Ramchandar Gomajee

Fabienne Elkhouryjudith Van Der Waerden

Laura Pryor

Maria Melchiorthe

Eden Mother-Child

Early life childcare and later behavioral difficulties: a causal relationship? Data from the French EDEN study

INTRODUCTION

1.1.Background

The first years of life are crucial in terms of long-term cognitive and behavioral development [START_REF] Shonkoff | Investment in early childhood development lays the foundation for a prosperous and sustainable society[END_REF]. Therefore, early interventions limiting the impact of adverse environments early on in life may help reduce inequalities in health, cognitive and social development in the short and long-term (Doyle et al., 2009). The most famous research in this area, conducted by James Heckman, shows that the earlier the intervention the higher return on investment and reaped benefits [START_REF] Carneiro | Human capital policy[END_REF]. Childcare prior to school entry varies in type, duration as well as quality. Some children are looked after in their own home, others by relatives or in the home of another caregiver, still others in collective settings with other children. Childcare therefore includes different environments which may have different consequences for children's cognitive and non-cognitive development.

Cognitive development plays a fundamental role with regard to later schooling, income, criminality and success in many facets of social and economic life [START_REF] Cunha | Investing in our young people[END_REF]. Over the last 30 years, many studies have therefore studied the influence of early childcare on cognitive development. They have shown that compared to parental care, formal care is associated with increased school readiness, improved receptive vocabulary and better reading skills [START_REF] Geoffroy | Closing the gap in academic readiness and achievement: the role of early childcare[END_REF][START_REF] Li | Timing of highquality child care and cognitive, language, and preacademic development[END_REF] as well as improved language skills [START_REF] Luijk | Hours in non-parental child care are related to language development in a longitudinal cohort study[END_REF]. This may be particularly true for children who come from a disadvantaged socioeconomic background [START_REF] Geoffroy | Closing the gap in academic readiness and achievement: the role of early childcare[END_REF] or children of immigrants [START_REF] Dustmann | Does Universal Child Care Matter? Evidence from a Large Expansion in Pre-School Education[END_REF]. Long terms benefits of early subsidized childcare include higher educational attainment, labor market participation and reduced welfare dependency especially for individuals from low socioeconomic backgrounds [START_REF] Havnes | No Child Left Behind: Subsidized Child Care and Children's Long-Run Outcomes[END_REF].

Similarly, non-cognitive characteristics including perseverance, self-esteem, self-control and motivation influence educational performance, as well as social and economic success (Borghans et al., 2008;[START_REF] Heckman | The Effects of Cognitive and Noncognitive Abilities on Labor Market Outcomes and Social Behavior[END_REF]. There has been much debate regarding the possible role of early life childcare arrangements in potentially increasing or offsetting developmental and behavioral risks. Studies undertaken in the 1980's came to the conclusion that early and extensive non-parental childcare is related to an increased level of disobedience and aggression in children later in life [START_REF] Baydar | Effects of maternal employment and child-care arrangements on preschoolers' cognitive and behavioral outcomes: Evidence from the Children of the National Longitudinal Survey of Youth[END_REF][START_REF] Belsky | Nonmaternal Care in the First Year of Life and the Security of Infant-Parent Attachment[END_REF][START_REF] Haskins | Public School Aggression among Children with Varying Day-Care Experience[END_REF][START_REF] Rubenstein | A Two-Year Follow-up of Infants in Community-Based Day Care[END_REF]. Some studies show that children who are in non-parental care have more psychological difficulties than those who stay at home [START_REF] Belsky | Are There Long-Term Effects of Early Child Care?[END_REF][START_REF] Belsky | Nonmaternal Care in the First Year of Life and the Security of Infant-Parent Attachment[END_REF][START_REF] Gialamas | Time spent in different types of childcare and children's development at school entry: an Australian longitudinal study[END_REF][START_REF] Mccartney | Testing a series of causal propositions relating time in child care to children's externalizing behavior[END_REF], and the strength of this association appears to be directly related to the number of hours spent out of home [START_REF] Belsky | Are There Long-Term Effects of Early Child Care?[END_REF][START_REF] Datta Gupta | Non-cognitive child outcomes and universal high quality child care[END_REF]. Other studies show either no negative impact of childcare on children [START_REF] Barnes | Experiences of childcare in England and socio-emotional development at 36 months[END_REF][START_REF] Beijsterveldt | Short-and Long-Term Effects of Child Care on Problem Behaviors in a Dutch Sample of Twins[END_REF][START_REF] Gialamas | Quality of childcare influences children's attentiveness and emotional regulation at school entry[END_REF], or even a beneficial effect, especially among children whose mother is depressed or who come from a disadvantaged family [START_REF] Love | The effectiveness of early head start for 3-year-old children and their parents: lessons for policy and programs[END_REF][START_REF] Yamauchi | Which children benefit from non-parental care?[END_REF].

Countries in which the impact of childcare arrangements on children's development have been studied vary much in terms of early childhood policies (e.g. affordable access to childcare, parental leave). If the quality of childcare based on UNICEF standards is taken into account (UNICEF Innocenti Research Centre, 2008), studies showing that early childcare is associated with increased levels of behavioral difficulties in children are mostly conducted in countries where the quality of childcare is considered to be low, such as the USA [START_REF] Belsky | Are There Long-Term Effects of Early Child Care?[END_REF][START_REF] Mccartney | Testing a series of causal propositions relating time in child care to children's externalizing behavior[END_REF] and Australia [START_REF] Gialamas | Time spent in different types of childcare and children's development at school entry: an Australian longitudinal study[END_REF]. In studies which take the quality of childcare into consideration, an increase in the level of children's behavioral difficulties was only observed in case of extended hours spent in low quality childcare settings [START_REF] Mccartney | Testing a series of causal propositions relating time in child care to children's externalizing behavior[END_REF], while children spending extended hours in high quality childcare have low levels of behavioral problems [START_REF] Vandell | Do Effects of Early Child Care Extend to Age 15 Years? Results From the NICHD Study of Early Child Care and Youth Development[END_REF].

The interpretation of results in this area can be complicated because different countries have different policies concerning the age of school entry. In most countries children officially start school at the age of six (Canada, Germany, Netherlands, Norway, USA) but in some they start at the age of five (Australia, Ireland, New Zealand, United Kingdom) or seven (Finland, South Africa, Sweden, Switzerland) (UNESCO Institute for Statistics, 2016). The scientific literature on childcare usually refers to care received by children up to the age of school entry.

Furthermore, most evaluated interventions in this area were conducted among children from deprived families, and only a few were conducted in countries that have a universal early childcare policy such as the Nordic countries.

In France, the official age for school entry is 6 years. However, 97% of children aged 3 and 99% aged 4 attend preschool ("école maternelle") which is free of charge [START_REF] Blanpain | Scolarisation et modes de garde des enfants âgés de 2 à 6 ans[END_REF].

Therefore, in France the difference in access to early childcare refers to childcare received before entering preschool, i.e. prior to age three. The French childcare system is based on the principle of universality: public childcare facilities are not aimed at a particular segment of the population but accept everyone irrespective of socioeconomic background [START_REF] Fagnani | Equal access to quality care: Lessons from France on providing high quality and affordable childcare and early education[END_REF]. One of the pillars of the French social protection system, which is largely funded by mandatory employer and employee contributions, is the family policy. The state allocates a budget to the "Caisse Nationale d'Allocations Familiales" (CNAF) (National Office for Family Allowances), which funds and regulates the distribution of subsidies to families as well as the financing of childcare facilities via decentralized "Caisse d'Allocations

Familiales" (CAF) (Office of Family Allowances) (Fraisse L et al., 2011). Each municipality, which is the administrative division of a town or a village, has a budget for the development of the territory under its jurisdiction and is free to decide how much money is invested in center-based facilities depending mainly on political orientation, population size and the degree of urbanization. The predominant role of municipalities is to run center-based childcare while the General Council ("le Département") is responsible for childminder agreements and center-based childcare quality control (Fraisse L et al., 2011).

Center-based childcare and childminders are widely used in France, but the selection generally depends on availability (demand exceeds supply) rather than parents' choice. In center-base childcare facilities, regulations concerning the supervision of children are strict: child-caregiver ratio, timetable of children's activities during the day, ensuring high quality childcare [START_REF] Choné | Female labor supply and child care in France[END_REF]. On the other hand, the quality of childminder's care may vary from one childminder to another, even though they are required to complete mandatory training in early child development before being delivered a 5-year renewable permit.

Furthermore, there are no pre-established activities [START_REF] Choné | Female labor supply and child care in France[END_REF] and childminders have a lower level of education when compared to educators in center-based childcare settings [START_REF] David-Alberola | Investing in early human development: Timing and economic efficiency[END_REF]. Nevertheless, childminders are the main type of out-of-home childcare in France, because the number of places available in center-based childcare is limited [START_REF] Micheau | Les modes d'organisations des crèches collectives et les métiers de la petite enfance[END_REF].

On average, French families spend 411 euro/month on childcare but this amount varies with the type of childcare, the number of hours the child spends in childcare, or and the number of children in the family [START_REF] Villaume | Combien dépensent les familles pour la garde de leurs enfants de moins de 3 ans ? DREES Numéro 930[END_REF]. Parents pay on a sliding scale according to their income. Through tax deductions and special allowances, childcare is subsidized by the state but can nevertheless be too costly for the most economically disadvantaged families [START_REF] Fagnani | Equal access to quality care: Lessons from France on providing high quality and affordable childcare and early education[END_REF]. Before tax deductions and special allowances, the average cost per month for 35 hours of presence is 334 euro per child for center-based childcare and 625 euro per child for a childminder [START_REF] Villaume | Combien dépensent les familles pour la garde de leurs enfants de moins de 3 ans ? DREES Numéro 930[END_REF].

There is a high demand for center-based childcare, especially from middle class families in which both parents are often employed and who view early socialization in a positive way.

However, there are often long waiting lists and the demand far exceeds the supply [START_REF] Fagnani | Equal access to quality care: Lessons from France on providing high quality and affordable childcare and early education[END_REF]. The exogenous allocation of places to center-based childcare creates a quasiexperimental situation whereby, among families with young children who wish to have them looked after, the type of childcare (center-based or child-minder) that is allocated does not depend on family characteristics.

In the present investigation, we examine the association between early childcare arrangements, between ages 0 and 3 years, and children's behavior up to age 8, using data from the longitudinal mother-child EDEN cohort study in France. Most studies assessed whether childcare is associated with behavioral difficulties at one time point or successive time points separately. In the present study we model patterns of behavioral difficulties from age 3 to age 8 using group-based trajectory modeling [START_REF] Jones | A SAS Procedure Based on Mixture Models for Estimating Developmental Trajectories[END_REF]. We then compare children's trajectories of behavioral difficulties depending on whether they are in informal vs.

formal care and also depending on the type of childcare they are in and its duration.

Additionally, because boys [START_REF] Chaplin | Gender differences in emotion expression in children: a meta-analytic review[END_REF]Miner and Clarke-Stewart, 2008), children whose mother is depressed [START_REF] Agnafors | Early predictors of behavioural problems in pre-schoolers -a longitudinal study of constitutional and environmental main and interaction effects[END_REF][START_REF] Tompson | Maternal depression, maternal expressed emotion, and youth psychopathology[END_REF], as well as those who grow up with parents who experience financial difficulties [START_REF] Barazzetta | Childhood Circumstances and Young Adulthood Outcomes: The Effects of Mothers' Financial Problems[END_REF][START_REF] Miller | The Role of Family Income Dynamics in Predicting Trajectories of Internalizing and Externalizing Problems[END_REF][START_REF] Rijlaarsdam | Economic Disadvantage and Young Children's Emotional and Behavioral Problems: Mechanisms of Risk[END_REF] are especially prone to behavioral difficulties, we also test whether the type of childcare has a different influence on the behavior of children depending on these characteristics.

METHODS

2.1.Study design, setting and participants

The EDEN mother-child cohort study primary aims to assess pre-and postnatal determinants of child growth, development and health. Recruitment of pregnant women (before 24 weeks of amenorrhea (WA) was conducted in two French university hospitals (Nancy andPoitiers, 2003-2005) during the prenatal visit at the Departments of Obstetrics and Gynecology.

Exclusion criteria were: multiple pregnancies, pre-pregnancy diabetes, illiteracy, and plans to move outside the region in the following 3 years. Mothers were examined clinically 3 times: between the 24 th and 26 th WA, at delivery and 5-6 years after delivery. Self-administered or midwife-administered questionnaires were filled out during the clinical examinations and additionally at 4 months, 8 months, 1 year, 2 years, 3 years, 5.5 years and at 8 years postdelivery [START_REF] Heude | Cohort Profile: The EDEN mother-child cohort on the prenatal and early postnatal determinants of child health and development[END_REF].

Among the 2002 women included in the EDEN cohort, 1,432 participants reported their children's behavioral score in at least one of the 3 possible study waves at 3, 5.5 and 8 years of age, while 1,428 also provided information about childcare between ages 0 and 3 years. These 1,428 mother-child dyads constitute our study population. Compared to the studied population, non-participating mothers are more likely to participate in the Nancy research center (61.4 vs. 47.8%; p<0.0001), to be younger (mean age = 28.2 years, SD = 5.1 years vs. 29.9 years, SD = 4.7 years; p<0.0001), have lower education (mean number of years of education = 12.4 years, SD = 2.3 years vs. 13.6 years, SD = 2.3 years; p<0.0001), smoke cigarettes before pregnancy (49.4 vs. 32.3 %; p<0.0001), less likely to be employed (63.9 % vs. 81.5 %; p < 0.0001), less likely to receive social support (62.5 vs. 71.4%; p=0.0002) and more likely to have a history of mental health problems (21.8 vs. 14.0%; p<0.0001).

Participating children, as compared to non-participants, were less likely to be first-born (38.9 vs. 46.5%; p=0.0047) and more likely to be born with a low birth-weight (7.3 vs. 4.8%; p=0.0352).

2.2.Variables

Childcare

From the age of 4 months to 3 years, age at which children generally enter preschool in France, mothers were asked to report the type of childcare primarily used by their child (crèche/daycare, day nursery, childminder, relatives, neighbors and friends, a babysitter or whether the parents looked after the child themselves).

Childcare type was then divided into 3 groups:

1. Childminder, CM (N = 636, 45.6%): a professional caregiver with a formal degree in early childhood education, with a 5-year renewable permit allowing her to take care of up to 6 children in her own house. Furthermore, by taking into account the type of childcare in the successive waves of questionnaires from 4 months to 2 years, we were able to divide children who were in center-

1 All children belonged to only one of the groups. In fact, very few children either switched from a childminder's care to center-based childcare or vice-versa (25) or were in both types simultaneously (19). Children stayed in informal care (mostly parental care) until they attended center-based or a childminder. Therefore, children who were never in either childminder's care or center-based care were in the "informal care" group; those who were never in center-based but had been in a childminder's care were in the "childminder" group and all those who had experienced center-based childcare at any point in the study were in the "center-based childcare" group. Thus children who experienced both childminder's care and center-based care were classified in the latter group (this situation concerns only a small percentage of all children). There are two main reasons behind this classification. Firstly, we assumed that center-based childcare would have the greatest effect on children, positive or negative. Secondly, in France center-based childcare is highly regarded by parents and some parents put their child in childminder's care while they are on a waiting list for a place in a center-based childcare. In this case the child may spent a few weeks with a childminder before entering center-based childcare. Hence, even the 5 groups based on type and duration of childcare which we studied were mutually exclusive. A child who spent 3 months with a child-minder before entering center-based care for 2 years is classified only in the group 'center-based childcare for at least one year'.

based and childminder groups into 2 more groups each, depending on the total time spent in care:

1. Childminder < 1 year, 2. Childminder >= 1 year, 3. Center-based care < 1 year, 4. Center-based care >= 1 year 5. Informal care.

Childhood behavioral difficulties: SDQ score

The French version of the Strength and Difficulty Questionnaire (SDQ) was used to measure children's behavior at ages 3, 5.5 and 8 years. Translated into several languages, the SDQ is a validated screening instrument for emotional and behavioral problems in children and adolescents from 3 years of age onwards [START_REF] Algorta | Diagnostic efficiency of the SDQ for parents to identify ADHD in the UK: a ROC analysis[END_REF][START_REF] Goodman | The Strengths and Difficulties Questionnaire: a research note[END_REF][START_REF] Shojaei | The strengths and difficulties questionnaire: validation study in French school-aged children and cross-cultural comparisons[END_REF].

It contains 25 items divided into 5 scales of 5 items each: one positive scale (pro-social behavior) and four negative scales (emotional symptoms, behavior/conduct problems, hyperactivity/inattention, and peer relationship problems). Each scale is scored from 0 to 10.

The 4 negative scales are summed to calculate the total SDQ score, which ranges from 0 to 40 [START_REF] Youthinmind | Information for researchers and professionals about the Strengths & Difficulties Questionnaires[END_REF].

Covariates

Covariates include variables potentially associated either with the type of childcare or children's behavioral difficulties. Only measures obtained between baseline and age 3 years, which is the age when children in France enter preschool, were considered.

Among these variables were participants' socio-demographic characteristics: study center (Nancy or Poitiers); family situation (married/cohabiting or single); mother's age at delivery; parents' education level (number of years of education dichotomized at 14 years, which is equivalent to the French High School Certificate (baccalauréat) + 2 years of higher education); parents' employment status at each study wave; and the presence of financial difficulties (difficulties to buy food or clothes for the family, difficulties to pay the rent, heating or electricity, regular overdraft of bank account, subsidized health coverage, a family income of less than 1500 euros at least once during the study period).

Maternal psychological difficulties:

The Centre for Epidemiological Studies Depression (CESD) questionnaire was used to detect mother's depressive symptoms at 24 weeks of amenorrhea [START_REF] Radloff | The CES-D Scale A Self-Report Depression Scale for Research in the General Population[END_REF]. To identify individuals at risk of clinical depression, the score was cut-off at 23 or above (Fuhrer and Rouillon, 1989). The Edinburgh Postnatal Depression Scale (EPDS), a 10-item questionnaire devised to identify postnatal depression was used during the first year after birth (4, 8 and 12 months) to screen for the mothers' depression symptoms, with a score ranging from 0 to 30 [START_REF] Cox | Detection of postnatal depression. Development of the 10-item Edinburgh Postnatal Depression Scale[END_REF]. A threshold score of 13 or above was used to identify women at risk of clinical depression [START_REF] Matthey | Variability in use of cut-off scores and formats on the Edinburgh Postnatal Depression Scale: implications for clinical and research practice[END_REF].

Other variables pertaining to the mothers' psychological status include the use of psychological help during pregnancy and after birth (e.g. anxiolytics, sleeping pills, counselling) and self-reported mental health problems prior to pregnancy.

Other variables include parents' own behavioral difficulties during childhood, parents' use of psychoactive drugs (alcohol, cigarettes, cannabis), and the frequency of different parent-child activities (e.g. reading a story, feeding, playing, bathing, etc.).

2.3.Statistical methods

Mean SDQ score

We tested whether there was a significant difference in behavioral symptoms across the different childcare groups by comparing mean SDQ scores: (i) children in formal vs. informal childcare; (ii) children with a childminder or in informal care vs. those in center-based childcare.

Group-based trajectory modelling -Behavioral trajectories

Using data from participants with at least one of the three SDQ measures, we estimated participating children's behavioral trajectories with group-based trajectory models [START_REF] Jones | A SAS Procedure Based on Mixture Models for Estimating Developmental Trajectories[END_REF][START_REF] Nagin | Age, Criminal Careers, and Population Heterogeneity: Specification and Estimation of a Nonparametric, Mixed Poisson Model*[END_REF][START_REF] Nagin | What Has Been Learned from Group-Based Trajectory Modeling? Examples from Physical Aggression and Other Problem Behaviors[END_REF]. This semi-parametric modeling procedure enables the identification of clusters of individuals who follow a similar developmental trajectory of a variable of interest [START_REF] Nagin | Group-Based Trajectory Modeling in Clinical Research[END_REF], in this case overall behavioral scores.

Unlike hierarchical modeling and latent curve analysis which assume that the population of trajectories varies continuously across individuals and in a fashion that can be explained by a multivariate normal distribution of population parameters, group-based trajectory modeling assumes that there may be distinctive developmental trajectories that reflect distinct etiologies [START_REF] Nagin | Group-Based Modeling of Development[END_REF]. Furthermore, the definition of groups is based on formal statistical modeling rather than subjective classification rules and can be verified by testing the posterior probability of group membership [START_REF] Nagin | Group-Based Modeling of Development[END_REF].

The method considers that data are missing at random [START_REF] Broadbent | Trajectory Patterns of Dental Caries Experience in the Permanent Dentition to the Fourth Decade of Life[END_REF] and individuals with missing data are assigned to their most likely group.

Several models were tested examining an increasing number of developmental trajectories and different shapes (linear, quadratic, cubic) -the best model was determined using the Bayesian Information Criterion (BIC) [START_REF] Jones | A SAS Procedure Based on Mixture Models for Estimating Developmental Trajectories[END_REF][START_REF] Jones | Advances in Group-Based Trajectory Modeling and an SAS Procedure for Estimating Them[END_REF][START_REF] Nagin | Group-Based Trajectory Modeling in Clinical Research[END_REF] and maximizing the probability of group membership (>=0.7 ) [START_REF] Nagin | Group-Based Modeling of Development[END_REF][START_REF] Nagin | Group-Based Trajectory Modeling in Clinical Research[END_REF].

Multiple Data Imputation

Complete case analysis, i.e. the analysis of subjects with no missing data may lead to the exclusion of large numbers of participants especially when examining measures repeated over time. It may also sometimes result in biased results [START_REF] Sterne | Multiple imputation for missing data in epidemiological and clinical research: potential and pitfalls[END_REF]. One approach to deal with missing data is multiple imputation (MI). MI does not seek to replace a missing value with a simulated one but rather produces a random sample of possible values that represent the unpredictability of the missing value (Yuan, n.d.). MI involves 3 steps:

1. Multiple copies of the dataset are created with missing values filled in by imputed values based on a Bayesian approach [START_REF] Sterne | Multiple imputation for missing data in epidemiological and clinical research: potential and pitfalls[END_REF]. 3. The results are combined to draw an inference.

Only covariates were imputed. We assume that data were missing at random, i.e. the probability of a variable being missing depends on observed variables but not on the value of the missing variable. A wide range of variables predictive of missing data were included in the imputation model, even if they did not form part of the substantive analyses [START_REF] Sterne | Multiple imputation for missing data in epidemiological and clinical research: potential and pitfalls[END_REF].

Full Conditional Specification (FCS) Multiple Imputation with 10 imputations of each missing value was used. FCS MI is flexible and allows the specification of a model for each type of variable, e.g. linear regression for continuous variables, logistic regression for binary variables [START_REF] Bartlett | Multiple imputation of covariates by fully conditional specification: Accommodating the substantive model[END_REF].

PROC MIANALYZE in SAS V9.4 was then used to combine results when analyzing the complete imputed dataset.

Inverse Probability Treatment Weights (IPW) adjusted analyses

One of the issues in examining the relationship between childcare type and children's development using observational data is that children who attend childcare differ from those who do not (e.g. high socioeconomic position, positive interactions with parents).

A propensity score (PS) is the probability of treatment assignment conditional on observed baseline covariates (Z = the indicator for the observed treatment exposure, X = the vector of covariates) [START_REF] Austin | An Introduction to Propensity Score Methods for Reducing the Effects of Confounding in Observational Studies[END_REF][START_REF] Rosenbaum | The central role of the propensity score in observational studies for causal effects[END_REF]): e(X) = P (Z = 1|X), 0 < e(X) < 1.

It is a covariate summary score. Subjects having the same PS, whether they are treated (exposed) or not, have, on average, the same distribution of covariates used to estimate the PS. Therefore, individuals with the same propensity score share a similar distribution of X regardless of their treatment exposure.

Several methods have been used to reduce selection bias by using PS: covariate adjustment, matching, stratification and Inverse Probability Weight of treatment (IPW) [START_REF] Austin | An Introduction to Propensity Score Methods for Reducing the Effects of Confounding in Observational Studies[END_REF][START_REF] Xu | Use of stabilized inverse propensity scores as weights to directly estimate relative risk and its confidence intervals[END_REF]. In our study we calculated propensity scores for each subject and used the Inverse Probability Weight of treatment to account for selection bias. In IPWs, each observation is weighted by the reciprocal of the predicted probability of receiving the treatment that was observed for each patient, which can be estimated using propensity scoring. "With the appropriate weights, the weighted study population will be balanced across the treatment groups on the confounding factors. This balance is what allows for unbiased estimates of treatment effects in randomized controlled trials, thus IPW can be thought of as simulating randomization in observational studies" [START_REF] Crowson | The Basics of Propensity Scoring and Marginal Structural Models[END_REF].

By using the inverse probability of treatment received to weigh the subjects, an artificial sample is created whereby treatment selection is independent of measured baseline covariates, making it possible to measure average treatment effects [START_REF] Austin | Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies[END_REF]. In the synthetic sample created after PS are calculated and IPW applied, the treated and untreated population will have similar baseline characteristics, the main difference between the two groups being the treatment.

The IPW are calculated from the PS and are applied only once in the model. They are used to balance covariates between the 3 childcare groups but do not modify the treatment exposure or outcomes. The IPW procedure serves to make counterfactual inference. Individuals with a high probability of being treated that are actually treated bring no counterfactual information whereas individuals with a low probability of receiving treatment who actually receive treatment are much more informative on treatment effects. Similarly for individuals with a high probability of being exposed who actually are not exposed are more informative.

Multinomial multivariate logistic regression was used to calculate the probability of being in each childcare group and hence the PS [START_REF] Crowson | The Basics of Propensity Scoring and Marginal Structural Models[END_REF]. Variables related to the outcome were included in the model for calculating propensity scores [START_REF] Austin | A comparison of the ability of different propensity score models to balance measured variables between treated and untreated subjects: a Monte Carlo study[END_REF][START_REF] Austin | An Introduction to Propensity Score Methods for Reducing the Effects of Confounding in Observational Studies[END_REF][START_REF] Brookhart | Propensity Score Methods for Confounding Control in Nonexperimental Research[END_REF]. We made sure that PS distributions in the different childcare groups overlapped [START_REF] Lanehart | Propensity Score Analysis and Assessment of Propensity Score Approaches Using SAS® Procedures[END_REF] (supplementary figure 1).

Weights were stabilized by dividing by the mean weight in order to minimize extreme weights and reduce type 1 error and weights above the 99 th percentile were curtailed to the 99 th percentile before verifying that the stabilized mean weight was close to 1. With IPWs, each observation is weighted by the inverse of the PS. Extreme weights may result from observations with a PS close to 1 or 0 and consequently result in rejecting the null hypothesis too frequently [START_REF] Xu | Use of stabilized inverse propensity scores as weights to directly estimate relative risk and its confidence intervals[END_REF]. Finally, we verified that confounding factors were more evenly distributed among the 3 childcare groups post-than pre-weighing (supplementary

Interactions

We tested for statistical interactions and performed stratified analyses to test whether the effect of childcare varied with the child's gender, maternal depression during the perinatal period, maternal education or low family income (income < 1500 euros) between pregnancy and the child's 3-year assessment. To maximize the statistical power of these analyses, we combined children with intermediate and high trajectories of behavioral or emotional difficulties yielding a binary outcome: low vs. intermediate/high SDQ score trajectory.

RESULTS

Descriptive analysis

Table 1 shows children's and families' characteristics depending on the type of childcare between ages 0 and 3 years (childminder: 44.5%, N = 636; center-based care: 25.7%, N = 367; informal care: 29.8%, N = 425). The average proportion of missing values among covariates was 6.4%. All the covariates had less than 20% of missing values except the mother's EPDS score at 12 months (22.2%). Overall, children who were in out-of-home care came from families with higher levels of education and employment, and lower levels of financial difficulties and parental mental health problems than those who were in informal care. We observed relatively few differences with regard to these baseline characteristics between children who were cared for by a child-minder and those who were in center-based care. In terms of trajectories of behavioral difficulties, children in the informal childcare group had the highest SDQ score levels, those in center-based care the lowest.

Behavioral and emotional symptoms levels

Table 2 shows the mean SDQ scores at 3, 5.5 and 8 years across the different childcare groups when compared successively to the informal group and the center-based childcare group. Children in informal care had the highest mean SDQ scores in each of the three periods while those in center-based had the lowest.

Behavioral and emotional trajectories

The final model for group-based trajectories consisted of 3 distinct groups (Figure 1

Bivariate analysis

In bivariate analyses, children who were looked after by a childminder or in informal care had higher levels of emotional and behavioral difficulties than those who were in center-based childcare, and children looked after by a childminder's care had lower levels than those in informal care (Table 3a).

Examining the time spent in childcare (Table 3b), we observed that compared to children who spent at least 1 year in center-based childcare, those who were in a childminder's care, irrespective of the time spent, as well as those who were in informal care were more likely to be in the high or intermediate trajectories of behavioral problems.

IPW-adjusted analysis

In IPW-adjusted multivariate analyses, the ORs associated with childcare attendance decreased and lost statistical significance (Table 4a).

However, compared to children who were in center-based childcare for at least one year (14.0%, N= 200), those who spent less than 1 year in a childminder's care (12.2%; N = 174) and those in informal care (29.8%, N = 425) had a higher likelihood of being in the intermediate and high trajectories of total difficulties while those who spent at least 1 year in a childminder's care (32.4%; N = 462) or less than 1 year in center-based childcare (11.7%; N = 167) were associated with the high trajectory of behavioral difficulties (Table 4b).

Subgroup analyses

Table 5 shows the results of subgroup analyses in which trajectories that correspond to intermediate and high symptom levels of behavioral difficulties were merged and compared with the low symptom trajectory.

Gender

In comparison to girls who were in informal childcare (28.3%, N =194), girls who were looked after by a childminder (43.9%, N =301) had a decreased likelihood of having a high trajectory of emotional or behavioral difficulties as did those who were in center-based childcare (27.7%, N =190), although the latter result did not reach statistical significance.

Spending more than 1 year in a childminder's care (31.4%, N = 215) or in center-based childcare (15.9%, N = 109) was associated with a decreased likelihood of having a higher trajectory of behavioral difficulties when compared to informal care while informal care. Girls looked after by a childminder for less than 1 year was associated with an increased likelihood of high level behavioral difficulties compared to girls who spent more than 1 year in centerbased childcare.

Compared to boys in informal care, those in a childminder's care (45.1%, N = 355), especially for less than 1 year (11.8%, N = 88), had a higher likelihood of high behavioral difficulties.

Mother's education level

Among children whose mother had a high educational level (N = 840), compared to those who spent at least 1 year in center-based childcare, those who were in a childminder's care for less than 1 year and those who were in informal care had higher levels of behavioral symptoms. On the other hand, among children whose mother had a low educational level (N = 588), only a childminder's care for less than 1 year was significantly associated to a higher trajectory of behavioral problems when compared to center-based childcare for at least 1 year.

Mother's depression status

Overall, there was no significant association between childcare type or time spent in childcare and behavioral trajectories for children whose mother was depressed during the perinatal period (23.9%, N = 343). To the contrary, among children whose mother did not have depression (76.1%, N = 1085), compared to those who were in center-based childcare for at least 1 year, those who were in informal childcare or in a childminder's care for less than 1 year were more likely to have a higher trajectory of behavioral difficulties.

Low income

Overall, we found no evidence that children growing up in low income families gained from being in childcare between ages 0 to 3. However, among families who had intermediate/high income (N = 1 159), compared to children were in center-based childcare, those who were in informal childcare or in a childminder's care were more likely to be in a trajectory of high emotional or behavioral difficulties.

DISCUSSION

Main findings

Using data from 1428 mother-child dyads participating in the EDEN mother-child cohort, we identified three trajectories of children's emotional and behavioral difficulties from ages 3 to 8: low (50.4%), intermediate (40.2%) and high (9.3%).

In bivariate analyses, children who were looked after by a childminder or who were in centerbased childcare were less likely to have intermediate or high levels of emotional or behavioral difficulties up to age 8 years. However, after taking into account a large number of sociodemographic, parental and child characteristics, we did not observe statistically significant differences between the three childcare groups. Nevertheless, when children who spent at least 1 year in center-based care were compared to children in other settings, we found that those in informal childcare as well as those who were in a childminder's care, irrespective of the time spent there, were more likely to have intermediate or high levels of behavioral difficulties.

Secondary analyses showed that levels of emotional or behavioral difficulties were generally lowest among children who were in center-based childcare, particularly if they stayed there for more than 1 year. Girls were more likely to benefit from formal childcare. There were no significant differences in levels of behavioral difficulties for boys who were in informal care or center-based childcare, but child-minders' care, especially for a duration of less than 1 year was associated with higher levels of behavioral difficulties symptoms. In the context of debates about the role of early childcare with regard to children's behavioral development, our study indicates that center-based childcare attendance in France does not cause children's psychological difficulties and may even have a protective effect.

Strengths and limitations

Our study has several strengths. First, we used a prospective design which allowed us to ascertain study exposure and outcome as well as multiple covariates prospectively, between the time of the child's birth to school entry, yielding rich information on children's situation, behavior as well as their family context. Second, we used group-based trajectory modeling of children's SDQ scores to define developmentally sound groups in a more precise way using a single measure. Third, we used propensity scores and inverse probability weights (IPW) to account for selection factors and confounders. Studies showed that in the case of fewer than eight classes per confounder, and with several confounders, analyses based on propensity scores yield estimates that are less biased, more robust, and more precise than a logistic regression based approach [START_REF] Cepeda | Comparison of logistic regression versus propensity score when the number of events is low and there are multiple confounders[END_REF]. Third, given the specific context prevailing in France, we were able to directly compare two groups that are quite similar in terms of background characteristics yet imply different educational content: childminder vs. centerbased childcare. This situation strengthens our ability to draw causal inferences from our findings.

However, we also acknowledge limitations. First, the EDEN study is not representative of children in France. Indeed, the study was set up in two cities and, as in other cohort studies, participants were selected. However, the sample we studied is heterogeneous from the point of view of family background characteristics and levels of childcare attendance are comparable to national estimates. Second, our measure of early childcare may be relatively rough. Indeed, children can change childcare type during the 3 years before they enter school and it may be that childcare transitions are relevant with regard to later development.

However, though transitions from informal childcare to formal childcare are frequent, especially during the first year of life, once a child is in a formal childcare type he/she generally does not go back to informal care. In our study, 25 children either moved from center-based childcare to a childminder or vice-versa while 19 children attended both types of formal childcare simultaneously at least once during the study period. Third, children's behavior was ascertained by their mothers, and may be sensitive to reporting bias.

Nevertheless, mothers are generally reliable in assessing their children's behavior in middle childhood [START_REF] Mieloo | Validity and reliability of the strengths and difficulties questionnaire in 5-6 year olds: differences by gender or by parental education?[END_REF] and our longitudinal measure is more valid than a single ascertainment at one point in time.

Possible mechanisms linking early childcare to children's behavior

In the context of controversy regarding the consequences of early childcare, our findings are in line with data from the NICHD Early Child Care Research Network study (The NICHD Early Child Care [START_REF] Research | Early Child Care and Self-Control, Compliance, and Problem Behavior at Twenty-Four and Thirty-Six Months[END_REF] as well as a large Norwegian cohort study (N = 75,271) [START_REF] Zachrisson | Little evidence that time in child care causes externalizing problems during early childhood in Norway[END_REF] which showed that early, extensive and continuous childcare was not related to an increased likelihood of childhood behavioral problems. In fact, quality nonparental care was reported to be positively associated to children's social capabilities and behavior (The NICHD Early Child Care [START_REF] Research | Early Child Care and Self-Control, Compliance, and Problem Behavior at Twenty-Four and Thirty-Six Months[END_REF]. However, in the first study only a small percentage of center-based childcare settings could be classified as 'good quality'; e.g. only 36 % of centers had enough caregivers at 6 months, 20% at 15 months and 26% at 24 months (The NICHD Early Child Care [START_REF] Research | Early Child Care and Self-Control, Compliance, and Problem Behavior at Twenty-Four and Thirty-Six Months[END_REF]. In Norway, on the other hand, childcare quality as high as in France (UNICEF Innocenti Research Centre, 2008) and very widespread (97% of children attend) [START_REF] Zachrisson | Little evidence that time in child care causes externalizing problems during early childhood in Norway[END_REF]. Our findings extend this prior knowledge by showing that center-based childcare may be especially beneficial if children attend for at least one year.

Our results can be explained by several mechanisms. First, children who attend childcare differ from those who stay at home until preschool entry with regard to many characteristics including family socioeconomic factors and parental history of mental health problems which can both influence children's behavioral development. While this is the case in our study as in other settings, we were able to control for multiple family background characteristics and compare children who were in childminder care with children who attended a center-based childcare setting, which reduces the influence of this type of bias. Second, center-based care enables children's early socialization and respect of societal rules [START_REF] Bernal | Subsidized childcare and child development in Colombia: Effects of Hogares Comunitarios de Bienestar as a function of timing and length of exposure[END_REF]. Exposure to positive stimulation through praise, play and reading [START_REF] Klasen | What works where? A systematic review of child and adolescent mental health interventions for low and middle income countries[END_REF] can have long-term beneficial consequences on brain, neuroendocrine, cognitive or psychosocial development [START_REF] Wachs | Issues in the timing of integrated early interventions: contributions from nutrition, neuroscience and psychological research[END_REF]. Fewer symptoms of depression and social inhibition as well as reduced violent behaviors have been observed among children who are stimulated from a young age onwards [START_REF] Walker | Early childhood stimulation benefits adult competence and reduces violent behavior[END_REF]. This type of stimulation in a formal setting may also favor children's cognitive development, with, in the long term, better general knowledge, academic achievement, and higher IQ. This may explain why some studies have shown that center-based childcare is especially beneficial children growing up in disadvantaged families [START_REF] Geoffroy | Closing the gap in academic readiness and achievement: the role of early childcare[END_REF][START_REF] Love | The effectiveness of early head start for 3-year-old children and their parents: lessons for policy and programs[END_REF]. In our study, we did not find evidence of such an effect, and to the contrary, children who did not experience hardship were those who were most likely to benefit from being in childcare. It may be that the universal approach which is applied in French early education programs does not suffice to help children overcome the difficulties that can result from deficits in early life stimulation and caretaking. However, it has to be noted that our sample is not representative of children in France, and additional research using nationally representative data will need to verify this hypothesis. Overall, we observed that at least one year spent in childcare was necessary to observe protective effects, which will also need to be verified using nationally representative data.

Social and economic policies

In 2013, France spent approximately 1.3% of its GDP on provision of childcare and preprimary education. Comparatively, the USA spent around 0.4 %, Norway 1.3% and Iceland 1.8% (OECD average: approximately 0.8%)(OECD, 2016). However, while public policies in the 1980s aimed to provide childcare facilities so that mothers could work, policies launched in the 1990's created the 'allocation parentale d'éducation (APE)', a financial aid for mothers who wish to stop working in order to take care of their children. This benefit is mainly used by women who are employed in low-wage jobs [START_REF] Fagnani | La version française de l'échelle CES-D (Center for Epidemiologic Studies-Depression Scale). Description et traduction de l'échelle d'autoévaluation. [The French version of the CES-D (Center for Epidemiologic Studies-Depression Scale)[END_REF]. Overall, in 2010, formal childcare use was used by less than 20% of families in the lowest income quintile, and 60% of families in the two highest income quintiles [START_REF] Mills | Trajectories of externalizing behavior from age 2 to age 9: relations with gender, temperament, ethnicity, parenting, and rater[END_REF]. Our results, along with other studies showing positive influences of center-based childcare on children's cognitive development, suggest that public policies should aim to give access to high quality centerbased childcare to a higher proportion of children under the age of 3 years [START_REF] Choné | Female labor supply and child care in France[END_REF].

Our study shows that center-based childcare in a country were childcare is strictly regulated, subsidized and of high quality has a positive impact on children's psychological development.

The impact of childminder's care, on the other hand, is more equivocal. This implies that public policies that increase access to high quality center-based childcare are most likely to have a positive role on children's long-term psychological development. Further studies are necessary to find out ways in which boys, as well as children whose mother is depressed or who grow up in low-income families can attend childcare in higher proportions and fully benefit from it. Table 1: Characteristics of children participating in the EDEN mother-child cohort in relation to childcare between 0-3 years (n = 1428(n = , 2003(n = -2011, France), France). % (N) or mean (standard deviation), p-value.

Variables

2.

  Standard procedures are used to analyze the different datasets producing different estimated associations and reflecting the uncertainty associated with the imputed missing values.

  ): a low SDQ score trajectory (50.4%, n = 721) (low symptoms throughout), an intermediate SDQ score trajectory (40.2%, n = 574) and a high SDQ score trajectory (9.3%, n = 133) (high symptoms throughout). The 3 trajectories were somewhat flat and parallel showing that children with a comparatively high level of behavioral or emotional symptoms at 3 years also had high levels at 5.5 and 8 years.Children's mean probabilities of being in each respective trajectory group were 0.78 (SD = 0.14) for the low trajectory, 0.84 (SD = 0.14) for the intermediate trajectory and 0.76 (SD = 0.16) for the high trajectory. We used the low trajectory group as the reference. Mean SDQ scores in the low, intermediate and high trajectory groups were respectively: 6.70 (SD = 2.81), 12.18 (SD = 3.15) and 17.91 (SD = 4.07) at 3.5 years; 5.10 (SD = 2.61), 11.06 (SD = 3.06) and 18.60 (SD = 4.18) at 5.5 years and 5.14 (SD = 2.72), 11.08 (SD = 3.42) and 18.07 (SD = 4.13) at 8 years.

  Health (IRESP: TGIR cohorte santé 2008 program), French Ministry of Health (DGS), French Ministry of Research, Inserm Bone and Joint Diseases National Research (PRO-A) and Human Nutrition National Research Programs, Paris-Sud University, Nestlé, French National Institute for Population Health Surveillance (InVS), French National Institute for Health Education (INPES), the European Union FP7 programmes (FP7/2007-2013, HELIX, ESCAPE, ENRIECO, Medall projects), Diabetes National Research Program (through a collaboration with the French Association of Diabetic Patients (AFD)), French Agency for Environmental Health Safety (now ANSES), Mutuelle Générale de l'Education Nationale (MGEN), French National Agency for Food Security, and the French-speaking association for the study of diabetes and metabolism (ALFEDIAM). The funders had no role in design and conduct of the study; collection, management, analysis, and interpretation of the data; preparation, review, and approval of the manuscript; or the decision to submit the manuscript for publication. Competing interests: All authors declare no support from any organization for the submitted work, no financial relationships with any organizations that might have an interest in the submitted work, no other relationships or activities that could appear to have influenced the submitted work.

N = 367, 25.7%): school

	2. Center-based childcare, CBC (-like structures such as
	crèches, daycare, center-based cribs and day nurseries with professional caregivers.
	3. Informal childcare, IC

(N = 425, 29.8%): parental

	care only (71.3%, n = 303) or
	parental care combined with other non-professional caregivers (e.g. grand-parents,
	aunts and uncles, friends) (28.7%, n = 122) grouped together because of shared
	characteristics.
	If the child attended center-based care at any one point in time, she/he was included in the
	'center-based' group; if the child never attended center-based care but with a childminder at
	any one point in time she/he was classified in the latter group. If the child was neither in
	center-based care nor with a childminder, she/he was allotted to the informal care group. The
	3 groups are mutually exclusive 1 .

Table 2 )

 2 [START_REF] Austin | Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies[END_REF][START_REF] Lanehart | Propensity Score Analysis and Assessment of Propensity Score Approaches Using SAS® Procedures[END_REF]. Multinomial logistic regression

models were used to analyze whether childcare type predicted children's SDQ score trajectory group (intermediate or high, vs. low group).

Table 2 :

 2 Comparison of SDQ total scores at 3, 5.5 and 8 years among children from childminder, center-based childcare and informal childcare, with successively children in informal care and children in center-based childcare as reference (at least 1 year in center-based childcare for comparison of time in childcare).

	Age	Childcare type	Child-minder	Center-based	Informal
						childcare	childcare
	3 years	N		593		337	374
		Mean SDQ score (SD)	9.98 (4.56)	8.94 (4.72)	10.77 (4.94)
		p (compared to IC)		0.0111	<0.0001	-
		p (compared to CBC)		0.0013	-	<0.0001
	5.5 years	N		537		300	346
		Mean SDQ score (SD)	8.97 (5.42)	7.76 (4.97)	9.28 (4.99)
		p (compared to IC)		0.8340	0.0038	-
		p (compared to CBC)		0.0013	-	0.0002
	8 years	N		401		235	238
		Mean SDQ score (SD)	8.95 (5.24)	7.66 (5.08)	9.03 (5.08)
		p (compared to IC)		0.3799	0.0002	-
		p (compared to CBC)		0.0024	-	0.0038
	Age	Time in childcare	CM < 1	CM >= 1	CBC < 1	CBC >= 1	IC
			YR		YR	YR	YR
	3 years	N	151		442	154	183	374
		Mean SDQ score (SD)	10.50 (4.61)	9.81 (4.53)	9.34 (4.95)	8.61 (4.51)	10.77 (4.94)
		p (compared to IC)	0.5432	0.0035	0.0015	<0.0001	-
		p (compared to CBC)	0.0003	0.0040	0.1589	-	<0.0001
	5.5 years	N	141		396	129	171	346
		Mean SDQ score (SD)	9.82 (5.39)	8.66 (5.40)	8.53 (5.52)	7.19 (4.45)	9.28 (4.99)
		p (compared to IC)	0.2999	0.1053	0.1578	<0.0001	-
		p (compared to CBC)	<0.0001	0.0018	0.0263	-	<0.0001
	8 years	N	80		321	98	137	238
		Mean SDQ score (SD)	10.24 (4.82)	8.62 (5.30)	7.89 (5.44)	7.49 (4.82)	9.03 (5.08)
		p (compared to IC)	0.0705	0.351	0.0638	0.0052	.
		p (compared to CBC)	0.0002	0.0310	0.5581	-	0.0052
	CM = childminder; CBC = center-based childcare; IC = informal childcare
	CM < 1 YR: child-minder for less than 1 year; CM >= 1 YR: child-minder for at least 1
	year; CBC < 1 YR: center-based childcare for less than 1 year; CBC >= 1 YR: center-based
	childcare for at least 1 year; IC: informal childcare.

Table 3a :

 3a Association between childcare type (age 0-3 years) and children's behavioral trajectories (age 3-8 years) in the French EDEN study. Bivariate analysis with multinomial logistic regression (Informal care, IC, as reference childcare OR in the left column and centerbased childcare, CBC, as reference in the right) -95% Confidence Interval.

	Unadjusted			IC as reference		CBC as reference
	Childcare type Trajectories	OR	95% CI	p	OR	95% CI	p
	CM	Intermediate	0.81	0.62 -1.04	0.1011	1.58 1.20 -2.08	0.0012
		High	0.85	0.56 -1.30	0.4528	1.92 1.17 -3.16	0.0105
	CBC	Intermediate	0.51	0.38 -0.69	<0.0001	-	-	-
		High	0.44	0.26 -0.76	0.0027	-	-	-
	IC	Intermediate	-	-	-	1.96 1.46 -2.65 <0.0001
		High	-	-	-	2.26 1.33 -3.84	0.0027
	CM = childminder; CBC = center-based childcare; IC = informal childcare

Table 3b :

 3b Association between time spent in childcare (age 0-3 years) and children's behavioral trajectories (age 3-8 years) -low trajectory as reference-in the French EDEN study. Bivariate analysis using multinomial logistic regression (informal childcare as reference group in the left column and center-based childcare for at least 1 year as reference in the left column). OR-95% Confidence Interval.

	Unadjusted		IC as reference		CBC >= 1 YR as reference
	Time in childcare Trajectories OR	95% CI	p	OR	95% CI	p
	CM < 1 YR (n=174) Intermediate	1.14 0.78 -1.66 0.5053 2.45 1.58 -3.80 <0.0001
	High	1.24 0.69 -2.23 0.4772 3.84 1.71 -8.61 0.0011
	CM >= 1 YR (n=462) Intermediate	0.71 0.54 -0.94 0.0167 1.53 1.06 -2.19 0.0216
	High	0.75 0.47 -1.19 0.2277 2.28 1.11 -4.70 0.0247
	CBC < 1 YR (n=167) Intermediate	0.57 0.39 -0.84 0.0043 1.23 0.78 -1.92 0.3715
	High	0.61 0.32 -1.17 0.1376 1.87 0.80 -4.38 0.1518
	CBC >= 1 YR (n=200) Intermediate	0.46 0.32 -0.67 <0.0001	-	-	-
	High	0.32 0.16 -0.66 0.0022	-	-	-
	IC (n=425) Intermediate	-	-	-	2.15 1.50 -3.10 <0.0001
	High	-	-	-	3.10 1.51 -6.37 0.0021
	CM < 1 YR: childminder for less than 1 year; CM >= 1 YR: childminder for at least 1 year; CB < 1 YR:
	center-based childcare for less than 1 year; CB >= 1 YR: center-based childcare for at least 1 year; IC:
	informal childcare.

Table 4a :

 4a Association between childcare type (age 0-3 years) and children's behavioral trajectories-low trajectory as reference-(age 3-8 years) in the French EDEN study. IPWsadjusted multinomial logistic regression 2 (informal childcare-IC-as reference group in the left table and center-based childcare-CBC-as reference in the right column). OR-95% Confidence Interval.

	Adjusted			IC as reference			CBC as reference
	Childcare type Trajectories	OR	95% CI	p	OR	95% CI	p
	CM (n=636) Intermediate	0.84	0.59 -1.19	0.3204	1.21	0.87 -1.70	0.2568
		High	1.52	0.89 -2.61	0.1268	1.44	0.46 -4.54 0.5011
	CBC (n=367) Intermediate	0.69	0.46 -1.04	0.0756	-	-	-
		High	1.05	0.31 -3.54	0.9294	-	-	-
	IC (n=425)	Intermediate -	-	-	1.45	0.96 -2.18 0.0756
		High	-	-	-	0.95	0.28 -3.20 0.9294

CM = child-minder; CBC = center-based childcare; IC = informal childcare

Table 4b :

 4b Association between time spent in childcare (age 0-3 years) and children's behavioral trajectories (age 3-8 years) -low trajectory as reference -in the French EDEN study. IPW-adjusted multinomial logistic regression 3 (using informal childcare as reference group in the left column and center-based childcare for at least one year in the right column). OR-95% Confidence Interval.

	Adjusted			IC as reference		CBC >= 1 YR as reference
	Time in childcare Trajectories	OR	95% CI	p	OR	95% CI	p
	CM < 1 YR (n=174) Intermediate	1.04	0.71 -1.53 0.8394	1.55	1.04-2.33 0.0328
		High	2.53	1.26 -5.07 0.0098	5.96	2.93-12.15 <0.0001
	CM >= 1 YR (n=462) Intermediate	0.77	0.52 -1.13 0.1871	1.15	0.76-1.73 0.4987
		High	1.20	0.62 -2.32 0.5817	2.83	1.38-5.82 0.0046
	CBC < 1 YR (n=167) Intermediate	0.70	0.48-1.04 0.0809	1.05	0.67-1.65 0.8223
		High	1.06	0.47 -2.36 0.8911	2.49	1.07-2.16 0.0337
	CBC >= 1 YR	Intermediate	0.67	0.46 -0.97 0.0330	-	-	-
	(n=200)	High	0.42	0.19-0.93 0.0331	-	-	-
	IC (n=425) Intermediate	-	-	-	1.49	1.03-2.16 0.0330
		High	-	-	-	2.36	1.07-5.19 0.0331
	CM < 1 YR: child-minder for less than 1 year; CM >= 1 YR: child-minder for at least 1 year; CB < 1 YR:
	center-based childcare for less than 1 year; CB >= 1 YR: center-based childcare for at least 1 year; IC:
	informal childcare.

Table 5 :

 5 Childcare type/time spent in childcare and children's emotional and behavioral symptoms in the French EDEN study: stratified analyses 4 by sex 5 ; mother's number of years of education; mother's depression status and family's financial difficulties status. Successive comparison to informal care (IC) and center-based care (CBC); OR of having a trajectory of high symptom levels -95% CI.

	CHILDCARE		IC as reference		CBC as reference			IC as reference	CBC as reference
	TYPE	OR 95% CI	p	OR 95% CI	p	OR 95% CI	p	OR 95% CI	p
			GIRLS (N = 685)				BOYS (N = 743)
	CM	0.54 0.35-0.82 0.0039 1.01 0.49-2.08 0.9718 1.59 1.03-2.46 0.0351 1.54 0.98-2.42 0.0583
	CBC	0.53 0.26-1.09 0.0828	-	-	-	1.03 0.60-1.70 0.9019 -	-	-
	IC	-	-	-	1.89 0.91-3.92 0.0828	-	-	-	0.97 0.56-1.66 0.9019
			< BAC + 2 YEARS (N = 588)			>= BAC + 2 YEARS (N = 840)
	CM	1.08 0.67-1.75 0.7505 1.38 0.56-3.40 0.4643 0.93 0.63-1.37 0.7057 1.21 0.83-1.76 0.3156
	CBC	0.78 0.29-2.15 0.6134	-	-	-	0.77 0.52-1.13 0.1744 -	-	-
	IC	-	-	-	1.28 0.47-3.50 0.6134	-	-	-	1.31 0.89-1.92 0.1744
			NO DEPRESSION (N = 1085)			DEPRESSION (N = 343)
	CM	0.89 0.64-1.24 0.5001 1.21 0.88-1.67 0.2353	1.1 0.47-2.60 0.8251 1.48 0.72-3.02 0.2851
	CBC	0.74 0.52-1.04 0.0787	-	-	-	0.74 0.31-1.80 0.5028 -	-	-
	IC				1.36 0.96-1.92 0.0787	-	-	-	1.34 0.55-3.25 0.5028
			NO FINANCIAL DIFFICULTIES (N = 1159)		FINANCIAL DIFFICULTIES (N = 269)
	CM	1.09 0.78-1.51 0.6134 1.65 1.16-2.36 0.0059 1.03 0.29-3.65 0.9585 0.52 0.18-1.47 0.2162
	CBC	0.66 0.44-0.98 0.0408	-	-	-	1.99 0.40-9.97 0.3759 -	-	-
	IC	-	-	-	1.52 1.02-2.27 0.0408	-	-	-	0.5 0.10-2.52 0.3759
	TIME									CBC>= 1 year as
	SPENT		IC as reference		CBC>= 1 year as reference		IC as reference	reference
				GIRLS					BOYS
	CM < 1 YR 0.63 0.38-1.06 0.0847 1.86 1.08-3.18 0.0244 2.36 1.42-3.90 0.0009 2.33 1.34-4.06 0.0029
	CM >= 1 YR 0.42 0.25-0.72 0.0014 1.24 0.71-2.15 0.4468 1.44 0.89-2.35 0.1403 1.43 0.83-2.45 0.1953
										1-
	CBC < 1 YR 0.53 0.32-0.87 0.0128 1.55 0.88-2.73 0.1301 1.01 0.58-1.76 0.9645	00 0.54-1.87 0.9933
	CBC >= 1 YR 0.34 0.20-0.59 0.0001	-	-	-	1.01 0.63-1.62 0.9676 -	-	-
	IC	-	-	-	2.93 1.71-5.04 0.0001	-	-	-	0.99 0.62-1.59 0

.9676 < BAC + 2 YEARS >= BAC + 2 YEARS CM < 1 YR 1.73 0

  .95-3.18 0

.0750 2.34 1

  .22-4.48 0

.0109 1.04 0

  .65-1.66 0

.8693 1.81 1

  .10-2.98

0.0205 CM >= 1 YR 0.89 0

  .50-1.58 0

.6834 1.19 0

  .66-2.18 0

.5619 0.81 0

  .51-1.29 0

.3760 1.41

	Supplementary
										0.85-2.33 0.1803
	CBC < 1 YR 0.59 0.30-1.16 0.1232 0.79 0.41-1.55 0.4939 0.94 0.59-1.48 0.7745 1.62 0.95-2.76 0.0732
	CBC >= 1 YR 0.74 0.42-1.33 0.3143	-	-	-	0.58 0.36-0.93 0.0252 -	-	-
	IC	-	-	-	1.35 0.75-2.41 0.3143	-	-	-	1.74 1.07-2.81 0.0252
				NO DEPRESSION					DEPRESSION
	CM < 1 YR 1.12 0.74-1.68 0.5933 2-00 1.28-3.14 0.0025 1.45 0.52-4.08 0.4720 1.33 0.52-3.41 0.5579
	CM >= 1 YR 0.81 0.53-1.24 0.3320 1.45 0.94-2.24 0.0918 0.78 0.28-2.15 0.6269 0.71 0.29-1.74 0.4576
	CBC < 1 YR 0.77 0.52-1.13 0.1821 1.37 0.84-2.26 0.2060 0.47 0.16-1.37 0.1614 0.43 0.17-1.09 0.0760

Table 1 :

 1 Childcare type and other covariates (ages 0 to 3 years) and children's behavioral trajectories (ages 3.5-8 years) in the French EDEN mother-child cohort. Bivariate multinomial logistic regression analyses (informal care serving as the reference group). OR-95% Confidence Interval.

	Variables	Low (n = 721)	Intermediate (n = 574)		High (n = 133)
		OR		OR 95% IC	p	OR 95% IC	p
	Type of childcare
	Informal care	1.00	ref	1.00	ref		1.00	ref
	Child-minder	1.00	ref	0.81 0.62-1.04 0.1011 0.85 0.56-1.30 0.4528
	Center-based care	1.00	ref	0.51 0.38-0.69 <0.0001 0.44 0.26-0.76 0.0027
	Child's characteristics
	Center of birth: Poitiers with Nancy as reference	1.00	ref	1.24 1.00-1.55 0.0551 1.17 0.81-1.69 0.4125
	Gender: Male	1.00	ref	1.22 0.98-1.52 0.0735 2.34 1.58-3.46 <0.0001
	Maternal age at delivery (years)	1.00	ref	1.36 1.13-1.65 0.0015 1.74 1.26-2.40 0.0008
	Low birth weight: yes	1.00	ref	1.22 0.71-2.10 0.4678 2.68 1.35-5.32 0.0048
	Prematurity: yes	1.00	ref	1.23 0.75-2.02 0.4050 2.00 1.01-3.98 0.0470
	Duration of breastfeeding (months)	1.00	ref	1.46 1.20-1.77 0.0001 1.49 1.07-2.06 0.0168
	Only child at the time of birth	1.00	ref	1.29 1.02-1.64 0.0350 1.47 1.00-2.19 0.0524
	Age of entry in school (years)	1.00	ref	1.32 1.06-1.65 0.0133 1.24 0.86-1.80 0.2570
	Parents' employment status
	Mother working at 24 weeks of pregnancy	1.00	ref	0.71 0.53-0.94 0.0178 0.49 0.32-0.75 0.0012
	Mother working at 4 months after birth	1.00	ref	1.01 0.80-1.27 0.9209 0.89 0.60-1.31 0.5458
	Mother working at 8 months after birth	1.00	ref	0.73 0.57-0.93 0.0114 0.54 0.37-0.81 0.0030
	Mother working at 12 months	1.00	ref	0.67 0.50-0.90 0.0077 0.38 0.25-0.58 <0.0001
	Mother working at 24 months	1.00	ref	0.70 0.52-0.95 0.0213 0.48 0.31-0.76 0.0015
	Father working at 24 weeks of pregnancy	1.00	ref	1.26 0.65-1.61 0.9037 1.39 0.34-1.22 0.1805
	Father working at 12 months	1.00	ref	0.57 0.37-0.88 0.0114 0.57 0.29-1.11 0.1008
	Father working at 24 months	1.00	ref	0.66 0.44-1.00 0.0513 0.56 0.30-1.06 0.0757
	Single parent status
	At 24 weeks of pregnancy	1.00	ref	1.52 1.01-2.27 0.0427 1.78 0.97-3.29 0.0641
	At 4 months after birth	1.00	ref	1.40 0.74-2.62 0.3002 1.44 0.53-3.91 0.4798
	At 8 months after birth	1.00	ref	1.45 0.80-2.64 0.2181 1.58 0.62-3.98 0.3365
	At 12 months after birth	1.00	ref	1.14 0.62-2.11 0.6730 0.98 0.33-2.89 0.9718
	At 24 months after birth	1.00	ref	1.12 0.64-1.95 0.6976 1.65 0.74-3.68 0.2199
	Parents' education level ¶
	Mother with higher education	1.00	ref	0.54 0.43-0.67 <0.0001 0.47 0.32-0.68 <0.0001
	Father with higher education	1.00	ref	0.60 0.49-0.74 <0.0001 0.49 0.33-0.72 0.0002
	Financial difficulties
	Low family income (< 1500 E) during pregnancy 1.00	ref	1.68 0.98-2.88 0.0599 2.40 1.10-5.23 0.0274
	>=1 financial difficulty during pregnancy	1.00	ref	1.42 1.14-1.76 0.0020 1.91 1.31-2.79 0.0008
	Low family income (< 1500 E) during 1st year	1.00	ref	1.83 1.43-2.35 <0.0001 2.45 1.66-3.62 <0.0001
	>=1 financial difficulty felt in 1st year	1.00	ref	1.58 1.14-2.19 0.0057 2.40 1.50-3.84 0.0003
	Low family income (< 1500 E) during 2nd year	1.00	ref	1.41 1.13-1.76 0.0027 2.60 1.72-3.93 <0.0001

2,2 Control variables include among others: center of study, children's characteristics (sex, prematurity, breastfeeding), parents' education level, financial status, mothers' psychological status, children's activities with parents, parental childhood problems, and mothers' substance abuse.

Control variables include among others: center of study, children's characteristics (sex, prematurity, breastfeeding), parents' education level, financial status, mothers' psychological status, children's activities with parents, parental childhood problems, and mothers' substance abuse.

For each stratified analysis a model was constructed excluding the variable of stratification. For example, for analyses stratified by sex, one model was constructed for boys and one for girls, both containing the same control variables but did not contain the variable for sex.

Acknowledgements

Funding: The EDEN cohort study was supported by Foundation for Medical Research (FRM), National Agency for Research (ANR), National Institute for Research in Public

Ethical approval: All cohort participants signed a written informed consent at enrolment.

Further details on the study can be found on the EDEN website: http://eden.vjf.inserm.fr/index.php/en/


Keywords: strain, ultrafast, ultrasound, landmark, abdomen

    Estimation of 2D strain in abdominal organs during an impact based on ultrafast ultrasound images: a physical landmark-based approach

Estimation of 2D strain in abdominal organs during an impact based on ultrafast ultrasound images: a physical landmark-based approach

Introduction

During an automotive accident, the abdomen can be subjected to large deformations leading to internal organ injuries. Although the external response of the abdomen has been studied extensively, internal organ kinematics during an impact is largely unknown. This makes the validation of the internal response of human abdominal models difficult. Typical impacts duration last a few tens of milliseconds and standard imaging tools such as MRI are too slow for such durations. For that purpose, besides X-ray based methods [START_REF] Howes | Kinematics of the thoracoabdominal contents under various loading scenarios[END_REF], methods based on ultrafast ultrasound imaging have been recently proposed. Two recent experimental studies provided results regarding the internal response of isolated porcine kidneys during an impact [START_REF] Helfenstein-Didier | Observation of the internal response of the kidney during compressive loading using ultrafast ultrasonography[END_REF] as well as the internal kinematics of in-situ organs during an impact [START_REF] Beillas | A new method to assess the deformations of internal organs of the abdomen during impact[END_REF]. However image processing is currently limited in these studies. Natural targets were tracked semi-automatically because of the poor quality of the images and, in the case of the kidney, uniaxial strains were computed between targets. This is a limitation as the manual selection of targets can lead to an operator bias and the tracking of numerous targets would be time consuming. Moreover through-plane motions may occur and make the natural targets appear and disappear. Many processing methods based on deformable image registration have been developed in particular to study cardiac motion from ultrasound images [START_REF] Sotiras | Deformable Medical Image Registration: A Survey[END_REF]. Such methods must respect properties of the field computed such as invertibility, topology preservation or diffeomorphicity. For that purpose, the deformation shape can be inspired from a physical model such as a finite element model [START_REF] Weiss | Strain measurement using deformable image registration[END_REF] or derived by interpolation or regression. Most of the time, they are based either on Radial Basis Functions (RBFs) or on free-form deformation (FFD) combined with spatial conditions to force the displacement field to be locally smooth, as well as temporal conditions in order to ensure the temporal continuity [START_REF] Ledesma-Carbayo | Cardiac motion analysis from ultrasound sequences using non-rigid registration[END_REF]. Such methods have not however been applied to ultrafast imaging data. The aim of this study is to develop a method that estimates 2D strains in organs subjected to impact based on ultrafast ultrasound images.

Methods

Process

This method is based on an automatic tracking algorithm developed for very noisy images using geometric local descriptor matching. Landmarks are first detected on each image by extracting local minima in a Difference of Gaussian (DoG) space. Geometric local descriptors are defined from each landmark and its closest neighbors. Correspondences from an image to another are established first for each descriptor. Then to ensure that correspondences are physically plausible for a descriptor, it is checked if an affine transformation between images can be found using a Random Sample Consensus (RANSAC) process [START_REF] Preibisch | Software for bead-based registration of selective plane illumination microscopy data[END_REF]. The remaining landmarks whose kinematic parameters are not compatible with experimental boundary conditions, impact velocity or possible acceleration fields are rejected. A smooth velocity field is then derived by spline interpolation between subsequent images. Displacement fields are obtained for each pixel by integration along the time axis. Strain fields are finally derived from Green-Lagrange equations.

Validation methodology

The algorithm was already subjected to two validation steps. 1) Ultrafast ultrasound images showing the interior of porcine kidneys were processed using the proposed method. The computed displacement field was compared with landmark trajectories derived from expert tracking. 2) Synthetic images were generated by applying a known inhomogeneous time-varying strain field. The proposed method was then applied to the stack of synthetic images in order to check its ability to predict the applied strain.

Results and discussion

For the two validation steps: 1) Vertical displacements computed using the proposed method and points manually tracked in an ultrasound sequence are compared in Figure 1. The points were chosen because they are located on the boundary of anatomic structures (cortex/pelvis). In that case, the displacements predicted by the proposed method were found to be consistent with those obtained by expert tracking within less than 10%.

2) The proposed method predicted the inhomogeneity of the strain field applied to the image stack (Figure 2). It was found that a random deformation field up to 30% applied to an original image can be predicted through the proposed method, although with some artefacts near the edge of the image. Although these validation steps are necessary, they are not sufficient for example to show the robustness of the algorithm towards the ultrafast ultrasound type noise and limited through-plane motions. An experimental validation method was therefore developed. A random pattern is painted on the two surfaces of a thin sample of bovine muscle. The sample, partially immerged and laying on a transparent plate, is subjected to manual compression along its long axis. Two stereo camera rigs compute the external strains based on the paint patterns (stereo-correlation software, Vic-3D®).

Simultaneously, an ultrasound probe is used to image the mid-plane between the external surfaces. The proposed method is then applied to the ultrasound images and predicted strains are compared to those obtained by stereo-correlation. Preliminary experiments have been conducted and the assessment of the proposed method is still ongoing.

Conclusions

This study describes a method to estimate 2D strains of abdominal organs subjected to an impact based on ultrafast ultrasound images. The method is based on an automatic tracking algorithm combined with additional spatial and temporal conditions. The preliminary steps of numerical validation have shown the interest of the approach, and could already allow expanding upon the current analyses. An experimental validation is ongoing to evaluate the robustness of the method to noise in the ultrafast ultrasound images and to assess its limits in terms of strain amplitude.

Figure 1 :

 1 Figure 1: Left: vertical displacement of points A (blue), B (red) and C (green) expertly tracked (solid) and assessed (dotted). Right: positions of the points over an ultrasound image (interior of porcine kidney compressed at 0.3m/s, 2000 images/sec)

Figure 2 :

 2 Figure 2: Original (a) and deformed (b) image. Known (c) and assessed (e) vertical displacement field. Known (d) and assessed (f) vertical strain field.

  Université de Lyon, F-69622, Lyon, France b. Université Claude Bernard Lyon 1, Villeurbanne, France c. IFSTTAR, UMR_T9406, LBMC Laboratoire de Biomécanique et Mécanique des Chocs, F69675, Bron, France -anicet.le-ruyet@ifsttar.fr ; philippe.beillas@ifsttar.fr

	Enfin, dans le but d'évaluer la
	robustesse de la méthode notamment en présence de bruit échographique, un protocole de validation
	expérimentale a été développé. Il permet de comparer les déformations prédites par la méthode à
	l'intérieur d'un échantillon fin à celles mesurées par stéréo-corrélation sur la surface externe. Les
	essais sont en cours.

A. LE RUYET abc , P.BEILLAS abc a.

Résumé

Dans un contexte de validation de modèles humains en situation d'accident, l'échographie ultrarapide rend possible l'observation du comportement interne d'organes abdominaux, avec toutefois une qualité d'image qui rend difficile l'utilisation de techniques de post-traitement classiques. Cette étude présente une méthode d'estimation de déformations 2D d'organes abdominaux imagés par échographie ultrarapide. Dans cette méthode, des marqueurs sont détectés sur chaque image du film échographique à l'aide d'un logiciel développé initialement pour le suivi automatique de cellules biologiques puis, via des étapes de filtrage, d'interpolation et d'intégration, des champs de déformations sont calculés. Pour l'instant, les prédictions ont été vérifiées (1) en comparant les déplacements estimés à ceux issus d'un suivi expert et (2) en appliquant la méthode à une image déformée numériquement un champ de déformation inhomogène connu.

Acknowledgements

The Aixplorer ultrasonic scanner was partly funded by the Region Rhône-Alpes.


L Stainier

Consistent incremental approximation of dissipation pseudo-potentials in the variational formulation of thermo-mechanical constitutive updates

Keywords:

Consistent incremental approximation of dissipation pseudo-potentials in the variational formulation of thermo-mechanical constitutive updates

Introduction

Thermo-mechanical coupling effects are of importance in a number of engineering problems: e.g. manufacturing processes such as forging or welding, or when considering the dynamic behavior of materials such as metals or polymers. Coupling between mechanical and thermal responses can occur through different mechanisms: temperature dependence of elastic and visco-plastic properties, heating from dissipative micromechanisms, phase change,. . . These effects can be introduced in constitutive models and balance equations (momentum, energy), but this breaks the mathematical symmetry of the problem. As a result, numerical models derived by standard approaches such as finite elements [START_REF] Oden | Finite Elements of Nonlinear Continua[END_REF][START_REF] Argyris | On the natural formulation and analysis of large deformation coupled thermo-mechanical problems[END_REF] are thus non-symmetric as well, which incurs higher computational cost and possibly some loss in robustness of solvers. In order to avoid these difficulties, partitioned or staggered approaches have been proposed [START_REF] Farhat | An unconditionally stable staggered algorithm for transient finite element analysis of coupled thermoelastic problems[END_REF]Armero andSimo, 1992,1993;[START_REF] Simo | Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation[END_REF], which retain the symmetry properties of isolated mechanical and thermal problems.

An energy-based variational formulation of general coupled thermo-mechanical boundary-value problems, including nonlinear and dissipative behaviors, has recently been proposed by [START_REF] Yang | A variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids[END_REF]. In that formulation, incremental balance equations (conservation of momentum and energy) are derived as Euler-Lagrange equations of a global optimization problem. This result in a fully coupled, yet symmetric, generic framework for coupled thermomechanical problems. This formulation has for example been successfully applied to finite thermo-visco-plasticity [START_REF] Stainier | Study and validation of a variational theory of thermomechanical coupling in finite viscoplasticity[END_REF]. We refer the reader to these two papers, and references therein, for a detailed description of the variational formulation and a general discussion about solving coupled thermo-mechanical boundary value problems and integrating associated constitutive equations. We will focus here on a specific aspect of the variational formulation in its incremental form.

The variational formulation is based on a thermodynamic approach to constitutive modeling, closely following the Generalized Standard Materials approach of [START_REF] Halphen | Sur les matériaux standard généralisés[END_REF]. In this thermodynamic approach, the constitutive behavior is described through potentials (state functions) and dissipation pseudo-potentials. These pseudo-potentials are not state functions, but are introduced in order to guarantee, through their convexity, a priori satisfaction of the second principle of thermodynamics (non-negative total dissipation). The functional which is optimized by the variational formulation is incremental by nature, and is obtained by integrating over a time step some combination of potentials (actually rate thereof) and pseudo-potentials, yielding a quantity homologous to an energy. The state functions can be integrated exactly, but not the dissipation pseudo-potentials, resulting in some level of error, which is inherent to most incremental approximations. This approximation error is acceptable as long as the formulation remains consistent with the continuous rate equations of the original constitutive model as the time step (or more generally load increment) tends to zero.

In the case of thermo-mechanical constitutive models including thermal softening effects on the yield or viscous stress, for exam-ple, deriving a consistent approximation to associated dissipation pseudo-potentials proves to be not as trivial as initially expected. In this paper, we illustrate this problem and then propose a consistent approximate expression for such incremental dissipation pseudo-potentials. The results obtained are illustrated on a simple example.

Variational constitutive updates

Simplified framework

Let us consider a simple elasto-visco-plastic model, without any hardening, and with a Perzyna-type rate-dependency. For the sake of conciseness, we will further limit ourselves to the unidimensional case, with linearized kinematics and assuming monotonous loading. We first introduce a Helmholtz free energy density

W (ε, Â, ε p ) = W e (ε -ε p , Â) + W h (Â) (1)
where ε is the total strain, Â the relative temperature (the absolute temperature is given by T = T 0 + Â, where T 0 is a reference temperature), ε p the plastic strain. The elastic and thermal stored energy densities W e and W h are typically taken as quadratic expressions of their (first) argument, but the actual choice has no effect on the subsequent discussion. Note that, for the sake of clarity and conciseness, we have made the simplifying assumption that there is no plastic stored energy. Accounting for such mechanisms through an additional plastic stored energy density W p (ε p , Â) does not modify the following discussion. Thermodynamic forces conjugate to strain and temperature are respectively the stress and the entropy:

≡ W ,ε (ε, Â, ε p ) (2a) Á ≡ -W , (ε, Â, ε p ) (2b)
The set of constitutive equations is closed by introducing a kinetic relation for the internal variable ε p through the convex dissipation pseudo-potential * ( εp ; Â):

W ,ε p (ε, Â, ε p ) + * , εp ( εp ; Â) = 0 (3)
The use of a semi-column in the expression of * denotes a potential parametric dependence on temperature. In order to fix ideas, we will assume that the dissipation pseudo-potential consists of a rate-independent and a rate-dependent part:

* ( εp ; Â) = y (Â) εp + m m+1 v (Â) ε0 εp ε0 (1/m)+1 (4)
where y (Â) and v (Â) are respectively temperature-dependent yield and viscous stresses, while ε0 is a reference strain-rate and m ∈ [1, + ∞ [ a rate-dependency exponent.

Variational update

Consider a time increment [t n , t n+1 ], for which the initial state {ε n ,  n , ε p n } is known as well as the final values ε n+1 and  n+1 . In order to update the plastic strain ε p n+1 , the following minimization problem can be solved:

W n (ε n+1 , Â n+1 ; ε n , Â n , ε p n ) ≡ inf ε p n+1 W (ε n+1 , Â n+1 , ε p n+1 ) -W (ε n , Â n , ε p n ) +Á n (Â n+1 -Â n ) + t n+1 tn * T n+1 T n ε p t ; Â(t) dt (5)
where

Á n = -W , (ε n ,  n , ε p n ),
W ,ε p (ε n+1 , Â n+1 , ε p n+1 ) + 1 t t n+1 tn * , εp T n+1 T n ε p t ; Â(t) dt = 0 (6)
If there is no explicit dependence of dissipation pseudo-potential * to temperature, (6) reduces to a consistent incremental approximation to (3):

W ,ε p (ε n+1 , Â n+1 , ε p n+1 ) + T n+1 T n * , εp T n+1 T n ε p t = 0 (7)
In the general case the second term in the left-hand side is replaced by an average over the time step. Consistency with (3) is also clearly verified in that case.

Adiabatic heat equation

A significant advantage of the variational formulation is that the incremental energy resulting from the minimization ( 5) acts itself as a potential for total strain and temperature:

dW n dε n+1 (ε n+1 ,  n+1 ; ε n ,  n , ε p n ) = n+1 (8a) dW n d n+1 (ε n+1 ,  n+1 ; ε n ,  n , ε p n ) = -Á n+1 + Á n + t d d n+1 * T n+1 T n ε p t ; Â(t) t (8b)
where n+1 and Á n+1 have to be understood as evaluations of ( 2a) and (2b) at time t n+1 , while the brackets denote an average over the time step. Using (8b), the local adiabatic heat equation can then be written, in incremental form, as

dW n d n+1 (ε n+1 ,  n+1 ; ε n ,  n , ε p n ) = 0 (9)
In order to see this, consider first the case where there is no explicit temperature dependence of *. Noting that = -W ,ε p (ε, Â, ε p ), relation ( 9) then indeed yields

-Á + 1 T n+1 n+1 ε p = 0 (10)
which is clearly consistent with the continuous adiabatic heat equation (in entropy form) T Á = εp . In the general case, it is not always possible to obtain a closed-form expression for the average of the dissipation pseudo-potential appearing in (8b). It is then necessary to resort to some incremental approximation, which is required to lead to a consistent incremental heat equation.

Consistent incremental approximation of dissipation

The most intuitive approximation to the average dissipation pseudo-potential may be to take a fully implicit approach:

* T n+1 T n ε p t ; Â(t) t ≈ * T n+1 T n ε p t ; Â n+1 (11)
Considering the specific expression given in (4), the incremental adiabatic heat Eq. ( 9) yields

Á = ε p T n ( y (Â n+1 ) + y (Â n+1 )T n+1 ) + v (Â n+1 ) + m m+1 (Â n+1 )T n+1 T n+1 T n ε p t ε0 1/m (12)
This incremental expression is not consistent with the continuous heat equation, because of the terms in x (Â n+1 )T n+1 . The problem is not solved by taking a generalized mid-point rule (except maybe for a fully explicit approximation, which is too restrictive). An alternative expression thus has to be found.

Let us now consider the following approximation:

* T n+1 T n ε p t ; Â(t) t ≈ T n T n+1 * T n+1 T n ε p t ; Â n + T T n+1 * T n+1 T n ε p t ; Â n+˛ (13)
where ˛∈ [0, 1] is an algorithmic parameter. The stationarity equation of the minimization problem then writes:

W ,ε p (ε n+1 , Â n+1 , ε p n+1 ) + * , εp T n+1 T n ε p t ; Â n + T T n * , εp T n+1 T n ε p t ; Â n+˛ = 0 (14)
which is consistent with (3) as t → 0. The incremental adiabatic heat Eq. ( 9) yields

Á = 1 T n+1 n+1 ε p + t T n T n+1 * T n+1 T n ε p t ;  n+˛ - * T n+1 T n ε p t ;  n + ˛ T * , T n+1 T n ε p t ;  n+˛ (15)
where we have used ( 14). It is easily checked that this expression is consistent with the continuous adiabatic heat equation when t → 0.

Example

In order to illustrate the above developments, we will consider an uniaxial tension test, using tridimensional elastovisco-plasticity model accounting for finite strains. This model is

W e (C e , T) = K 0 2 (tr[ε e ]) 2 + G 0 (dev[ε e ]) 2 - 3 2 ˇ(T -T 0 )K 0 log(det[C e ]) (17a) W h (T ) = 0 c 0 T -T log T T 0 (17b)
with the natural elastic strain defined as ε e = 1 2 log(C e ). Values for material parameters K 0 , G 0 , ˇ, 0 c 0 are given in Table 1. The dissipation pseudo-potential is given by expression (4) with εp =

y (Â) = y0 (1 -ω y Â) (18a) v (Â) = v0 (1 -ω v Â) (18b)
Reference temperature is the initial temperature T 0 = 293 K.

A series of virtual adiabatic tensile tests were performed at strain rates of ε = 10 -1 s -1 (slow) and ε = 10 2 s -1 (fast), varying algorithmic parameters such as ˛and the time step t. Fig. 1 illustrates stress-strain curves obtained using very small time steps and relatively large time steps. Fig. 1 also illustrates associated temperature increase under adiabatic conditions.

Considering the solutions obtained with very small time steps (500,000 increments for a relative elongation of 100%) as the refer-

K0 [GPa] G0 [GPa] ˇ[K -1 ] y0 [MPa] ωy [K -1 ]
58.333 26.923 23.8 × 10 -6 70.0 0.002

0 c0 [J m -3 K -1 ] ε0 [s -1 ] m v0 [MPa] ωv [K -1 ]
2.43 × 10 6 0.1 10.0 100.0 0.002 ence (convergence to 7 significative digits was observed), we have computed errors on final values of stress and temperature for various time step sizes, in order to analyze the influence of algorithmic parameter ˛on convergence behavior. Fig. 2 shows the evolution of the relative errors on stress and temperature for slow and fast loading rates, in semi-logarithmic plots, while Fig. 3 shows the evolution of the same relative errors, but this time in absolute value, on fully logarithmic plots. From Fig. 2, it can be seen that the relative error on temperature is typically one order of magnitude smaller than the error on stress. It also clearly appears that the error on stress monotonically converges to zero as the number of increments increases. In the case of the error on temperature, the situation is less clear: when the algorithmic parameter ˛= 1, a non-monotonous convergence is observed. Due to the highly non-linear nature of the thermo-mechanical coupling, no clear explanation for this behavior could be identified. Logarithmic plots in Fig. 3 show linear convergence for all quantities, independently of the choice of algorithmic parameter. The apparent super-convergence observed on temperature for a fast loading rate and ˛= 1 is related to the non- monotonous convergence and the change of sign of the error. Finally, the effect of algorithmic parameter ˛on the precision can be analyzed. As can be seen from Figs. 3 and4, this parameter shows a much less significant effect on stress than on temperature. This is specially visible for the case of fast loading rates (Fig. 3b). The error on stress slightly increases with increasing ˛, while the error on temperature decreases with ˛and is thus minimal for ˛= 1. In all cases, the dependence is very close to linear. From these results, it seems that the choice of a fully implicit approach for the temperature-dependence of visco-plastic parameters (˛= 1) could be recommended for obtaining the best precision on temperature without degrading too much the quality of stress values.

Conclusion

In this paper, we have shown how to construct a consistent incremental approximation of dissipation pseudo-potentials for the energy-based variational formulation of coupled thermo-viscoplasticity proposed in [START_REF] Yang | A variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids[END_REF] and [START_REF] Stainier | Study and validation of a variational theory of thermomechanical coupling in finite viscoplasticity[END_REF]. We have seen that the most intuitive approximation did not lead to fully consistent results in the case of temperaturedependent yield and/or viscous stresses, and that an alternative approximation was thus needed. We have proposed such an approximation, ensuring fully consistent results, and have shown that it exhibits linear asymptotic convergence (with respect to time increment) of errors on stress and temperature. Such a linear convergence behavior was expected, since the variational incremental approach is globally an implicit backward-Euler scheme. The proposed consistent incremental approximation is parameterized by scalar coefficient ˛, for which an optimal value (˛= 1) could be identified. This value indeed minimizes the error on the adiabatic temperature variation, while not significantly impacting associated values of stress. Due to the highly non-linear nature of thermomechanical coupling effects considered here, no real explanation could be obtained as for why this is the case (in more classical contexts, a mid-point rule often proves to be the most precise). A lead would be that this is probably linked to the explicit-implicit nature of factor T n+1 /T n which enters the formulation, and which is actually key to ensure its variational nature.

The example considered here was limited to a simple uniaxial tensile test, under adiabatic conditions. But the proposed consistent incremental approximation remains valid and applicable to more complex boundary-value problems, including conduction and other heat exchange mechanisms such as convection, within the variational framework described in [START_REF] Stainier | A variational finite element approach for thermo-mechanical coupling in computational structural mechanics[END_REF]. And although we have considered in this paper the case of thermovisco-plasticity, the incremental approximation is also valid for other dissipative models, such as thermo-visco-elasticity, for example by extending the isothermal variational approach proposed in [START_REF] Fancello | A variational formulation of constitutive models and updates in non-linear finite viscoelasticity[END_REF].

Finally, note that an alternative incremental variational formulation has recently been derived by [START_REF] Canadija | On the thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening by means of incremental energy minimization[END_REF], starting from the continuous formulation proposed in [START_REF] Yang | A variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids[END_REF]. In a direct parallel to the continuous formulation, their approach uses an internal temperature field which is determined independently of the external temperature field T. Factors T n+1 /T n are then replaced by T n+1 / n+1 , and n+1 is also used as temperature parameter in the dissipation pseudo-potential. Consistency is then trivially verified. On the other hand, this formulation requires to use the internal energy instead of the free energy, and establishing analytical expressions for internal energy is significantly more complex in general cases than for free energy. The consistent incremental approach described here thus keeps all of its interest in such cases.

  and ε p = ε p n+1ε p n . The stationarity (or Euler-Lagrange) equation corresponding to minimization principle (5) writes:

Fig. 1 .

 1 Fig.1. Stress-strain (solid line) and temperature-strain (dashed line) curves under uniaxial tension (lines: 500,000 steps, symbols: 10 steps).

Fig. 2 .

 2 Fig. 2. Evolution of relative errors on final values of stress and temperature.

Fig. 3 .

 3 Fig. 3. Evolution of relative errors (in absolute value) on final values of stress and temperature.

23

  D p • D p , where D p = sym Ḟp F p -1 . The yield stress and viscous stress vary linearly with temperature:

Fig. 4 .

 4 Fig. 4. Evolution of relative errors on final values of stress and temperature (obtained with 100 increments).

Table 1

 1 Material parameters.