Lucie Rigaill

Cécile Garcia

Does male mate choice select for female colouration in a promiscuous primate species?

Keywords: male mate choice, female ornament, colourful trait, sexual signalling, Japanese macaques

The traditional view of sex roles and sexual selection emphasises the evolution of male ornaments as a result of female mate choice and male-male competition. Female ornaments are now receiving more attention, although their roles in mating decision are still less well understood, especially considering cases in which colourful ornaments are expressed by both sexes. In this study, we analysed whether female skin colouration (luminance and redness of the face and hindquarters) influenced male mate choice and sexual behaviours in relation to intra-cycle (cycle phase), inter-cycle (number of consecutive cycles, conceptive nature of the cycle), and inter-individual (age, social rank, weight, and parity) variation in a captive population of Japanese macaques (Macaca fuscata). Males did not preferentially choose darker/redder females. Moreover, males did not appear to use female skin colouration to apportion their mating efforts on the most fertile period of the menstrual cycle or during cycles that lead to conception, or to discriminate between females. To our knowledge, our study is among the few to report a lack of male choice for female colouration in a species where both sexes potentially display ornamentation. While female colouration appeared to contain information about intra-cycle, inter-cycle, and inter-individual variation in fecundity, this study further demonstrates that this trait may not have been sexually selected and that males mated regardless of such variation across females. This study adds to the growing research on the role and evolution of female colouration in the context of sexual signalling and mate attraction.

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Introduction

Mate choice is a crucial process in reproduction. Individuals have to simultaneously choose and avoid potential mates based on the expected benefits (e.g., good genes, social support, reproductive success) and costs (e.g., disease transmission, energy costs, intra-sexual competition) associated with mating [START_REF] Beltran-Bech | Impact of infection on mate choice[END_REF][START_REF] Møller | Parasitism, host immune function, and sexual selection[END_REF][START_REF] Thrall | Sexually transmitted diseases in polygynous mating systems: Prevalence and impact on reproductive success[END_REF][START_REF] Trivers | Parental investment and sexual selection[END_REF]. The traditional view of sex roles and sexual selection has emphasised the importance of female mate choice and male competition, and, as a consequence, the function of male ornaments has been well studied across taxa [START_REF] Andersson | Sexual selection and mate choice[END_REF], 2006;[START_REF] Cunningham | Sex roles and sexual selection[END_REF][START_REF] Fitzpatrick | Male mate choice, male quality, and the potential for sexual selection on female traits under polygyny[END_REF]. Perhaps partly as a result, the role of male mate choice on the evolution of female traits has been relatively less investigated. However, males can also face costs associated with mating in the form of energetically demanding courtship displays, mate guarding, and the production of ejaculates [START_REF] Balshine | Why is mutual mate choice not the norm? Operational sex ratios, sex roles and the evolution of sexually dimorphic and monomorphic signalling[END_REF][START_REF] Dewsbury | Ejaculate cost and male choice[END_REF][START_REF] Harshman | The cost of reproduction: The devil in the details[END_REF][START_REF] Lukas | Costs of mating competition limit male lifetime breeding success in polygynous mammals[END_REF]. Similarly, female competition can also arise when females vary in reproductive quality or when the number of females exceeds male mating or mate-guarding capacities such as in cases with a high ovarian cycle synchrony or with a female-biased operational sex ratio, i.e., the ratio of the number of fertile adult males to the number of potentially fertile females in a group at a given time [START_REF] Amundsen | Why are female birds ornamented?[END_REF][START_REF] Bonduriansky | The evolution of male mate choice in insects: A synthesis of ideas and evidence[END_REF][START_REF] Edward | The evolution and significance of male mate choice[END_REF][START_REF] Kvarnemo | The dynamics of operational sex ratios and competition for mates[END_REF]. Thus, male mate choice or mutual mate choice can evolve in some species to balance the ratio of costs/benefits inherent in mating [START_REF] Clutton-Brock | Sexual selection in males and females[END_REF][START_REF] Fitzpatrick | The evolution of male mate choice and female ornamentation: A review of mathematical models[END_REF][START_REF] Johnstone | Mutual mate choice and sex differences in choosiness[END_REF][START_REF] Kokko | The evolution of mate choice and mating biases[END_REF].

There is increasing evidence that males discriminate among possible mates in species with traditional and reversed-sex roles and with different mating systems (from monogamy to polygynandry). Males may choose older females that have already successfully reproduced and raised offspring (spiny-footed lizards, Acanthodactylus erythrurus: [START_REF] Belliure | Male mate choice based on female coloration in a lizard: The role of a juvenile trait[END_REF] chimpanzees, Pan troglodytes: [START_REF] Muller | Male chimpanzees prefer mating with old females[END_REF]; brown widow spiders, Latrodectus geometricus: [START_REF] Waner | Male mate choice in a sexually cannibalistic widow spider[END_REF]. Males may also choose females of higher social ranks compared with lower-ranking ones. This may be particularly adaptative in species where social rank is maternally inherited: higher-ranking offspring may achieve better condition and higher survival than lower-ranking offspring (e.g., mandrills, Mandrillus sphinx: [START_REF] Setchell | Mate choice in male mandrills (Mandrillus sphinx)[END_REF]. Males may also choose to mate with unmated vs. mated females to reduce sperm competition and increase their reproductive success (house crickets, Acheta domesticus: [START_REF] Assis | Male mate choice based on chemical cues in the cricket Acheta domesticus (Orthoptera: Gryllidae)[END_REF] wolf spiders, Schizocosa ocreata: [START_REF] Roberts | Information content of female chemical signals in the wolf spider, Schizocosa ocreata: Male discrimination of reproductive state and receptivity[END_REF]. To mate selectively, males may thus rely on female signals and/or express preferences for female traits that correlate with reproductive quality, such as pheromones (widow spiders, Lactrodectus hesperus:

Baruffaldi & Andrade, 2015; humans, Homo sapiens: [START_REF] Gildersleeve | Body odor attractiveness as a cue of impending ovulation in women: Evidence from a study using hormone-confirmed ovulation[END_REF]; wolf spiders: [START_REF] Roberts | Information content of female chemical signals in the wolf spider, Schizocosa ocreata: Male discrimination of reproductive state and receptivity[END_REF], body size (long-tailed dance flies, Rhamphomyia longicauda: [START_REF] Funk | Courtship role reversal and deceptive signals in the long-tailed dance fly, Rhamphomyia longicauda[END_REF][START_REF] Lebas | Nonlinear and correlational sexual selection on 'honest' female ornamentation[END_REF]stalk-eyed flies, Diasemopsis meigenii: Harley et al., 2013;rainbow darters, Etheostoma caeruleum: Soudry et al., 2020;hermit crabs, Pagurus middendorffii: Wada et al., 2011), or vocal pitch (humans: Pisanski et al., 2018).

Among female traits, colourful ornaments have received more attention than others. In some species, females express sex-specific colourful ornaments whose colour and/or size are attractive to males. Such colourful ornaments are likely to be sexually selected traits as they correlate with indices of fecundity, such as sexual or egg maturity, readiness to spawn, brood size, or offspring condition (two-spotted gobies, Gobiusculus flavescens: [START_REF] Amundsen | Male mate choice selects for female coloration in a fish[END_REF]collared lizards, Crotaphytus collaris: Baird, 2004; cichlid fishes, Pelvicachromis taeniatus: [START_REF] Baldauf | Female nuptial coloration and its adaptive significance in a mutual mate choice system[END_REF]blue crabs, Calinnectes sapidus: Baldwin & Johnsen, 2012;spiny-footed lizards: Belliure et al., 2018;cichlid fishes, Mikrogeophagus ramizeri: LaPlante & Delaney, 2020;pygmy halbeaks, Dermogenys collettei: Ogden et al., 2020;striped plateau lizards, Sceloporus virgatus: Weiss, 2002[START_REF] Weiss | Ornamentation, age, and survival of female striped plateau lizards, Sceloporus virgatus[END_REF]. Comparatively, studies on the role of non-sex-specific colourful ornaments (i.e., ornaments expressed by both sexes) have yielded a contrasting picture of the extent to which these ornaments modulate mate choice.

Female colouration may indeed represent a genetically correlated response of selection for male traits with no clear adaptive value in females. Males may thus express no preference for such traits (three spined sticklebacks, Gasterosteus aculeatus: [START_REF] Wright | Conspicuous Female Ornamentation and Tests of Male Mate Preference in Threespine Sticklebacks (Gasterosteus aculeatus)[END_REF] or preference may be merely explained by sensory exploitation rather than sexual selection (sockeye salmons, Oncorhynchus nerka: [START_REF] Foote | Female colour and male choice in sockeye salmon: Implications for the phenotypic convergence of anadromous and nonanadromous morphs[END_REF]. However, there is some evidence that males can show preferences toward non-sex-specific colourful ornaments that reflect female reproductive state (rhesus macaque, Macaca mulatta: [START_REF] Dubuc | Sexual skin color contains information about the timing of the fertile phase in free-ranging Macaca mulatta[END_REF][START_REF] Higham | Color signal information content and the eye of the beholder: A case study in the rhesus macaque[END_REF][START_REF] Higham | Familiarity affects the assessment of female facial signals of fertility by free-ranging male rhesus macaques[END_REF]agamid lizards, Ctenophorus ornatus: LeBas & Marshall, 2000) or reproductive success (blue tits, Cyanistes caeruleus: [START_REF] Doutrelant | Female coloration indicates female reproductive capacity in blue tits[END_REF][START_REF] Doutrelant | Female plumage coloration is sensitive to the cost of reproduction. An experiment in blue tits[END_REF][START_REF] Mahr | Female attractiveness affects paternal investment: Experimental evidence for male differential allocation in blue tits[END_REF]rhesus macaques: Dubuc, Winters, et al., 2014;rock sparrows, Petroni petronia: Pilastro et al., 2003;[START_REF] Griggio | Female ornamentation and directional male mate preference in the rock sparrow[END_REF][START_REF] Griggio | Males prefer ornamented females: A field experiment of male choice in the rock sparrow[END_REF]; brown boobies, Sula leucogaster: [START_REF] Montoya | Repeatability of a dynamic sexual trait: Skin color variation in the Brown Booby (Sula leucogaster)[END_REF]blue-footed boobies, Sula nebouxii: Torres & Velando, 2005). Therefore, in these species, female colourful ornaments can have a crucial impact on the mating strategies and reproductive success of both males and females.

Here, we investigate the potential role of female colouration on male mate choice in a promiscuous primate species, the Japanese macaque (Macaca fuscata). Female mate choice has been suggested as an important process modulating mating strategies in this species [START_REF] Fujita | Interaction between male and female mating strategies and factors affecting reproductive outcome[END_REF]. However, this does not mean that males should not choose or discriminate among potential mates. While their parental investment is mostly restricted to gamete contribution, male Japanese macaques deal with high costs associated with mating. Japanese macaques are seasonal breeders and multi-mount ejaculators and thus invest a non-negligible portion of their daily energy expenditure for somatic maintenance and mating itself [START_REF] Fooden | Systematic review of Japanese macaques, Macaca fuscata (Gray, 1870)[END_REF][START_REF] Matsubara | Costs of Mate Guarding and Opportunistic Mating Among Wild Male Japanese Macaques[END_REF][START_REF] Thomsen | How costly are ejaculates for Japanese macaques? Primates[END_REF]. Like males of other species, male Japanese macaques also face physical and physiological costs associated with extended consortships and mateguarding, reduced feeding activity, and male-male aggression leading to increased vigilance, injuries, or chronic stress (e.g., in several mammal species including primates: [START_REF] Alberts | Mate guarding constrains foraging activity of male baboons[END_REF][START_REF] Girard-Buttoz | Mate-guarding constrains feeding activity but not energetic status of wild male long-tailed macaques (Macaca fascicularis)[END_REF][START_REF] Higham | The costs of reproductive success in male rhesus macaques (Macaca mulatta) on Cayo Santiago[END_REF][START_REF] Lukas | Costs of mating competition limit male lifetime breeding success in polygynous mammals[END_REF][START_REF] Thompson | The high price of success: Costs of mating effort in male primates[END_REF]. Males would thus benefit from discriminating among females to focus their mating effort on mates that are fertile and of "good quality". In Japanese macaques, both sexes express red skin colouration (luminance: how light/dark the skin appears, redness: at least partially reflecting the colour of the blood itself) of the face and hindquarters that peaks during the mating season with sex hormones [START_REF] Fooden | Systematic review of Japanese macaques, Macaca fuscata (Gray, 1870)[END_REF]. Recent studies provided more insight into the potential signalling function of female colouration in this species and characteristics that males may be selecting for. Hindquarter colouration may signal the intracycle probability of ovulation while concealing its exact timing and face colouration may indicate early pregnancy [START_REF] Rigaill | Multimodal advertisement of pregnancy in free-ranging female Japanese macaques (Macaca fuscata)[END_REF][START_REF] Rigaill | The redder the better? Information content of red skin coloration in female Japanese macaques[END_REF]. Hindquarter colouration may also contain information about the inter-cycle probability of conception [START_REF] Rigaill | The redder the better? Information content of red skin coloration in female Japanese macaques[END_REF]. Lastly, there is some evidence that skin colouration may reflect inter-individual variation (e.g., social rank, weight, but not parity, [START_REF] Rigaill | Testing for links between face color and age, dominance status, parity, weight, and intestinal nematode infection in a sample of female Japanese macaques[END_REF][START_REF] Rigaill | The redder the better? Information content of red skin coloration in female Japanese macaques[END_REF]. Pflüger and colleagues (2014) showed that males increased attention toward an experimentally intense red-coloured version of an unknown female face. Thus, female colouration may be a sexually selected trait involved in mate choice in Japanese macaques if males appropriately use this colourful trait to inform their mating decisions. Yet, to our knowledge, no study has tested this hypothesis in a non-forced-choice discrimination situation.

Our study thus aims at identifying the potential role of female Japanese macaque face and hindquarter colouration (luminance and redness) on the likelihood that a male will engage or maintain sexual interactions with a female. If female colouration acts as a fertility signal modulating mate choice, we hypothesised that males should display more sexual behaviours toward darker/redder (i.e., stronger signal) females. We further hypothesised that the relationship between male response and female colouration should vary according to a) intracycle (timing of the fertile phase), b) inter-cycle (number of consecutive cycles and probability of conception across cycles), and/or c) inter-individual (age, social rank, weight, and parity) variation. If so, we predicted a positive relationship between male sexual behaviours and darker/redder female colouration a) when the likelihood of ovulation is the highest (i.e., during pre-and fertile phases vs. the post-fertile phase), b) during cycles that lead to conception (i.e., during conceptive cycles and the first cycles of a given mating season), and/or c) with older, higher ranking, heavier, and possibly multiparous females.

Methods

Subjects and housing

We collected data during the 2011-2012 mating season, from early November to late January, from a captive population of Japanese macaques living in a 1,210 m 2 outdoor enclosure at Kyoto University Primate Research Institute (KUPRI, Inuyama, Japan). The group was composed of 39 individuals consisting of 13 adult females (> five years, mean ± SD = 10.64 ± 6.84 years, range = 5-27), 6 sexually immature females (2.79 ± 0.81 years, range = 1-4), 3 adult males (> six years, 10.67 ± 3.06 years, range = 8-14), 12 sexually immature males (2.99 ± 1.05 years, range = 1-4), and 5 infants less than one year old. All adult and sexually active females and males in this group were included in this study. The ratio of adult males to adult females is similar to other captive and wild populations from studies of sexual behaviours and mating strategies [START_REF] Fooden | Systematic review of Japanese macaques, Macaca fuscata (Gray, 1870)[END_REF][START_REF] Fujita | Interaction between male and female mating strategies and factors affecting reproductive outcome[END_REF]. We did not include subadult males/females to avoid confounding factors due to maturational age. Females were naturally cycling, i.e., with no hormonal contraceptive treatment. Female age, weight at the beginning of the mating season (to the nearest 100g), and parity (total number of infants born divided by the number of years post-adulthood) were provided by the Center for Human Modeling Research of KUPRI.

Animals were fed twice daily between 11:00 AM and 12:00 PM and between 4:00 PM and 5:00

Behavioral observations

One observer collected behavioural data using focal animal sampling distributed from 7:00 AM to 5:00 PM, 7 days a week for 3 months. We used a female-centered approach because of the need to assess female trait variation in order to address the question of male responses. We followed each focal female 2 hours per week in 4 blocks of 30-min continuous focal samples equally distributed across days and females. During focal observations, we recorded all social and sexual behaviours between males and females, along with the direction of the behaviour between the focal female and any identified adult male. We recorded all occurrences of female presentations, female approaches, female mounts, female vocalisations (oestrus and copulation calls), male approaches, male holding behaviours, male inspections of female genitals (i.e., visual and/or tactile and/or olfactory inspections of the female genital area), and male mounts (i.e., ejaculatory and non-ejaculatory). The number of observed records for each behaviour is given in Table 1. Female and male social ranks were assessed by transcribing agonistic interactions for which a clear win/loss outcome was identified into an agonistic interaction matrix. We then calculated the Normalized David's Score (NDS) to assess female social rank positions [START_REF] De Vries | Measuring and testing the steepness of dominance hierarchies[END_REF]. Given that there were only three adult males in the group, male ranks were assessed as alpha (winning all agonistic interactions), beta (defeating all males except the alpha male), and gamma (losing all agonistic interactions) directly from the agonistic interaction matrix.

Sum

Female colouration

One experimenter took digital images of female face and hindquarters every two days in the morning between 11:00 AM and 12:00 PM. We used a Canon EOS 350D camera with an 8 megapixel CMOS censor and an EF28-135 mm f/3.5-5.6 IS USM lens. Images were standardized by daily manual setting of the white balance using an X-Rite White Balance Card and a Gretag X-Rite Color Checker (GretagMacbeth ColorChecker). For each female's photo, we assessed colour from the whole face and hindquarter areas excluding the eyes, nose, forehead, and tail. A technician who was blind to females' identity and characteristics extracted reflectance spectra using Colourworker software (Chrometics Ltd. Website, available: http://www.chrometrics.com). These spectra were converted into quantal catches, i.e., the stimulation of Japanese macaque photoreceptors [START_REF] Rigaill | The redder the better? Information content of red skin coloration in female Japanese macaques[END_REF] using equations and macaque cone ratio and spectral sensitivity values given elsewhere [START_REF] Higham | Color signal information content and the eye of the beholder: A case study in the rhesus macaque[END_REF][START_REF] Stevens | Studying primate color: Towards visual system-dependent methods[END_REF]. Based on the quantal catch data for both face and hindquarter colouration, we calculated luminance (how light/dark the skin appears) and redness (variation in chromatic parameters) data.

Female reproductive status

To determine each female's reproductive status, we collected a total of 381 faecal samples (mean ± SD per female = 29.31 ± 3.17, range = 24-34) on average every 1.59 days (range = 0-4 days) to accurately assess the ovarian cycle [START_REF] Hodges | Field endocrinology: Monitoring hormonal changes in free-ranging primates[END_REF]. We collected faecal samples in their entirety immediately after voiding, with most samples collected between 8:00 AM and 12:00 PM. All samples were then stored at -20°C until processing. Faecal samples were analysed for pregnanediol-3-glucuronide (PdG) using enzyme immunoassays as previously described by [START_REF] Garcia | Relationship between sexual interactions and the timing of the fertile phase in captive female Japanese macaques (Macaca fuscata)[END_REF] to assess the presumed day of ovulation and the periovulatory period (15-day period around the estimated ovulation) of each ovarian cycle. The onset of the luteal phase was determined as the sample with a faecal PdG concentration which was at least two standard deviations greater than the mean PdG concentration of the 3-4 preceding baseline values [START_REF] Hodges | Field endocrinology: Monitoring hormonal changes in free-ranging primates[END_REF]. We determined the 2-day window for ovulation as days -2 and -3 relative to the faecal PdG rise. The fertile phase was defined as a period of five days (covering a two-day window for ovulation plus three preceding days to account for sperm longevity in the reproductive tract [START_REF] Behboodi | The use of a urinary estrone conjugates assay for detection of optimal mating time in the cynomolgus macaque (Macaca fascicularis)[END_REF]. The five-day period preceding the fertile phase represented the pre-fertile phase, while the five-day period following the fertile phase was defined as the post-fertile phase. We recorded a total of 24 ovarian cycles (mean ± SD per female = 2.0 ± 0.83, range = 1-3) and the majority of these cycles were non-conceptive (19/24). One female failed to cycle during the study and was thus excluded from the analysis.

Data analysis

We restricted our analyses to the pre-fertile, fertile, and post-fertile phases. We used 173 focal observations with matching female photo (i.e., focal and photo on the same observation day)

for the analyses (mean ± SD = 7.53 ± 1.21 per female and per cycle). We built our dataset based on the 30-min focal observations and attributed one line to each focal female -male dyad per focal observation (i.e., three lines per focal female), which was the unit of analysis.

We carried out our analyses in R version 3.3.3. Before running the statistical analyses, we checked for correlations between our variables of interest. Female weight correlated with parity (rs = -0.50, P < 0.001) and age (rs = 0.73, P < 0.001). Parity correlated with age (rs = -0.42, P < 0.001). Parity also weakly correlated with female social rank (rs = -0.25, P < 0.001).

We thus excluded female age and weight from our analyses. None of the female behaviours (i.e., presentations, approaches, and mounts) were correlated with each other (all P > 0.05).

None of the male behaviours (i.e., approaches, holding behaviours, genital inspections, and mounts) were correlated with each other (all P > 0.05).

We aimed at testing whether the likelihood of a male behavioural response (binary variable: 1 = occurrence, 0 = non-occurrence) toward a female was predicted by female colouration, i.e., whether males where more likely to engage in or maintain sexual interactions with darker/redder females. The little occurrences of copulatory event alone prevented us from analysing the effect of female colouration on this behaviour only. We thus computed a binary response variable that corresponded to a male behavioural response as the occurrence of at least one of the behaviours involved in establishing and maintaining consortship that were displayed by the interacting male of the dyad (i.e., approaches, holding behaviours, genital inspections, and copulations). Similarly, we also computed a binary variable that corresponded to a female behavioural display (hereafter, female behaviours), as the occurrence of at least one of the behaviours involved in establishing and maintaining consortship that were displayed by the interacting female of the dyad (i.e., presentations, approaches, and mounts).

We built a series of generalized linear mixed models with a binomial error structure and logit link function with the function glmer from the lme4 package [START_REF] Bates | Fitting linear mixed-effects models using lme4[END_REF] to investigate the relationship between male behavioural response (binary response variable) and female hindquarter and face luminance and redness (continuous predictor variables). Inspection of the cumulative distribution functions revealed good fits to the normal distribution for hindquarter luminance and to the lognormal distribution for face luminance, face redness, and hindquarter redness. We first constructed three different models to test the relationship between male behaviours and female face colouration (luminance and redness), male behaviours and female hindquarter colouration (luminance and redness), and male behaviours and female face and hindquarter colouration (luminance and redness). To account for a) intra-cycle, b) intercycle, and c) inter-individual effects on the potential relationship between female colouration and male response, our three models included interactions between female colouration and a)

the cycle phase (categorical variable, three levels: pre-fertile, fertile, and post-fertile phases), b) the number of consecutive cycles (continuous variable) and the conceptive nature of the cycle (binary variable: 1 = conceptive, 0 = non-conceptive cycles, hereafter conception), and c) female social rank (continuous variable) and parity (continuous variable). We also included female behaviours (binary variable) and male social rank (categorical variable, three levels: alpha, beta, gamma) as covariates to account for the potential effects of female direct solicitation and male social attributes on mating decisions. Finally, the date of the focal, female identity, and male identity were included as random factors. We also constructed a null model in which the predictor variables and all covariates were removed but the random effects structure was maintained. All continuous variables were standardized to mean = 0 and SD = 1

prior to modelling to improve model performance and interpretability [START_REF] Harrison | A brief introduction to mixed effects modelling and multi-model inference in ecology[END_REF].

We checked for variance inflation (GVIF (1/(2xDf)) < 3) using the car package [START_REF] Fox | An {R} Companion to Applied Regression[END_REF]; we excluded the number of consecutive cycles from the models due to variance inflation.

We ensured that all relevant model assumptions (homogeneity of the residuals and stability of estimates) were met by visually inspecting histograms of the residuals and plots of residuals against fitted values.

We used an information-theory approach to objectively compare and rank the four candidate models (i.e., null model, face colouration model, hindquarter colouration model, and face and hindquarter colouration model) in terms of how well they fitted the existing data. This method allowed us to assess the likelihood that one or more models among the candidates is/are best supported by the data [START_REF] Burnham | AIC model selection and multimodel inference in behavioral ecology: Some background, observations, and comparisons[END_REF][START_REF] Symonds | A brief guide to model selection, multimodel inference and model averaging in behavioural ecology using Akaike's information criterion[END_REF]. We used the function model.sel of the MuMIn package [START_REF] Bartoń | Contact pheromones mediate male preference in black widow spiders: Avoidance of hungry sexual cannibals?[END_REF] to rank models based on their respective Akaike's information criterion corrected for small sample size (AICc values). We considered models with ΔAICc (the difference in AICc values between the smallest AICc and the other AICc values) below four as competitive models with the plausibly best model in the set of candidate models [START_REF] Burnham | Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach[END_REF]. We reported the weight of the models which indicates to what extent one candidate model is more likely than another to provide a reasonable explanation of the variance in the data, along with the evidence ratio (ER) which indicates the extent to which the higher-ranked model is more parsimonious than another candidate model. We also reported the marginal (fixed effects alone) and conditional (fixed effects and random structures) coefficients of determination (R 2 m and R 2 c) of all models using the function r.squaredGLMM [START_REF] Nakagawa | A general and simple method for obtaining R2 from generalized linear mixed-effects models[END_REF]. We extracted weighted parameter estimates (β), standard errors (SE), and 95% confidence intervals (95% CIs) of model intercepts and predictors from conditional averaging of the 4 candidate models (function model.avg).

Results

Among the candidate models, the model including female hindquarter colouration (luminance and redness) and covariates had the highest weight (0.998) and more than 499 times more empirical support than the competing models (face colouration: ΔAICc = 12.3; face and hindquarter colouration: ΔAICc = 17.6; null model: ΔAICc = 49.8, Table 2). The fixed and random effects explained around 28% of the observed variance in the occurrence of male behavioural responses.

There was no effect of cycle phase (fertile vs. prefertile and post-fertile phases) on the interaction between male behavioural responses and female colouration (luminance and redness of the face and hindquarters, all P > 0.50, Table 3, Figure 1A). There was a slight tendency for a higher probability of male behavioural responses when hindquarter luminance was high in non-conceptive vs. conceptive cycles (ß = -0.96 ± 0.52 SE, P = 0.068, 95% CI =-1.98; 0.07, Table 3, Figure 1B). In other words, our results suggest that during conceptive cycles, there was a higher probability of male behavioural responses when the signal was strong (lower luminance = darker hindquarters). However, female face and hindquarter colouration per se did not explain variation in the occurrence of male behavioural responses (luminance and redness, all P > 0.05, Table 3). Similarly, the likelihood of observing male sexual behaviours was not influenced by cycle phase or the conceptive nature of the cycle (all P > 0.50, Table 3).

Males were more likely to display sexual behaviours toward females that have solicited or interacted with them (ß = 2.99 ± 0.42 SE, P < 0.001, 95% CI = 2.15; 3.82, Table 3). Males were also more likely to display sexual behaviours toward females of lower parity (ß = -1.17 ± 0.53 SE, P = 0.026, 95% CI = -2.20; -0.14, Table 3) and of higher social rank (ß = 0.60 ± 0.28 SE, P = 0.031, 95% CI = 0.05, 0.15, Table 3). Male social rank was positively related to the occurrence of male sexual behaviours (alpha vs. beta: ß = -0.97 ± 0.37 SE, P = 0.010, 95% CI = -1.71; -0.24, alpha vs. gamma: ß = -1.45 ± 0.41 SE, P < 0.001, 95% CI = -2.26; -0.65, Table 3).

Models k logLik

Discussion

We examined male choice toward a non-sex-specific colourful ornament in a promiscuous primate species with traditional sex roles and demonstrated that males did not bias their sexual behaviours toward darker/redder females. Our results also showed that male Japanese macaques did not seem to use female colouration (luminance and redness of the face and hindquarters) to focus their mating effort on the most fertile period of the menstrual cycle, or during cycles that lead to conception, or on "good quality" mates.

The lack of effect of female colouration on male sexual behaviours -or in other words, the lack of male choice for darker/redder females-is surprising given that previous studies suggested a potential role of female colouration on male mate choice in Japanese macaques

Hindquarter luminance Male Behavioural response

Cycle

Non-conceptive

Conceptive over-estimated due to the binary choice, the lack of familiarity with the stimuli which significantly matters in communication (e.g., visual trait: [START_REF] Higham | Familiarity affects the assessment of female facial signals of fertility by free-ranging male rhesus macaques[END_REF]odorant trait: Setchell et al., 2010;auditory trait: Townsend et al., 2011), or the lack of precise measure of attention such as tracking fine-scale eye movements [START_REF] Setchell | The application of noninvasive, restraint-free eye-tracking methods for use with nonhuman primates[END_REF][START_REF] Yorzinski | Through their eyes: Selective attention in peahens during courtship[END_REF].

Our present study demonstrated that female colouration is unlikely to be a sexual signal since males did not appear to use this trait to discriminate when and with whom to mate to reproduce efficiently. Skin colouration may thus correlate with female characteristics without having a signalling function (correlation without causation). This highlights the importance of studying traits suspected to play a role in sexual communication and mating strategies from both signaller (what information may be contained) and perceiver (how the trait may affect behaviours) perspectives. Alternatively, we cannot completely rule out that mate choice for female colouration occurs in this species but that our data did not capture such effect. One may argue that males of this group may not rely -or need to rely -on this colouration to make their mating decisions as a result of the lower male-male competition due to the lack of peripheral males and the monopolization of mating from the alpha male. However, if female colouration indeed plays a role in male sexual strategies in this species where females vary in quality and female-female competition is high, one may have expected the alpha male to express mate choice as he has priority of access to mating. However, the alpha male did not appear to do so (see supplementary material).

In the absence of male mate choice for a female trait and of a clear ovulatory signalling in this species [START_REF] Garcia | Relationship between sexual interactions and the timing of the fertile phase in captive female Japanese macaques (Macaca fuscata)[END_REF][START_REF] O'neill | Relationship between ovarian cycle phase and sexual behavior in female Japanese macaques (Macaca fuscata)[END_REF][START_REF] Rigaill | The redder the better? Information content of red skin coloration in female Japanese macaques[END_REF], does this mean that males mate randomly? This is highly unlikely considering the energetic costs of extended sperm production, prolonged consortship, and male-male competition [START_REF] Matsubara | Costs of Mate Guarding and Opportunistic Mating Among Wild Male Japanese Macaques[END_REF][START_REF] Thomsen | How costly are ejaculates for Japanese macaques? Primates[END_REF]. While female colouration may not inform males about the probability of ovulation or female characteristics, this traits appears to be involved in the communication of pregnancy status (information content: [START_REF] Rigaill | Multimodal advertisement of pregnancy in free-ranging female Japanese macaques (Macaca fuscata)[END_REF]. Moreover, males have been found to reduce or avoid mating with pregnant females in populations facing higher energetic constrains [START_REF] Fujita | Interaction between male and female mating strategies and factors affecting reproductive outcome[END_REF][START_REF] Rigaill | Multimodal advertisement of pregnancy in free-ranging female Japanese macaques (Macaca fuscata)[END_REF]. However, whether males indeed use female colouration to avoid wasting energy in post-conceptive mating calls for further investigation. We also found in the present study that males were more likely to respond to female solicitations that could be used to apportion their mating effort toward receptive females vs. non-cycling females. Males may thus adapt their mating strategies according to female behaviour and to their access to mate by either intensifying mate guarding (higher-ranking males) or seizing a mating opportunity (lower-ranking males). Monitoring female behaviours would ultimately benefit to both higherand lower-ranking males by balancing the costs and benefits of mate searching, direct competition, and mating. Alternatively, male sexual behaviours may be a by-product response of female mate choice which has been reported as an active reproductive strategy in Japanese macaques [START_REF] Huffman | Mate selection and partner preferences in female Japanese macaques[END_REF][START_REF] Huffman | Influences of female partner preference on potential reproductive outcome in Japanese macaques[END_REF][START_REF] Soltis | Sexual selection in Japanese macaques II: Female mate choice and male-male competition[END_REF]. We also found that males may bias their mate choice toward females at the beginning of their reproductive history and toward higherranking females. This may be related to parental investment as lower-ranking females and/or those toward the end of their reproductive history may have less energy to invest in future offspring or be less capable to balance the costs of reproduction with other costs (e.g., competition, health) than higher-ranking females and/or those at the beginning of their reproductive history (e.g., mandrills, [START_REF] Setchell | Mate choice in male mandrills (Mandrillus sphinx)[END_REF]. More precise data on the interaction between female behaviours and characteristics and male mate choice is needed to better understand the process of male mate choice in Japanese macaques.

Studies of the role of non-sex-specific colourful ornaments displayed on feather or skin have been relatively limited in comparison with sex-specific colourful ornaments in both males and females. From this rather incomplete picture, it appears that male choice and preferences for female traits may evolve because of the costs associated with reproduction. One factor may be the cost of sexual signalling itself, when female ornamentations are costly to produce or maintain and thus impact the energy allocated to reproduction and ultimately, offspring quality and survival [START_REF] Doutrelant | Female coloration indicates female reproductive capacity in blue tits[END_REF][START_REF] Doutrelant | Female plumage coloration is sensitive to the cost of reproduction. An experiment in blue tits[END_REF][START_REF] Griggio | Female ornamentation and directional male mate preference in the rock sparrow[END_REF][START_REF] Peters | No evidence for general condition-dependence of structural plumage colour in blue tits: An experiment[END_REF][START_REF] Pilastro | Male rock sparrows adjust their breeding strategy according to female ornamentation: Parental or mating investment[END_REF], although it is complicated to accurately assess such signalling costs. Another factor may be the shared costs of rearing offspring in socially monogamous species displaying extrapair copulations [START_REF] Griggio | Males prefer ornamented females: A field experiment of male choice in the rock sparrow[END_REF][START_REF] Mahr | Female attractiveness affects paternal investment: Experimental evidence for male differential allocation in blue tits[END_REF][START_REF] Torres | Male preference for female foot colour in the socially monogamous blue-footed booby, Sula nebouxii[END_REF]. Under such conditions, both males and females would benefit from traits indicative of one's fecundity to adjust their mating and parental effort according to the costs and prospective benefits of signalling to and of reproducing with a particular mate. Alternatively, male choice for a colourful trait has been also described in at least one promiscuous species with traditional sex roles. In the territorial agamid lizards, female colourful traits appear to convey information about receptivity: male choice may thus also evolve to maximize reproductive success when mating decisions are limited or constrained [START_REF] Lebas | The role of colour in signalling and male choice in the agamid lizard Ctenophorus ornatus[END_REF].

To our knowledge, our study is among the few to report a lack of male choice for female colouration (sockeye salmons: [START_REF] Foote | Female colour and male choice in sockeye salmon: Implications for the phenotypic convergence of anadromous and nonanadromous morphs[END_REF]threespine sticklebacks: Wright et al., 2015). A possible explanation could involve the operational sex ratio. A female-biased operational sex ratio, i.e., when males are a limited resource, could favour female-female competition and ornamentation. However, in seasonal breeders, it is likely that the number of cycling, receptive, and defendable females within a group or location decreases rapidly during the mating season as a consequence of reproductive synchrony [START_REF] Ims | The ecology and evolution of reproductive synchrony[END_REF][START_REF] Thompson | Reproductive seasonality[END_REF]. Male-male competition for the access to mates may thus strengthen which would select for male ornaments, especially in species where the cost of reproduction is heavier on females than males. Such scenario may explain the lack of male choice for female colouration in Japanese macaques.

Females are indeed more likely to conceive at the beginning of the mating season and to invest high energy on gestation and maternal care while facing energetic stress from decreased food availability and thermoregulation [START_REF] Fujita | Interaction between male and female mating strategies and factors affecting reproductive outcome[END_REF][START_REF] Garcia | Seasonal and reproductive variation in body condition in captive female Japanese macaques (Macaca fuscata)[END_REF][START_REF] Garcia | Energetic consequences of seasonal breeding in female Japanese macaques (Macaca fuscata)[END_REF]. Thus, female colouration may only represent a correlated response of selection for ornamentation in males [START_REF] Lande | Sexual dimorphism, sexual selection, and adaptation in polygenic characters[END_REF]. The signalling function of male colouration and its role on mate choice in Japanese macaques is still unknown. However, evidence from other primate species suggests that male colouration can correlate with some aspects of male-male competition (geladas, Theropithecus gelada: [START_REF] Bergman | Chest color and social status in male geladas (Theropithecus gelada)[END_REF]snub-nosed monkeys, Rhinopithecus bieti: Grueter et al., 2015;drills, Mandrillus leucophaeus: Marty et al., 2009;rhesus macaques: Petersdorf et al., 2017;mandrills: Setchell & Dixson, 2001) and may be used in female mate choice (rhesus macaques: Dubuc, Allen, et al., 2014;[START_REF] Dubuc | Who cares? Experimental attention biases provide new insights into a mammalian sexual signal[END_REF][START_REF] Waitt | Evidence from rhesus macaques suggests that male coloration plays a role in female primate mate choice[END_REF].

In conclusion, we found no clear evidence that female colouration (luminance and redness of the face and hindquarters) influence male mate choice in Japanese macaques. Female colouration does not appear to be a sexually selected trait conveying relevant information about intra-cycle, inter-cycle, and inter-individual differences in fecundity. However, future studies should investigate the potential role of both female and male colouration in the communication of reproductive status (e.g., cycling vs. non cycling, pregnancy), male-male competition, and female mate choice to draft any definite conclusions. Whether reproductive seasonality and reproductive costs influence the evolution of male mate choice and female traits remains an open question, especially in primates for which little data is available. Further studies focusing on the relationship between female/male colouration and male/female sexual behaviours and mate choice in species facing different reproductive constraint should help to improve our understanding of the role and evolution of non-sex-specific colourful ornaments in primates.

  PM. Water was supplied ad libitum. The Center for Human Evolution Modeling Research of the Primate Research Institute reviewed and approved our research protocol in agreement with the Guidelines for the Care and Use of Nonhuman Primates of the Kyoto University Primate Research Institute.

Figure 1 .

 1 Figure 1. Interaction plot of the generalized linear mixed model exploring the relationship

(

  face redness:[START_REF] Pflüger | Colour and contrast of female faces: Attraction of attention and its dependence on male hormone status in Macaca fuscata[END_REF] face and hindquarter luminance and redness:[START_REF] Rigaill | The redder the better? Information content of red skin coloration in female Japanese macaques[END_REF]. While we investigated changes in male sexual behaviours toward group-member females varying in skin colouration and characteristics,[START_REF] Pflüger | Colour and contrast of female faces: Attraction of attention and its dependence on male hormone status in Macaca fuscata[END_REF] investigated male responses toward two visual stimuli of an unknown female face. A red effect may have been

Table 1 .

 1 Total

	Female behaviours
	Presentations	10
	Approaches	86
	Mounts	86
	Male behaviours
	Approaches	153
	Holding behaviors	54
	Genital inspections	39
	Mounts	75
	Female vocalisations	30

number of female behaviours, male behaviours, and female vocalisations recorded.

AICc ΔAICc weight ER R 2 m -R 2 c

	Hindquarter colouration	24 -140.1 330.6	0.0	0.998	-0.26 -0.28
	Face colouration	24 -146.2 342.9 12.31 0.002 499 0.19 -0.23
	Face and hindquarter colouration	36 -135.3 348.2 17.59	0.00	-0.29 -0.31
	Null	4	-186.2 380.4 49.84	0.00	-0.00 -0.14

Table 2 .

 2 Characteristics of candidate models.

	ß	SE z-value	P	95% CI
	Intercept -1.20	0.53	2.25	0.024	-2.45; -0.16
	Face luminance 0.21	0.44	0.47	0.64	-0.66; 1.08
	Face redness 0.39	0.60	0.64	0.52	-0.79; 1.57
	Hindquarter luminance 0.50	0.37	1.36	0.17	-0.22; 1.23
	Hindquarter redness -0.45	0.38	1.16	0.25	-1.20; 0.31
	Female behaviour 2.99	0.42	7.02	< 0.001	2.15; 3.82
	Cycle phase (fertile vs. prefertile) -0.09	0.42	0.21	0.84	-0.90; 0.73
	Cycle phase (fertile vs. post-fertile) -0.25	0.40	0.63	0.53	-1.03; 0.53
	Conception -0.25	0.56	0.44	0.66	-1.35; 0.86
	Parity -1.17	0.53	2.22	0.026	-2.20; -0.14
	Female rank 0.60	0.28	2.15	0.031	0.05; 1.15
	Male rank (alpha vs. beta) -0.97	0.37	2.60	0.009	-1.71; -0.24
	Male rank (alpha vs. gamma) -1.45	0.41	3.54	< 0.001 -2.26; -0.65
	Interaction: face luminance x

k: number of predictors, AICc: corrected Akaike's information criterion, ΔAICc: AICc difference between higher-ranked models and other candidate models., weight: model probabilities, ER: evidence ratios, R 2 m -R 2 c: marginal and conditional coefficients of determination. Candidate models are presented from higher-ranked to lower-ranked ones.

Table 3 .

 3 Model

averaged parameters estimates (ß) ± adjusted standard errors (SE), z-value, probability (P), and confidence intervals (95% CIs) from conditional averaging of all candidate models. For each categorical comparison, the first level indicate baseline. Results for which CI does not include zero are presented in bold, trends are presented in italic.

Acknowledgments

The authors thank The Center for Human Evolution Modeling Research of the Primate Research Institute and M.A. Huffman for use of the primate facilities. We thank A. Chimènes

for her help with the preparation of the digital photography. We also thank J. Duboscq and AJJ.

MacIntosh for helpful discussion on the analyses and comments on an earlier version of this manuscript. This work was financially supported by the Centre National de la Recherche Scientifique (France) to CG.


Marta Avalos

Hélène Touchais

Marcela Henríquez-Henríquez

Optimising criteria for manual smear review following automated blood count analysis: A machine learning approach

Keywords: Machine Learning for Healthcare Applications, Data mining, Feature selection, Interpretability, Categorisation of continuous variables, Lasso, GAM

The complete blood count (CBC) performed by automated haematology analysers is the most common clinical procedure in the world. Used for health checkup, diagnosis and patient follow-up, the CBC impacts the majority of medical decisions. If the analysis does not fit an expected setting, the laboratory staff manually reviews a blood smear, which is highly time-consuming. Criteria for reviewing CBCs are based on international consensus guidelines and locally adjusted to account for laboratory resources and populations characteristics. Our objective is to provide a clinical laboratory decision support tool to identify which CBC variables are linked to an increased risk of abnormal manual smear and at which threshold values. Thus, we treat criteria adjustment as a feature selection problem. We propose a cost-sensitive Lasso-penalised additive logistic regression combined with stability selection, adapted to the peculiarities of data and context: class-imbalance, categorisation of continuous predictors, required stability and enhanced interpretability. Using simulated and real CBC data, we show that our proposal is competitive in terms of predictive performance (compared to deep neural networks) and model selection performance (provided that there is sufficient data in the neighbourhood of the true thresholds). The R code is publicly available as an open source project.

Introduction

The complete blood count (CBC) with leukocyte differential count obtained by automated haematology analysers is often included as part of a routine checkup. It is one of the most ordered laboratory tests used to monitor overall health, medical condition or medical treatment and to help detect a variety of disorders. When the analysis does not fit an expected setting, the machine triggers a warning flag. Then, laboratory staff manually prepares and reviews a blood smear using a microscope. The International Consensus Group for Haematology Review published in 2005 criteria for reviewing CBCs [START_REF] Barnes | The international consensus group for hematology review: suggested criteria for action following automated CBC and WBC differential analysis[END_REF]. In general, adjustments are needed to account for local populations characteristics and laboratory resources [START_REF] Trimoreau | Harmonisation of full blood count reports, recommendations of the French-speaking cellular haematology group (GFHC)[END_REF][START_REF] Comar | Evaluation of criteria of manual blood smear review following automated complete blood counts in a large university hospital[END_REF][START_REF] Palur | Effectiveness of the International Consensus Group criteria for manual peripheral smear review[END_REF][START_REF] Baccini | Platelet counting: Ugly traps and good advice. proposals from the French-Speaking Cellular Hematology Group (GFHC)[END_REF]. While the manual technique is more effective identifying nuances in cell shapes, it incurs a substantial time and cost burden on laboratory staff. Also, it requires a high degree of technical skill. There is great interest in optimising human and financial resources (i.e. minimising the number of useless reviews) whilst ensuring the quality of patient care (i.e. minimising the number of omitted useful reviews).

Motivation example

The manual smear is also used as a quality control tool to evaluate and calibrate the automated haematology analysers. Routinely, all samples (with or without a warning flag) analysed in a given period of time are reviewed to evaluate the need for adjustments in criteria for manual smear review. As a secondary use of these large-scale data we propose to train machine learning algorithms.

CBCs and corresponding blood smear reviews for 10 000 patients were performed at the clinical laboratory of the Pontificia Universidad Catolica de Chile in 2016 in the framework of the quality control procedure. After excluding hospitalised patients and patients with a non-consensual smear result, a total of 9 594 data were available. The response variable, normal/abnormal smear, is imbalanced: only 7% of the smears were abnormal.

CBC data consisted in haemoglobin (g/dL), haematocrit (%), mean corpuscular haemoglobin (pg/cell), mean corpuscular volume (f L), erythrocytes (10 6 /µL), red blood cell distribution width (%), platelets (10 3 /µL), leukocytes (10 3 /µL), immature granulocytes (%) and the leukocyte differential count which includes neutrophils, basophils, eosinophils, monocytes and lymphocytes (10 3 /µL and %). The presence or absence of alarms of suspected alterations of blood cells (binary), sex (binary) and age (years) were also reported.

Objective

Machine learning tools for decision support have already been applied in the field of haematology [START_REF] Sivapalaratnam | Artificial intelligence and machine learning in haematology[END_REF][START_REF] Shouval | Machine learning and artificial intelligence in haematology[END_REF][START_REF] Radakovich | Artificial intelligence in hematology: Current challenges and opportunities[END_REF]. However, they are essentially limited to predict a given disease onset from CBC data. We aimed to provide a machine learning based-decision support tool for adjusting abnormal levels in the CBC tests for use at laboratory level. We treated criteria adjustment as a feature selection problem, thus the objective was to identify which CBC variables are linked to an increased risk of abnormal manual smear and at which threshold values.

Our proposal is based on cost-sensitive Lasso-penalised additive logistic regression, which represents a trade-off between predictive performance and interpretability. Cost-sensitive learning and data sampling are the main methods in imbalanced binary classification [START_REF] Leevy | A survey on addressing highclass imbalance in big data[END_REF][START_REF] Fernandez Hilario | Learning from Imbalanced Data Sets[END_REF]. The first is a better choice for large sample sizes. Weighting individuals' contributions to account for the degree of imbalance is one of the simplest cost-sensitive approaches. Categorisation of continuous variables, though not recommended, is usual in clinical studies since the analysis and results interpretation are greatly simplified. Thus, we considered additive functions to belong to the space of piece-wise constant functions. On the other hand, the natural sparsity encouraged by the Lasso penalty is combined with a stability selection procedure to enhance model stability.

Related work

When no prior knowledge is available to guide the choice of the categories, continuous variables are typically categorised into quantiles. Also, additive models fitted by cubic splines are used as an exploratory data analysis tool [START_REF] Barrio | A new approach to categorising continuous variables in prediction models: Proposal and validation[END_REF]. Instead of using piece-wise cubic polynomials which are next transformed into constant functions, we propose to use directly piece-wise constant polynomials. A crucial question is how to select the optimal number and/or location of threshold values. That could be viewed as a variable selection problem. Indeed, once the basis functions have been set, we have moved from a non-linear to a linear space, so that a variable selection methods for linear settings can be used. Standard variable selection procedures have already been applied [START_REF] Barrio | A new approach to categorising continuous variables in prediction models: Proposal and validation[END_REF][START_REF] Liquet | CPMCGLM: an R package for p-value adjustment when looking for an optimal transformation of a single explanatory variable in generalized linear models[END_REF], but only when very few quantitative predictors have to be categorised. Yet, CBC data consist in more than twenty blood cell measurements.

Penalised methods leading to parsimonious models, such as the Lasso [START_REF] Hastie | Statistical Learning with Sparsity[END_REF], have emerged as convenient alternative approaches. Extensions to parsimonious additive models have been proposed based on a reformulation of the estimation problem in terms of group Lasso [START_REF] Amato | Additive model selection[END_REF], sparse group Lasso, overlapping group Lasso, or fused Lasso [START_REF] Petersen | Data-adaptive additive modeling[END_REF]. Several R packages have been developed, e.g. COSSO, GAMBoost, SAM, GAMSEL, hgam, MSGLasso, FLAM [START_REF] Petersen | Data-adaptive additive modeling[END_REF]. However, these packages are limited in their capability to support the logistic model or a mixture of qualitative and quantitative predictors or piece-wise constant basis functions or to handle large data sets.

Method

Let (X 1 , . . . , X p , Y ) be a random vector with Y a binary variable, coded 0 for normal and 1 for abnormal blood smears. Let {(X i1 , . . . , X ip , Y i )} n i=1 be an i.i.d. sample of size n and {(x i1 , . . . , x ip , y i )} n i=1 a realization of the sample. Let us note x j = (x 1j , . . . , x nj ) the vectors of size n × 1. Consider the additive logistic model in terms of the logit function:

logit(P (Y i = 1|x i1 , . . . , x ip )) = ln P (Y i = 1|x i1 , . . . , x ip ) 1 -P (Y i = 1|x i1 , . . . , x ip ) = β 0 + p j=1 f j (x ij ) , (1)
where f j are unknown univariate real (centred) functions and β 0 is the unknown intercept.

Let {χ k j (x j )} pj k=1 be a fixed basis of functions and denote χ k j = χ k j (x j ). Any f j can be expanded in terms of these basis functions: f j (x j ) = pj k=1 β k j χ k j where {β k j } pj k=1 are unknown parameters. By considering the block matrix (χ 1 , . . . , χ p ) and vector β = (β 1 , . . . , β p ) , we can write: f j (x j ) = χ j β j . Consider now the particular case of piece-wise constant polynomials: χ k ij = 1 if x ij > q k j and 0 otherwise, with q k j the k-th value of a collection of K j fixed values, e.g. the ordered percentiles of X j .

We consider the problem of estimating a high-dimensional additive model using sparse methods. The number of candidate explanatory variables could be large but only a small set of them contribute to the response. In addition, the relationships between explanatory variables and the outcome are possibly non linear. Through the logistic likelihood function, we can write the weighted penalised optimisation problem:

max β n i=1 ω i ln e yi p j=1 χ ij β j 1 + e p j=1 χ ij β j -λ β 1 , (2)
where to simplify, the intercept is omitted, λ > 0 controls the amount of regularisation, the weights ω i = ω > 1 if y i = 1 and 1 if y i = 0 are used to account for the degree of imbalance in the minority class. Indeed, when facing with an imbalanced learning problem, a simple strategy consists in weighting individuals' contributions to the likelihood function [START_REF] Leevy | A survey on addressing highclass imbalance in big data[END_REF][START_REF] Fernandez Hilario | Learning from Imbalanced Data Sets[END_REF].

Practical implementation

The lasso estimator β that maximises (2) depends on λ and ω. When the primary goal of the study is prediction, K-fold cross-validation (e.g. based on the area under the receiver operating characteristic curve -ROC AUC) is routinely applied to estimate λ. When the primary goal is to infer the set of truly relevant variables, the combination of cross-validation with stability selection [START_REF] Meinshausen | Stability selection[END_REF] is a better option [START_REF] Wang | High-dimensional regression in practice: an empirical study of finite-sample prediction, variable selection and ranking[END_REF]. The stability selection procedure calculates the empirical selection probability for each χ k j , that we use as a continuous measure of the stability associated to the Lasso estimates.

Class weighting can be determined on the basis of expert judgement, using the inverse of the empirical class distribution in the learning data, or evaluating a performance criterion on a grid of ω-values with test data. We applied the following heuristic as a previous step: we determined a grid of ω-values around the inverse of the observed class distribution, the optimal value of ω was that maximising the cross-validated ROC AUC under the constraint that recall cannot be inferior to a given fixed value (to be determined by laboratory clinicians, 0.85 here).

To avoid overoptimistic results, we implement cross-validation with stratified folds (i.e. each fold contains roughly the same proportion of abnormal smears as in the whole sample) and using weights only in the training step [START_REF] Kong | Hyperparameter optimisation for improving classification under class imbalance[END_REF]. We developed a simple R function to construct features based on percentiles for each variable. Equation ( 2) and ROC AUC cross-validation can be resolved by calling the R glmnet package [START_REF] Friedman | Regularized paths for generalized linear models via coordinate descent[END_REF]. Another R function allows to visualise the Lasso estimates and corresponding selection probabilities represented by a light to dark gradient of grey.

Evaluation

The purpose of this evaluation was manifold, we aimed to answer the following questions:

-Is the predictive performance of our model comparable to that of powerful models such as deep neural networks (NN), which explore a larger nonlinear feature space? -Is our procedure able to detect relevant predictors and relevant thresholds? -Could our procedure provide sound and informative findings?

To answer these questions we used real CBC data which also served as the core of a simulation study.

Predictive performance: Lasso VS NN

We developed a NN (Fig. 1). The 22 predictor variables were the input variables and it returned the probability of abnormal smear as output. Continuous variables were previously standardised and non-linearly transformed into the network. To account for class imbalance, we adopted again the weighting strategy [START_REF] Johnson | Survey on deep learning with class imbalance[END_REF]. The NN was trained in 100 epochs using cross entropy as loss function. Other options were evaluated but only the architecture providing the best performance is presented. The NN model was implemented in Python with Keras.

We split the real data into 20 stratified folds. We iteratively fitted the weighted logistic Lasso and NN with the data from 19 folds and calculated precision, recall, specificity, ROC AUC and the area under the precision-recall curve (PR AUC) on the remainder fold. The weight ω was set to 20 for both of the methods.

Detection of relevant predictors and thresholds

We performed a simulation study: the input data matrix and the features χ j , were those defined by the original CBC data, ensuring the same structure. Y was generated from the features, assuming the logistic model. We considered 4 scenarios. In scenarios #1 and #2, among 19 blood cell measurements, two were relevant. They showed low correlation. In scenarios #3 and #4, five predictors were relevant. They showed moderate to high correlation between them and low correlation with all the other variables.

One β k j per relevant variable was non-zero (all set to 1.4). They were chosen such that thresholds were located: at the extreme percentiles (in #1 and #4), where observations are sparse or at not-extreme percentiles (in #2 and #3), where observations are frequent. β 0 was calibrated such that 7% of smears were abnormal.

For each scenario, Y was generated 100 times, then selection probabilities were estimated by the Lasso with stability selection.

Applicability of the procedure

Finally, we illustrate how the proposed procedure could help laboratory clinicians to make decisions about fine-tuning abnormal levels. The Lasso estimates are graphically presented. A light to dark gradient of grey represent the selection probability estimated from stability selection: the more the estimation is dark, the more stable the results are. Clinicians could explore at which values the data show a probable change in the risk of abnormal smear compared to the recommended threshold values and thus adjust the local criteria. 3) present predictive performance results of the Lasso compared to Deep neural network. NN presented very high recall values and performed slightly better than the Lasso. Inversely, the Lasso presented the best precision and specificity values. As a consequence, ROC and PR AUC were comparable and high in both cases. The consensus criterion led to inferior results in terms of precision and recall. Table [START_REF] Amato | Additive model selection[END_REF] shows simulation study results. When the true features were correlated and located at the extreme percentiles (#4), targets were often missed. The five true values were correctly identified in 14%-57% of the cases. Indeed, the relevant predictors, being redundant, may not be simultaneously selected. When the true thresholds have to be learn from sparse observations but relevant predictors are not correlated, neither among themselves nor with noisy variables (#1), the true values were were correctly identified in 52%-68% of the cases. In scenarios #2 and #3, relevant predictors were very frequently selected (more than 90% on average). Interestingly, values surrounding the true thresholds are more often selected than distant values. This could explain that the true threshold is not selected more often than about 90%. Noisy variables were rarely selected.

Figures (2)-(

Figure [START_REF] Barrio | A new approach to categorising continuous variables in prediction models: Proposal and validation[END_REF] shows Lasso estimates in terms of variation of OR. By convention, we choose the reference category as the one with lowest coefficient estimate. The variations of OR can thus be interpreted as OR increments. The selection probability estimated from stability selection is represented by the intensity of the grey colour. Green lines (with arrows indicating the direction) indicate new thresholds suggested by the Lasso estimates.

The consensus recommendation (in blue) is sometimes close to those suggested by the Lasso, e.g. mean corpuscular volume, or this seems to be enough, given the incertitude around the lasso estimates, e.g. neutrophils as counts and the superior threshold of lymphocytes as counts. Other times, the Lasso estimates suggest thresholds more conservative (i.e. leading to more smear reviews) than those of the consensus, e.g. haemoglobin, mean corpuscular haemoglobin, platelets, red blood cell distribution width and basophils as counts (for the later, the recommended threshold is greater than the observed values). In addition, a threshold seems to be relevant in some cases in which no recommendation exists, e.g. immature granulocytes, lymphocytes as percentage and the inferior limit of lymphocytes as counts. Inversely, some consensus thresholds seems to be useless, e.g. monocytes as counts and the inferior thresholds of leukocytes.

The presence of a machine alarm as well as mean corpuscular volume higher than 106f L appear to be stable strong predictors of abnormal smear. Red blood cell distribution width-CV higher than 16% and lymphocytes higher than 51% appears to be strongly associated with the presence of abnormal smear. In the case of lymphocytes this association is stronger and stable for values higher than 71%.

Discussion

In this paper we propose an machine learning tool to fine-tune abnormal levels in the CBC tests from data routinely produced in the internal laboratory quality control. This procedure, tailored to the particularities of the data, represents a non-linear baseline for a variety of tasks and could be of high importance for resource constrained environments. Offsetting imbalance by a weight appears to be an intuitive solution. In addition, the automatic choice of variables, as well as their functional form, can be managed by additive models with a penalty encouraging sparsity. Indeed, this corresponds to automatising the common practice of categorisation in the biomedical field. On the other hand, the practical implementation is simplified since standard efficient R packages that resolve the plain Lasso could be applied. Our proposal is close to existing approaches [START_REF] Amato | Additive model selection[END_REF][START_REF] Petersen | Data-adaptive additive modeling[END_REF]. Nevertheless, by choosing the piece-wise constant function basis, that conveniently categorises CBC data, interpretation, estimation, and computational issues are substantially simplified. This simplicity allows us to consider the extension to more complex models as a perspective, for instance, by including pairwise interactions between blood cell measures and age and sex.

As expected, deep neural network outperformed the Lasso in terms of recall. Indeed, the Lasso explores a reduced feature space compared to neural networks. Both approaches showed high values, though (the median (SD) recall values were 0.91 (0.04) and 0.87 (0.06) for deep neural network and the Lasso, respectively). This is all the more important since errors inherent to the subjectivity of manual blood smear tend to be biased: in case of doubt, the result of the worst case is assumed. Thus, a fraction of manual smears labelled as abnormal may actually be normal, potentially inflating false negative results. The Lasso presented the best precision and specificity. The weighting may have a greater impact on the neural network to the detriment of its specificity and its precision (by overestimating positives). The consensus criterion led to inferior results in terms of both, precision and recall.

AUC scores, balancing recall and precision/specificity, were comparable for NN and the Lasso. By means of recall and specificity, ROC AUC accounts for both true positives and true negatives, while PR AUC focuses on positives i.e. the minority class. Thus, PR AUC is recommended for predictive performance evaluation in imbalanced classification [START_REF] Saito | Precrec: fast and accurate precision-recall and ROC curve calculations in R[END_REF]. However ROC AUC is well-established in biomedical and other disciplines since easier to interpret [START_REF] Carrington | A new concordant partial auc and partial c statistic for imbalanced data in the evaluation of machine learning algorithms[END_REF]. We used ROC and PR AUC for evaluation but only ROC AUC to tune hyperparameters. A future perspective is the exploration of the PR AUC for hyperparameter tuning.

A wide variety of methods whose non-linearity/complexity lies somewhere between Deep neural networks and logistic regression exists. However, our goal was not to find the best prediction method, but to show that the proposed simple approach doesn't imply an important deterioration of the predictive performance.

The simulation study showed that our procedure was able to detect true thresholds with high probability provided that enough data are available in the thresholds' neighbourhood. On the other hand, the conclusions arisen from the visualisation of Lasso estimates and their relative importance are sound and relevant for a particular clinical laboratory [START_REF] Vellido | The importance of interpretability and visualization in machine learning for applications in medicine and health care[END_REF]. They generalise neither to other machines nor to other populations. However, the procedure itself appears to be an useful tool for decision support at laboratory-level. Our R code to implement it is publicly available on request from the corresponding author.

More standard measures of uncertainty are confidence intervals and p-values. Some R packages allow to compute these measures for high-dimensional data with binary response. However, we failed to use them with our CBC data. Alternatively, we implemented resampling-based percentile confidence intervals. Nevertheless, their coverage rate was poor and thus we don't recommend their use.

From the perspective of an extended decision-making tool, it would be interesting to provide a deeper analysis of errors, for example using as outcome the agreement/disagreement between manual smear and CBC results according to the locally fine-tuned thresholds. This approach would allow to directly deal with the question of how to decrease laboratory burden by reducing useless microscopic reviews.

Fig. 1 .

 1 Fig. 1. Retained NN architecture.

Fig. 2 .

 2 Fig. 2. Predictive performance. The boxplots represent distributions over the 20 validations sets from real CBC data. Dashed horizontal lines are plotted at median values.

Fig. 4 .

 4 Fig. 4. Lasso estimates of the real CBC data in terms of variation of the OR. The light to dark gradient of grey indicates the selection probability estimated by stability selection: the darker, the more stable. Normal observations are indicated in black and the abnormal ones in red. The recommended thresholds [3] are indicated by blue lines). Green lines indicate new thresholds suggested by the Lasso estimates.

Table 1 .

 1 Selection probability (%) obtained by stability selection combined with the Lasso in the four scenario simulations. Values correspond to the range of mean probabilities (over 100 simulations) for relevant and irrelevant predictors and thresholds. Predictive performance. The boxplots represent distributions over the 20 validations sets from real CBC data. Dashed horizontal lines are plotted at median values.

	Scenarios

Acknowledgements

The calculations were carried out on the CURTA cluster of MCIA (Mesocenter for Intensive Calculation in the Aquitaine French region).

We would like to thank:

-C.S. Ong (Data61, CSIRO, Canberra) and A.K. Menon (currently, Google, New York, formerly, Data61, CSIRO) for sharing their knowledge on imbalanced classification, -T. Quiroga (Department of Clinical Laboratories, School of Medicine, the Pontifical Catholic University of Chile, Santiago) for advice and data contribution, -E. Lagarde, L. Bourdois and J. Naredo (the "deep learning guys" from Inserm BPH U1219, France) and A. Carlier (INP-ENSEEIHT) for sharing their experience in implementing deep learning methods, -A. Rouanet (Inserm BPH U1219, France) for assistance in using the CURTA cluster.