatment[END_REF] and air pollution [START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF]. Two investigators (MGr, MGi) evaluated all the eligible publications and the results were shared with all the authors.

Results

Cohort studies, ecological studies, case-crossover studies, daily time-series, observational studies, and big data evaluated effects of acute exposure to air pollutants. Cohorts and case-control studies evaluated chronic exposure [START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF][START_REF] Maheswaran | Impact of outdoor air pollution on survival after stroke : population-based cohort study[END_REF][START_REF] Maheswaran | Air pollution and stroke. An overview of the evidence base[END_REF] (Table 1).

Cohort studies

Three studies are the most robust. Ljungman and Mittleman [START_REF] Ljungman | Ambient air pollution and stroke[END_REF] demonstrated a strong association between long-term exposure to air pollutants and stroke mortality, from 6 cohorts studies. NO2 was associated to stroke mortality in 3 of the 6 studies. NO2 and PM were associated to hospitalization for stroke induced by long-term exposure to air pollutants. Scheers et al [START_REF] Scheers | Long-term exposure to particulate matter air pollution is a risk factor for stroke : meta-analytical evidence[END_REF], from 15 cohort studies observed that stroke risk was significantly associated to the rise of PM2.5 levels. The urban multiethnic cohort of Kulick et al [START_REF] Kulick | Residential proximity to major roadways and risk of incident ischemic stroke in NOMAS (The Northern Manhattan Study)[END_REF] found evidence that within-city variation in residential proximity to major roadway was associated with higher risk of ischemic stroke among non current smoker.

Ecological studies

Nine studies [START_REF] Maheswaran | Impact of outdoor air pollution on survival after stroke : population-based cohort study[END_REF][START_REF] Hu | Linking stroke mortality with air pollution, income and greenness in northwest Florida : an ecological geographical study[END_REF][START_REF] Zhang | Acute effects of particulate air pollution on ischemic stroke and hemorrhagic stroke mortality[END_REF][START_REF] Maheswaran | Stroke mortality associated with living near main roads in England and Wales-a geographical study[END_REF][START_REF] Tsai | Evidence for an association between air pollution and daily stroke admissions in Kaohsiung, Taiwan[END_REF][START_REF] Wellenius | Air pollution and hospital admissions for ischemic and hemorrhagic stroke among Medicare beneficiaries[END_REF][START_REF] Iribarren | Cohort study of exposure to environmental tobacco smoke and risk of first ischemic stroke and transient ischemic attack[END_REF] confirmed the positive association between air pollution and stroke and the decrease of out-door air pollution levels in developed countries contrasting with the increase of outdoor air pollution in developing countries [3 et 11]. The recent work of Zhang et al [START_REF] Zhang | Acute effects of particulate air pollution on ischemic stroke and hemorrhagic stroke mortality[END_REF] demonstrated that the rise of PM2.5 was associated with the rise of both ischemic and hemorrhagic stroke mortality while the rise of PM10 was associated with only ischemic stroke mortality.

Meta-analyses, time-series and case-crossover studies

Henrotin et al [START_REF] Henrotin | Short-term effects of ozone air pollution on ischemic stroke occurrence : a case-crossover analysis from a 10-year population-based study in Dijon, France[END_REF][START_REF] Henrotin | Evidence of the role of shortterm exposure to ozone on ischemic cerebral and cardiac events : the Dijon Vascular Project (DIVA)[END_REF] provided new data showing that the association of ischemic stroke with air pollution significantly rose with the number of vascular risk factors (tobacco consumption, hypertension, hypercholesterolemia, and diabetes). Yang et al [START_REF] Yang | An evidence-based appraisal of global association between air pollution and risk of stroke[END_REF] observed from a meta-analysis of 34 studies found another results with the association of the risk of ischemic strokes with several air pollutants (PM, SO2, CO and NO2). The meta-analysis of 94 studies by Shah et al [START_REF] Shah | Short term exposure to air pollution and stroke : systematic review and meta-analysis[END_REF] observed an increase of hospital admissions and mortality correlated with the rise of the pollutant levels. The meta-analysis including 45 studies by Wang et al [START_REF] Wang | Short-term changes in ambient particulate matter and risk of stroke : a systematic review and meta-analyses[END_REF] observed a significant association between the PM2.5 and PM10 levels and stroke mortality. Guo et al [START_REF] Guo | Ambient air pollution and risk for ischemic stroke : a short-term exposure assessment in South China[END_REF] demonstrated in 2017 a borderline significant association between NO2, short-term exposure modeled as an averaged lag effect and ischemic stroke risk. Maheswaran et al [START_REF] Maheswaran | Air pollution and subtypes, severity and vulnerability to ischemic stroke. A population-based case-crossover study[END_REF] demonstrated from the population-based case-crossover study in South London an association between an increased NO2 exposure with a higher risk of cerebrovascular small vessel disease, within the period 1995-2006.

Contrasting with this positive association, Butland et al [START_REF] Butland | Air pollution and the incidence of ischaemic and haemorrhagic stroke in South London Stroke Register : a casecross-over analysis[END_REF] did not find any association in 2017. It is a time-stratified case-cross over study, using data from the population-based study of South London from 2005 to 2012. They included 1 311 incident ischemic strokes and 256 haemorrhagic strokes. The authors found no evidence for an association between all strokes or ischemic strokes and same day exposure to PM2.5, PM10, O3, NO2. For haemorrhagic strokes, they found a negative association with PM10 suggestive of a 14.6 % fall in risk per 10 µg/m 3 increase of pollutant. The change in air pollutants distribution may explain the discrepancy. The time-series of Dong et al [START_REF] Dong | Acute effects of air pollution on ischaemic stroke onset and deaths : a time-series study in Changzhou, China[END_REF] demonstrated that short-term exposure to ambient NO2 was associated with increased ischemic stroke risk. In addition, SO2 was associated with increased ischemic stroke onset and death. The role of traffic-related air pollution has been well demonstrated with the cross-over study of Vivanco-Hidalgo et al [START_REF] Vivanco-Hidalgo | Short-term exposure to traffic-related air pollution and ischemic stroke onset in Barcelone, Spain[END_REF]. Some geographical differences may explain these differences [START_REF] Scheers | Long-term exposure to particulate matter air pollution is a risk factor for stroke : meta-analytical evidence[END_REF] or measurement errors [START_REF] Dong | Acute effects of air pollution on ischaemic stroke onset and deaths : a time-series study in Changzhou, China[END_REF]. Timestratified case-cross-over design may lead to reduced statistical power compared with a Poisson regression time-series approach [START_REF] Butland | Air pollution and the incidence of ischaemic and haemorrhagic stroke in South London Stroke Register : a casecross-over analysis[END_REF].

Big data

Big data on stroke are now available in stroke disease and they deserve a special place. The Global Burden of Disease 2013 study [START_REF] Feigin | Global Burden of stroke and risk factors in 188 countries, during 1990-2013 : a systematic analysis for the Global Burden of Disease Study 2013[END_REF][START_REF] Brauer | Ambient air pollution exposure estimation for the Global Burden of Disease 2013[END_REF] evaluated the population-attribuable risk of stroke-related disability-adjusted life-years associated with possible risk factors in 188 developed and developing countries from 1990 through 2013. They observed a significant higher proportion of stroke associated with air pollution, close to 30 %, higher in low-income and middle-income (33.7 %) compared with high-income countries (10.2 %). They observed that only the low income and middle income countries, close to a fifth of stroke risk, was associated with solid fuel combustion in 2013 [START_REF] Feigin | Global Burden of stroke and risk factors in 188 countries, during 1990-2013 : a systematic analysis for the Global Burden of Disease Study 2013[END_REF], confirming previous studies [START_REF] Maheswaran | Impact of outdoor air pollution on survival after stroke : population-based cohort study[END_REF][START_REF] Brauer | Ambient air pollution exposure estimation for the Global Burden of Disease 2013[END_REF]. They used local available data in low and middle-income countries, and reported that ambient and house-hold air pollution was a new modifiable risk factor for stroke and a new public health problem, leading to new worldwide public health policies.

The mechanisms of action of air pollution

Beside the classical role of air pollution on inflammation, formation and rupture of atheroma plaque [START_REF] Bai | Changes in atherosclerotic plaques induced by inhalation of diesel exhaust[END_REF][START_REF] Bauer | Study. Investigative Group. Urban particulate matter air pollution is associated with subclinical atherosclerosis results from the HNR (Heinz Nixdorf Recall) study[END_REF], endothelium dysfunction, cardiac arrhythmia, hypertension, activation of thrombosis [START_REF] Newby | ESC working group on thrombosis. European association for Cardio vascular prevention and rehabilitation. ESC heart failure association. Expert position paper on air pollution and cardiovascular disease[END_REF] and vasoconstriction [START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF], new fundamental mechanisms may explain the association between air pollution and stroke [START_REF] Lee | Air pollution and stroke[END_REF].

Translocation of nanoparticles

The fact that nanoparticles fraction of PM pollution are able to cross the alveolar-capillary barrier towards the systemic circulation, they can directly impact on circulation blood cells [START_REF] Oberdörster | Extrapulmonary translocation of ultrafine carbon particles following whole-body inhalation exposure of rats[END_REF], from a proinflammatory activation on endothelial cells [START_REF] Aung | Comparative gene responses to collected ambient particles in vitro: endothelial responses[END_REF]. Combusion-derived nanoparticles may regulate the adhesion of several molecules on the endothelial cells [START_REF] Lee | Motorcycle exhaust particles upregulate expression of vascular adhesion molecule-1 and intercellular adhesion molecule-1 in human umbilical vein endothelial cells[END_REF][START_REF] Montiel-Dávalos | PM2.5 and PM10 induce the expression of adhesion molecules and the adhesion of monocytic cells to human umbilical vein endothelial cells[END_REF]. These multiple steps may explain the activation of monocytes and macrophages and increase endothelial cell permeability [START_REF] Chao | Diesel exhaust particle exposure causes redistribution of endothelial tube VE-cadherin[END_REF][START_REF] Li | Diesel exhaust particles modulate vascular endothelial cell permeability: implication of ZO-1 expression[END_REF]. The research on this new mechanism is difficult due to the small size and the number of small particles that will translocate, and to the technical difficulties to detect carbon-based nanoparticles in cells. The robust information is that nanoparticles can translocate quickly from lungs towards the blood circulation and areas of vascular inflammation [START_REF] Miller | Inhaled nanoparticles accumulate at sites of vascular disease[END_REF].

Autonomic dysfunction

Inhaled small particles may disturb cardiovascular homeostasis via changes in autonomic function induced by activation of neural sensory receptors of the alveolar surface [START_REF] Lee | Air pollution and stroke[END_REF]. Some evidence is provided by reduction of heart rate variability after exposure du PM 2.5 [START_REF] Pieters | An epidemiological appraisal of the association between heart rate variability and particulate air pollution: a meta-analysis[END_REF], leading to atrial fibrillation [START_REF] Wang | Particulate matter induces cardiac arrhythmias via dysregulation of carotid body sensitivity and cardiac sodium channels[END_REF][START_REF] Nadziejko | Effect of particulate and gaseous pollutants on spontaneous arrhythmias in aged rats[END_REF][START_REF] Chung | Air pollution is associated with ischemic stroke via cardiogenic embolism[END_REF] that is a major risk factor for stroke [START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF].

Epigenetic mechanisms

Epigenetic mechanisms such as post-translational modifications of DNA, RNA-related processes and histones may play an important role in the regulation of many vascular mechanisms [START_REF] Hu | The emerging role of epigenetics in cerebral ischemia[END_REF], including endothelial stem/progenitor cell functions [START_REF] Fraineau | Epigenetic regulation of endothelial-cellmediated vascular repair[END_REF]. The clinical consequence is that blood DNA hypomethylation could be considered as a biomarker of atheroma [START_REF] Lee | Air pollution and stroke[END_REF] allowing to identify persons at risk of atheroma [START_REF] Baccarelli | Ischemic heart disease and stroke in relation to blood DNA methylation[END_REF]. The role of DNA methylation on numerous genes contributing to stroke such as endothelial function [START_REF] Fraineau | Epigenetic regulation of endothelial-cellmediated vascular repair[END_REF][START_REF] Bind | Quantile regression analysis of the distributional effects of air pollution on blood pressure, heart rate variability, blood lipids, and biomarkers of inflammation in elderly American men: the normative aging study[END_REF][START_REF] Dai | Fine particles, genetic pathways, and markers of inflammation and endothelial dysfunction: analysis on particulate species and sources[END_REF], the cytokine-mediated inflammatory reactions and coagulation [START_REF] Bind | Quantile regression analysis of the distributional effects of air pollution on blood pressure, heart rate variability, blood lipids, and biomarkers of inflammation in elderly American men: the normative aging study[END_REF][START_REF] Dai | Fine particles, genetic pathways, and markers of inflammation and endothelial dysfunction: analysis on particulate species and sources[END_REF], are another new way of research. Flavonoïds, methyl nutrients, folic acid, vitamin B12, may act on DNA methylation and may have a protective effect on prevention of stroke [START_REF] Zhong | Cardiac autonomic dysfunction: particulate air pollution effects are modulated by epigenetic immunoregulation of toll-like receptor 2 and dietary flavonoid intake[END_REF][START_REF] Byrne | Epigenetic modulation in the treatment of atherosclerotic disease[END_REF].

We can identify several limitations of epidemiological studies, provided from most of developed countries where indoor exposure is not evaluated, individual exposures may be generalized to a larger area and it may be difficult to identify the cause of death that may be multiple. Earlier studies evaluated the combined effects of acute and chronic exposure to air pollutions.

Screening people at risk

Children and elderly people are more vulnerable to air pollution [START_REF] Desikan | Effect of exhaust-and non-exhaust-related components of particulate matter on long-term survival after stroke[END_REF][START_REF] Maheswaran | Air pollution and subtypes, severity and vulnerability to ischemic stroke. A population-based case-crossover study[END_REF][START_REF] Sacks | Particulate matterinduced health effects: who is susceptible?[END_REF][START_REF] Sade | Air pollution and ischemic stroke among young adults[END_REF], such as Hispanics with diabetes [START_REF] O'donnell | Fine particulate air pollution (PM2.5) and the risk of acute ischemic stroke[END_REF], patients with previous vascular risk factors [START_REF] Henrotin | Short-term effects of ozone air pollution on ischemic stroke occurrence : a case-crossover analysis from a 10-year population-based study in Dijon, France[END_REF][START_REF] Henrotin | Evidence of the role of shortterm exposure to ozone on ischemic cerebral and cardiac events : the Dijon Vascular Project (DIVA)[END_REF], people with recurrent stroke [START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF], patients with neighborhood deprivation living close to roadways or factories [START_REF] Desikan | Effect of exhaust-and non-exhaust-related components of particulate matter on long-term survival after stroke[END_REF][START_REF] Maheswaran | Air pollution and stroke. An overview of the evidence base[END_REF][START_REF] Shah | Short term exposure to air pollution and stroke : systematic review and meta-analysis[END_REF], and people lining in low-income countries [START_REF] Feigin | Global Burden of stroke and risk factors in 188 countries, during 1990-2013 : a systematic analysis for the Global Burden of Disease Study 2013[END_REF][START_REF] Brauer | Ambient air pollution exposure estimation for the Global Burden of Disease 2013[END_REF] . Finally, greater exposure to air pollution on previous cardiovascular risk factors is associated with a greater risk of stroke [START_REF] Maheswaran | Air pollution and subtypes, severity and vulnerability to ischemic stroke. A population-based case-crossover study[END_REF].

Possible recommendations

Therefore, epidemiology and experimental evidence demonstrate that there is no safe air pollution and that it is necessary to promote public health policy to eradicate both indoor and outdoor air pollution worldwide [START_REF] Lee | Air pollution and stroke[END_REF][START_REF] Feigin | Global Burden of stroke and risk factors in 188 countries, during 1990-2013 : a systematic analysis for the Global Burden of Disease Study 2013[END_REF], reducing individual and collective combustion of fossil fuels and developing future urban planning including robust monitoring and protection from air pollution [START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF]. For medical practice, it is necessary to recall that epidemiological studies have shown that people at risk during an acute air pollution exposures are elderly people, as well as people with previous vascular risk as blood hypertension, diabetes, hypercholesterolemia, atrial fibrillation, sleep apnea or tobacco consumption [START_REF] Béjot | A review of epidemiological research on stroke and dementia and exposure to air pollution[END_REF].

Conclusion

A new modifiable risk factor for stroke is in the air. Epidemiological studies have demonstrated the significant association between air pollution and stroke. New fundamental mechanisms described suggest urgent effective strategy. • Maheswaran et al [START_REF] Maheswaran | Impact of outdoor air pollution on survival after stroke : population-based cohort study[END_REF]. South London (UK) • Hu et al [START_REF] Hu | Linking stroke mortality with air pollution, income and greenness in northwest Florida : an ecological geographical study[END_REF]. Florida (USA) • Zang et al [START_REF] Zhang | Acute effects of particulate air pollution on ischemic stroke and hemorrhagic stroke mortality[END_REF] in Beijing (China) • Maheswaran and Elliot [START_REF] Maheswaran | Stroke mortality associated with living near main roads in England and Wales-a geographical study[END_REF]. England and Wates (UK) • Tsai et al [START_REF] Tsai | Evidence for an association between air pollution and daily stroke admissions in Kaohsiung, Taiwan[END_REF] in Kaohsiung (Taiwan) • Wellenius et al [START_REF] Wellenius | Air pollution and hospital admissions for ischemic and hemorrhagic stroke among Medicare beneficiaries[END_REF] in 9 US cities • Iribarren et al [START_REF] Iribarren | Cohort study of exposure to environmental tobacco smoke and risk of first ischemic stroke and transient ischemic attack[END_REF] in San Francisco (USA)

• Guo et al [START_REF] Guo | Ambient air pollution and risk for ischemic stroke : a short-term exposure assessment in South China[END_REF] in Guangzhou (China) • Henrotin et al [START_REF] Henrotin | Short-term effects of ozone air pollution on ischemic stroke occurrence : a case-crossover analysis from a 10-year population-based study in Dijon, France[END_REF][START_REF] Henrotin | Evidence of the role of shortterm exposure to ozone on ischemic cerebral and cardiac events : the Dijon Vascular Project (DIVA)[END_REF] • Feigin et al [START_REF] Feigin | Global Burden of stroke and risk factors in 188 countries, during 1990-2013 : a systematic analysis for the Global Burden of Disease Study 2013[END_REF] in worldwide • Brauer et al [START_REF] Brauer | Ambient air pollution exposure estimation for the Global Burden of Disease 2013[END_REF] in worldwide

  = 1 cross-over study in Dijon (France) • Yang et al [31] = 1 meta-analysis from 34 studies • Shah et al [32] = 1 meta-analysis from 94 studies • Wang et al [33] = 1 meta-analysis of 45 studies • Maheswaran R et al [34] in South London (UK) • Butland et al [35] = 1 cross-over study from a community-based study in London (UK) • Dong et al [36] = 1 time-series in Changzhou (China) • Vivanco-Hidalgo et al [37] = 1 cross-over study from a hospital-based study in Barcelone (Spain)

Table 1 .

 1 Clinical studies on stroke and air pollutions

		Cohort Studies	Ecological studies	Meta-analyses, time-series and case-crossover	Big-data
				studies
	Clinical	• Ljungman and Mittleman [20] = 6
	studies on	cohort studies (UK, USA, Denmark,
	stroke and air	Japan) • Scheers et al [21] = 20 cohort studies
	pollutions	(USA, Europe, Asia) • Kulick et al [22] = 1 population-based
		cohort in Northern Manhattan (USA)

Disclosure of interest

No Financial or personnal conflicts or relationship.


Florian Privé
email: florian.prive.21@gmail.com

Bjarni J Vilhjálmsson

Hugues Aschard

Michael G B Blum
email: michael.blum@univ-grenoble-alpes.fr

Making the Most of Clumping and Thresholding for Polygenic Scores

Polygenic prediction has the potential to contribute to precision medicine. Clumping and thresholding (C+T) is a widely used method to derive polygenic scores. When using C+T, several p value thresholds are tested to maximize predictive ability of the derived polygenic scores. Along with this p value threshold, we propose to tune three other hyper-parameters for C+T. We implement an efficient way to derive thousands of different C+T scores corresponding to a grid over four hyper-parameters. For example, it takes a few hours to derive 123K different C+T scores for 300K individuals and 1M variants using 16 physical cores. We find that optimizing over these four hyper-parameters improves the predictive performance of C+T in both simulations and real data applications as compared to tuning only the p value threshold. A particularly large increase can be noted when predicting depression status, from an AUC of 0.557 (95% CI: [0.544-0.569]) when tuning only the p value threshold to an AUC of 0.592 (95% CI: [0.580-0.604]) when tuning all four hyper-parameters we propose for C+T. We further propose stacked clumping and thresholding (SCT), a polygenic score that results from stacking all derived C+T scores. Instead of choosing one set of hyper-parameters that maximizes prediction in some training set, SCT learns an optimal linear combination of all C+T scores by using an efficient penalized regression. We apply SCT to eight different case-control diseases in the UK biobank data and find that SCT substantially improves prediction accuracy with an average AUC increase of 0.035 over standard C+T.

*bxs

PRSice and PLINK) or 500 kb, and test several values for p T ranging from 1 to 10 8 . 2,4,5,7 Moreover, to compute the PRS, the target sample genotypes are usually imputed to some degree of precision in order to match the variants of summary statistics. The inclusion of imputed variants with relatively low imputation quality is common, assuming that using more variants in the model yields better prediction. Here, we explore the validity of this approach and suggest an additional INFO T threshold on the quality of imputation (often called the INFO score) as a fourth parameter of the C+T method.

Introduction

The ability to predict disease risk accurately is a principal aim of modern precision medicine. As more population-scale genetic datasets become available, polygenic risk scores (PRS) are expected to become more accurate and clinically relevant. The most commonly used method for computing polygenic scores is clumping and thresholding (C+T), also known as pruning and thresholding (P+T). The C+T polygenic score is defined as the sum of allele counts (genotypes), weighted by estimated effect sizes obtained from genome-wide association studies, where two filtering steps have been applied. [1][2][3][4][5][6] More precisely, the variants are first clumped (C) so that only variants that are weakly correlated with one another are retained. Clumping selects the most significant variant iteratively, computes correlation between this index variant and nearby variants within some genetic distance w c , and removes all the nearby variants that are correlated with this index variant beyond a particular value, 2 c r . Thresholding (T) consists in removing variants with a p value larger than a chosen level of significance (p > p T ). Both steps, clumping and thresholding, represent a statistical compromise between signal and noise. The clumping step prunes redundant correlated effects caused by linkage disequilibrium (LD) between variants. However, this procedure may also remove independently predictive variants in LD. Similarly, thresholding must balance between including truly predictive variants and reducing noise in the score by excluding null effects.

When applying C+T, one has three hyper-parameters to select, namely the squared correlation threshold 2 c r and the window size w c of clumping, along with the p value threshold p T . Usually, C+T users assign default values for clumping, such as 2 c r of 0.1 (default of PRSice), 0.2 or 0.5 (default of PLINK), and w c of 250 kb (default of We implement an efficient way to compute C+T scores for many different parameters (LD, window size, p value, and INFO score) in R package bigsnpr. 8 Using a training set, one could therefore choose the best predictive C+T model among a large set of C+T models with many different parameters, and then evaluate this model in a test set. Moreover, instead of choosing one set of parameters that corresponds to the best prediction, we propose to use stacking, i.e., we learn an optimal linear combination of all computed C+T scores using an efficient penalized regression to improve prediction beyond the best prediction provided by any of these scores. 9 We call this method SCT (stacked clumping and thresholding). Using the UK Biobank data 10 and external summary statistics for simulated and real data analyses, we show that testing a larger grid of parameters consistently improves predictions as compared to using some default parameters for C+T. We also show that SCT consistently improves prediction compared to any single C+T model when sample size of the training set is large enough.

Material and Methods

Clumping and Thresholding (C+T) and Stacked C+T (SCT)

We compute C+T scores for each chromosome separately and for several parameters: Threshold on imputation INFO score INFO T within {0.3, 0.6, 0.9, 0.95}. Squared correlation threshold of clumping 2 c r within {0.01, 0.05, 0.1, 0.2, 0.5, 0.8, 0.95}.

Base size of clumping window within {50, 100, 200, 500}. The window size w c is then computed as the base size divided by 2 c r . For example, for 2 0.2 c r 

, we test values of w c within {250, 500, 1000, 2500} (in kb). This is motivated by the fact that linkage disequilibrium is inversely proportional to genetic distance between variants. 11 A sequence of 50 thresholds on p values between the least and the most significant p values, equally spaced on a log-log scale. Overall, we compute 22  4  7  4  50 = 123,200 vectors of polygenic scores. Then, we stack all these polygenic scores (for individuals in the training set) by using these scores as explanatory variables and the phenotype as the outcome in penalized regression. 9 In other words, we fit weights for each C+T scores using an efficient penalized logistic regression available in R package bigstatsr. 12 This results in a linear combination of C+T scores, where C+T scores are linear combinations of variants, so that we can derive a single vector of variant effect sizes to be used for prediction in the test set. We refer to this method as SCT (stacked clumping and thresholding) in the rest of the paper.

From this grid of 123,200 vectors of polygenic scores, we also derive two C+T scores for comparison. First, "stdCT" is the standard C+T score using some default parameters, i.e., with 2 c r = 0.2, w c = 500, a liberal threshold of 0.3 on imputation INFO score, and choosing the p value threshold (10 8 ) maximizing the AUC on the training set. 4 Second, "maxCT" is the C+T score maximizing the AUC on the training set among the 5,600 (123,200/22) C+T scores corresponding to all different sets of parameters tested. Although the set used to choose optimal hyper-parameters is commonly called "validation set," we refer to this set as "training set" here because it is also the set used for learning weights of stacking. Also note that stdCT and maxCT use the same set of parameters for all chromosomes, i.e., for one set of the four hyper-parameters, they are defined as

    1 22
VV  L

. In contrast, SCT uses the whole matrix of 123,200 vectors.

Simulations

We use variants from the UK Biobank (UKBB) imputed dataset that have a minor allele frequency larger than 1% and an imputation INFO score larger than 0.3. There are almost 10M such variants, of which we randomly choose 1M in two different ways. First, we randomly sample 1M from these 10M variants, so we use variants that are mainly well imputed (Figure S1A). Second, we sample variants according to the inverse of INFO score density, so we use variants that are globally not as well imputed as before (Figure S1B).

To limit population structure and family structure, we restrict individuals to the ones identified by the UK Biobank as British with only subtle structure and exclude all second individuals in each pair reported by the UK Biobank as being related. 10 This results in a total of 335,609 individuals that we split into three sets: a set of 315,609 individuals for computing summary statistics (GWAS), a set of 10,000 individuals for training hyperparameters, and a test set of 10,000 individuals for evaluating models.

We read the UKBB BGEN files using function snp_readBGEN from package bigsnpr. 8 For simulating phenotypes and computing summary statistics, we read UKBB data as hard calls by randomly sampling hard calls according to reported imputation probabilities. For the training and test sets, we read these probabilities as dosages (expected values). This procedure enables us to simulate phenotypes using hard calls and then to use the INFO score (imputation accuracies) reported by the UK Biobank to assess the quality of the imputed data used for the training and test sets.

We simulate binary phenotypes with a heritability h 2 = 0.5 using a liability threshold model (LTM) with a prevalence of 10%. 13 We vary the number of causal variants (100, 10K, or 1M) to match a range of genetic architectures from low to high polygenicity. Liability scores are computed from a model with additive effects only: we compute the liability score of the i th individual as ² causal , ,

i j i j i jS y w G   
where S causal is the set of causal variants, w j are weights generated from a Gaussian distribution

  2 causal 0, / | | N h S , G i,j is the allele count of individual i for variant j, ² , ij
G corresponds to its standardized version (zero mean and unit variance), and ϵ follows a Gaussian distribution

  2 0,1 Nh  .
We explore three additional simulation scenarios with more complex architectures. In scenario "2chr," 100 variants of chromosome 1 and all variants of chromosome 2 are causal with half of the heritability for both chromosomes; it aims at assessing predictive performance when disease architectures are different for different chromosomes. In scenario "err," we sample 10K random causal variants but 10% of the GWAS effects (causal or not) are reported with an opposite effect in the summary statistics; it aims at assessing whether methods are able to partially correct for errors or mere differences in effect sizes between GWAS and the target data. In scenario "HLA," 7,105 causal variants are chosen in one long-range LD region of chromosome 6; it aims at assessing whether methods can handle strong correlation between causal variants.

To compute summary statistics, we use Cochran-Armitage additive test. 14 Given that we restricted the data to have minimal population structure, this test based on contingency tables is much faster than using a logistic regression with ten principal components as covariates (a few minutes versus several hours) while providing similar effect sizes and Z-scores (Figure S2).

In simulations, we compare five methods: stdCT, maxCT, SCT (defined in Clumping and Thresholding (C+T) and Stacked C+T (SCT)), lassosum, 15 and LDpred. 16 Each simulation scenario is repeated 10 times and the average AUC is reported. We prefer to use AUC over Nagelkerke's R 2 because AUC has a desirable property of being independent of the proportion of cases in the validation sample; one definition of AUC is the probability that the score of a randomly selected case subject is larger than the score of a randomly selected control subject. 17 An alternative to AUC would be to use a better R 2 on the liability scale. 18,19

Real Summary Statistics

We also investigate predictive performance of C+T, SCT, lassosum, and LDpred using the UK Biobank. We first pick existing external summary statistics from published GWAS of real diseases. 20 We then divide the UK Biobank dataset into one training set and one test set. The training set is used to choose optimal hyperparameters in C+T, LDpred, and lassosum, and to learn stacking weights in SCT. The test set is used to evaluate the final model. Training SCT and choosing optimal hyper-parameters for C+T (stdCT and maxCT) and lassosum use 63%-90% of the UK Biobank individuals reported in Table 1. Therefore, the training set can contain as many as 250K individuals. LDpred is excluded from this analysis due to computational infeasibility.

To assess how sample size affects predictive performance of methods, we also compare these methods Publisher: CELL; Journal: ajhg:American Journal of Human Genetics; Copyright: Volume: <txtVolume>; Issue: <txtIssue>; Manuscript: <txtManuscript>; DOI: 10.1016/; PII: TOC Head: ; Section Head: ; Article Type:

Page 4 of 11 (including LDpred) using a much smaller training set of 500 case subjects and 2,000 control subjects only. We repeat this random split six times for each phenotype, and the average AUC is reported.

As in simulations, we restrict individuals to the ones identified by the UK Biobank as British with only subtle structure and exclude all second individuals in each pair reported by the UK Biobank as being related. 10 Table 1 summarizes the number of case and control subjects in the UKBB after this filtering and for each phenotype analyzed, as well as the number of individuals and variants used in the GWAS. For details on how we define phenotypes in the UKBB, please refer to our R code (see Reproducibility). Briefly, we use self-reported illness codes (field #20001 for cancers and #20002 otherwise) and ICD10 codes (fields #40001, #40002, #41202, and #41204 for all diseases and field #40006 specifically for cancers).

We keep all variants with a GWAS p value lower than 0.1, except for prostate cancer (0.05) and asthma (0.5). This way, we keep around 1M variants for each phenotype, deriving all C+T scores and stacking them in SCT in less than 1 day for each phenotype, even when using 300K individuals in the training set. To match variants from summary statistics with data from the UK Biobank, we first remove ambiguous alleles [A/T] and [C/G].

We then augment the summary statistics twice: first by duplicating each variant with the complementary alleles, then by duplicating variants with reverse alleles and effects. Finally, we include only variants that we match with UKBB based on the combination of chromosome, position, and the two alleles. Note that, when no or very few alleles are flipped, we disable the strand flipping option and therefore do not remove ambiguous alleles; this is the case for all phenotypes analyzed here.

Reproducibility

The code to reproduce the analyses and figures of this paper is available as R scripts online (see Web Resources). 21 To execute these scripts, you need to have access to the UK Biobank data that we are not allowed to share. An introduction to SCT is also available online (see Web Resources).

Results

Simulations

We test six different simulations scenarios. In all these scenarios, maxCT-that tests a much larger grid of hyper-parameters values for C+T on the training set-consistently provides higher AUC values on the test set as compared to stdCT that tests only several p value thresholds while using default values for the other parameters (Figure 1). The absolute improvement in AUC of maxCT over stdCT is particularly large in the cases of 100 and 10,000 causal variants, where causal effects are mostly independent of one another. In these cases, using a very stringent 2 0.01 c r 

threshold of clumping provides higher predictive performance than using a standard default of 2 0.2 c r  (Figures S6A andS6B). However, 2 0.2 c r 

provides best predictive performance when simulating 1M causal variants. Still, in this scenario, using a large window size w c of 2,500 kb increases AUC as compared to using default values of either 250 or 500 kb (Figure S6C).

As for SCT, it provides equal or higher predictive performance than maxCT in the different simulation scenarios (Figure 1). In the first three simple scenarios simulating 100, 10K, or 1M causal variants anywhere on the genome, predictive performance of SCT are similar to maxCT. In the "2chr" scenario where there are large effects on chromosome 1, small effects on chromosome 2 and no effect on other chromosomes, mean AUC is 78.7% for maxCT and 82.2% for SCT. In the "err" scenario where we report GWAS summary statistics with 10% opposite effects (errors or mere differences), mean AUC is 70.2% for maxCT and 73.2% for SCT. In these simulations, lassosum and LDpred do not provide the best predictive performance, expect when simulating all variants as causal (1M).

Results are similar when using less well imputed variants in the simulations. Globally in these simulations, including a broad range of imputed variants with INFO score as low as 0.3 often maximizes prediction (Figures S6 andS7).

Effects resulting from SCT (Figure S5) are mostly comprised between the GWAS effects and 0. For the simulation with only 100 causal variants, resulting effects are either nearly the same as in the GWAS or near 0 (or exactly 0). When there are some correlation between causal predictors (scenarios "1M" and "HLA") or when reporting GWAS effects with some opposite effect ("err"), some effects resulting from SCT are in the opposite direction as compared to the GWAS effects.

Real Summary Statistics

In terms of AUC, maxCT outperfoms stdCT for all eight diseases considered with a mean absolute increase of 1.3% (Figures 2 andS4). A particularly large increase can be noted when predicting depression status (MDD), from an AUC of 55.7% (95% CI: [54.4-56.9]) with stdCT to an AUC of 59.2% (95% CI: [58.0-60.4]) with maxCT. For MDD, a liberal inclusion in clumping ( 2 c r = 0.8) and a stringent threshold on imputation accuracy (INFO T = 0.95) provides the best predictive performance (Figure S9f). For all eight diseases, predictions were optimized when choosing a threshold on imputation accuracy of at least 0.9, whereas optimal values for 2 c r where very different depending on the architecture of diseases, with optimal selected values over the whole range of tested values for 2 c r (Table 2). Furthermore, when training size uses a large proportion of the UK Biobank data, SCT outperforms maxCT for all eight diseases considered with an additional mean absolute increase of AUC of 2.2%, making it 3.5% as compared to stdCT (Figure 2 and Table S2). Predictive performance improvement of SCT versus maxCT is particularly notable for coronary artery disease (2.8%), type 2 diabetes (3.1%), and asthma (3.4%).

Effects resulting from SCT have mostly the same sign as initial effects from GWAS, with few effects being largely unchanged and others having an effect that is shrunk to 0 or equals to 0, i.e., variants not included in the final model (Figure S8).

When training size is smaller (500 case subjects and 2,000 control subjects only instead of 120K-250K individuals), SCT is not as good as when training size is large, yet SCT remains a competitive method expect for depression for which maxCT performs much better than all other methods (Figure S4). Performance of C+T and lassosum, methods that use the training set for choosing optimal hyper-parameters only, as opposed to SCT that learns new weights, are little affected by using a smaller training size. However, even though lassosum can provide more accurate prediction for T2D and CAD, it can also perform very poorly for other diseases such as BRCA, PRCA, and MDD (Figures 2 andS4). LDpred performs much better than the other methods for predicting MDD, yet maxCT and SCT perform equally or better than LDpred for the seven other traits (Figure S4).

Discussion Predictive Performance Improvement of C+T

C+T is an intuitive and easily applicable method for obtaining polygenic scores trained on GWAS summary statistics. Two popular software packages that implement C+T, namely PLINK and PRSice, have further contributed to the prevalence of C+T. 5,7,22 Usually, C+T scores for different p value thresholds are derived, using some default values for the other three hyper-parameters. In R package bigsnpr, we extend C+T to efficiently consider more hyper-parameters (four by default) and enable the user to define their own qualitative variant annotations to filter on. Using simulated and real data, we show that choosing optimized hyperparameter values rather than default ones can substantially improve the performance of C+T, making C+T a very competitive method. Indeed, in our simulations (Figure 1), we found that optimizing C+T (maxCT) performed on par with more sophisticated methods such as lassosum and LDpred.

Moreover, it is possible to rerun the method using a finer grid in a particular range of these hyper-parameters. For example, it might be useful to include variants with p values larger than 0.1 for predicting rheumatoid arthritis and depression (Figures S9B andS9F). Another example would be to focus on a finer grid of large values of 2 c r for coronary artery disease (Figure S9G) or to focus on a finer grid of stringent imputation thresholds only (Table 2). One can of course also optimize lassosum and LDpred, which could result in better relative performance. One limitation of LDpred is that its non-infinitesimal version is a Gibbs sampler, which can become unstable if the data do not fit the model well. In practice, this affects the robustness of LDpred and becomes more severe when applied to summary statistics with large sample sizes. Indeed, LDpred has been reported to not adequately adjust for LD structure when sample size of summary statistics is large and that using alternative methods such as PRS-CS may solve this issue. 23 Alternatively, recent work also suggests that excluding long-range LD regions when applying LDpred can improve its predictive performance. 24,25 It may therefore be possible to improve the robustness of the LDpred Gibbs sampler by handling LD and heritability estimation more carefully. When applying LDpred and lassosum, we did not explore non-standard parameter settings or extensions of LDpred. Table 2 shows that the optimal parameters of C+T are very different depending on which trait is analyzed. For instance, T1D and PRCA have similar optimal parameters (a stringent clumping 2 c r and p value threshold p T of 10 5 ), while RA, T2D, MDD, and CAD perform also similarly (a lenient clumping 2 c r and p value threshold).

The first group captures a small set of variants with relatively large effects while the second group captures a multitude of small effects. A possible explanation is that traits of the second group are much more polygenic than in the first group. Another possible explanation is that variants associated with the second group have smaller effects that are more difficult to detect and that including more variants (possibly a lot being noise) provides higher predictive performance.

Using a large grid of C+T scores for different hyper-parameters, we show that stacking these scores instead of choosing the best one improves prediction further. 9 Combining multiple PRSs is not a new idea, 26,27 but we push this idea to the limit by combining 123,200 polygenic scores. This makes SCT more flexible than any C+T model, but it of course also requires a larger training with individual-level genotypes and phenotypes to learn the weights in stacking.

Normally, cross-validation should be used to prevent overfitting when using stacking. 9 However, crossvalidation is not necessary here since building C+T scores does not make use of the phenotype of the training set that is later used for stacking. Moreover, although it has been suggested to use positivity constraints in stacking, 9 we allow C+T scores to have negative weights in the final model for three reasons. First, because C+T scores are overlapping in the variants they use, using some negative weights allows us to weight groups of variants that correspond to the difference of two sets of variants. Second, because of LD, variants may have different effects when learned jointly with others (Figures S5C andS5F). Third, if reported GWAS effects are heterogenous between the GWAS dataset and the validation or target dataset, then having variants with opposite effects can help to adjust the effects learned during GWAS.

Limitations of the Study

In this study, we limited the analysis to eight common diseases and disorders, as these all had substantial number of case subjects and publicly available GWAS summary statistics based on substantial sample sizes. For example, for psychiatric disease, we include only depression (MDD) because diseases such as schizophrenia and bipolar disorder have very few case subjects in the UK Biobank; dedicated datasets should be used to assess effectiveness of maxCT and SCT for such diseases. We also do not analyze many automimmune diseases because summary statistics are often outdated (2010-2011 1 ) and because there are usually large effects in regions of chromosome 6 with high LD, methods that use individual-level data instead of summary statistics are likely to provide better predictive models. 12 We also chose not to analyze any continuous trait such as height or BMI because there are many individual-level data available in UKBB for such phenotypes and methods directly using individual-level data are likely to provide better predictive models for predicting in UKBB than the ones using summary statistics. 12,28 Phenotypes with tiny effects such as educational attainment for which huge GWAS summary statistics are available might be an exception. 29 The principal aim of this work is to study and improve the widely used C+T method. The idea behind C+T is simple as it directly uses weights learned from GWASs; it further removes variants as one often does when reporting hits from GWASs, i.e., only variants that pass the genome-wide threshold (p value thresholding) and that are independent association findings (clumping) are reported. Yet, there are two other established methods based on summary statistics, LDpred and lassosum. 15,16,30 Several other promising and more complex methods such as NPS, PRS-CS, and SBayesR are currently being developed. 23,25,31 One could also consider other variants of C+T such as choosing a different set of hyper-parameters for each chromosome separately, which would be beneficial in the "2chr" simulation scenario where genetic architecture differs between chromosomes. Here, we include lassosum in the comparisons since no other method has yet shown that they provide any improvement over lassosum. In addition, we found lassosum to be easy to use. When applied to real data, lassosum yields mixed results because it achieves almost the largest AUC for some diseases (CAD, T2D) whereas it is less discriminative than standard C+T for other diseases (BRCA, MDD, PRCA) (Figure 2). This lack of robustness of lassosum may be explained by the presence of large effects in the latter diseases, which can be an issue for methods that model LD. This may also be due to the initial filtering on p value that we use for real data to keep disk usage and computation time manageable when analyzing the UK Biobank dataset. We also include LDpred in some of the comparisons as it is a widely used method

Extending maxCT and SCT

The stacking step of SCT can be used for either binary or continuous phenotypes. Yet, for some diseases, it makes sense to include age in the models, using for example Cox proportional-hazards model to predict age of disease onset, with possibly censored data. 32 Cox regression has already proven useful for increasing power in GWAS. 33 Currently, we support linear and logistic regressions in our efficient implementation of package bigstatsr, but not Cox regression. This is an area of future development; for now, if sample size is not too large, one could use R package glmnet to implement stacking based on Cox model. 34 One might also want to use other information such as sex or ancestry (using principal components). Indeed, it is easy to add covariates in the stacking step as (possibly unpenalized) variables in the penalized regression. Yet, adding covariates should be done with caution (see the example of CAD at the end of Supplemental Data).

Finally, note that there is an extra parameter in the SCT pipeline that makes possible for a user to define their own groups of variants. This allows to extend the grid of computed C+T scores and opens many possibilities for SCT. For example, we could derive and stack C+T scores for two (or more) different GWAS summary statistics, e.g., for different ancestries or for different phenotypes. This would effectively extend SCT as a multivariate method. We could also add other hyper-parameters to C+T, such as a threshold on the minor allele frequency (MAF). Indeed, it has been shown that imputation accuracy depends on both INFO score and MAF. 35 For MDD, we tested to add a threshold on MAF as a fifth hyper-parameter of C+T. This improved AUC of maxCT from 59.2% to 60.7% (Figure S10). For maxCT, the optimal threshold on MAF among   6 5 1 10 ,10 , ,10     was 0.01 (Figure S10). Note that this threshold on MAF now makes maxCT more predictive than LDpred that internally uses a MAF threshold of 0.01 by default. This raises questions about the ability of current methods to capture the effects of rare variants, and developing methods that can efficiently use both common and rare variants is a direction of future work. Note that it might be more efficient to construct one hyper-parameter combining INFO score and MAF instead of treating them as two distinct hyper-parameters; this is a direction of future work.

Conclusion

In this paper, we focused on understanding and improving the widely-used C+T method by testing a wide range of hyper-parameters values. More broadly, we believe that any implementation of statistical methods should come with an easy and effective way to choose hyper-parameters of these methods well. We believe that C+T will continue to be used for many years as it is both simple to use and intuitive. Moreover, as we show here, when C+T is optimized using a large grid of hyper-parameters, it becomes a very competitive method since it can adapt to many different disease architectures by tuning a large set of hyper-parameters.

Moreover, instead of choosing one set of hyper-parameters, we show that stacking C+T predictions improves predictive performance further, as long as training size is not too small. SCT has many advantages over any single C+T prediction: first, it can learn different architecture models for different chromosomes, it can learn a mixture of large and small effects and it can more generally adapt initial weights of the GWAS in order to maximize prediction. Moreover, SCT remains a linear model defined by a single vector of coefficients as it is a linear combination (stacking) of linear combinations (C+T scores).

Number of case and control subjects in UK Biobank (UKBB) for several disease phenotypes, along with corresponding published GWAS summary statistics. Summary statistics are chosen from GWASs that did not include individuals from UKBB. For depression, we remove UKBB individuals from the pilot release since they were included in the GWAS from which we use summary statistics. Mean and 95% CI of 10 4 non-parametric bootstrap replicates of the mean AUC of 10 simulations for each scenario. The blue dotted line represents the maximum achievable AUC for these simulations (87.5% for a prevalence of 10% and an heritability of 50%; see Equation 3 of Wray et al. 44 ). See corresponding values in Table S1. Training SCT and choosing optimal hyper-parameters for C+T and lassosum use 63%-90% of the individuals reported in Table 1. See corresponding values in Table S2.

  Thus, for individual i, chromosome k, and the four hyper-parameters INFO T , 2 c r , w c , and p T , we compute C+T predictions the effect sizes (p values) estimated from the GWAS, G i,j is the dosage for individual i and variant j, and the set to first restricting to variants of chromosome k with an INFO score  INFO T and that further result from clumping with parameters 2 c r and w c .

  Publisher: CELL; Journal: ajhg:American Journal of Human Genetics; Copyright: Volume: <txtVolume>; Issue: <txtIssue>; Manuscript: <txtManuscript>; DOI: 10.1016/; PII: TOC Head: ; Section Head: ; Article Type: Page 6 of 11

Figure 2 .

 2 Figure 2. Results of the Real Data Applications with Large Training Size AUC values on the test set of UKBB (mean and 95% CI from 10 4 bootstrap samples).Training SCT and choosing optimal hyper-parameters for C+T and lassosum use 63%-90% of the individuals reported in Table1. See corresponding values in TableS2.

  For example, for type 2 diabetes, there are 1,408,672 variants in summary statistics (p < 0.1), of which 215,821 are ambiguous SNPs. If we remove these ambiguous SNPs, 1,145,260 variants are matched with UKBB, of which only 38 are actually flipped. So, instead, we do not allow for flipping and do not remove ambiguous alleles, and then 1,350,844 variants are matched with UKBB.

  . A full comparison of methods (including individuallevel data methods), including binary and continuous traits with different architectures, using different sizes of summary statistics and individual-level data for training, and in possibly different populations would be of great interest, but is out of scope for this paper. Indeed, we believe that different methods may perform very differently in different settings and that understanding what method is appropriate for each case is of paramount interest if the aim is to maximize prediction accuracy to make PRS clinically useful.

	Publisher: CELL; Journal: ajhg:American Journal of Human Genetics; Copyright:
	Volume: <txtVolume>; Issue: <txtIssue>; Manuscript: <txtManuscript>; DOI: 10.1016/; PII:
	TOC Head: ; Section Head: ; Article Type:
	Page 7 of 11

Table 2 .

 2 Optimal Choices of C+T Parameters Choice of C+T parameters is based on the maximum AUC in the training set. Hyper-parameters of C+T are the squared correlation threshold 2 c r and the window size w c of clumping, the p value threshold p T and the threshold on the quality of imputation INFO T . Choosing optimal hyper-parameters for C+T use 63%-90% of the individuals reported inTable 1. Resulting predictions of maxCT in the test set are reported in Figure 2. Figure 1. Results of the Six Simulation Scenarios with Well-Imputed Variants Publisher: CELL; Journal: ajhg:American Journal of Human Genetics; Copyright: Volume: <txtVolume>; Issue: <txtIssue>; Manuscript: <txtManuscript>; DOI: 10.1016/; PII: TOC Head: ; Section Head: ; Article Type: Page 11 of 11 Scenarios are (100) 100 random causal variants; (10K) 10,000 random causal variants; (1M) all 1M variants are causal variants; (2chr) 100 variants of chromosome 1 are causal and all variants of chromosome 2, with half of the heritability for both chromosomes; (err) 10,000 random causal variants, but 10% of the GWAS effects are reported with an opposite effect; (HLA) 7,105 causal variants in a long-range LD region of chromosome 6.

	Trait	w c	2 c r	INFO T p T
	Breast cancer (BRCA)	2,500	0.2	0.95	2.2e04
	Rheumatoid arthritis (RA)	200	0.5	0.95	7.5e02
	Type 1 diabetes (T1D)	10K-50K 0.01	0.90	2.6e05
	Type 2 diabetes (T2D)	625	0.8	0.95	1.1e02
	Prostate cancer (PRCA)	10K-50K 0.01	0.90	4.2e06
	Depression (MDD)	625	0.8	0.95	1.0e01
	Coronary artery disease (CAD)	526	0.95	0.95	3.5e02
	Asthma	2,500	0.2	0.90	2.2e04

https://www.immunobase.org/downloads/protected_data/GWAS_Data/

Acknowledgments

We thank Shing Wan Choi for helpful discussions about lassosum. We are grateful to the two anonymous reviewers who contributed to improving this paper. F.P. and M.G.B.B. acknowledge LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) and ANR project FROGH (ANR-16-CE12-0033). F.P. and M.G.B.B. also acknowledge the Grenoble Alpes Data Institute that is supported by the French National Research Agency under the "Investissements d'avenir" program (ANR-15-IDEX-02). F.P. and B.J.V. acknowledge Niels Bohr Professorship from the Danish National Research Foundation to Prof. John J. McGrath, and the Lundbeck Foundation Initiative for Integrative Psychiatric Research, iPSYCH (R248 -2017-2003). This research has been conducted using the UK Biobank Resource under Application Number 25589.

Declaration of Interests

The authors declare no competing interests.