Raphaëlle Eydoux

Frederic Castinetti

Magali Authier

Manon Vialle

Frédérique Albarel

Thierry Brue

Blandine Courbiere

Raphaëlle Eydoux

Fréderic Castinetti

Raphaelle Eydoux
email: raphaelle.eydoux@gmail.com

Women's perceptions of femininity after craniopharyngioma: a qualitative study

Keywords: Craniopharyngioma, qualitative study, femininity, sexuality, infertility, body image, intimacy

come   L'archive ouverte pluridisciplinaire

Introduction

Craniopharyngioma (CP) is a rare benign slow-growing epithelial tumor of the sellar and suprasellar region that is derived from embryonic remnants of Rathke's pouch. 1 CP accounts for 3 to 5% of all intracranial tumours, corresponding to 0.5 to 2 cases per 1 million person-years, with a bimodal age distribution at 5-14 and 50-74 years. [START_REF] Bunin | The descriptive epidemiology of craniopharyngioma[END_REF][START_REF] Müller | Diagnostics, Treatment, and Follow-Up in Craniopharyngioma[END_REF] Due to its location and treatment, CP can be responsible for visual, neuropsychological and endocrine sequelae that can impair patient quality of life. [START_REF] Zoicas | Craniopharyngioma in adults[END_REF][START_REF] Erfurth | Endocrine aspects and sequel in patients with craniopharyngioma[END_REF][START_REF] Puget | Treatment strategies in childhood craniopharyngioma[END_REF][START_REF] Bogusz | Childhood-onset craniopharyngioma: latest insights into pathology, diagnostics, treatment, and follow-up[END_REF][START_REF] O'steen | Advances in the management of craniopharyngioma[END_REF][START_REF] Puget | Pediatric craniopharyngiomas: classification and treatment according to the degree of hypothalamic involvement[END_REF] The most frequent endocrine impairments are anterior pituitary insufficiency, diabetes insipidus and hypothalamic obesity. [START_REF] Mortini | Craniopharyngiomas: a life-changing tumor[END_REF][START_REF] Karavitaki | Craniopharyngiomas[END_REF][START_REF] Müller | Functional capacity, obesity and hypothalamic involvement: cross-sectional study on 212 patients with childhood craniopharyngioma[END_REF] Pubertal induction and medically assisted reproduction (MAR) may be required in CP patients as gonadotrophin deficiency affects puberty, sexuality and fertility, depending on the gender and age at which CP treatment occurs. [START_REF] Geffner | Hypopituitarism in childhood[END_REF][START_REF] Thompson | Management of hypothalamic disease in patients with craniopharyngioma[END_REF] As a result, the intimate relationships of women with CP may be affected and they may also experience reproductive difficulties.

Several quantitative studies have previously demonstrated a reduced quality of life in adults who underwent surgery for CP in childhood. [START_REF] Heinks | Quality of life and growth after childhood craniopharyngioma: results of the multinational trial KRANIOPHARYNGEOM[END_REF][START_REF] Dekkers | Quality of life in treated adult craniopharyngioma patients[END_REF][START_REF] Yano | Quality of Life and Clinical Features of Long-Term Survivors Surgically Treated for Pediatric Craniopharyngioma[END_REF][START_REF] Patel | Long-term quality of life after endonasal endoscopic resection of adult craniopharyngiomas[END_REF] Only a few studies have evaluated the intimacy of patients treated during childhood or adolescence, [START_REF] Poretti | Outcome of craniopharyngioma in children: Long-term complications and quality of life[END_REF][START_REF] Clopper | Postsurgical IQ and behavioral data on twenty patients with a history of childhood craniopharyngioma[END_REF][START_REF] Hidalgo | Quality of life, hypothalamic obesity, and sexual function in adulthood two decades after primary gross-total resection for childhood craniopharyngioma[END_REF] reporting decreased psychosexual function and a lower rate of sexual activity compared to the general population. In these quantitative studies, standardized questionnaires may have limited the evaluation of sexual quality of life [START_REF] Arrington | Questionnaires to measure sexual quality of life[END_REF] and have thus failed to allow an in depth exploration of patient experiences. A qualitative study, which has the benefit of greater understanding rather than systematically describing [START_REF] Glaser | The discovery of grounded theory: strategies for qualitative research[END_REF] appears to be the best approach to explore intimacy and, to our knowledge, has not to date been carried out in women with a history of CP. Moreover, we considered that it would be relevant to study women who were treated during childhood, adolescence and childbearing years, since these issues may vary with age at diagnosis. The aim of our study was to assess the impact of CP on femininity and couple relationships in adult women, with the goal of finding ways to improve the overall care of these patients.

Subjects and methods

We conducted 15 in-depth, semi-structured interviews between March 2019 and April 2020. Interviews were carried out in French. Women were recruited from the Reference Centre for Rare Pituitary Diseases and the Centre of Reproductive Medicine at a University Teaching Hospital (Marseille, France) during a follow-up consultation. We included women operated on for CP from paediatric age to 40 years-old. Sample size was determined by the principle of data saturation. The interviews were conducted either at the hospital or at their home depending on their patients' personal convenience. A phone interview was proposed for women who were unable to meet face to face. Interviews were conducted by one researcher (RE) using a guide composed of open-ended questions. This guide evolved, being reworked after each interview to improve its relevance, and consisted of three parts: -Family and socio-professional situation, -History of CP: age at diagnosis, post-operative sequelae, treatments implemented and body changes.

-Emotional and intimate life: women who were operated on during childhood or adolescence were asked about their experience of puberty and their first sexual intercourse. The rest of the interview was conducted in a very personalized way according to each person's profile: the impact on couple relationships and sexuality, any experience of infertility and patient access to parenthood. We ended the interview by asking women to rate their general well-being on a scale of 0 to 10. Finally, women gave us feedback on their personal care. The interviews were audio recorded then transcribed verbatim by the same researcher (RE) and were anonymized before commencing the analysis. The interviews lasted between 25 and 75 min (mean 39 min +/-12.1). Data analysis was carried out using an inductive approach according to the grounded theory method which aims to produce theories from empirical material. [START_REF] Lejeune | Analyser sans compter ni classer[END_REF] First, a microanalysis was carried out separately by two researchers (RE and MA): codes were compared and did not differ between the two investigators. Codes were then pooled and organized to create categories. Finally, these categories were connected to each other in order to bring out the main themes of this analysis, new theories anchored in the stories of the women interviewed. Results were translated from French to English after analysis.

The study was approved by the local ethical committee of Aix-Marseille University and participants gave their informed consent.

Results

Interviews were conducted with 15 women. Mean age was 38 years (from 25 to 51 years +/-7.5), 6 women underwent surgery during childhood, 4 during adolescence and 5 between the age of 28 and 40 years. 13 women had panhypopituitarism, the remaining two had at least gonadotropic deficiency thus every women received hormone replacement therapy (HRT). Subjects' clinical characteristics are summarized in Table 1.

Three main themes emerged from interviews: (1) Apparent changes leading to altered self-perception; (2) Managing the hidden disabilities of the disease; (3) Building parenthood and a couple relationship: coping with infertility and sexual dysfunction.

Apparent changes leading to altered self-perception

Apparent changes. During the postoperative period, women may have been confronted with multiple changes. Body changes such as significant weight gain or ophthalmic impairments were the first to be addressed by women as these were the most visible signs of the disease: P3 "People ask questions really quickly because it's physical." Women expressed difficulties in accepting these changes, especially in the case of obesity which was characterized as an uncontrollable burden: P8 "Gaining weight without eating is really difficult to manage, it doesn't make any sense". Behavioral changes were also reported, such as a loss of self-control, which can affect the subject's relationship with food due to lack of satiety or the management of emotions, leading to misunderstandings between the subject and their family: P13 "And then I'm going to have mood swings, I don't know why. My relatives don't understand very well". In addition, most women noticed a loss of energy and increased tiredness resulting in a need to change their normal habits: P11 "I was a very dynamic person. Now when I wake up, I'm already tired."

Alterations in self-perception. In the face of these changes, some women developed an altered self-perception, with the specific nature of the alteration dependent on the age at diagnosis. Women who were operated on as adults were particularly affected by the sudden change from their previous condition, which they often compared to a metamorphosis:

P14 "We are a little bit left alone in front of this metamorphosis. It was very difficult to see myself changing mentally and physically."

For some subjects, these changes may have resulted in a loss of self-recognition: P15 "Sometimes I passed in front of the mirror and I was scared because I felt like it wasn't me. I felt like I was trapped in a body that did not belong to me." This loss of self-recognition has also been reported in some cases by women with pituitary deficits as they felt that their treatments took control of their feelings: P12 "I feel like everything is distorted, it's like I don't have feelings anymore."

The specific issue of "developing a woman's body" was also raised by subjects who required induction of puberty. Some considered it as good fortune, allowing them to become women: P2 "For me, the craniopharyngioma made me become the woman I was meant to be." In one case, HRT resulted in the development of an 'artificial body':

P1 "The problem is that I don't feel natural. For me, my body is artificially made." Finally, one woman expressed disappointment with the failure to obtain satisfactory secondary sexual characteristics: P4 "I had almost no chest and zero pubic hair. It really didn't work." Overall, most women reported having a poor body image, which may have been responsible for low self-esteem or ill-being, with a mean well-being score of 5.7+/-1.9 out of 10. This poor body image in some cases impacted on their femininity and may also have constituted an obstacle to forming relationships: P4 "I preferred not having a relationship rather than showing my body as it was. I got used to the idea of being single."

Managing the hidden disabilities of the disease

In addition to the body changes previously noted, women raised issues of the "hidden disabilities" caused by CP treatment which may have equally impacted their quality of life and their relationships. For some women, the disease engendered a need for permanent control resulting in them no longer being able to be carefree: P10 "There is no longer any lightness of life." This feeling was reported in some cases by women who developed obesity and who emphasized the difficulty of losing weight despite dietary restrictions: P12 "Weight is a constant struggle for me." These restrictions may also have resulted in social barriers, impacting their relationships: P1 "Going to restaurants is becoming a social problem for me." This has also been reported by women with pituitary deficits, who have characterized their daily treatments as an "invisible disability" since they require permanent forethought and constant situational adjustment: P1 "The thing with my disease is that I can't accept the unexpected. Everything has to be planned." For some, the need for taking medication is a daily reminder of their illness: P9 "It's like something hovering over my life that I'm trying to forget but always reminds me that it's there." Finally, these daily requirements in some cases created apprehension about frightening a potential partner: P2 "I was afraid that all my treatments would scare someone else."

Building parenthood and couple relationships: coping with sexual dysfunction and infertility.

Sexual dysfunction. Women raised the issues of vaginal dryness and lack of libido and complained about the difficulty in finding adequate HRT. While a few women were satisfied with the management of vaginal dryness, most of them expressed dissatisfaction or even poor tolerance of these treatments and reported no improvement in their libido: P7 "There is one treatment that I can't tolerate, and the pill dries me up completely." As a result, vaginal dryness and lack of libido may have affected their well-being and the couple's sexual health: P9 "There is a bit of an obligation within the couple, you know, that there will be sexual intercourse."

Nevertheless, different approaches to sexuality were noted. Women who were operated on during childhood reported that they were not informed about what "normal sexuality" is and that they therefore had no basis for comparison, unlike those subjects who were operated on in adulthood: P1 "People who have had sex before, well, they know what it is. I don't even know what libido is." For women who underwent surgery during adolescence, the disease may have caused a delay in the usual course of events leading to sexual initiation being later when compared to other adolescents: P10 "When you are young you have flirtations, but everything was blocked for me." Therefore, among the 10 women operated on during childhood or adolescence, four were still virgins (aged 25, 41, 43 and 51 years old), and the mean age of first sexual intercourse was 20+/-2.5 years old. Among all the women interviewed, only a few reported having developed a fulfilling sexuality. For them it was a real co-learning experience, requiring patience from their partner: P7 "Finally, I learned to discover my own pleasure when I met someone great." For women who were unsatisfied despite HRT, lubricants could have been a useful alternative but also require planning that might impact on the spontaneity of sexual intercourse. Finally, most women expressed resignation in the search for a fulfilling sexuality, mainly because of a lack of libido, and reported having learned to do "without" and to develop their sexuality for the benefit of their partner: P13 "We do it anyway, we have to relieve him."

Coping with infertility. Infertility induced by hypogonadotropic hypogonadism was experienced in very different ways. Most of the women interviewed considered motherhood as an intrinsic element of being a woman and felt their infertility as a disability that impacted their femininity: P11 "You know, a woman is a life bearer, she gives life." As a result, infertility may have created a feeling of guilt and an apprehension concerning frightening their partner:

P5 "That's what scared me a little bit, to think that maybe he wouldn't stay with me if I couldn't bear a child."

Resorting to MAR represented a coping strategy for some women, who were hoping for parenthood, and was perceived in different ways. Women operated on during childhood or adolescence reported that they were psychologically prepared for infertility, and that this helped them to accept the idea of undergoing MAR, unlike those who underwent surgery later: P5 "Well, I've known it for so long, I thought it was like that." However, some women felt time pressure and the fear of not being able become a parent: P3 "We know that we have to find someone, then build a case and that if our situation is not considered stable enough, the case won't be successful." Finally, for women who planned a pregnancy, MAR was a difficult but optimistic project for the couple: P11 "It's not easy for us to pursue MAR as an option but when you are well taken care of, it's comforting, and it gives you hope." In the case of live birth after MAR, some women considered this as having achieved a personal accomplishment that also aided in strengthening their femininity:

P6 "It's quite a victory over the disease to have had babies. And I don't know if it's being a mom, but I feel more feminine."

Different experiences of parenthood. Approaching the issue of infertility allowed some women to question their relationship to parenthood. Indeed, women in some cases expressed doubts about their ability to take care of a child because of their disease: P10 "I wondered if I could really support this role. Sometimes I even find it hard to manage myself, so with a child, it's not easy." Other women simply expressed that they had never felt the desire to have children. Women who had children before their intervention considered themselves fortunate to have had the time to become mothers which allowed them to distance themselves from infertility. For one of them, the illness may also have been motivation for a new pregnancy: P11 "It's for my daughter, she must have someone to lean on. I lived it with my disease, it's good to have siblings." Women who expressed a sense of helplessness regarding their disease may have developed compensatory attitudes, for example creating another form of parenthood: P4 "We live together with my mother; she means everything to me. And maybe I felt the desire to have children when I saw my nieces and nephews. But at the same time, I took care of them so much that I consider them a bit like my own children."

Adjustments to couple relationship. Women who established a couple relationship agreed that it was necessary to explain the disease to their partner in order for them to accept the burden and to show support and understanding. In the end, some women considered that the disease had actually contributed to strengthening their relationship: P15 "You know, between everyday life, work, my little girl, we had other priorities. With what happened, I think we realized that we could have lost each other and then, our relationship became our priority." For others, the disease created an imbalance within the couple, with the feeling of being in a position of weakness: P12 "He's not very forgiving, he always asks for just as much. And it's complicated for me to be the one to tell him to stop because it reminds me of the fact that I can't give any more." Finally, some expressed that there were misunderstandings within the couple and a lack of support which could have led to relationship breakdown: P14 "I didn't have the support I needed and there wasn't that level of attention, that tenderness."

Discussion

To our knowledge, we report here the first qualitative study about femininity and couple relationships in adult women treated for CP. Previous quantitative studies that have been conducted on CP patient intimacy have essentially focused on sexuality. In a study using the Medical Outcomes Study sexual functioning survey, Hildalgo et al., reported a lower proportion of sexually active individuals, compared to the general population, in 22 young adults (9 women, 13 men) with a history of CP in childhood [START_REF] Hidalgo | Quality of life, hypothalamic obesity, and sexual function in adulthood two decades after primary gross-total resection for childhood craniopharyngioma[END_REF] . However, global sexual functioning was not different compared to the general population, though its assessment may have been biased by the use of a standard questionnaire in a small study cohort of both women and men [START_REF] Arrington | Questionnaires to measure sexual quality of life[END_REF] . In our study we chose to focus on women, since we hypothesized that intimacy sequelae are most likely gender specific. Our qualitative study allowed a deeper analysis of women's intimacy, focusing on the factors that may have impacted their femininity and their relationships.

We first identified a phenomenon of an altered self-perception in response to physical and behavioural changes engendered by CP treatment. This feeling was reported by women who were operated in adulthood, who compared these changes to a traumatic metamorphosis. This change from the existing condition, with a "before and after" the disease, has also been described in other endocrine diseases as leading to a loss of selfrecognition [START_REF] Estcourt | The impact of thyroid eye disease upon patients' wellbeing: a qualitative analysis[END_REF][START_REF] Sibeoni | Patients' perspectives on acromegaly diagnostic delay: a qualitative study[END_REF] . Notably, the development of obesity emerged as a major factor in decreased self-esteem, which is in agreement with the results of other studies conducted on obesity [START_REF] Doll | Obesity and physical and emotional well-being: associations between body mass index, chronic illness, and the physical and mental components of the SF-36 questionnaire[END_REF] and on quality of life of patients with CP [START_REF] Yano | Quality of Life and Clinical Features of Long-Term Survivors Surgically Treated for Pediatric Craniopharyngioma[END_REF] . Behavioural disturbances, previously reported in other studies of CP patients, [START_REF] Poretti | Outcome of craniopharyngioma in children: Long-term complications and quality of life[END_REF][START_REF] Pascual | Craniopharyngiomas Primarily Involving the Hypothalamus: A Model of Neurosurgical Lesions to Elucidate the Neurobiological Basis of Psychiatric Disorders[END_REF] mainly concerned a loss of self-control with unexpected reactions leading to misunderstandings in their relationships. Alterations in self-perception were also noticed by women who required induction of puberty, reporting very different experiences of the process leading to developing a "woman's body". Quantitative studies have previously been conducted in patients with congenital hypogonadism in order to assess the influence of pubertal management on sexuality, [START_REF] Carel | Self-esteem and social adjustment in young women with Turner syndrome--influence of pubertal management and sexuality: population-based cohort study[END_REF] reporting that age at first intercourse was related to age at onset of puberty. In our study, women were not always able to recall precisely age at onset of puberty but age at first intercourse seemed to be associated with body image, with obesity having a major negative impact. Finally, among the 10 women who were operated on during childhood or adolescence, four were still virgins at the time of the study and the mean age of the first sexual intercourse was 20+/-2.5 years old, which is delayed compared to the latest data on first intercourse for girls in France, which is reported to be 17.6 years. [START_REF] Bajos | Genre et sexualité. D'une décennie à l'autre[END_REF] In addition to the influence of body image, women emphasized the difficulty in managing vaginal dryness and a lack of libido despite HRT, which has also been reported in women with congenital hypogonadotropic hypogonadism. [START_REF] Oyarzún | Sexuality and quality of life in congenital hypogonadisms[END_REF] Lack of libido, which had been noted in a previous study on CP patients, [START_REF] Larijani | Presentation and outcome of 93 cases of craniopharyngioma[END_REF] was the most frequently reported item and had the most impact on the couple's sexual health.

Finally, women have also addressed their experience of infertility. While some of them may have felt that this impacted on their femininity, others accepted it more easily, either because they already had children or because they did not have plans to conceive. For women who did wish to have children, resorting to MAR may have been difficult but was also a source of hope, with the real possibility of having a live-birth. [START_REF] Hayashi | Successful pregnancy following gonadotropin therapy in a patient with hypogonadotropic hypogonadism resulting from craniopharyngioma[END_REF][START_REF] Volz | Conception and spontaneous delivery after total hypophysectomy[END_REF] Considering the subject of infertility also allowed women to reflect on parenthood. While for some women the disease may have been perceived as a hindrance to caring for a child, others saw it as motivation for a new pregnancy.

In this study we interviewed women who underwent surgery at different ages: in childhood, adolescence or adulthood. On the one hand, this may have constituted a bias, by interviewing populations that, in some aspects, were not comparable. On the other hand, this allowed the emergence of common themes while also identifying specificities related to the subject's age at intervention. Moreover, satisfaction with HRT and the evaluation of its efficacy may also have been biased, as some women acknowledged that they had discontinued treatment early. Though the recruitment was consecutive from the clinic (avoiding the risk of patients' selection on a positive or negative outcome), the monocentric nature of the study did not allow to determine a potential impact of ethnicity on the outcome. Finally, all the women interviewed in our study were heterosexuals, therefore we did not study the impact of CP on the intimate life of women with a different sexual orientation.

In conclusion, in this study we showed that women with CP are affected in terms of both their femininity and in their relationships, as CP modifies body image, sexuality and affects women's reproductive potential. Qualitative studies offer a new approach to determine the expectations of a population exposed to a pathology, the results of which could help to meet their needs and improve their therapeutic education. In fact, women interviewed in our study stated that they had different expectations of medical professionals, especially during transitional states such as puberty, initiation to sexuality and childbearing years, which is consistent with women's feelings reported in other endocrine pathologies such as congenital adrenal hyperplasia. [START_REF] Engberg | The experience of women living with Congenital Adrenal Hyperplasia: impact of the condition and the care given[END_REF] They notably insisted on the difficulty of addressing sexuality issues during education sessions and suggested a more individualized care. Our study emphasizes the importance of these issues in women's lives and reinforces the need for offering comprehensive, multidisciplinary but also personalized care for women treated for CP. Similar studies should now be conducted in men with CP since they are equally concerned by the issues raised here and may have different experiences of CP.

Legend to table 1

Clinical characteristics of participants. MAR= Medically Assisted Reproduction, NR= Not requested. Well-being score: Participants were asked to rate their general well-being on a scale of 0 to 10. a Participants P4, P9 and P14 discontinued HRT early. b None of the childless women had initiated MAR procedures.

Acknowledgements:

The authors wish to thank Ian Darby (Editing biomed) for English editing.

Data availability statement: Data collected for the study, including individual participant data and a data dictionary defining each field in the set, will be made available upon reasonable request to the corresponding author.


Olivier David
email: olivier.david@inrae.fr

Arnaud Le Rouzic

Christine Dillmann

Optimization of sampling designs for pedigrees and association studies

Keywords: Bayesian statistics, genetic algorithm, high-dimensional statistics, optimal designs, quantitative genetics

In many studies, related individuals are phenotyped in order to infer how their genotype contributes to their phenotype, through the estimation of parameters such as breeding values or locus effects. When it is not possible to phenotype all the individuals, it is important to properly sample the population to improve the precision of the statistical analysis. This article studies how to optimize such sampling designs for pedigrees and association studies. Two sampling methods are developed, stratified sampling and D optimality. It is found that it is important to take account of mutation when sampling pedigrees with many generations: as the size of mutation effects increases, optimized designs sample more individuals in late generations. Optimized designs for association studies tend to improve the joint estimation of breeding values and locus effects, all the more as sample size is low and the genetic architecture of the trait is simple. When the trait is determined by few loci, they are reminiscent of classical experimental designs for regression models and tend to select homozygous individuals. When the trait is determined by many loci, locus effects may be difficult to estimate, even if an optimized design is used.

Introduction

Quantitative genetics studies the inheritance of quantitative traits such as height or weight, which vary continuously. Experimental studies in quantitative genetics use phenotypic and genetic data to address various fundamental or applied issues such as the selection of plants or animals in breeding programs [START_REF] Jannink | Genomic selection in plant breeding: from theory to practice[END_REF], the identification of individuals at increased risk of disease [START_REF] Vazquez | A comprehensive genetic approach for improving prediction of skin cancer risk in humans[END_REF], the evaluation of genetic resources stored in gene banks [START_REF] Crossa | Genomic prediction of gene bank wheat landraces[END_REF], the inference of the genetic architecture of traits in association studies (Barton et al., 2007, Chap. 14;[START_REF] Rau | Exploring drivers of gene expression in the cancer genome atlas[END_REF], the study of the genetic mechanisms underlying the response to selection [START_REF] Durand | Standing variation and new mutations both contribute to a fast response to selection for flowering time in maize inbreds[END_REF][START_REF] Durand | Dearth of polymorphism associated with a sustained response to selection for flowering time in maize[END_REF] or the inference of the strength of evolutionary forces such as drift, mutation or selection (Gillespie, 1998, Chap. 5;Barton et al., 2007, Chap. 19;[START_REF] David | Bayesian inference of natural selection from spatiotemporal phenotypic data[END_REF]. Such studies require a sampling step and the choice of a sampling strategy when it is not possible to phenotype all the individuals of the studied population [START_REF] Cochran | Sampling Techniques[END_REF].

Several studies have investigated how to optimize sampling designs for genomic prediction [START_REF] Isidro | Training set optimization under population structure in genomic selection[END_REF][START_REF] Rincent | Predicting genomic selection efficiency to optimize calibration set and to assess prediction accuracy in highly structured populations[END_REF][START_REF] Akdemir | Design of training populations for selective phenotyping in genomic prediction[END_REF]. In such experiments, the population is genotyped with genome-wide markers. The individuals belonging to a candidate set are available for phenotyping, while the other individuals cannot be phenotyped because they may be dead, for example. Since phenotyping is costly, some individuals are sampled in the candidate set and are phenotyped. The genetic and phenotypic data are then used to predict the breeding values of target individuals. Designing such surveys consists of choosing the individuals to be phenotyped in the candidate set in order to predict the target breeding values as precisely as possible.

Sampling designs for genomic prediction have been generated using stratified sampling (SS) [START_REF] Isidro | Training set optimization under population structure in genomic selection[END_REF]. This method first divides the population into homogeneous subpopulations by clustering the genetic data. Individuals are then sampled in each subpopulation. This method accounts for the population structure and ensures a high genetic diversity in the sample. Alternatively, sampling designs for genomic prediction have been optimized using optimal design theory [START_REF] Atkinson | Optimum experimental designs[END_REF][START_REF] Chaloner | Bayesian experimental design: A review[END_REF][START_REF] Isidro | Training set optimization under population structure in genomic selection[END_REF][START_REF] Rincent | Predicting genomic selection efficiency to optimize calibration set and to assess prediction accuracy in highly structured populations[END_REF][START_REF] Akdemir | Design of training populations for selective phenotyping in genomic prediction[END_REF]. In this approach, a criterion that quantifies the merit of a design is first defined, for example, a criterion that quantifies the variance of the estimators of interest. Efficient designs are then generated by optimizing this criterion over an appropriate class of designs. Design criteria may be optimized using combinatorial optimization methods such as genetic algorithms or simulated annealing [START_REF] Reeves | Modern heuristic techniques for combinatorial problems[END_REF][START_REF] David | Optimal complete block designs to adjust for interplot competition with a covariance analysis[END_REF][START_REF] Van Groenigen | The influence of variogram parameters on optimal sampling schemes for mapping by kriging[END_REF][START_REF] Müller | Optimal Bayesian design by inhomogeneous Markov chain simulation[END_REF][START_REF] Amzal | Bayesian-optimal design via interacting particle systems[END_REF][START_REF] Brus | Optimization of sample patterns for universal kriging of environmental variables[END_REF][START_REF] Butler | On the design of field experiments with correlated treatment effects[END_REF][START_REF] Del Moral | Particle methods: An introduction with applications[END_REF][START_REF] Akdemir | Design of training populations for selective phenotyping in genomic prediction[END_REF]. In these studies, optimized designs generally performed better than simple random sampling (SRS) and improved the precision of genomic prediction.

Quantitative genetics includes other types of experiments than genomic prediction, for example, surveys involving pedigrees or association studies. It is not clear how to sample pedigrees, if more individuals should be sampled in early or in late generations, and how mating system and mutation influence sampling designs. The experimental setup of an association study is similar to that of genomic prediction, however, their objective is to estimate the effects of loci on traits rather than breeding values. Likewise, it is not clear how to sample individuals in association studies and how the genetic architecture of the trait influences sampling designs.

This article develops sampling methods for pedigrees and association studies. Observations are assumed to follow a classical infinitesimal model, in which the phenotype can be partitioned into independent genetic and environmental components (Section 2). Two sampling methods are developed, SS and D optimality (DO) (Section 3). A real maize experiment is used to study how to sample pedigrees, and the influence of mating system and mutation on sampling (Section 4). Finally, simulations are carried out to study how to sample individuals in association studies, and the influence of sample size and the genetic architecture of the trait on sampling (Section 5).

Models

We consider a population of N diploid individuals. This population contains a candidate set C, whose individuals are available for phenotyping for some quantitative trait. A sample of n < N individuals to be phenotyped is chosen in C. Let X be the n × N incidence matrix whose entry in row j and column i satisfies Xji = 1 if the jth observation is collected on individual i, and Xji = 0 otherwise. For example, if the second observation is collected on individual 69, then X2 69 = 1 and X2 i = 0 for i = 69.

Trait values are assumed to follow an infinitesimal model [START_REF] Wray | Accounting for mutation effects in the additive genetic variance-covariance matrix and its inverse[END_REF][START_REF] Gillespie | Population genetics: a concise guide[END_REF][START_REF] Lynch | Genetics and Analysis of Quantitative Traits[END_REF][START_REF] Barton | The infinitesimal model: Definition, derivation, and implications[END_REF]. They are assumed to be the sum of a genetic component, referred to as the breeding value, and an environmental component. They are observed with measurement errors: for 1 ≤ j ≤ n, the jth observation y j is

y j = µ + N i=1 Xjiui + ej + ξj,
where µ is the grand mean, ui is the breeding value of individual i, ej is an environmental effect, and ξj is a measurement error. The residuals εj = ej + ξj are assumed to be independent and normally distributed with mean 0 and variance σ 2 ε , that is, εj ∼ N 0, σ 2 ε . They are assumed to be independent of breeding values, that is, there are no genotype-environment interactions.

Modeling breeding values using the pedigree

The survey is assumed to involve some genetic data in addition to the phenotypic data. These data may consist of a pedigree dataset, that indicates the parents and the date of birth of each individual. Without loss of generality, dates are expressed here in generations, like in [START_REF] Wray | Accounting for mutation effects in the additive genetic variance-covariance matrix and its inverse[END_REF]. This is more appropriate for populations with discrete generations, but any other time unit such as day or week could be used instead. The only restriction is that parents must appear in earlier time units than their offspring. The pedigree is assumed to have G + 1 generations, denoted by 0, . . . , G.

In order to study the sensitivity of designs to mutation, breeding values are modeled taking account of mutation. The mutations arising at generation 0 are not taken into account since they are considered as being part of the alleles present at this generation. New mutations are assumed to arise independently in the individuals of generation 1 and of subsequent generations. The alleles present at generation 0 and new mutations are assumed to have additive effects on breeding values [START_REF] Wray | Accounting for mutation effects in the additive genetic variance-covariance matrix and its inverse[END_REF]: ui = G g=0 αgi, where α0i is the component of the breeding value of individual i that is the result of alleles inherited from ancestors born in generation 0, and for g ≥ 1, αgi is the component that is the result of mutations arising in generation g. The vector α0 of genetic effects inherited from generation 0 is assumed to be normally distributed [START_REF] Wray | Accounting for mutation effects in the additive genetic variance-covariance matrix and its inverse[END_REF][START_REF] Lynch | Genetics and Analysis of Quantitative Traits[END_REF][START_REF] Barton | The infinitesimal model: Definition, derivation, and implications[END_REF]:

α0 | σ 2 0 ∼ N 0, σ 2 0 A0
, where σ 2 0 is the additive genetic variance, and A0 is an N × N kinship (or relationship, similarity) matrix. This matrix can be computed in various ways. For the sake of simplicity, it is assumed here that α0i is under the control of a single locus with many alleles and that it is the sum of the contributions from the two alleles carried by individual i, that is, there is no dominance (Gillespie, 1998, Chap. 5). Alleles are transmitted from parents to offspring by Mendelian inheritance. The entry A 0ii is then equal to twice the coefficient of kinship between individuals i and i [START_REF] Wright | Coefficients of inbreeding and relationship[END_REF]Gillespie, 1998, Chap. 5). This coefficient is the probability that two alleles, one sampled from individual i and one sampled from individual i independently, are identical by descent. It is also assumed here that the individuals of generation 0 are not related and are not inbred, but this assumption can easily be relaxed if information on their relatedness and their inbreeding is available.

For g ≥ 1, the vector αg of mutation effects is assumed to be normally distributed [START_REF] Wray | Accounting for mutation effects in the additive genetic variance-covariance matrix and its inverse[END_REF]:

αg | σ 2 m ∼ N 0, σ 2 m Ag , where σ 2
m is the mutational variance, and Ag is an N × N kinship matrix. Since αgi is the result of mutations arising at generation g, αgi = 0 if individual i was born before generation g, with the convention that all the elements of Ag involving this individual are equal to 0, that is, A gii = 0 for 1 ≤ i ≤ N . Since new mutations are transmitted from parents to offspring by Mendelian inheritance and are assumed to have additive effects, the elements of Ag involving individuals born after generation g -1 are equal to twice their coefficient of kinship. Since mutations are assumed to arise independently in gametes, those arising at generation g do not cause any inbreeding in the individuals born at generation g.

Since mutations arise independently over time, the vector of breeding values is normally distributed with mean 0 and conditional variance matrix

Var u | σ 2 ε , γ 2 0 , γ 2 m = σ 2 ε V , where V = γ 2 0 A0 + γ 2 m G g=1
Ag,

and the variance components γ 2 0 and γ 2 m are equal to

γ 2 0 = σ 2 0 /σ 2 ε and γ 2 m = σ 2 m /σ 2 ε .

Modeling breeding values using genotyping data

Alternatively, the genetic data may consist of a genotyping dataset, giving the genotype of all the individuals at L biallelic loci. Let M be the N × L matrix that specifies which alleles each individual carries at each locus. Specifically, the genotype M il of individual i at locus l is set to 1 (respectively, -1) if individual i carries two copies of allele 1 (respectively, 2) at locus l, and to 0 if individual i is heterozygous at locus l [START_REF] Vanraden | Efficient methods to compute genomic predictions[END_REF]. The columns of M are often standardized [START_REF] Vanraden | Efficient methods to compute genomic predictions[END_REF][START_REF] Rincent | Predicting genomic selection efficiency to optimize calibration set and to assess prediction accuracy in highly structured populations[END_REF][START_REF] Gianola | Inferring trait-specific similarity among individuals from molecular markers and phenotypes with Bayesian regression[END_REF]. As explained in the next paragraph, they are divided here by the square root of the average diagonal element of M M T , where T denotes transposition, yielding the matrix

Z = M / √ z, where z = trace M M T /N .
Alleles are assumed to have additive effects on breeding values, that is, there is no dominance and no epistasis:

ui = L l=1 Z il β l ,
where β l is a parameter that quantifies the effect of locus l. As L increases, the genetic architecture of the trait is more complex. The vector β of locus effects is assumed to be normally distributed [START_REF] Gianola | Inferring trait-specific similarity among individuals from molecular markers and phenotypes with Bayesian regression[END_REF]:

β | σ 2 β ∼ N 0, σ 2 β IL , where σ 2
β is the additive genetic variance and IL is the identity matrix of size L. Thus, the vector of breeding values is normally distributed with mean 0 and conditional variance matrix

Var u | σ 2 ε , γ 2 β = σ 2 ε V , where V = γ 2 β ZZ T and the variance component γ 2 β is equal to γ 2 β = σ 2 β /σ 2 ε .
Thanks to the scaling of the columns of Z mentioned above, the prior variance of breeding values is approximately equal to σ 2 ε γ 2 β , which facilitates the specification of the prior distribution of γ 2 β . The columns of Z are not centered here so that the grand mean µ represents the average trait value of a heterozygous individual at all the loci.

Prior distributions This article uses a Bayesian framework. A vague normal prior distribution is placed on the intercept

µ: µ | σ 2 ε ∼ N µ, σ 2 ε v
, where µ is a known constant and v → +∞. The residual variance is assumed to follow a prior distribution with mean σ 2 ε . Variance components are assumed to follow prior distributions with means γ 2 0 , γ 2 m and γ 2 β . The prior means γ 2 0 and γ 2 β could range from 0.25 to 4, leading to heritabilities ranging from 0.2 to 0.8 [START_REF] Johnson | Theoretical models of selection and mutation on quantitative traits[END_REF]Gillespie, 1998, Chap. 5;Barton et al., 2007, Chap. 14). The prior mean γ 2 m could range from 0.0001 to 0.04 [START_REF] Wray | Accounting for mutation effects in the additive genetic variance-covariance matrix and its inverse[END_REF]Barton et al., 2007, Chap. 14).

The unconditional prior variance of breeding values is Var

(u) = σ 2 ε E (V ) = σ 2 ε V ( γ 2 0 , γ 2 m )
when the genetic data are pedigree data, or Var (u) = σ 2 ε V ( γ 2 β ) when the genetic data are genotyping data.

Sampling methods

The objective of the sample survey is to estimate the breeding values of the individuals belonging to some target set T , or to estimate locus effects. Without loss of generality, the population is assumed to be the union of C and T . Choosing the design of the survey consists of choosing the sample, that is, n individuals in C to be phenotyped.

Stratified sampling

The population may be sampled using SS. First, the individuals of the candidate set are divided into n clusters using a hierarchical clustering. When the genetic data are pedigree data, the dissimilarity δ ii between individuals i and i is equal to

δ ii = √ V iiV i i -V ii , where V = V ( γ 2 0 , γ 2 m ).
When the genetic data are genotyping data, it is the Euclidian distance calculated from M [START_REF] Isidro | Training set optimization under population structure in genomic selection[END_REF]. The design is then generated by sampling one individual in each cluster independently and uniformly.

D optimality

3.2.1 D criteria. Sampling designs may be generated by maximizing the D criterion for breeding values (Atkinson and Donev, 1992, Chap. 6;[START_REF] Chaloner | Bayesian experimental design: A review[END_REF][START_REF] Verdinelli | A note on Bayesian design for the normal linear model with unknown error variance[END_REF]. This criterion quantifies estimation errors. When breeding values are estimated by their posterior mean, the matrix of squared errors is equal to {u -

E (u | y)}{u -E (u | y)} T .
Since the true breeding values are unknown and since observations are not available when the survey is designed, squared errors are averaged over the possible values of these variables, yielding the matrix

{u -E (u | y)}{u -E (u | y)} T p(u, y)dudy,
where p(u, y) is the joint density of u and y. This matrix is equal to the average posterior variance matrix of breeding values:

{u -E (u | y)}{u -E (u | y)} T p(u | y)p(y)dudy = Var (u | y) p(y)dy.
The D criterion for breeding values is based on the determinant of this matrix:

Du = ln[det{ Var (u | y) p(y)dy}].
When variance components have low prior variance, it is easier to compute, since it is approximately equal to (online Appendix A)

Du ≈ ln{det(V -1 + X T QX)},
where Q = (In -J n/n), J n is the square matrix of ones of size n, and V is equal to V ( γ 2 0 , γ 2 m ) or to V ( γ 2 β ) depending on the experimental setup. This expression assumes that the prior variance matrix of breeding values is invertible, which is often the case. It may be singular when, for example, two individuals are identical twins (or have the same genotype at all the loci), when the individuals born at generation 0 have been selected, or when L < N [START_REF] Henderson | Best linear unbiased prediction using relationship matrices derived from selected base populations[END_REF][START_REF] Henderson | Estimation of singular covariance matrices of random effects[END_REF][START_REF] Vanraden | Efficient methods to compute genomic predictions[END_REF][START_REF] Fernando | An efficient exact method to obtain GBLUP and single-step GBLUP when the genomic relationship matrix is singular[END_REF].

Alternatively, sampling designs may be generated by maximizing the D criterion for locus effects, which is equal to

D β = ln[det{ Var (β | y) p(y)dy}].
When the prior variance of γ 2 β is low, this criterion is approximately equal to (online Appendix A)

D β ≈ ln{det(γ -2 β IL + Z T X T QXZ)}.
When variance components have low prior variance, maximizing Du or D β is equivalent to maximizing the following version of the D criterion (online Appendix A):

D = ln{det(In + QXV X T Q)}.
Thus, the D criteria for estimating all of the breeding values or all of the locus effects can be optimized simultaneously.

Genetic algorithm.

The D criterion can be maximized using the genetic algorithm STPGA [START_REF] Akdemir | Optimization of genomic selection training populations with a genetic algorithm[END_REF][START_REF] Akdemir | Design of training populations for selective phenotyping in genomic prediction[END_REF]. This algorithm makes a population of designs evolve so that D values increase in the population over iterations. At each iteration, designs with large D values are selected in the population and produce offspring that forms the next generation. Offspring may not be identical to their parents because of mutation and recombination. Inefficient designs that were visited recently may be removed from the next generation (Tabu search). Designs generated using regression methods may be introduced in the next generation.

This algorithm can be implemented using the function GenAlgForSubsetSelectionNoTest of the R package STPGA [START_REF] Akdemir | STPGA: Selection of Training Populations by Genetic Algorithm[END_REF]. In this article, each design of the initial population is generated by SS independently (Section 3.1). Maximizing D using STPGA is computationally more efficient than maximizing Du since D involves a matrix of dimension n × n only; it is more efficient than maximizing D β when n < L.

Sampling designs for pedigrees 4.1 Explicit result

When the genetic data are pedigree data and when the prior variances of γ 2 0 and γ 2 m are low, if the breeding values of the selected individuals are not correlated a priori and have maximal prior variance among C, then the design is D-optimal over the set of all designs given C and n (online Appendix B).

Simulation study

A simulation study was carried out to check that STPGA can optimize the D criterion. Its R code can be found in Supporting Information. In this study, the population consisted of a theoretical pedigree comprising 80 individuals, which reproduced by outcrossing and which were divided into 10 unrelated families (see online Appendix C for more information). The candidate and target sets both comprised all the individuals. Sample size was either equal to 10 or to 70. Breeding values were modeled using the pedigree. In total, 80 design searches were carried out with different values of variance components in cases when optimal/efficient designs were known. For example, when n = 10, sampling the individuals of the last generation was optimal (Section 4.1). STPGA performed rather well since it always generated designs as efficient as the reference designs, except in two cases.

4.3 Sampling a maize pedigree 4.3.1 Experimental setup. The influence of mating system and mutation on sampling was studied using a real maize population. To study the genetic mechanisms underlying the response to selection, a divergent selection experiment for maize flowering time was carried out in France [START_REF] Durand | Dearth of polymorphism associated with a sustained response to selection for flowering time in maize[END_REF]. It yielded four early and four late subpopulations for flowering time. For the sake of simplicity, we only considered one early subpopulation and one late subpopulation here, that were derived from a seed lot of the maize inbred line F252 by successively selecting and selfing the earliest and the latest plants at each generation. Each subpopulation had a single and different ancestor. Both subpopulations formed the target set that comprised 163 plants. These plants were dead and could not be phenotyped. However, they had been selfed and the resulting seeds could be sown to produce plants that could be phenotyped. These seeds, one for each plant of the target set, formed the candidate set of plants that could be phenotyped, which comprised 163 plants. The population included a total of N = 326 plants that were distributed in G + 1 = 17 generations (Figure 1). It was structured since it comprised the early and late subpopulations. The pedigree of the population and some genotyping data were available, while other genotyping data remained to be collected. Scientists were interested in estimating both breeding values and locus effects. We studied how to sample n = 50 plants in the candidate set.

In order to study the influence of mating system, a theoretical pedigree with outcrossing was considered in addition to the real pedigree with selfing. In this scenario, each plant that was not a founder of the pedigree had two parents. The first one was its parent in the original pedigree with selfing. The second one was an additional plant placed in generation 0, which differed between subpopulations. Thus, all the plants of generations 1 to 16 of a given subpopulation had the same second parent (Figure 2). 4.3.2 Methods. Kinship matrices were generated using the pedigree. The prior means of variance components took two possible values: γ 2 0 = 0.25 or γ 2 0 = 4, γ 2 m = 0.0001 or γ 2 m = 0.04. The explicit result of Section 4.1 did not allow the construction of optimal designs in this application. For each combination of values of variance components, a design was generated by maximizing the D criterion using STPGA, and 100 designs were generated by SS (see online Appendix D for more information). In addition, 100 designs were generated by SRS.

Sampling methods were compared using the D criterion. This criterion assumed that the target set consisted of the whole population, which was not the case in this application. To better assess designs, the maximal and average variance criteria were also computed: M Vu = maxi∈T { Var (ui | y) p(y)dy}, Au = 1

|T | i∈T

Var (ui | y) p(y)dy, where |T | was the cardinal of T . These criteria took the target set into account. As they were based on posterior variances, they quantified estimation errors (Section 3.2). Their computation assumed that variance components had low prior variance (online Appendix D).

4.3.3

Results. Designs were sensitive to genetic variances and mating system. As the mutational variance increased, SS and DO sampled more plants in late generations (Figures 1, 2 andonline Figures D.1-D.4). In addition, this effect was stronger when plants reproduced by outcrossing or when the prior mean of the additive genetic variance component γ 2 0 was low. For example, when plants reproduced by selfing and γ 2 0 = 0.25, DO sampled more plants in early (respectively, late) generations when γ 2 m = 0.0001 (respectively, γ 2 m = 0.04) (Figure 1). SS and DO tended to select individuals whose breeding values were weakly correlated a priori and had large prior variances (online Tables D.1-D.3). For example, when plants reproduced by selfing and when γ 2 m = 0.0001, they sampled more plants in early generations, presumably because the correlation between breeding values was high in late generations.

SS and DO had better D, M Vu and Au values than SRS, except for the Au criterion when plants reproduced by selfing, γ 2 0 = 0.25 and γ 2 m = 0.0001 (Figure 3, online Figures D.5 and D.6). SS had a lower efficiency variability than SRS. SS and DO efficiencies were larger in the presence of selfing.

Sampling designs for association studies

Explicit result

When the genetic data are genotyping data, when the prior variance of γ 2 β is low, and when L ≤ n, if at each locus, half of the selected individuals has genotype -1 and the other half has genotype 1, and if locus effects are not correlated a posteriori (when L > 1), then the design is D-optimal over the set of all designs given C and n (online Appendix B). Locus effects are q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q a q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q b q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q c q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q d Figure 1. Sampling designs for the early maize subpopulation when plants reproduce by selfing. Each line corresponds to a generation, with generation 0 (respectively, 16) at the top (respectively, bottom) of the pedigree. Vertices are plants, and edges connect parents and their offspring. The candidate set comprises the tips of ancestral lines. Sampled (respectively, not sampled) plants are in black (respectively, gray). The designs were generated by DO with parameter values γ 2 0 = 0.25 and γ 2 m = 0.0001 (a), γ 2 0 = 0.25 and γ 2 m = 0.04 (b), γ 2 0 = 4 and γ 2 m = 0.0001 (c), γ 2 0 = 4 and γ 2 m = 0.04 (d). The corresponding designs for the late subpopulation can be found in online Appendix D. not correlated a posteriori when the matrix Z T X T QXZ is diagonal, that is, when the genotypes of the selected individuals at different loci are not correlated.

Simulation study

5.2.1 Experimental setup. The influence of sample size and the genetic architecture of the trait on sampling was studied by means of simulations. The R code of this study can be found in Supporting Information. For the sake of simplicity, the experimental setup had a smaller size than real association studies: the population was composed of 16 individuals, genotyped at 18 loci. The candidate and target sets both consisted of the whole population.

The population was divided into 4 subpopulations comprising 4 individuals. Allele frequencies varied between subpopulations (see online Appendix E for more information). Each locus was assumed to be at Hardy-Weinberg equilibrium in each subpopulation. Genotypes were simulated independently between individuals and between loci, that is, there was no linkage disequilibrium.

Sample size took three possible values: n = 4, n = 8 or n = 12. The trait was assumed to be affected either by L = 2 loci or by all the loci (L = 18). The prior mean of the additive genetic variance component γ 2 β was either equal to 0.25 or to 4.

Methods.

For each sample size, 100 designs were generated by SRS. For each L value and each sample size, 100 designs were generated by SS. When L = 2, and when n = 4 or 8, DO designs were generated using the explicit result of q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q a q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q b q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q c q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q d Figure 2. Sampling designs for the early maize subpopulation when plants reproduce by outcrossing. Each line corresponds to a generation, with generation 0 (respectively, 16) at the top (respectively, bottom) of the pedigree. Vertices are plants, and edges connect parents and their offspring. The candidate set comprises the tips of ancestral lines. Sampled (respectively, not sampled) plants are in black (respectively, gray). The designs were generated by DO with parameter values γ 2 0 = 0.25 and γ 2 m = 0.0001 (a), γ 2 0 = 0.25 and γ 2 m = 0.04 (b), γ 2 0 = 4 and γ 2 m = 0.0001 (c), γ 2 0 = 4 and γ 2 m = 0.04 (d). The corresponding designs for the late subpopulation can be found in online Appendix D. Section 5.1. For the other parameter values, DO designs were generated by optimizing the D criterion using STPGA (see online Appendix E for more information).

Sampling methods were compared using the D criterion. To better quantify estimation errors for breeding values and locus effects, the maximal and average variance criteria (M Vu, Au, M V β and A β ) were also computed for these parameters. For example, the Au criterion was equal to Au = 1

N i

Var (ui | y) p(y)dy. The computation of these criteria assumed that the additive genetic variance component had low prior variance (online Appendix E).

Estimation errors when n = 4 and γ 2 β = 4 were also quantified for each sampling method by simulating trait values. The mean squared error of estimates and the mean correlation between estimated and true parameter values were estimated for breeding values and locus effects (online Appendix E). q q q 0.25 1e-04 outcrossing D0 SRS SS q q 4 1e-04 outcrossing D0 SRS SS q q q q q q q q q q 0.25 0.04 outcrossing D0 SRS SS 4 0.04 outcrossing q q q q q q q q q q q q q q q q q q q q q q q q q q 0.25 1e-04 selfing q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 4 1e-04 selfing q q q q q q q q q q 0.25 0.04 selfing 0.6 0.8 1.0 1.2 1.4 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 4 0.04 selfing Figure 3. Efficiency of designs for sampling a maize population. Sampling methods are SRS, SS or DO. The prior mean of the additive genetic variance component γ 2 0 is either equal to 0.25 or to 4 (lower strip above each plot). The prior mean of the mutational variance component γ 2 m is either equal to 0.0001 or to 0.04 (middle strip above each plot). Plants reproduce either by selfing or by outcrossing (upper strip above each plot). Designs are compared using the M Vu criterion, whose values are divided by the median M Vu values of SRS.

However, there were a few cases when optimized designs were less efficient. When L = 18, all the sampling methods showed similar estimation errors for locus effects, with correlations between estimated and true parameters lower than 0.3 (Figure 6, online Figure E.6 and online Table E.1). In this case, estimation errors were lower for breeding values than for locus effects, with correlations between estimated and true parameters larger than 0.6 (online Table E.1).

SS was less efficient than SRS when L = 2 and n > 4 (Figures 5,). Unlike DO, SS tended to avoid sampling some genotypes several times by sampling individuals in different clusters, which was not very efficient when L = 2 and n > 4.

Discussion

Sampling designs for pedigrees

This article investigated how to sample pedigrees. Mating system and mutation influenced designs through the prior variance matrix of breeding values. In the maize application, it was preferable to sample more individuals in early generations when the mutational variance component was low and when individuals reproduced by selfing, because individuals from late generations provided similar information since their breeding values were highly correlated. On the contrary, when the mutational variance component was larger, breeding values were less correlated in late generations so that more individuals could be sampled in these generations. It is often important to take account of mutation when analyzing long-term experiments, as, for example, in selection experiments where mutation can explain a continued response [START_REF] Wray | Accounting for mutation effects in the additive genetic variance-covariance matrix and its inverse[END_REF][START_REF] Durand | Standing variation and new mutations both contribute to a fast response to selection for flowering time in maize inbreds[END_REF]. Our results show that it is also useful to take account of mutation at the design stage.

In the association study, the additive genetic variance component did not greatly influence sampling designs, in agreement with earlier results [START_REF] Akdemir | Optimization of genomic selection training populations with a genetic algorithm[END_REF]. On the contrary, in the maize application, it influenced designs, with low values strengthening the influence of mutation. This is because when breeding values are modeled using the pedigree, variance components determine the relative importance of standing genetic diversity, mutation and information provided by the data.

Sampling designs for association studies

This article investigated how to sample populations in association studies. Optimized designs tended to improve the joint estimation of breeding values and locus effects, all the more as sample size was low [START_REF] Akdemir | Design of training populations for selective phenotyping in genomic prediction[END_REF]. Both objectives were consistent when all breeding values and all locus effects were of interest, but they may be inconsistent when the survey is aimed at inferring some particular breeding values or locus effects.

When the genetic architecture of the trait was rather simple, optimized designs were reminiscent of classical experimental designs. Since they tended to sample homozygous individuals, they were similar to optimal designs for linear regression (Atkinson and Donev, 1992, Chap. 6). Since they tended to reduce the correlation between genotypes at different loci, they were similar to orthogonal factorial designs (Atkinson and Donev, 1992, Chap. 7;[START_REF] Kobilinsky | Automatic generation of generalised regular factorial designs[END_REF]. This is because our model can be considered as a linear model in which loci play the role of independent variables or factors.

When the genetic architecture of the trait was rather complex, optimized designs were not more efficient than SRS for estimating locus effects, and the correlation between estimated and true locus effects was low. This is why the number of loci was kept not much larger than population size in our association study. On the contrary, optimized designs were more efficient than SRS for estimating breeding values in this case, and estimation errors were lower for breeding values than for locus effects. Locus effects were difficult to estimate because the number of model parameters was larger than sample size. Our results raise the issue of identifying when high-dimensional regression methods provide reliable results [START_REF] Verzelen | Minimax risks for sparse regressions: Ultra-high dimensional phenomenons[END_REF]Giraud, 2014, Chap. 1).

Stratified sampling

This article developed an SS method for sampling related individuals. This method does not require an optimization step. It can be used with both informative or weakly-informative prior distributions for variance components when kinship is modeled using the pedigree. In previous SS methods, the population was divided into large subpopulations and several individuals were sampled in each subpopulation [START_REF] Isidro | Training set optimization under population structure in genomic selection[END_REF]. The population was divided here into smaller clusters and one individual was sampled in each cluster. This may more effectively capture structures in the population and reduce the efficiency variability of sampling designs. However, efficiency variability may remain high with our method when clusters are large, as, for example, when sample size is much smaller than population size.

SS performed rather well in our applications. This may be because populations were structured in these examples [START_REF] Isidro | Training set optimization under population structure in genomic selection[END_REF]. SS was not very efficient when the genetic architecture of the trait was simple and when sample size was not low. In this case, SS tended to sample more heterozygous individuals than DO to increase the genetic diversity in the sample, which was not optimal. However, including heterozygous individuals in the sample could be useful to check model assumptions, in particular the absence of dominance (Atkinson and Donev, 1992, Chap. 20). The prior mean of the additive genetic variance component γ 2 β is equal to 4. The trait is determined by 2 or 18 loci (upper strip above each plot). Sample size is 4, 8 or 12 (lower strip above each plot). Designs are compared using the M V β criterion, whose values are divided by the median M V β values of SRS.

D optimality

This article also implemented optimal design theory using the D criterion. This criterion quantifies the precision of the analysis. It is more relevant when experimenters are interested in estimating all the breeding values or all the locus effects. When the objective is to estimate some particular breeding values, locus effects or contrasts, criteria that quantify precision for these parameters are more appropriate [START_REF] Rincent | Predicting genomic selection efficiency to optimize calibration set and to assess prediction accuracy in highly structured populations[END_REF][START_REF] Akdemir | Design of training populations for selective phenotyping in genomic prediction[END_REF]. In the maize application, the D criterion remained relevant even if the target set did not include all the plants, because the candidate and target sets were both large and sampled the pedigree in a similar manner [START_REF] Akdemir | Design of training populations for selective phenotyping in genomic prediction[END_REF]. The D criterion allowed the derivation of explicit results, which helped to assess the genetic algorithm, to understand the properties of optimal designs, and to study the link between designs for estimating breeding values and designs for estimating locus effects.

Optimal design theory requires an optimization step. The D criterion reduces the corresponding computing time since its computation only involves the determinant of an n × n matrix. The optimizations carried out in this article typically lasted a few minutes. On the contrary, other criteria, such as the CD (coefficient of determination), P EV (prediction error variance), Au or M Vu criteria, require the inversion of an N × N matrix for their evaluation when L ≥ N [START_REF] Rincent | Predicting genomic selection efficiency to optimize calibration set and to assess prediction accuracy in highly structured populations[END_REF][START_REF] Akdemir | Design of training populations for selective phenotyping in genomic prediction[END_REF]. Computing times ranging from a few seconds to a few days have been reported for the optimization of these criteria [START_REF] Isidro | Training set optimization under population structure in genomic selection[END_REF][START_REF] Rincent | Predicting genomic selection efficiency to optimize calibration set and to assess prediction accuracy in highly structured populations[END_REF].

DO was implemented here in the presence of prior information on variance components. This made the design criterion easier to compute. Prior information on variance components is sometimes available: it can rely on general estimates of phenotypic variance or estimates of heritability from earlier parent-offspring studies in the same or in close populations or species [START_REF] Butler | On the design of field experiments with correlated treatment effects[END_REF]. For example, in the application, some phenotypic data have already been collected and could be used to specify prior distributions for variance components [START_REF] Durand | Standing variation and new mutations both contribute to a fast response to selection for flowering time in maize inbreds[END_REF][START_REF] Durand | Dearth of polymorphism associated with a sustained response to selection for flowering time in maize[END_REF]. Our implementation of DO is expected to work better when some reliable prior information on variance components is available. When this is not the case, it would be more satisfactory to more effectively take the uncertainty on variance components into account, but the design criterion would then have a more complicated expression [START_REF] Bueno Filho | Block designs for random treatment effects[END_REF][START_REF] Chaloner | Bayesian experimental design: A review[END_REF], or to implement a sequential approach.

  5.2.3 Results. Unlike SRS, SS and DO always sampled individuals in all the subpopulations (Figure 4, online Figures E.1 and E.2). When L = 2 and n = 4, DO sampled the genotypes (1, 1), (1, -1), (-1, 1) and (-1, -1) (online Figure E.1). When L = 2 and n = 8, it sampled these genotypes twice. It was not very sensitive to γ 2 β (Figure 4 and online Figure E.2). Optimized designs tended to improve the estimation of both breeding values and locus effects (Figures 5, 6, online Figures E.3-E.6 and online TableE.1). As sample size increased and the genetic architecture of the trait was more complex, the advantage of optimized designs over SRS diminished. Criterion values were less dispersed for SS than for SRS. The performance of design methods was not very sensitive to γ 2 β (online Figures E.3 and E.4).

Figure 4 .

 4 Figure4. Sampling designs for the theoretical association study. The genetic data are represented using a multidimensional scaling. There are 4 subpopulations, which are located near plot corners. The trait is determined by 18 loci. Sample size is 4, 8 or 12 (lower strip above each plot). Sampling methods are SRS, SS or DO with γ 2 β = 4 (upper strip above each plot). Dots (respectively, crosses) represent sampled (respectively, not sampled) individuals.

Figure 5 .

 5 Figure 5. Efficiency of sampling methods in the theoretical association study. Sampling methods are SRS, SS or DO. The prior mean of the additive genetic variance component γ 2 β is equal to 4. The trait is determined by 2 or 18 loci (upper strip above each plot). Sample size is 4, 8 or 12 (lower strip above each plot). Designs are compared using the M Vu criterion, whose values are divided by the median M Vu values of SRS.

Figure 6 .

 6 Figure 6. Efficiency of sampling methods in the theoretical association study. Sampling methods are SRS, SS or DO. The prior mean of the additive genetic variance component γ 2β is equal to 4. The trait is determined by 2 or 18 loci (upper strip above each plot). Sample size is 4, 8 or 12 (lower strip above each plot). Designs are compared using the M V β criterion, whose values are divided by the median M V β values of SRS.

Acknowledgements

This study was funded by the French BASC project Itemaize. The authors are grateful to the INRAE MIGALE bioinformatics facility (MIGALE, INRAE, 2020. Migale bioinformatics Facility, doi: 10.15454/1.5572390655343293E12) for providing computing resources. The authors thank the associate editor and the three referees for their helpful comments. They also thank Gail Wagman for improving the English of the article. This is the accepted version of the following article: David, O., Le Rouzic, A., Dillmann, C. ( 2021) Optimization of sampling designs for pedigrees and association studies. Biometrics, which has been published in final form at https://doi.org/10.1111/biom.13476. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy (http://www.wileyauthors.com/self-archiving).

Data Availability Statement

The data that support the findings in this paper are available from the corresponding author upon reasonable request.

Supporting Information

Web Appendices, Tables, Figures, and Codes referenced in Sections 3, 4 and 5, are available with this paper at the Biometrics website on Wiley Online Library.