Pierre-Louis Guillot

Sebastien Destercke

Preference Elicitation with Uncertainty: Extending Regret Based Methods with Belief Functions

Keywords: Belief Functions, Preference Elicitation, Multicriteria Decision

Preference elicitation is a key element of any multi-criteria decision analysis (MCDA) problem, and more generally of individual user preference learning. Existing efficient elicitation procedures in the literature mostly use either robust or Bayesian approaches. In this paper, we are interested in extending the former ones by allowing the user to express uncertainty in addition of her preferential information and by modelling it through belief functions. We show that doing this, we preserve the strong guarantees of robust approaches, while overcoming some of their drawbacks. In particular, our approach allows the user to contradict herself, therefore allowing us to detect inconsistencies or ill-chosen model, something that is impossible with more classical robust methods.

Introduction

Preference elicitation, the process through which we collect preference from a user, is an important step whenever we want to model her preferences. It is a key element of domains such as multicriteria decision analysis (MCDA) or preference learning [START_REF] Fürnkranz | Preference learning[END_REF], where one wants to build a ranking model on multivariate alternatives (characterised by criteria, features, . . . ). Our contribution is more specific to MCDA, as it focuses on getting preferences from a single user, and not a population of them.

Note that within this setting, preference modelling or learning can be associated with various decision problems. Such problems most commonly include the ranking problem that consists in ranking alternatives from best to worst, the sorting problem that consists in classifying alternatives into ordered classes, and finally the choice problem that consists in picking a single best candidate among available alternatives. This article only deals with the choice problem but can be extended towards the ranking problem in a quite straightforward manner -commonly known as the iterative choice prodecure -by considering a ranking as a series of consecutive choices [START_REF] Benabbou | Incremental elicitation of choquet capacities for multicriteria choice, ranking and sorting problems[END_REF].

In order for the expert to make a recommendation in MCDA, she must first restrict her search to a set of plausible MCDA models. This is often done accordingly to a priori assumptions on the decision making process, possibly constrained by computational considerations.

In this paper, we will assume that alternatives are characterised by q real values, i.e. are represented by a vector in R q , and that preferences over them can be modelled by a value function f : R q → R such that a b iff f (a) > f (b). More specifically, we will look at weighted averages. The example below illustrate this setting. Our results can straightforwardly be extended to other evaluations functions (Choquet integrals, GAI, . . . ) in theory, but would face additional computational issues that would need to be solved.

Example 1 (choosing the best course). Consider a problem in which the DM is a student wanting to find the best possible course in a large set of courses, each of which has been previously associated a grade from 0 to 10 -0 being the least preferred and 10 being the most preferred -according to 3 criteria: usefulness, pedagogy and interest. The expert makes the assumption that the DM evaluates each course according to a score computed by a weighted sum of its 3 grades. This is a strong assumption as it means for example that an increase of 0.5 in usefulness will have the same impact on the score regardless of the grades in pedagogy and interest. In such a set of models, a particular model is equivalent to a set of weights in R 3 . Assume that the DM preferences follow the model given by the weights (0.1, 0.8, 0.1), meaning that she considers pedagogy to be eight time as important as usefulness and interest which are of equal importance. Given the grades reported in Table 1, she would prefer the Optimization course over the Machine learning course, as the former would have a 5.45 value, and the later a 3.2 value. Beyond the choice of a model, the expert also needs to collect or elicit preferences that are specific to the DM, and that she could not have guessed according to a priori assumptions. Information regarding preferences that are specific to the DM can be collected by asking them to answer questions in several form such as the ranking of a subset of alternatives from best to worst or the choice of a preferred candidate among a subset of alternatives.

Example 2 (choosing the best course (continued)). In our example, directly asking for weights would make little sense (as our model may be wrong, and as the user cannot be expected to be an expert of the chosen model). A way to get this information from her would therefore be to ask her to pick her favorite out of two courses. Let's assume that when asked to choose between Optimization and Graph theory, she prefers the Optimisation course. The latter being better than the former in pedagogy and worse in interest, her answer is compatible with weights (0.05, 0.9, 0.05) (strong preference for pedagogy over other criteria) but not with (0.05, 0.05, 0.9) (strong preference for interest over other criteria). Her answer has therefore given the expert additional information on the preferential model underlying her decision. We will see later that this generates a linear constraint over the possible weights.

Provided we have made some preference model assumptions (our case here), it is possible to look for efficient elicitation methods, in the sense that they solve the decision problem we want to solve in a small enough, if not minimal number of questions. A lot of work has been specifically directed towards active elicitation methods, in which the set of questions to ask the DM is not given in advance but determined on the fly. In robust methods, this preferential information is assumed to be given with full certainty which leads to at least two issues. The first one is that elicitation methods thus do not account for the fact that the DM might doubt her own answers, and that they might not reflect her actual preferences. The second one, that is somehow implied by the first one, is that most robust active elicitation methods will never put the DM in a position where she could contradict either herself or assumptions made by the expert, as new questions will be built on the basis that previous answers are correct and hence should not be doubted. This is especially problematic when inaccurate preferences are given early on, or when the preference model is based on wrong assumptions. This paper presents an extension of the Current Solution Strategy [START_REF] Boutilier | Constraint-based optimization and utility elicitation using the minimax decision criterion[END_REF] that includes uncertainty in the answers of the DM by using the framework based on belief functions presented in [START_REF] Destercke | A generic framework to include belief functions in preference handling and multi-criteria decision[END_REF]. Section 2 will present necessary preliminaries on both robust preference elicitation based on regret and uncertainty management based on belief functions. Section 3 will present our extension and some of the associated theoretical results and guarantees. Finally Section 4 will present some first numerical experiments that were made in order to test the method and its properties in simulations.

Preliminaries

Formalization

Alternatives and models: We will denote X the space of possible alternatives, and X ⊆ X the subset of available alternatives at the disposal of our DM and about which a recommendation needs to be made. In this paper we will consider alternatives summarised by q real values corresponding to criteria, hence X ⊆ R q . For any x ∈ X and 1 ≤ i ≤ q, we denote by x i ∈ R the evaluation of alternative x according to criterion i. We also assume that for any x, y ∈ X such that x i > y i for some i ∈ {1, . . . , q} and x l ≥ y l , ∀l ∈ {1, . . . , q} \ {i}, x will always be strictly preferred to y -meaning that preferences respect ceteris paribus monotonicity, and we assume that criteria utility scale is given.

X is a finite set of k alternative such that X = {x 1 , x 2 , . . . , x k } with x j the j-th alternative of X. Let P(X) be a preference relation over X, and x, y ∈ X be two alternatives to compare. We will state that x P y if and only if x is strictly preferred to y in the corresponding relation, x P y if and only if x and y are equally preferred in the corresponding relation, and x P y if and only if either x is strictly preferred to y or x and y are equally preferred.

Preference modelling and weighted sums: In this work, we focus on the case where the hypothesis set Ω of preference models is the set of weighted sum models1 . A singular model ω will be represented by its vector of weights in R q , and ω will be used to describe indifferently the decision model and the corresponding weight vector. Ω can therefore be described as:

Ω = ω ∈ R q : ω i ≥ 0 and q i=1 ω i = 1 .
Each model ω is associated to the corresponding aggregating evaluation function

f ω (x) = q i=1 ω i x i ,
and any two potential alternatives x, y in X can then be compared by comparing their aggregated evaluation:

x ω y ⇐⇒ f ω (x) ≥ f ω (y) (1)
which means that if the model ω is known, P ω (X) is a total preorder over X, the set of existing alternatives. Note that P ω (X) can be determined using pairwise relations ω . Weighted averages are a key model of preference learning whose linearity usually allows the development of efficient methods, especially in regret-based elicitation [START_REF] Benabbou | Minimax regret approaches for preference elicitation with rank-dependent aggregators[END_REF]. It is therefore an ideal starting point to explore other more complex functions, such as those that are linear in their parameters once alternatives are known (i.e., Choquet integrals, Ordered weighted averages).

Robust preference elicitation

In theory, obtaining a unique true preference model requires both unlimited time and unbounded cognitive abilities. This means that in practice, the best we can do is to collect information identifying a subset Ω of possible models, and act accordingly. Rather than choosing a unique model within Ω , robust methods usually look at the inferences that hold for every model in Ω . Let Ω be the subset of models compatible with all the given preferential information, then we can define P Ω (X), a partial preorder of robust preferences over X, as follows:

x

Ω y ⇐⇒ ∀ω ∈ Ω f ω (x) ≥ f ω (y). (2)
The research question we address here is to find elicitation strategies that reduce Ω as quickly as possible, obtaining at the limit an order P Ω (X) having only one maximal element 2 . In practice, one may have to stop collecting information before that point, explaining the need for heuristic indicators of the fitness of competing alternatives as potential choices.

Regret based elicitation: Regret is a common way to assess the potential loss of recommending a given alternative under incomplete knowledge. It can help both the problem of making a recommendation and finding an efficient question. Regret methods use various indicators, such as the regret R ω (x, y) of choosing x over y according to model ω, defined as

R ω (x, y) = f ω (y) -f ω (x). (3)
From this regret and a set Ω of possible models, we can then define the pairwise max regret as

PMR(x, y, Ω ) = max ω∈Ω R ω (x, y) = max ω∈Ω (f ω (y) -f ω (x)) (4)
that corresponds to the maximum possible regret of choosing x over y for any model in Ω . The max regret for an alternative x defined as

MR(x, Ω ) = max y∈X PMR(x, y, Ω ) = max y∈X max ω∈Ω (f ω (y) -f ω (x)) (5)
then corresponds to the worst possible regret one can have when choosing x. Finally the min max regret over a subset of models Ω is

mMR(Ω ) = min x∈X MR(x, Ω ) = min x∈X max y∈X max ω∈Ω (f ω (y) -f ω (x)) (6)
Picking as choice x * = arg min mMR(Ω ) is then a robust choice, in the sense that it is the one giving the minimal regret in a worst-case scenario (the one leading to max regret).

Example 3 (choosing the best course (continued)). Let X = [0, 10] 3 be the set of valid alternatives composed of 3 grades from 0 to 10 in respectively pedagogy, usefulness and interest. Let X = {x 1 , x 2 , x 3 , x 4 } be the set of available alternatives in which x 1 corresponds to the Machine learning course, x 2 corresponds to the Optimization course, x 3 corresponds to the Linear algebra course and x 4 corresponds to the Graph theory course, as reported in table 1. Let x, y ∈ X be two alternatives and Ω the set of weighted sum models, PMR(x, y, Ω) can be computed by optimizing max ω∈Ω ω 1 (x 1 -y 1 ) + ω 2 (x 2 -y 2 ) + ω 3 (x 3 -y 3 ) . As this linear function of ω is optimized over a convex polytope Ω, it can easily be solved exactly using linear programming (LP). Results of PMR(x, y, Ω) and MR(x, Ω) are shown in Table 2. In this example, x 1 is the alternative with minimum max regret, and the most conservative candidate to answer the choice problem according to regret. Regret indicators are also helpful for making the elicitation strategy efficient and helping the expert ask relevant questions to the DM. Let Ω and Ω be two sets of models such that mMR(Ω ) < mMR(Ω ). In the worst case, we are certain that x * Ω the optimal choice for Ω is less regretted than x * Ω the optimal choice for Ω , which means that we would rather have Ω be our set of models than Ω . Let I, I be two pieces of preferential information and Ω I , Ω I the sets obtained by integrating this information. Finding which of the two is the most helpful statement in the progress towards a robust choice can therefore be done by comparing mMR(Ω I ) and mMR(Ω I ). An optimal elicitation process (w.r.t. minimax regret) would then choose the question for which the worst possible answer gives us a restriction on Ω that is the most helpful in providing a robust choice. However, computing such a question can be difficult, and the heuristic we present next aims at picking a nearly optimal question in an efficient and tractable way.

The Current Solution Strategy: Let's assume that Ω is the subset of decision models that is consistent with every information available so far to the expert. Let's restrict ourselves to questions that consist in comparing pairs x, y of alternatives in X. The DM can only answer with I 1 = x y or I 2 = x y. A pair helpful in finding a robust solution as fast as possible can be computed as a solution to the following optimization problem that consists in finding the pair minimizing the worst-case min max regret :

min (x,y)∈X 2 WmMR({x, y}) = min (x,y)∈X 2 max mMR(Ω ∩ Ω x y ), mMR(Ω ∩ Ω x y ) (7)
The current solution strategy (referred to as CSS) is a heuristic answer to this problem that has proved to be efficient in practice [START_REF] Boutilier | Constraint-based optimization and utility elicitation using the minimax decision criterion[END_REF]. It consists in asking the DM to compare x * ∈ arg mMR(Ω ) the least regretted alternative to y * = arg max y∈X PMR(x * , y, Ω ) the one it could be the most regretted to (its "worst opponent"). CSS is efficient in the sense that it requires the computation of only one value of min max regret, instead of the O(q 2 ) required to solve [START_REF] Fürnkranz | Preference learning[END_REF].

Example 4 (Choosing the best course (continued).). Using the same example, according to table 2, we have mMR(Ω) = MR(x 1 , Ω) = PMR(x 1 , x 2 , Ω), meaning that x 1 is the least regretted alternative in the worst case and x 2 is the one it is most regretted to. The CSS heuristic consists in asking the DM to compare x 1 and x 2 , respectively the Machine learning course and the Optimization course.

Uncertain preferential information

Two key assumptions behind the methods we just described are that (1) the initial chosen set Ω of models can perfectly describe the DM's choices and (2) the DM is an oracle, in the sense that any answer she provides truly reflects her preferences, no matter how difficult the question. This certainly makes CSS an efficient strategy, but also an unrealistic one. This means in particular that if the DM makes a mistake, we will just pursue with this mistake all along the process and will never question what was said before, possibly ending up with sub-optimal recommendations.

Example 5 (choosing the best course (continued)). Let's assume similarly to Example 2 that the expert has not gathered any preference from the DM yet, and that this time she asks her to compare alternatives x 1 and x 2 -respectively the Machine learning course and the Optimization course. Let's also assume similarly to Example 1 that the DM makes decisions according to a weighted sum model with weights ω = (0.1, 0.8, 0.1).

f ω (x 2 ) = 5.45 > 3.2 = f ω (x 1 )
, which means that she should prefer the Optimization course over the Machine learning course. However for some reason -such as her being unfocused or unsure about her preference -assume the DM's answer is inconsistent with ω and she states that x 1 x 2 rather than x 2 x 1 .

Then Ω the set of model consistent with available preferential information is such that

Ω = Ω x1 x2 = {ω ∈ Ω : 3 i=1 ω i x i 1 -x i 2 ≥ 0} = {ω ∈ Ω : ω 2 ≤ 2 3 -5
24 ω 1 }, as represented in Figure 1. It is clear that ω ∈ Ω : subsequent questions will only ever restrict Ω and the expert will never get quite close to modelling ω . A similar point could be made if ω * , the model according to which the DM makes her decision, does not even belong to Ω the set of weighted sums models that the expert chose.

As we shall see, one way to adapt min-max regret approaches to circumvent the two above difficulties can be to include a simple measure of how uncertain an answer is.

The belief function framework In classical CSS, the response to a query by the DM always implies a set of consistent models Ω such that Ω ⊆ Ω. Here, we allow the DM to give alongside her answer a confidence level α ∈ [0, 1], interpreted as how confident she is that this particular answer matches her preferences. In the framework developed in [START_REF] Destercke | A generic framework to include belief functions in preference handling and multi-criteria decision[END_REF], such information is represented by a mass function on Ω , referred to as m Ω α and defined as :

m Ω α (Ω) = 1 -α, m Ω α (Ω ) = α.
Such mass assignments are usually called simple support [START_REF] Shafer | A mathematical theory of evidence[END_REF] and represent elementary pieces of uncertain information. A confidence level of 0 will correspond to a vacuous knowledge about the true model ω * , and will in no way imply that the answer is wrong (as would have been the case in a purely probabilistic framework). A confidence level of 1 will correspond to the case of certainty putting a hard constraint on the subset of models to consider.

Remark 1. Note that values of α do not necessarily need to come from the DM, but can just be chosen by the analyst (in the simplest case as a constant) to weaken the assumptions of classical models. We will see in the experiments of Section 4 that such a strategy may indeed lead to interesting behaviours, without necessitating the DM to provide confidence degrees if she thinks the task is too difficult, or if the analyst thinks such self-assessed confidence is meaningless.

Dempster's rule Pieces of information corresponding to each answer will be combined through non-normalized Dempster's rule + ∩ . At step k, m k the mass function capturing the current belief about the DM's decision model can thus be defined recursively as :

m 0 = m Ω 1 . . . m k = m k-1 + ∩ m Ω k α k . (8)
This rule, also known as TBM conjunctive rule, is meant to combine distinct pieces of information.

It is central to the Transferable Belief Model, that intends to justify belief functions without using probabilistic arguments [START_REF] Smets | The combination of evidence in the transferable belief model[END_REF].

Note that an information fusion setting and the interpretation of the TBM fit our problem particularly well as it assumes the existence of a unique true model ω * underlying the DM's decision process, that might or might not be in our predefined set of models Ω. Allowing for an open world is a key feature of the framework. Let us nevertheless recall that non-normalized Dempster's rule + ∩ can also be justified without resorting to the TBM [START_REF] Pichon | The unnormalized dempsters rule of combination: A new justification from the least commitment principle and some extensions[END_REF][START_REF] Klawonn | On the axiomatic justification of dempsters rule of combination[END_REF][START_REF] Klawonn | The dynamic of belief in the transferable belief model and specializationgeneralization matrices[END_REF].

In our case this independence of sources associated with two mass assignments m Ωi αi and m

Ωj αj

means that even though both preferential information account for preferences of the same DM, the answer a DM gives to the ith question does not directly impact the answer she gives to the jth question : she would have answered the same thing had their ith answer been different for some reason. This seems reasonable, as we do not expect the DM to have a clear intuition about the consequences of her answers over the set of models, nor to even be aware that such a setor axioms underlying it -exists. One must however be careful to not ask the exact same question twice in short time range. Since combined masses are all possibility distributions, an alternative to assuming independence would be to assume complete dependence, simply using the minimum rule [START_REF] Destercke | Idempotent conjunctive combination of belief functions: Extending the minimum rule of possibility theory[END_REF] which among other consequences would imply a loss of expressivity3 but a gain in computation 4 .

As said before, one of the key interest of using this rule (rather than its normalised version) is to allow m(∅) > 0, notably to detect either mistakes in the DM's answer (considered as an unreliable source) or a bad choice of model (under an open world assumption). Determining where the conflict mainly comes from and acting upon it will be the topic of future works. Note that in the specific case of simple support functions, we have the following result:

Proposition 1. If m Ω k
α k are simple support functions combined through Dempster's rule, then m(∅) = 0 ⇔ ∃ω, P l({ω}) = 1

with P l({ω}) = E⊆Ω,ω∈E m(E) the plausibility measure of model ω.

Proof. (Sketch) The ⇐ part is obvious given the properties of Plausiblity measure. The ⇒ part follows from the fact that if m(∅) = 0, then all focal elements are supersets of i∈{1,...,k} Ω i , hence all contains at least one common element.

This in particular shows that m(∅) can, in this specific case, be used as an estimate of the logical consistency of the provided information pieces.

Consistency with robust, set-based methods : when an information I is given with full certainty α = 1, we retrieve a so-called categorical mass m k Ω I = 1. Combining a set I 1 , ..., I k of such certain information will end up in the combined mass

m k   i∈{1,...,k} Ω i   = 1
which is simply the intersection of all provided constraints, that may turn up either empty or non-empty, meaning that inconsistency will be a Boolean notion, i.e.,

m k (Ø) = 1 if i∈{1,...,k} Ω i = ∅ 0 otherwise
Recall that in the usual CSS or minimax regret strategies, such a situation can never happen.

Extending CSS within the belief function framework

We now present our proposed extension of the Current Solution Strategy integrating confidence degrees and uncertain answers. Note that in the two first Sections 3.1 and 3.2, we assume that the mass on the empty set is null in order to parallel our approach with the usual one not including uncertainties. We will then consider the problem of conflict in Section 3.3.

Extending regret notions

Extending PMR: when uncertainty over possible models is defined through a mass function 2 Ω → [0, 1], subsets of Ω known as focal sets are associated to a value m(Ω ) that correspond to the knowledge we have that ω belongs to Ω and nothing more. The extension we propose averages the value of PMR on focal sets weighted by their corresponding mass :

EPMR(x, y, m) = Ω ⊆Ω m(Ω ).PMR(x, y, Ω ) (9)
and we can easily see that in the case of certain answers (α = 1), we do have

EPMR(x, y, m k ) = PMR   x, y,   i∈{1,...,k} Ω i     (10)
hence formally extending Equation (4). When interpreting m(Ω ) as the probability that ω belongs to Ω , EPMR could be seen as an expectation of PMR when randomly picking a set in 2 Ω .

Extending EMR: Similarly, we propose a weighted extension of maximum regret

EMR(x, m) = Ω ⊆Ω m(Ω ).MR(x, Ω ) = Ω ⊆Ω m(Ω ). max y∈X {PMR(x, y, Ω )} . (11)
EMR is the expectation of the maximal pairwise max regret taken each time between x and y ∈ X its worst adversary -as opposed to a maximum considering each y ∈ X of the expected pairwise max regret between x and the given y, described by MER(x, m) = max y EPMR(x, y, m). Both approaches would be equivalent to MR in the certain case, meaning that if

α i = 1 then EMR(x, m k ) = MER(x, m k ) = MR   x,   i∈{1,...,k} Ω i     . (12)
However EMR seems to be a better option to assess the max regret of an alternative, as under the assumption that the true model ω * is within the focal set Ω , it makes more sense to compare x to its worst opponent within Ω , which may well be different for two different focal sets. Indeed, if ω * the true model does in fact belong to Ω , decision x is only as bad as how big the regret can get for any adversarial counterpart y Ω ∈ X.

Extending mMR: we propose to extend it as

mEMR(m) = min x∈X EMR(x, m). (13)
mEMR minimizes for each x ∈ X the expectation of max regret and is different from the expectation of the minimal max regret for whichever alternative x is optimal, described by EmMR(m) = Ω m(Ω ) min x∈X MR(x, Ω ). Again, these two options with certain answers boil down to mMR as we have

mEMR(m) = EmMR(m) = mMR   i∈{1,...,k} Ω i   . (14)
The problem with EmMR is that it would allow for multiple possible best alternatives, leaving us with an unclear answer as to what is the best choice option, (arg min EmMR) not being defined.

It indicates how robust in the sense of regret we expect any best answer to the choice problem to be, assuming there can be an optimal alternative for each focal set. In contrast, mEMR minimizes the max regret while restricting the optimal alternative x to be the same in all of them, hence providing a unique argument and allowing our recommendation system and elicitation strategy to give an optimal recommendation.

Extending CSS: our Evidential Current Solution Strategy (ECSS) then amounts, at step k with mass function m k , to perform the following sequence of operations:

-Find x * = arg mEMR(m k ) = arg min x∈X EMR(x, m k ); -Find y * = arg max y∈X EPMR(x * , y, m k ); -Ask the DM to compare x * , y * and provide α k , obtaining m Ω k α k ; -Compute m k+1 := m k + ∩ m Ω k α k
-Repeat until conflict is too high (red flag), budget of questions is exhausted, or mEMR(m k ) is sufficiently low Finally, recommend x * = arg mEMR(m k ). Thanks to Equations (10), ( 12) and ( 14), it is easy to see that we retrieve CSS as the special case in which all answers are completely certain. Example 6. Starting with intial mass function m 0 such that m 0 (Ω) = 1, the choice of CSS coincides with the choice of ECSS (all evidence we have is commited to ω ∈ Ω). With the values of PMR reported in Table 2 the alternatives the DM is asked to compare are x 1 the least regretted alternative and x 2 its most regretted counterpart. In accordance with her true preference model ω * = (0.1, 0.8, 0.1), the DM states that x 2

x 1 , i.e., she prefers the Optimization course over the Machine learning course, with confidence degree α = 0.7. Let Ω 1 be the set of WS models in which x 2 can be preferred to x 1 , which in symmetry with example 5 can be defined as

Ω 1 = {ω ∈ Ω : ω 2 ≥ 2 3 -5
24 ω 1 }, as represented in Figure 2. The values of PMR and MR on Ω 1 can be computed using LP as reported in table 3. Values of PMR and MR and have been previously computed on Ω as reported in Table 2. Values of EPMR and EMR can be then be deduced by combining them according to Equations ( 9) and ( 11), as reported in Table 4. In this example x 3 minimizes both MR and EMR and our extension agrees with the robust version as to which recommendation is to be made. However the most regretted counterpart to which the DM has to compare x 3 in the next step differs, as ECSS would require that she compares x 3 and x 1 rather than x 3 and x 2 for CSS.

Preserving the properties of CSS

This section discusses to what extent is ECSS consistent with three key properties of CSS:

1. CSS is monotonic, in the sense that the minmax regret mMR reduces at each iteration. 2. CSS provides strong guarantees, in the sense that the felt regret of the recommendation is ensured to be at least as bad as the computed mMR. 3. CSS produces questions that are non-conflicting (whatever the answer) with previous answers.

We would like to keep the first two properties at least in the absence of conflicting information, as they ensure respectively that the method will converge and will provide robust recommendations. However, we would like our strategy to raise questions possibly contradicting some previous answers, so as to raise the previously mentioned red flags in case of problems (unreliable DM or bad choice of model assumption). As shows the next property, our method also converges. Proposition 2. Let m k-1 and m Ω k α k be two mass functions on Ω issued from ECSS such that

m k (Ø) = m k-1 + ∩ m Ω k α k (Ø) = 0, then 1. EPMR(x, y, m k ) ≤ EPMR(x, y, m k-1 ) 2. EMR(x, m k ) ≤ EMR(x, m k-1 ) 3. mEMR(m k ) ≤ mEMR(m k-1 )
Proof. (sketch) The two first items are simply due to the combined facts that on one hand we know [START_REF] Smets | The application of the matrix calculus to belief functions[END_REF] that applying + ∩ means that m k is a specialisation of m k-1 , and on the other hand that for any Ω ⊆ Ω we have f (x, y, Ω ) ≤ f (x, y, Ω ) for any f ∈ {PMR, MR}. The third item is implied by the second as it consists in taking a minimum over a set of values of EMR that are all smaller.

Note that the above argument applies to any combination rule producing a specialisation of the two combined masses, including possibilistic minimum rule [START_REF] Destercke | Idempotent conjunctive combination of belief functions: Extending the minimum rule of possibility theory[END_REF], Denoeux's family of w-based rules [START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence[END_REF], etc. We can also show that the evidential approach, if we provide it with questions computed through CSS, is actually more cautious than CSS: Proposition 3. Consider the subsets of models Ω 1 , . . . , Ω k issued from the answers of the CSS strategy, and some values α 1 , . . . , α k provided a posteriori by the DM. Let m k-1 and m Ω k α k be two mass functions issued from ECSS on Ω such that m k (∅) = 0. Then we have

1. EPMR(x, y, m k ) ≥ PMR(x, y, i∈{1,...,k} Ω i ) 2. EMR(x, m k ) ≥ MR(x, i∈{1,...,k} Ω i ) 3. mEMR(m k ) ≥ mMR( i∈{1,...,k} Ω i )
Proof. (sketch) The first two items are due to the combined facts that on one hand all focal elements are supersets of i∈{1,...,k} Ω i and on the other hand that for any Ω ⊆ Ω we have f (x, y, Ω ) ≤ f (x, y, Ω ) for any f ∈ {PMR, MR}. Any value of EPMR or EMR is a weighted average over terms all greater than their robust counterpart on i∈{1,...,k} Ω i , and is therefore greater itself. The third item is implied by the second as the biggest value of EMR is thus necessarily bigger than all the values of MR.

This simply shows that, if anything, our method is even more cautious than CSS. It is in that sense probably slightly too cautious in an idealized scenario -especially as unlike robust indicators our evidential extensions will never reach 0 -but provides guarantees that are at least as strong.

While we find the two first properties appealing, one goal of including uncertainties in the DM answers is to relax the third property, whose underlying assumptions (perfectness of the DM and of the chosen model) are quite strong. In Sections 3.3 and 4, we show that ECSS indeed satisfies this requirement, respectively on an example and in experiments.

Evidential CSS and conflict

The following example simply demonstrates that, in practice, ECSS can lead to questions that are possibly conflicting with each others, a feature CSS does not have. This conflict is only a possibility: no conflict will appear should the DM provide answers completely consistent with the set of models and what she previously stated, and in that case at least one model will be fully plausible 5 (see Proposition 1).

Example 7 (Choosing the best course (continued)). At step 2 of our example the DM is asked to compare x 1 to x 3 in accordance with Table 4. Even though it conflicts with ω * the model underlying her decision the DM has the option to state that x 1 x 3 with confidence degree α > 0, putting weight on Ω 2 the set of consistent model defined by Ω 2 = {ω ∈ Ω :

q i=1 ω i x i 1 -x i 3 ≥ 0} = {ω ∈ Ω : ω 2 ≤ 9
16 -3 8 ω 1 }. However as represented in figure 3, This means that x 1

Ω 1 ∩ Ω 2 = Ø. 0 ω 1 1 ω 2 1 2 3 11 19
8 19 0.1 0.8 Ω1 Ω ω 0 ω 1 1 ω 2 1 2/3 9/16
x 2 and x 3 x 1 are not compatible preferences assuming the DM acts according to a weighted sum model. This can be either because she actually does not follow such a model, or because one of her answers did not reflect her actual preference (which would be the case here). Assuming she does state that x 1 x 3 with confidence α = 0.6, information about the preference model at step 2 is captured through mass function m 2 defined as : m 2 : Ω → 0.12 Ω 2 → 0.18 Ω 1 → 0.28 Ø → 0.42 Meaning that ECSS detects a degree of inconsistency equal to m 2 (Ø) = 0.42. This illustrating example is of course not representative of real situations, having only four alternatives, but clearly shows that within ECSS, conflict may appear as an effect of the strategy. In other words, we do not have to modify it to detect consistency issues, it will automatically seek out for such cases if α i < 1. In our opinion, this is clearly a desirable property showing that we depart from the assumptions of CSS.

However, the inclusion of conflict as a focal element in the study raises new issues, the first one being how to extend the various indicators of regret to this situation. In other words, how can we compute PMR(x, y, Ø), MR(x, Ø) or MMR(Ø)? This question does not have one answer and requires careful thinking, however the most straightforward extension is to propose a way to compute PMR(x, y, Ø), and then to plug in the different estimates of ECSS. Two possibilities immediately come to mind:

-PMR(x, y, Ø) = max ω∈Ω R ω (x, y), which takes the highest regret among all models, and would therefore be equivalent to consider conflict as ignorance. This amounts to consider Yager's rule [START_REF] Yager | On the dempster-shafer framework and new combination rules[END_REF] in the setting of belief functions. This rule would make the regret increases when conflict appears, therefore providing alerts, but this clearly means that the monotonicity of Proposition 3 would not longer hold. Yet, one could discuss whether such a property is desirable or not in case of conflicting opinions. It would also mean that elicitation methods are likely to try to avoid conflict, as it will induce a regret increase. -PMR(x, y, Ø) = min ω∈Ω max(0, R ω (x, y)), considering ∅ as the limit intersection of smaller and smaller sets consistent with the DM answers. Such a choice would allow us to recover monotonicity, and would clearly privilege conflict as a good source of regret reduction.

Such distinctions expand to other indicators, but we leave such a discussion for future works.

On computational tractability

ECSS requires, in principle, to compute PMR values for every possible focal elements, which could lead to an exponential explosion of the computational burden. We can however show that in the case of weighted sums and more generally of linear constraints, where PMR has to be solved through a LP program, we can improve upon this worst-case bound. We introduce two simplifications that lead to more efficient methods providing exact answers:

Using the polynomial number of elementary subsets. The computational cost can be reduced by using the fact that if P j = {Ω i1 , . . . , Ω i k } is a partition of Ω j , then :

PMR(x, y, Ω j ) = max l∈{i1,...,i k } PMR(x, y, Ω l )
Hence, computing PMR on the partition is sufficient to retrieve the global PMR through a simple max. Let us now show that, in our case, the size of this partition only increases polynomially.

Let Ω 1 , . . . , Ω n be the set of models consistent with respectively the first to the nth answer, and Ω C i , . . . , Ω C n their respective complement in Ω. Due to the nature of the conjunctive rule + ∩ , every focal set

Ω of m k = m Ω 1 + ∩ m Ω1 α1 + ∩ . . . + ∩ m Ωn
αn is the union of elements of the partition P Ω = { Ω1 , . . . , Ωs }, with :

Ωk = Ω i∈U k Ω i i∈{1,...,n}\U k Ω C i , U k ⊆ {1, . . . , n}
Which means that for each Ω 's PMR can be computed using the PMR of its corresponding partition. This still does not help much, as there is a total of 2 n possible value of Ωk . Yet, in the case of convex domains cut by linear constraints, which holds for the weighted sum, the following theorem shows that the total number of elementary subset in Ω only increases polynomially.

Theorem 1. [12, P 39] Let E be a convex bounded subset of F an euclidean space of dimension q, and H = {η 1 , . . . , η n } a set of n hyperplanes in F such that ∀i ∈ {1, . . . , n}, η i separates F into two subsets F ηi 0 and F ηi 1 . To each of the 2 n possible U ⊆ {1, . . . , n} a subset

F H U = F i∈U F ηi 1 i∈{1,...,n}\U F ηi 0 can be associ- ated. Let Θ H = U ⊆ {1, . . . , n} : F H U ∩ E = Ø and B H = |Θ H |, then B H ≤ Λ n q = 1 + n + n 2 + • • • + n q
Meaning that at most Λ n q of the F H U subsets have a non empty intersection with E.

In the above theorem (the proof of which can be found in [START_REF] Schläfli | Theorie der vielfachen Kontinuität[END_REF], or in [10] for the specific case of E ⊂ R 3 ), the subsets B H are equivalent to Ωk , whose size only grow according to a polynomial whose power increases with q.

Using the polynomial number of extreme points in the simplex problem. Since we work with LP, we also know that optimal values will be obtained at extreme points. Optimization on focal sets can therefore ALL be done by maxing points at the intersection of q hyperplanes. This set of extreme point is

E = {ω = η i ∩ • • • ∩ η q : {i 1 , . . . , i q } ∈ {1, . . . , n}} (15)
with η i the hyper-planes corresponding to the questions. We have |E| = n q ∈ O(n q ) which is reasonable whenever q is small enough (typically the case in MCDA). The computation of the coordinate of extreme points related to constraints of each subset can be done in advance for each ω ∈ E and not once per subset and pair of alternatives, since E Ω the set of extreme points of Ω will always be such that E Ω ⊆ E. The computation of the dot products necessary to compute R ω (x, y) for all ω ∈ E, x, y ∈ X can also be done once for each ω ∈ E, and not be repeated in each subset Ω s.t. ω ∈ E Ω . Those results indicate us that when q (the model-space dimension) is reasonably low and questions correspond to cutting hyper-planes over a convex set, ECSS can be performed efficiently. This will be the case for several models such as OWA or k-additive Choquet integrals with low k, but not for others such as full Choquet integrals, whose dimension if we have k criteria is 2 k -2. In these cases, it seems inevitable that one would resort to approximations having a low numerical impact (e.g., merging or forgetting focal elements having a very low mass value).

Experiments

To test our strategy and its properties, we proceeded to simulated experiments, in which the confidence degree was always constant. Such experiments therefore also show what would happen if we did not ask confidence degrees to the DM, but nevertheless assumed that she could make mistakes with a very simple noise model.

The first experiment reported in Figure 4 compares the extra cautiousness of EMR when compared to MR. To do so, simulations were made for several fixed degrees of confidence -including 1 in which case EMR coincides with MR -in which a virtual DM states her preferences with the given degree of confidence, and the value of EMR at each step is divided by the initial value so as to observe its evolution. Those EMR ratios were then averaged over 100 simulations for each degree. Results show that while high confidence degrees will have a limited impact, low confidence degrees (< 0.7) may greatly slow down the convergence.

The second experiment reported in Figure 5 aims at finding if ECSS and CSS truly generate different question strategies. To do so, we monitored the two strategies for a given confidence degree, and identify the first step k for which the two questions are different. Those values were averaged over 300 simulations for several confidence degrees. Results show that even for a high confidence degree (α = 0.9) it takes in average only 3 question to see a difference. This shows that the methods are truly different in practice. The third experiment reported in Figure 6 is meant to observe how good m(Ø) our measure of inconsistency is in practice as an indicator that something is wrong with the answers given by a DM. In order to do so simulations were made in which one of two virtual DMs answers with a fixed confidence degree and the value of m(Ø) is recorded at each step. They were then averaged over 100 simulations for each confidence degree. The two virtual DMs behaved respectively completely randomly (RAND) and in accordance with a fixed weighted sum model (WS) with probability α and randomly with a probability 1 -α. So the first one is highly inconsistent with our model assumption, while the second is consistent with this assumptions but makes mistakes. Results are quite encouraging: the inconsistency of the random DM with the model assumption is quickly identified, especially for high confidence degrees. For the DM that follows our model assumptions but makes mistakes, the results are similar, except for the fact that the conflict increase is not especially higher for lower confidence degrees. This can easily be explained that in case of low confidence degrees, we have more mistakes but those are assigned a lower weight, while in case of high confidence degrees the occasional mistake is quite impactful, as it has a high weight.

Conclusion

In this paper, we have proposed an evidential extension of the CSS strategy, used in robust elicitation of preferences.

We have studied its properties, notably comparing them to those of CSS, and have performed first experiments to demonstrate the utility of including confidence degrees in robust preference elicitation. Those latter experiments confirm the interest of our proposal, in the sense that it quickly identifies inconsistencies between the DM answer and model assumptions. It remains to check whether, in presence of mistakes from the DM, the real-regret (and not the computed one) obtained for ECSS is better than the one obtained for CSS.

As future works, we would like to work on the next step, i.e., identify the sources of inconsistency (whether it comes from bad model assumption or an unreliable DM) and propose correction strategies. We would also like to perform more experiments, and extend our approach to other decision models (Choquet integrals and OWA operators being the first candidates).

Fig. 1 .

 1 Fig. 1. Graphical representation of Ω, Ω and ω

Fig. 2 .

 2 Fig. 2. Graphical representation of Ω, Ω1 and ω

Fig. 3 .

 3 Fig. 3. Graphical representation of Ω, Ω1, Ω2 and ω

Fig. 4 .

 4 Fig. 4. Average evolution of min max regret with various degrees of confidence

Fig. 5 .

 5 Fig. 5. Average position of the first different question in the elicitation process / degrees of confidence

Fig. 6 .

 6 Fig. 6. Evolution of average inconsistency with a DM fitting the WS model and a randomly choosing DM

Table 1 .

 1 Grades of courses

	Machine learning:	usefulness pedagogy interest 8.5 1.5 10	Optimization:	usefulness pedagogy interest 3 5.5 2
	Linear algebra:	usefulness pedagogy interest 7 5 5.5	Graph theory:	usefulness pedagogy interest 1 2 6

Table 2 .

 2 Values

of PMR(x, y, Ω) (left) and MR(x, Ω) (right)

Table 3 .

 3 Values of PMR(x, y, Ω1) (left) and MR(x, Ω1) (right)

								x	MR
								x1	4
	x3 -5 6 x4 109 38	-0.83 0.5 2.87 3.5 81 19	1.39 0 4.26	-11 6	-1 -1.83 0	x2 53 38 x3 x4 81 19	1.39 0.5 4.26
	@ x x1 @ @ y x2 x3 23 30 x4 809 190	x1 0 2.4 0.77 0.5 x2 4 0 827 390 4.26 3.5 909 190	x3 3.5 2.18 0 4.78	-68 60	x4 0.5 0.5 -1.13 0	x x1 x2 1283 MR 4 3.38 380 x3 23 0.7 30 x4 1389 380 5.23

Table 4 .

 4 Values of EPMR(x, y, m1) (left) and EMR(x, m1) (right)

In principle, our methods apply to any value function with the same properties, but may have to solve computational issue that depends on the specific chosen hypothesis.

Or in some cases a maximal set {x1, . . . , xp} of equally preferred elements s.t. x1 . . . xp

For instance, no new values of confidence would be created when using a finite set {α1, . . . , αM } for elicitation

The number of focal sets increasing only linearly with the number of information pieces.

This contrasts with a Bayesian/probabilistic approach, where no model would receive full support in non-degenerate cases.


MD-PhD Joe-Elie Salem
email: joe-elie.salem@aphp.fr

MD # † § Lee Nguyen

MD-PhD # § Nadjib Hammoudi

Gisèle Preud

' Homme

Jean-Sébastien Hulot

MD Monique Leban

Christian Funck-Brentano

Philippe Touraine

Richard Isnard

Anne Bachelot

Gisèle Preud'homme Md

MD-PhD † § Jean-Sebastien Hulot

Christian Funck-Brentano Md-Phd

Philippe Touraine Md-Phd

Richard Isnard Md-Phd

Anne Bachelot Md-Phd

Philippe Chanson

Sylvie Salenave

Sophie Christin Maitre

Nathalie Bourcigaux

Michel Polak

Juliane Leger

Complex Association of Sex Hormones on Left Ventricular Systolic Function: Insight into Sexual Dimorphism

Keywords: echocardiography, physiology, Sex Hormones, congenital adrenal hyperplasia, ventricular functions

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

Introduction

In the healthy population, women have different reference values for most cardiac morphological and functional parameters. [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF][START_REF] Cheng | Age-and sex-based reference limits and clinical correlates of myocardial strain and synchrony: the Framingham Heart Study[END_REF][START_REF] Hayward | Effect of sex hormones on cardiac mass[END_REF][START_REF] Kocabay | Normal left ventricular mechanics by two-dimensional speckle-tracking echocardiography. Reference values in healthy adults[END_REF][START_REF] Shi | Left Ventricular Longitudinal and Circumferential Layer-Specific Myocardial Strains and Their Determinants in Healthy Subjects[END_REF] Men have larger and thicker left ventricles (LV) than women. [START_REF] Hayward | Effect of sex hormones on cardiac mass[END_REF][START_REF] Shi | Left Ventricular Longitudinal and Circumferential Layer-Specific Myocardial Strains and Their Determinants in Healthy Subjects[END_REF] This is associated with a differential pattern of the radial, circumferential and longitudinal components of its systolic function. [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF][START_REF] Cheng | Age-and sex-based reference limits and clinical correlates of myocardial strain and synchrony: the Framingham Heart Study[END_REF][START_REF] Kocabay | Normal left ventricular mechanics by two-dimensional speckle-tracking echocardiography. Reference values in healthy adults[END_REF][START_REF] Shi | Left Ventricular Longitudinal and Circumferential Layer-Specific Myocardial Strains and Their Determinants in Healthy Subjects[END_REF] Women and men tend to experience different types of heart failure (HF). Men are more likely to suffer from systolic dysfunction leading to HF with reduced ejection fraction (HFrEF) while women have higher rates of diastolic dysfunction leading to HF with preserved ejection fraction (HFpEF). [START_REF] Komajda | Heart failure with preserved ejection fraction: a clinical dilemma[END_REF] Extensive data exist concerning the influence of sex hormones on heart repolarization [START_REF] Salem | Influence of steroid hormones on ventricular repolarization[END_REF] but data concerning their influence on cardiac morphology and function, are scarce and disparate. [START_REF] Smet | Sex steroids in relation to cardiac structure and function in men[END_REF][START_REF] Kloner | Testosterone and Cardiovascular Disease[END_REF] Large epidemiological studies showed that low serum total-testosterone is a biomarker of cardiovascular mortality [START_REF] Kloner | Testosterone and Cardiovascular Disease[END_REF][START_REF] Oskui | Testosterone and the cardiovascular system: a comprehensive review of the clinical literature[END_REF], [START_REF] Budoff | Testosterone Treatment and Coronary Artery Plaque Volume in Older Men With Low Testosterone[END_REF] and several small sample trials have shown promising results with testosterone for the treatment of HF. [START_REF] Caminiti | Effect of long-acting testosterone treatment on functional exercise capacity, skeletal muscle performance, insulin resistance, and baroreflex sensitivity in elderly patients with chronic heart failure a double-blind, placebo-controlled, randomized study[END_REF][START_REF] Iellamo | Testosterone therapy in women with chronic heart failure: a pilot double-blind, randomized, placebocontrolled study[END_REF][START_REF] Malkin | Testosterone therapy in men with moderate severity heart failure: a double-blind randomized placebo controlled trial[END_REF][START_REF] Toma | Testosterone supplementation in heart failure: a meta-analysis[END_REF] However, administration of higher supraphysiological levels of testosterone have been associated with impaired diastolic function and post-infarction remodeling in clinical and preclinical models. [START_REF] Smet | Sex steroids in relation to cardiac structure and function in men[END_REF][START_REF] Cavasin | Estrogen and testosterone have opposing effects on chronic cardiac remodeling and function in mice with myocardial infarction[END_REF][START_REF] Chung | Effects of testosterone and nandrolone on cardiac function: a randomized, placebo-controlled study[END_REF][START_REF] Marra | Cardiovascular abnormalities and impaired exercise performance in adolescents with congenital adrenal hyperplasia[END_REF] Preclinical studies also have suggested that estrogens could prevent deterioration of cardiac function and remodeling after experimental myocardial infarction. [START_REF] Cavasin | Estrogen and testosterone have opposing effects on chronic cardiac remodeling and function in mice with myocardial infarction[END_REF] The potential influence of other sex steroid hormones showing important role in heart repolarization such as progesterone, [START_REF] Salem | Influence of steroid hormones on ventricular repolarization[END_REF] or gonadotropins regulating their production [START_REF] Abehsira | Complex Influence of Gonadotropins and Sex Steroid Hormones on QT Interval Duration[END_REF] , has not been studied. To further investigate these issues, we studied the association of main sex steroid hormones and gonadotropins on left ventricular morphology and function in healthy subjects and in a unique cohort of patients with congenital adrenal hyperplasia (CAH) due to 21-hydroxylase deficiency, a rare disease, used in our study as a model of testosterone dysregulation. CAH corresponds to a group of inherited autosomal recessive disorders that arises from defective steroidogenesis and results from a deficiency in one or several of the enzymes of cortisol biosynthesis. [START_REF] Bachelot | MANAGEMENT OF ENDOCRINE DISEASE: Congenital adrenal hyperplasia due to 21-hydroxylase deficiency: update on the management of adult patients and prenatal treatment[END_REF] Deficiency of the 21-hydroxylase enzyme is the most common form of CAH and is characterized by cortisol and in some cases aldosterone deficiency associated with adrenal androgen excess and renin increase (Figure 1). [START_REF] Bachelot | MANAGEMENT OF ENDOCRINE DISEASE: Congenital adrenal hyperplasia due to 21-hydroxylase deficiency: update on the management of adult patients and prenatal treatment[END_REF] Cortisol deficiency results in the ACTH-induced accumulation of substrate precursors such as 17OH-progesterone and progesterone, and to increased secretion of adrenal androgens. [START_REF] Bachelot | MANAGEMENT OF ENDOCRINE DISEASE: Congenital adrenal hyperplasia due to 21-hydroxylase deficiency: update on the management of adult patients and prenatal treatment[END_REF] Goals of treatment are to replace deficient hormones by glucocorticoids and in some cases mineralocorticoids, and control adrenal androgen excess. The optimal dose of glucocorticoid is that which fails to fully suppress 17hydroxy(OH)-progesterone and maintains androgens in the mid-normal range. [START_REF] Bachelot | MANAGEMENT OF ENDOCRINE DISEASE: Congenital adrenal hyperplasia due to 21-hydroxylase deficiency: update on the management of adult patients and prenatal treatment[END_REF] There is variability in ACTH suppression by this supplementation, leading to inter-individual variability in residual secretion of adrenal androgen and ultimately testosterone levels. In some patients, this leads to insufficient control of their disease, resulting in high levels of progesterone and adrenal androgens, disturbing gonadal axis with low LH and FSH release by the pituitary gland. [START_REF] Bachelot | MANAGEMENT OF ENDOCRINE DISEASE: Congenital adrenal hyperplasia due to 21-hydroxylase deficiency: update on the management of adult patients and prenatal treatment[END_REF] Due to this variable response, we were expecting testosterone levels to be higher in CAH women and lower in CAH men than in sex-matched controls (see Figure 1 for further details).

The purpose of this work was first to determine if LV morphology and systolic function are different within and between men and women depending on CAH status. Secondly, our aim was to investigate for the first time the association of gonadotropins and sex steroid hormones with LV function to better decipher the normal structure and function of a beating heart. [START_REF] Buckberg | Left ventricular form and function: scientific priorities and strategic planning for development of new views of disease[END_REF]

Methods

Study design

Our work is ancillary to CARDIOHCS [NCT01807364], a multicenter prospective observational case-control study comparing early cardiovascular damages in adult men and women with CAH due to 21α-hydroxylase deficiency and healthy controls. All patients gave written informed consent to participate and the study was approved by each hospital ethics committee.

Study population

Eighty-four young adults (58 women and 26 men) with CAH and 84 controls matchedpaired for sex, age (±5years) and smoking status (non-smoking, past smoking, active smoking) were prospectively included in CARDIOHCS study between 05/2011 and 08/2015. CAH adults included were diagnosed during childhood, proven by genetic testing confirming a 21α-hydroxylase deficiency. Exclusion criteria for CAH and healthy subjects were: known history of cardiovascular disease and pregnancy. Estradiol and/or progesterone contraception in the previous month was an exclusion criterion in healthy subjects and interruption was promoted in CAH patients. CAH patients were treated, as needed by their standard of care, with hydrocortisone or dexamethasone and some of them also received fludrocortisone. In the morning, before 10a.m., participants had a clinical examination including past medical history, physical examination with blood pressure measurement (with a standard sphygmomanometer), weight (kg) and height (m) measurements (body mass index: weight / height 2 ), and electrocardiographic recording after at least 10 minutes of rest in the supine position. Included subjects were asked for chronic treatment intake. Blood samples for the determination of serum concentrations of 17-OH-progesterone, progesterone, testosterone, sex hormone-binding globulin (SHBG), estradiol, follicle-stimulating hormone (FSH), luteinizing hormone (LH) were collected in a dry tube and further assayed in the immunology laboratory of Pitié-Salpêtrière. Estradiol, progesterone, SHBG, testosterone, FSH and LH serum concentrations were assayed by chemi-luminescence (Modular-E170 Roche®) and 17-OHprogesterone by radioimmunoassay (KIP1409; DIAsource ImmunoAssays).

Study procedures and laboratory analysis

Hexokinase/Glucose-6-phosphate dehydrogenase was used for the determination of fasting plasma glucose levels (Roche®). The methodology of electrocardiographic acquisition and measurements have been described elsewhere. [START_REF] Abehsira | Complex Influence of Gonadotropins and Sex Steroid Hormones on QT Interval Duration[END_REF] Echocardiography acquisition and analysis

Transthoracic echocardiography was performed within 48 hours of clinical and biological evaluation. It was performed using a Vivid9 equipped with a M5S convex transducer (GE Healthcare; Horten, Norway) by one expert physician, blindly to clinical and biological data.

Two-dimensional images, acquired with appropriate sector size and depth settings to achieve optimal visualization of all LV myocardium at the optimal frame rate of 50-80 frames/sec, were transferred to a workstation equipped with the EchopacPC software version 113 (GE-Vingmed-Ultrasound®; Horten, Norway). All examinations were analyzed off-line by one blinded senior cardiologists. All measurements were averaged over 3 to 6 cardiac cycles.

The following measurements were performed according to current guidelines: [START_REF] Lang | Recommendations for cardiac chamber quantification by echocardiography in adults: an update from the American Society of Echocardiography and the European Association of Cardiovascular Imaging[END_REF] LV internal diameter, inter-ventricular septal and posterior wall thicknesses at end diastole from M-mode. LV mass was derived according to Devereux formula. LV relative wall thickness was defined as 2 times LV wall thickness divided by LV diastolic diameter, measured at end diastole. LV wall thickness was calculated from the mean of posterior and inter-ventricular septal wall thicknesses. Left atrium maximal volume was measured using biplane method from apical four and two-chamber views. LV volumes and LV ejection fraction (LVEF) were measured using the biplane Simpson method. Measurements were indexed to body surface area as appropriate. Trans-mitral E wave velocity (E) was determined by Doppler-echography.

Using pulsed tissue-Doppler mode, we acquired e', the mitral annular early diastolic velocity, on the four-chamber apical view, at septal level. Septal E/e' was then calculated and considered as a surrogate of left ventricular filling pressure. [START_REF] Salem | Impact of inodilator drugs on echocardiographic assessments of left ventricular filling pressure in patients with decompensated end-stage heart failure*[END_REF][START_REF] Salem | Impact of negative inotropic drugs on accuracy of diastolic stress echocardiography for evaluation of left ventricular filling pressure[END_REF] LV longitudinal, radial and circumferential strains were measured offline and blinded by one expert cardiologist, with the use of a standard commercially available software (Q-analysis, EchopacPC, GE-Vingmed-Ultrasound®; Horten, Norway). Radial strain (RS) and absolute-value of circumferential strain (CS) were assessed on LV short-axis images at the papillary muscle level (midventricular) with speckle-tracking analysis. For all subjects, global longitudinal strain (GLS) values were derived from the average of LV apical 4-chamber and 2-chamber views, using speckle-tracking analysis (Figure 2). Three apical views were missing, or of insufficient quality in 43/168 (25.6%) subjects, precluding use of 18 segments derived GLS in this study.

Intra-observer variabilities of LVEF, LV strains and E/e' measurements in our laboratory are excellent (intra-class correlation coefficient ≥0.83) and have been reported elsewhere. [START_REF] Salem | Impact of negative inotropic drugs on accuracy of diastolic stress echocardiography for evaluation of left ventricular filling pressure[END_REF][START_REF] Hammoudi | Subclinical left ventricular systolic impairment in steady state young adult patients with sickle-cell anemia[END_REF] Statistical analyses and power of the study This study was sufficiently powered to detect the expected sex difference in absolute value of LVEF and GLS (≈2-3%) within CAH and healthy subgroups. [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF][START_REF] Cheng | Age-and sex-based reference limits and clinical correlates of myocardial strain and synchrony: the Framingham Heart Study[END_REF][START_REF] Kocabay | Normal left ventricular mechanics by two-dimensional speckle-tracking echocardiography. Reference values in healthy adults[END_REF][START_REF] Shi | Left Ventricular Longitudinal and Circumferential Layer-Specific Myocardial Strains and Their Determinants in Healthy Subjects[END_REF] However, our study lacked power (<70%) to study sex differences for absolute values of RS and CS. [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF][START_REF] Cheng | Age-and sex-based reference limits and clinical correlates of myocardial strain and synchrony: the Framingham Heart Study[END_REF][START_REF] Kocabay | Normal left ventricular mechanics by two-dimensional speckle-tracking echocardiography. Reference values in healthy adults[END_REF][START_REF] Shi | Left Ventricular Longitudinal and Circumferential Layer-Specific Myocardial Strains and Their Determinants in Healthy Subjects[END_REF] In details, the study had a 82% power to detect a LVEF difference of at least 3% between healthy men and any other of the 3 other groups by ANOVA (alpha risk=0.05; LVEF standard-deviation=4%, expected mean LVEF=65%; considering n=25 in CAH or healthy men subgroups, and 50 in each women subgroup). This study had a 87% power to detect a GLS difference of at least |2%| between healthy men and any other of the 3 other groups by ANOVA (alpha-risk=0.05; GLS standard-deviation=|2.5%|, expected mean GLS=|20%|; considering n=25 in each men subgroup and 50 in each women subgroup).

In men with no exogenous sex hormone intake and delay <48 hours between hormonal sampling and echocardiography (n=41), the study had a power of 80% to detect a significant correlation (with r>|0.41|, alpha-risk=0.05) between hormones levels and LVEF or strain value.

We could not analyze the relations between sex hormone levels and echocardiographic parameter in the women subset, because they are characterized by very steep variation of hormonal levels during menstrual cycle, as opposed to men. [START_REF] Buffet | Regulation of the human menstrual cycle[END_REF] In fact, most echocardiography were performed at least 24 to 48 hours after blood sampling.

Results

Clinical and hormonal evaluations

The clinical characteristics of patients included in this study are shown in Table

Echocardiographic evaluations

Heart morphological parameters such as initial thoracic aortic diameter, indexed LV telediastolic volume and indexed LV mass were not different between healthy men and CAH men nor between healthy women and CAH women. However, these parameters were significantly higher in men than in women (p<0.0001, Table3).

LVEF and GLS were higher in healthy women than in healthy men (63.9±4.2% vs. 60.9±5.1%, p<0.05 and 20.0±1.9% vs. 17.9±2.4%, p<0.001, respectively, Table 3 and Figure 3) while there was no difference in LVEF or GLS between CAH women and CAH men.

LVEF was not significantly different between women (healthy or CAH) and CAH men (Figure3) while LVEF was 2.98±1.42% higher in women (healthy or CAH) than in healthy men (p<0.05, Figure3). E/e' was not significantly different among any subgroups (Table 3 andFigure3). Mean apical 4-chamber and 2-chamber LS were used to calculate GLS for all subjects in this study rather that the classical 18 segments derived GLS also taking into consideration apical 3-chamber, which was not computable in 43 patients. Of note, in the subset of patients in whom conventional 3 views derived GLS was available (n=125), results

presented in Table 3 were similar for those of GLS derived from mean 4 and 2 chambers.

Correlation between LV systolic function and sex hormones in men

Among the 52 men included in this study, 11 were excluded from analyses of correlations between echocardiographic parameters and hormonal levels, leaving 41 men (20 CAH) for final analysis. Reason of exclusion were exogenous intake of hormonal drugs (n=2)

and excessive delay (>48h) between echocardiography and hormonal assays (n=9). According to univariable analysis in all available men (Supplemental Figure1), LVEF and GLS were negatively correlated with bioavailable-testosterone (r=-0.3, p≤0.05, n=41 and r=-0.45, p<0.01, n=39, respectively), while mid-ventricular radial strain (RS) was positively correlated with bioavailable-testosterone (r=0.38, p<0.05, n=40). LVEF, GLS and RS were not significantly associated with any other hormones sampled in this study, including ACTH and renin levels (Supplemental Table 1). Absolute value of CS was positively correlated to FSH (r=0.65, p<0.0001, n=40) and negatively correlated to progesterone (r=-0.33, p<0.05, n=40) (Supplemental Table 1). LVEF, GLS, RS and CS were not significantly associated to age, neither to systolic or diastolic blood pressure.

Multivariable analysis (ANCOVA) was performed in these men, integrating hormonal levels (bioavailable-testosterone for LVEF, GLS and RS; progesterone and FSH for CS), and disease status (healthy or CAH). Bioavailable-testosterone remained negatively associated to LVEF (r=-0.27, p≤0.05) and GLS (r=-0.43, p<0.01) while bioavailable-testosterone was positively related to radial strain (r=0.35, p<0.05) with no significant influence of CAH or healthy status. Additionally, FSH was still related to CS (r=0.67, p<0.0001).

Discussion

The objective of this work was to investigate the association of sex, CAH disease and interactions between gonadotropins and sex steroid hormones on LV morphology and systolic function parameters in humans. Our study in adult healthy subjects and in CAH patients, overexpressing testosterone in women and under expressing testosterone in some men, is unique in that subjects were exposed to a wide range of sex steroid hormones and gonadotropins levels. We found that: 1/there is no major difference in heart function and morphology between well treated CAH and healthy adults; 2/In CAH patients, known sex differences persist for aortic size and LV volume but are blunted for LV systolic function. This finding seemed to be associated to sex hormones disturbance among CAH patients; 3/In men, we found that bioavailable-testosterone levels were correlated negatively with LVEF and absolute GLS, and positively with RS. Absolute CS correlated positively with FSH levels.

Complex hormonal system regulating cardiac function

Sex dimorphism concerning LV strain values reflecting the different component of cardiac systolic function have been well established by use of echocardiographic speckle-tracking technology [START_REF] Cheng | Age-and sex-based reference limits and clinical correlates of myocardial strain and synchrony: the Framingham Heart Study[END_REF][START_REF] Kocabay | Normal left ventricular mechanics by two-dimensional speckle-tracking echocardiography. Reference values in healthy adults[END_REF][START_REF] Shi | Left Ventricular Longitudinal and Circumferential Layer-Specific Myocardial Strains and Their Determinants in Healthy Subjects[END_REF] and later, by magnetic-resonance-imaging. [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF] Radial component of systolic function is higher in men than in women, while longitudinal and circumferential component of systolic function are higher in women than in men, [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF] resulting in lower LVEF in men than in women. [START_REF] Shi | Left Ventricular Longitudinal and Circumferential Layer-Specific Myocardial Strains and Their Determinants in Healthy Subjects[END_REF] Our findings confirm these data and suggest that the higher level of bioavailabletestosterone in men contribute to the observed sexual dimorphism, being associated to increased radial strain and decreased longitudinal strain.

Sex steroid hormones secretions are constantly regulated by the hypothalamic-pituitary axis and, as occurs in other organs such as ovaries, testes or bones, [START_REF] Wang | Follicle-Stimulating Hormone Increases the Risk of Postmenopausal Osteoporosis by Stimulating Osteoclast Differentiation[END_REF] the association of sex steroid hormones on LV function may be counterbalanced by the direct effects of gonadotropins. The rationale for a complex hormonal system regulating heart function is supported by the fact that RNA of sex steroids and gonadotropins receptors are expressed in the myocardium [START_REF] Salem | Influence of steroid hormones on ventricular repolarization[END_REF][START_REF] Smet | Sex steroids in relation to cardiac structure and function in men[END_REF] and it has been recently found positive associations between FSH and duration of cardiac repolarization in both sex. [START_REF] Abehsira | Complex Influence of Gonadotropins and Sex Steroid Hormones on QT Interval Duration[END_REF][START_REF] Canpolat | The association of premature ovarian failure with ventricular repolarization dynamics evaluated by QT dynamicity[END_REF] In our study, the positive association between FSH levels and CS is a new finding, further supporting the potential of gonadotrophins to alter myocardial function. We can hypothesize different effects of FSH and testosterone on cardiac systolic function depending on the multilayered cardiac fiber orientation constituted of cardiomyocytes with different electro physio-mechanical properties. [START_REF] Shi | Left Ventricular Longitudinal and Circumferential Layer-Specific Myocardial Strains and Their Determinants in Healthy Subjects[END_REF][START_REF] Antzelevitch | M cells in the human heart[END_REF][START_REF] Sengupta | Left ventricular structure and function: basic science for cardiac imaging[END_REF] M-cells, in mid myocardium, between the endocardial and epicardial layers of the heart, are histologically similar to epicardial and endocardial cells but electro physiologically and pharmacologically different. [START_REF] Antzelevitch | M cells in the human heart[END_REF] M-cells appear to be hybrid between Purkinje and ventricular cells and are involved in alteration of ventricular repolarization heterogeneity, a process known to be highly sex hormones sensitive. [START_REF] Salem | Influence of steroid hormones on ventricular repolarization[END_REF] Tpeak-Tend duration, a validated electrocardiographic index of repolarization heterogeneity, [START_REF] Salem | GENomE wide analysis of sotalol-induced IKr inhibition during ventricular REPOLarization, "GENEREPOL study": Lack of common variants with large effect sizes[END_REF][START_REF] Antzelevitch | T peak-Tend interval as an index of transmural dispersion of repolarization[END_REF] has been recently and repeatedly associated with alteration of cardiac mechanical function. [START_REF] Oskouie | Differences in Repolarization Heterogeneity Among Heart Failure With Preserved Ejection Fraction Phenotypic Subgroups[END_REF][START_REF] Sauer | Diastolic electromechanical coupling: association of the ECG T-peak to T-end interval with echocardiographic markers of diastolic dysfunction[END_REF] Leren et al.

have shown that patients with cardiac congenital electrophysiological defects, such as patients with long QT syndrome, have reduced systolic function, also evaluated by strain. [START_REF] Leren | Cardiac Mechanical Alterations and Genotype Specific Differences in Subjects With Long QT Syndrome[END_REF]

Effect of testosterone administration in HF

Several small sample studies have evaluated if, compared to placebo, testosterone improved exercise capacity in men [START_REF] Caminiti | Effect of long-acting testosterone treatment on functional exercise capacity, skeletal muscle performance, insulin resistance, and baroreflex sensitivity in elderly patients with chronic heart failure a double-blind, placebo-controlled, randomized study[END_REF] or in women [START_REF] Toma | Testosterone supplementation in heart failure: a meta-analysis[END_REF] with stable chronic HFrEF [START_REF] Iellamo | Testosterone therapy in women with chronic heart failure: a pilot double-blind, randomized, placebocontrolled study[END_REF][START_REF] Malkin | Testosterone therapy in men with moderate severity heart failure: a double-blind randomized placebo controlled trial[END_REF] . Testosterone appeared to be a promising therapy to improve peak-oxygen consumption, 6-minute and shuttle walk-test, but there was no significant variation in LVEF. [START_REF] Iellamo | Testosterone therapy in women with chronic heart failure: a pilot double-blind, randomized, placebocontrolled study[END_REF][START_REF] Chung | Effects of testosterone and nandrolone on cardiac function: a randomized, placebo-controlled study[END_REF] The overall beneficial effect of testosterone in HF is currently attributed to peripheral vasodilatation, improvement in general metabolism with increase of lean body and muscle mass, increase of hemoglobin and hematocrit levels and to a decrease in insulin resistance rather than to improvement of LV morphology and inotropy [START_REF] Caminiti | Effect of long-acting testosterone treatment on functional exercise capacity, skeletal muscle performance, insulin resistance, and baroreflex sensitivity in elderly patients with chronic heart failure a double-blind, placebo-controlled, randomized study[END_REF][START_REF] Iellamo | Testosterone therapy in women with chronic heart failure: a pilot double-blind, randomized, placebocontrolled study[END_REF][START_REF] Malkin | Testosterone therapy in men with moderate severity heart failure: a double-blind randomized placebo controlled trial[END_REF][START_REF] Toma | Testosterone supplementation in heart failure: a meta-analysis[END_REF] . Of note, supra-therapeutic doses of anabolic-steroids, with chemical structure and properties related to T, have been associated with a reduced cardiac function and adverse cardiac outcomes. [START_REF] Sullivan | The cardiac toxicity of anabolic steroids[END_REF][START_REF] Parker | Anabolic-androgenic steroids: worse for the heart than we knew?[END_REF][START_REF] Baggish | Long-term anabolic-androgenic steroid use is associated with left ventricular dysfunction[END_REF] Zwadlo et al., have recently demonstrated that finasteride, an anti-androgenic drug currently used in patients to treat prostate disease, reversed pathological cardiac hypertrophy and dysfunction in mice and was therefore proposed as a possible therapeutic option for heart failure in men. [START_REF] Zwadlo | Antiandrogenic therapy with finasteride attenuates cardiac hypertrophy and left ventricular dysfunction[END_REF] It was suggested that lower levels of androgens in women might explain why they have a better prognosis when experiencing aortic valve stenosis, hypertrophic cardiomyopathy, or heart failure, in comparison to men. [START_REF] Zwadlo | Antiandrogenic therapy with finasteride attenuates cardiac hypertrophy and left ventricular dysfunction[END_REF][START_REF] Wu | Influence of sex on ventricular remodeling after myocardial infarction in mice[END_REF] Our findings are in line with the absence of a direct positive effect of testosterone on LVEF and further support the extra-cardiac effect of testosterone to explain symptoms improvement in HF patients. [START_REF] Budoff | Testosterone Treatment and Coronary Artery Plaque Volume in Older Men With Low Testosterone[END_REF]

Cardiac risk in CAH patients

In the classical form of CAH, steroid supplementation is mandatory to prevent adrenal crisis and suppress androgen excess. [START_REF] Bachelot | MANAGEMENT OF ENDOCRINE DISEASE: Congenital adrenal hyperplasia due to 21-hydroxylase deficiency: update on the management of adult patients and prenatal treatment[END_REF] However, the therapeutic window is narrow, and CAH infants with steroid deficiency may develop a form of cardiomyopathy which is reversible after corticosteroid treatment. [START_REF] Minette | Cardiac function in congenital adrenal hyperplasia: a pattern of reversible cardiomyopathy[END_REF] Supra-physiological doses of corticosteroid are often used to suppress androgen overproduction, and this may lead to iatrogenic Cushing's syndrome, a condition that is associated with insulin resistance, hypertension, and increased cardiac morbidity. [START_REF] Tritos | Management of Cushing disease[END_REF],

In our cross-sectional cohort comparing CAH adults and age-matched healthy controls, we did not find any difference in heart morphology and systolic function at rest, between CAH and healthy subjects within sex groups. The sample size of this echocardiographic study in CAH patient is the largest reported so far. Our study supports the findings of other limited sample size studies [START_REF] Marra | Cardiovascular abnormalities and impaired exercise performance in adolescents with congenital adrenal hyperplasia[END_REF][START_REF] Green-Golan | Patients with classic congenital adrenal hyperplasia have decreased epinephrine reserve and defective glycemic control during prolonged moderate-intensity exercise[END_REF][START_REF] Weise | Patients with classic congenital adrenal hyperplasia have decreased epinephrine reserve and defective glucose elevation in response to high-intensity exercise[END_REF] that well treated and monitored CAH patients do not develop important morphological heart abnormalities at rest.

Limitations

Women of child-bearing age, as opposed to men, are characterized by steep variation of hormonal levels during menstrual cycle [START_REF] Buffet | Regulation of the human menstrual cycle[END_REF] , precluding the realization of any association study with any phenotypic evaluation if hormonal sampling is not realized concomitantly. A dedicated study is needed to evaluate whether and how hormones may influence the different component of LV function in women.

One could also wonder if sex differences in LVEF found in our study could be explained by sex-related differences in LV volumes; although the data in Table 3 and from other large cohorts such as MESA and Framingham show that stroke volume in males are proportionally higher than in females. [START_REF] Liu | Healthy aging of the left ventricle in relationship to cardiovascular risk factors: The Multi-Ethnic Study of Atherosclerosis (MESA)[END_REF][START_REF] Chuang | CMR reference values for left ventricular volumes, mass, and ejection fraction using computeraided analysis: the Framingham Heart Study[END_REF] Of note, correlations found between hormonals levels and left systolic function parameters were significant but such amplitude of correlations are not expected to be relevant from a diagnostic perspective, but they do support a physiologic concept.

Conclusion.

Our data support that existence of sex dimorphism concerning left ventricular systolic cardiac function is significantly associated to testosterone levels.

Figures legends

Tables

  CAH patients were referred to the study by three endocrinology unit (Pitié-Salpêtrière hospital, Paris; Saint-Antoine Hospital, Paris; Bicêtre Hospital, Le Kremlin-Bicêtre, France). CAH and healthy subjects were explored at the Centre d'Investigation Clinique Paris-Est (CIC-1421, Pitié-Salpétrière hospital, Paris).

Figure 1 .

 1 Figure 1. Schematic representation of patho-physiological processes (chronological steps

Figure 2 .

 2 Figure 2. Examples of measurement of apical 4 chamber longitudinal strain (A), apical 2

Figure 3 .

 3 Figure 3. Two-way ANOVA results as a function of sex and disease status (CAH and

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3.

  Multivariable analysis was performed by ANCOVA (XLSTAT-software, Addinsoft®). Explanatory variables were selected to be tested in multivariable analysis only if they were associated to the dependent variable in univariate analyses, except for CAH status which was always tested. Best model

	was selected driven by Schwarz's Bayesian criterion. Statistical significance was accepted for
	p≤0.05.

Data are reported as mean ± standard-deviations or median and interquartile range, as appropriate. Comparisons of quantitative variables were made by using Wilcoxon-paired ttest, Mann-Whitney tests, ANOVA (one-way, or two-way) with Tukey's or Sidak's post-test, as appropriate. Qualitative variables were compared by using Chi2-test. Correlations between linear variables were assessed by calculating Pearson's or Spearman's coefficient (r), as appropriate by using Prism6 (GraphPad-software®, San-Diego, USA).

Table 1 . Clinical characteristics of the patients included in the study Abbreviations

 1 * Significant compared to CAH females; ‡ Significant compared to healthy males; ‡ ‡ Significant compared to CAH males.

		CAH males	Healthy males CAH females	Healthy	p
					females
	Number of subjects	26	26	58	58
	Age (years)	28.8±7.9	28.7±8.5	30.7±9.1	30.9±9	ns
	Body surface area (m²)	1.80±0.19 **, ‡ 1.90±0.15 *,**, ‡ ‡ 1.63±0.17 *, ‡, ‡ ‡ 1.72±0.13 **, ‡ <0.0001
	BMI >35 kg/m²	0	0	3(5%)	0	ns
	SBP (mmHg)	117.4±9.3 *,**	118.9±8.6 *,**	110.0±14 ‡, ‡ ‡ 108.5±10.2 ‡, ‡ ‡ <0.0001
	DBP (mmHg)	69.1±9.8	69.6±7.2	68.2±12.3	68.3±7.9	ns
	Treated hypertension	0(0%)	0(0%)	0(0%)	0(0%)	ns
	Cardiovascular disease in family (%)	0(0%)	1(4%)	1(2%)	5(9%)	ns
	Current smoker / Past or non-	7 / 19	7 / 19	16 / 42	16 / 42	ns
	smoker	(27/73%)	(27/73%)	(31/69%)	(31/69%)
	Treated dyslipidemia or	0(0%)	0(0%)	2(3%)	0(0%)	ns
	diabetes
	Fasting glycemia	0(0%)	0(0%)	0(0%)	1(2%)	ns
	≥7mmol/l
	Fludrocortisone (µg/day)	100 [50-100] **	NA	50[0-94] ‡ ‡	NA	<0.01
	Equivalent of hydrocortisone	30[20-40] **	NA	20[10-25] ‡ ‡	NA	<0.001
	(mg/day)

: BMI: body mass index; bpm: beats per minute; CAH: Congenital adrenal hyperplasia; cm: centimeter; kg: kilograms; SBP: systolic blood-pressure; DBP: diastolic blood-pressure; HR: heart rate; msec: millisecond; NA: not applicable Statistics: ANOVA and Tukey's or Sidak's post-test or Mann-Whitney test for quantitative variables. Chi2 for qualitative variables. Not significant (ns) if p≥0.05. Quantitative variables are represented by the mean ± standard deviation or median and interquartile range [-]. * Significant compared to healthy females; *

Table 2 . Biological evaluations in females and males

 2

	FEMALES
		CAH females	Healthy females	p
		n: 58	n: 58
	17-OH progesterone (ng/mL)	14.7[4.4-50.8]	1.2[0.7-2.1]	<0.0001
	Progesterone (ng/mL)	2.7[1-8.5]	0.9[0.5-3]	<0.01
	SHBG (nmol/L)	63[31-91]	64[54 -87]	ns
	Bioavailable-testosterone (ng/mL)	0.16[0.04-0.3]	0.08[0.04-0.14]	<0.001
	Estradiol (pg/mL)	65[42-162]	87[40-170]	ns
	FSH (UI/L)	5[3.2-6.4]	6.4[4.1-8.3]	<0.01
	LH (UI/L)	5.5[2.8-8.3]	6.7[5.1-12.8]	<0.01
	Renin (pg/mL)	19.3[13.4-44]	11.3[7.6-16.8]	<0.001
	ACTH (pg/mL)	30.3[9-67.3]	16.5[10.8-24.6]	<0.0001
	MALES
		CAH males	Healthy males	p
		n: 26	n: 26
	17-OH progesterone (ng/mL)	15.1[4.0-44.3]	2[1.6-2.4]	<0.0001
	Progesterone (ng/mL)	1.0[0.4-3.6]	0.8[0.6-1]	<0.05
	SHBG (nmol/L)	36.2[28.2-55]	40.2[24.5-46.3]	ns
	Bioavailable-testosterone (ng/mL)	2.3[1.3-3]	2.9[2.5-3.4]	<0.05
	Estradiol (pg/mL)	29.5[19.8-32.6]	28.3[24.7-37.3]	ns
	FSH (UI/L)	3.9[2.3-7]	3.4[2.6-5.2]	ns
	LH (UI/L)	4.8[2-5.8]	4.5[3.5-5.3]	ns
	Renin (pg/mL)	26.0[15.9-44.6]	13.1[10-16.5]	<0.01
	ACTH (pg/mL)	44.9[18-161.8]	27.2[14.8-38]	<0.01

Abbreviations: ACTH: Adrenocorticotropic hormone; FSH: follicle-stimulating hormone; LH: luteinizing hormone; SHBG: Sex hormone-binding globulin Statistics: Wilcoxon-paired t-test for quantitative variables. Quantitative variables are represented by median and interquartile range [-]. Not significant (ns) if p≥0.05.

Table 3 . Echocardiographic characteristics of the patients included in the study

 3

		Healthy females	CAH females	CAH males	Healthy males	p interaction	p sex	p disease
		n=58	n=58	n=26	n=26	(sex-disease)	influence	influence
	Ascending aorta(mm)	29.2±3.9 ‡, ‡ ‡	28.8±2.7 ‡, ‡ ‡	32.6±2.7 *,**	33.0±3.1 *,**	ns	<0.0001	ns

This study was supported by the French institutional PHRC 2010-A00824-35 funding source and APHP, CIC Paris-Est.

LV morphology

LV end-diastolic volume index (ml/m²) 48.8±8.5 ‡, ‡ ‡ 47.3±10.7 ‡, ‡ ‡ 55.9±12.3 [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF] E/e' 6.6±1.9 [START_REF] Kocabay | Normal left ventricular mechanics by two-dimensional speckle-tracking echocardiography. Reference values in healthy adults[END_REF] 6.5±1.4 6.4±1.5 [START_REF] Hayward | Effect of sex hormones on cardiac mass[END_REF] 6.0±1.4 ns ns ns

LV systolic function

LV Ejection fraction (%) 63.9±4.2 ‡ 63.9±4.5 ‡ 63.0±4.6 [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF] 60.9±5.1 *,** ns <0.05 ns 4C longitudinal strain (%) -19.9±2.2 [START_REF] Cheng | Age-and sex-based reference limits and clinical correlates of myocardial strain and synchrony: the Framingham Heart Study[END_REF] ‡, ‡ ‡ -18.9±2.4 [START_REF] Kocabay | Normal left ventricular mechanics by two-dimensional speckle-tracking echocardiography. Reference values in healthy adults[END_REF] ‡ -17.9±2.0 [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF] * -17.2±2.6 [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF] *,** ns <0.0001 ns 2C longitudinal strain (%) -20.6±2.2 [START_REF] Hayward | Effect of sex hormones on cardiac mass[END_REF] ‡, ‡ ‡ -19.8±2.4 [START_REF] Kocabay | Normal left ventricular mechanics by two-dimensional speckle-tracking echocardiography. Reference values in healthy adults[END_REF] -18.7±2.1 [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF] * -18.8±2.7 [START_REF] Hayward | Effect of sex hormones on cardiac mass[END_REF] * ns <0.001 ns Global longitudinal strain (%) -20.0±1.9 [START_REF] Hayward | Effect of sex hormones on cardiac mass[END_REF] ‡, ‡ ‡ -19.4±2.2 [START_REF] Kocabay | Normal left ventricular mechanics by two-dimensional speckle-tracking echocardiography. Reference values in healthy adults[END_REF] ‡ -18.3±1.8 [START_REF] Andre | Age-and genderrelated normal left ventricular deformation assessed by cardiovascular magnetic resonance feature tracking[END_REF] * -17.9±2.4 [START_REF] Hayward | Effect of sex hormones on cardiac mass[END_REF]

Supplemental Figure 1. Correlations in men between levels of bioavailable-testosterone and LVEF (A), absolute value of global longitudinal strain (B), and mid-ventricular radial strain (C). Correlations between levels of FSH and mid-ventricular absolute value of circumferential strain (D).