Electre Tri-C , a multiple criteria decision aiding sorting model applied to assisted reproduction

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journal homepage: www.intl.elsevierhealth.com/journals/ijmi

Electre Tri-C, a multiple criteria decision aiding sorting model applied to assisted reproduction J.R. Figueira a,1, J. Almeida-Dias b,c,∗, S. Matias d, B. Roy b, M.J. Carvalho e, C.E. Plancha e,f a

INPL, École des Mines de Nancy, Laboratoire LORIA, Nancy, France LAMSADE, Université Paris-Dauphine, Paris, France c CEG-IST, Instituto Superior Técnico, Universidade Técnica de Lisboa, Lisbon, Portugal d Instituto Superior Técnico, Universidade Técnica de Lisboa, Lisbon, Portugal e CEMEARE, Centro Médico de Assistência à Reproduc¸ão, Lisbon, Portugal f Unidade Biologia da Reproduc¸ão, Instituto Histologia e Biologia do Desenvolvimento, Faculty of Medicine, University of Lisbon, Lisbon, Portugal b

a r t i c l e

i n f o

a b s t r a c t

Article history:

Objective: The aim of this paper is to apply an informatics tool for dealing with a medical

Received 9 May 2010

decision aiding problem to help infertile couples to become parents, when using assisted

Received in revised form

reproduction.

17 September 2010

Methods: A multiple criteria decision aiding method for sorting or ordinal classification prob-

Accepted 18 December 2010

lems, called Electre Tri-C, was chosen in order to assign each couple to an embryo-transfer category. The set of categories puts in evidence a way for increasing the single pregnancy rate, while minimizing multiple pregnancies. The decision aiding sorting model was co-

Keywords:

constructed through an interaction process between the decision aiding analysts and the

Infertility

medical experts.

Assisted reproductive technology

Results: According to the sample used in this study, the Electre Tri-C method provides a

Risk of multiple pregnancies

unique category in 86% of the cases and it achieves a sorting accuracy of 61%. Retrospec-

Multiple criteria decision aiding

tively, the medical experts do agree that some of their judgments concerning the number of

Sorting problems

embryos to transfer back to the uterus of the woman could be different according to these

Electre Tri-C

results. The current ART methodology achieves a single pregnancy rate of 47% and a twin pregnancy rate of 14%. Thus, this informatics tools may play an important role for supporting ART medical decisions, aiming to increase the single pregnancy rate, while minimizing multiple pregnancies. Limitations: Building the set of criteria comprises a part of arbitrariness and imperfect knowledge, which require time and expertise to be refined. Among them, three criteria are modeled by means of a holistic classification procedure by the medical experts. © 2011 Elsevier Ireland Ltd. All rights reserved.

∗ Corresponding author at: LAMSADE, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, F-75 775 Paris Cedex 16, France. Tel.: +33 1 44 05 42 87; fax: +33 1 44 05 40 91. E-mail address: [email protected] (J. Almeida-Dias). 1 Part of this Work was accomplished when J.R. Figueira was a permanent member of CEG-IST, Lisbon. 1386-5056/$ – see front matter © 2011 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.ijmedinf.2010.12.001

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1.

Introduction

The application of informatics tools for medical decision aiding problems is not new, and it has been increasing tremendously over the last years. There are applications in cancer care [1–5], decision tools for medical diagnosis [6–10], among others. For a comprehensive survey of the application of multiattribute techniques and other decision aiding tools in health care and medical treatment, see [11–14]. However, to our best knowledge there are no multiple criteria informatics tools applied to medical decision problems for dealing with Assisted Reproductive Technology (ART). When a couple fails to conceive, the source of the problem must be investigated in both man and woman. The problem may be related only to one of the couple members, to both or, apparently to none of them. According to several multidisciplinary medical evaluations, a procedure is chosen to try to overcome the problem. The most effective procedures in ART are in vitro fertilization (IVF) and intracytoplasmic sperm injection (ICSI). These techniques allow in most cases to obtain pre-implantation embryos in culture. A difficult task at this point is to decide on the adequate number of embryos to transfer back into the uterus. On the one hand, we want to obtain high pregnancy rates, but on the other hand, we want to minimize the risk of a multiple pregnancy. Thus, this decision is usually based on the clinician’s point of view, that includes the past history of the patient with such information as indications for ART, previous cycles of ART, prior pregnancies, use of donor gametes/embryos, as well as couple’s values, and based on the point of view of the embryologist that includes the quality of sperm (based on morphology, motility, and concentration of sperm in the ejaculate), the quality of oocytes, and the quality of embryos. Thus, we are dealing with a difficult decision aiding problem [15]. An essential part of this approach is related to the structuring process of building criteria, since each point of view is composed of a large set of heterogeneous elementary consequences (aspects, characteristics, attributes, exams, etc.), that should be associated in order to form homogeneous groups of dimensions (elementary consequences, “measured” through the application of a descriptor, a metric, or a “primary scale”). The emergence of these groups will then allow the construction of a coherent set of criteria for operationalizing the several points of view. The aim of this study is thus to apply a sorting procedure in order to reach a recommendation concerning the number of embryos to be transferred back to the uterus of the woman, taking into account the preferences of the embryologist when defining the parameters of the model. The choice of Electre Tri-C[16] method among the different tools available for sorting problems is related to the fact that this medical problem led to the definition of the categories by characteristic reference couples instead of boundary couples or other norms. Electre Tri-C was particularly designed for such a type of problems.

2.

263

Methods

This section presents the main concepts, definitions, and notation with respect to the chosen sorting method, Electre Tri-C[16], including the main options of the ART application structuring process.

2.1.

Concepts, definition, and notation

As stated by [17], over the last 15 years, the need for wellfounded methodologies is of the uttermost importance in medical informatics. In what follows, we will describe our methodology in a comprehensive way. This means that we try to overcome the drawbacks frequently found in medical informatics tools and decision support systems, where medical professionals have no knowledge about the underlying methodology [18]. In our case, a co-constructive interactive process between the analysts and the medical experts allowed to make the latter familiar with the methodological approach. Let A = {a1 , a2 , . . ., ai , . . . } be the set of potential actions, i.e. the set of couples (women/men) to be subjected to an ART treatment. This set can be completely known a priori or it may appear progressively during the decision aiding process. The couples are evaluated on a coherent set of criteria [19], denoted F = {g1 , g2 , . . ., gj , . . ., gn }, with n ≥ 3. Therefore, gj (a) represents the performance of the couple a according to the criterion gj , j = 1, . . ., n. When using the Electre Tri-C method [16], the objective is to assign the couples to a set of completely ordered categories, denoted {C1 , C2 , . . ., Ch , . . ., Cq }, with q ≥ 2. Assume that C1 represents the worst category and Cq represents the best category. Each criterion gj must be either associated with an increasing preference direction or a decreasing preference direction. In the latter case, it means that the preferences increase when the performances decrease, and in the former case, it means that the preferences increase when the performances increase too. Moreover, each criterion is also associated with a completely ordered preference scale which contains all the possible performances of a couple according to such a criterion taking into account the decision aiding ART sorting context. The criteria are also associated with two discriminating thresholds (called indifference and preference thresholds, denoted qj and pj , respectively, j = 1, . . ., n). These thresholds are relevant to take into account the imperfect character of the performances of each couple as well as some arbitrariness when building the set of criteria. The aggregation of the performances of each couple is obtained by making use of the so-called “power of the criteria”, which is defined by the relative importance coefficients, or weights, denoted wj , j = 1, . . ., n, and, optionally, the veto thresholds, denoted vj , j = 1, . . ., n, which is used to manage critical values on a certain criterion, which appear risky when submitted to a certain medical treatment.

2.2.

An overview of Electre Tri-C

Electre Tri-C[16] was designed to be used within the framework of a constructive approach. This decision aid-

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Table 1 – The set of criteria. Code

Description of the criteria

Nature of the scale of the criteria

g1 g2 g3

Age of the woman Duration of infertility Number of oocytes retrieved a

g4 g5

Evaluation of the women Sperm origin

g6 g7

Morphological quality of the four best embryos (in the transfer day) Developmental quality of the four best embryos

a b

c

d

Years (integer) Years (integer) (1) If Or = 0 (2) If Or = − 1 or Or = 1 (3) If Or = − 2 or Or ≥ 2 (4) If Or = − 3 or Or = − 4 (5) If Or = − 5 or Or = − 6 (6) If Or = − 7 or Or = − 8 (7) If Or ≤ − 9 Holistic classification model A b (1) Testicular biopsy cryopreserved sperm (2) Cryopreserved ejaculated sperm (3) Fresh ejaculated sperm Holistic classification model B c Holistic classification model C d

Preference Decreasing Decreasing Decreasing

Increasing Increasing

Increasing Increasing

Or is the effective number of oocytes retrieved minus the ideal number (which is 12). A holistic classification made by the gynaecologist/obstetrician according to seven categories, denoted {1, 2, 3, 4, 5, 6, 7}, where 1 is the worst category and 7 is the best one. An overall morphological classification of the four best embryos which is obtained by the sum of the individual score of each embryo previously evaluated by the embryologist according to five points, denoted {1, 2, 3, 4, 5}, where 1 is the worst value and 5 is the best one. A holistic classification of the whole set of four embryos by the embryologist according to five categories, denoted {1, 2, 3, 4, 5}, where 1 is the worst category and 5 is the best one.

ing sorting method must be applied in the contexts where the categories are completely ordered (from the worst to the best, for instance). Each category must be defined a priori to receive actions (e.g. couples), which will be or might be processed in the same way (at least in a first step). The definition of each category is based on a unique characteristic reference couple, which is the most representative taking into account the next processing operations (e.g. transfer the same number of embryos). The Electre Tri-C assignment results are based on the outranking credibility indices (see Appendix A) which are compared to a chosen credibility level, denoted . This level is a minimum degree of credibility, (a,a ), which is considered or judged necessary by the medical experts to validate or not the statement “a outranks a ” (meaning that a is at least as good as a ) taking all the criteria from F into account. In general, this minimum credibility level takes a value within the range [0.5,1] and it can roughly be interpreted as a majority level as in the voting theory. In order to preserve the role of the characteristic reference couples, Electre Tri-C makes use of a selecting function, denoted (a,a ). This function allows to choose between two consecutive selected categories and it can be defined in several ways. In this ART application the following selecting function is going to be used: (a,a ) = min {(a,a );(a ,a)}. Definitions 1 and 2 present the Electre Tri-C assignment procedure, which is composed of two joint rules, called the descending rule and the ascending rule, respectively (which must be used conjointly and not separately). Definition 1 (Descending rule). Choose a credibility level, (1/2 ≤  ≤ 1). Decrease h from (q + 1) until the first value, t, such that (a,bt ) ≥ :

(a) For t = q, select Cq as a possible category to assign action a. (b) For 0 < t < q, if (a,bt ) > (a,bt+1 ), then select Ct as a possible category to assign a; otherwise, select Ct+1 . (c) For t = 0, select C1 as a possible category to assign a. Definition 2 (Ascending rule). Choose a credibility level, (1/2 ≤  ≤ 1). Increase h from zero until the first value, k, such that (bk ,a) ≥ : (a) For k = 1, select C1 as a possible category to assign action a. (b) For 1 < k < (q + 1), if (a,bk ) > (a,bk−1 ), then select Ck as a possible category to assign a; otherwise, select Ck−1 . (c) For k = (q + 1), select Cq as a possible category to assign a. Each one of these rules selects only one category for a possible assignment of a couple. They are used conjointly in order to highlight the highest category and the lowest category, which can appear potentially appropriate to receive a couple. These two extreme categories can be the same. When they differ, this means that the assignment of such a couple remains illdetermined within a range of possible categories taking into account the way that the set of characteristic couples defines the categories.

2.3.

Modeling the set of criteria

In this ART application, the couples (women/men) were evaluated on a set of seven criteria (see Table 1) based on a set of chosen medical exams, co-constructed through an interaction process between the analysts and the medical experts (gynaecologists/obstetricians, embryologists, biologists, physicians, psychologists). For modeling the set of criteria, only biological data (the history of the couple and the quality of the embryos) are considered within the decision aiding assignment model. In addition to the main options presented in Table 1, according

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Table 2 – Performance of the characteristic actions/couples. Ch

Nature of the category

bh

Criteria g1

C1 C2 C3 C4

Transfer of four embryos Transfer of three embryos Transfer of two embryos Transfer of one embryo

b1 b2 b3 b4

g2

42 38 33 25

10 6 3 1

g3

g4

g5

g6

g7

7 5 4 1

1 2 5 6

1 2 3 3

6 10 16 19

1 2 4 5

Table 3 – Definition of the discriminating thresholds. Thresholds

Indifference Preference

Criteria g1

g2

g3

g4

g5

g6

g7

q1 (g1 (bh )) p1 (g1 (bh ))

q2 (g2 (bh )) p2 (g2 (bh ))

0 1

0 1

0 0

q6 (g6 (bh )) p6 (g6 (bh ))

0 0

to the set of criteria, we have [15]: – g1 : this criterion is relevant because the younger the woman is, the more favorable the infertility treatment prognostic; – g2 : this criterion is relevant because the longer the duration of infertility is, the worst the infertility treatment prognostic; – g3 : this criterion is relevant in order to take into account the Or (number of oocytes retrieved) differences. For the same difference, an effective number of oocytes retrieved lower than 12 is worst than an effective number of oocytes greater than 12; – g4 : this is a very complex criterion which was modeled by a holistic classification procedure made by the gynaecologist/obstetrician, when taking into account several dimensions as well as the experience of the medical experts; – g5 : this criterion is related to the quality of the sperm and it represents the only male input of the decision aiding assignment model; – g6 and g7 : these two criteria are also very complex ones and they were modeled through a holistic classification procedure made by the embryologist, when taking into account several dimensions as well as the experience of the medical experts. As stated above, the criteria g4 , g6 , and g7 were modeled through a holistic classification procedure (models A, B, and C) by simplifying a very complex task. Such a modeling simplification was intuitively validated by the medical experts from their large past medical experience.

2.4.

Modeling the set of categories

The objective of the decision aiding sorting model is to give a “recommendation” to the medical experts about the number of embryos to be transferred back to the uterus of the woman in order to obtain a pregnancy and, at the same time, to reduce the risk of multiple pregnancies. Of course the final decision should be made by the couples themselves, but the “medical experts’ recommendation” is very important. The nature of the set of categories is, therefore, related to the transfer of embryos: four embryos (denoted C1 ), three embryos (denoted C2 ), two embryos (denoted C3 ), and one embryo (denoted C4 ). These categories are clearly ordered from

the least risky category, C4 , to the most risky category, C1 . Furthermore, the less the number of embryos to transfer is, the better the category is in terms of the risk of multiple pregnancies, but the possibility of achieving a pregnancy can simultaneously decrease [15]. The set of categories was defined by characteristic reference couples, denoted bh , h = 1, . . ., 4 (see Table 2). These characteristic couples were defined according to the main characteristics of reference couples through a co-construction process between the analysts and the medical experts, while taking also into account the medical experience of the embryologists and the gynaecologists/obstetricians.

2.5. Modeling the imperfect knowledge and ill-determination As stated above, the set of criteria is built from a set of chosen medical exams and the characteristics of the couples. In this application variable discriminating thresholds are used for the criteria g1 , g2 , and g6 , and constant ones for the remaining criteria (see Table 3). The indifference threshold, qj , is the maximal advantage of a couple over another couple according to the criterion gj , which is compatible with the statement “the two couples are indifferent”. The preference threshold, pj , is the minimal positive advantage of a couple over another couple according to the criterion gj to be passed for being convincing for the strict preference of the first couple. When taking only the criterion g1 into account, we assume that a 28 years old woman should be subjected to an ART treatment more similar to a 25 years old reference woman than to a 33 years old reference woman. In such a case, one embryo instead of two embryos may be transferred back to the uterus of that woman. This means that the younger the woman is, the more favorable the ART treatment prognostic is, and less embryos must be transferred to the uterus of such a woman [15]. The variable discriminating thresholds presented in Table 4 must be interpreted as follows:

– Any woman and the younger 25 years old reference woman are indifferent if the difference of their ages is at most five years, q1 (g1 (b4 )) = 5. On the contrary, any woman and the older 25 years old reference woman are indifferent if the difference of their ages is at most three years, q1 (g1 (b4 )) = 3.

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Fig. 1 – Illustration of the variable thresholds on criterion g1 . Indifference zone with respect to the characteristic couple bh . Strict preference zone (frontier and exterior) with respect to the characteristic couple bh .

Table 4 – Discriminating thresholds on the criterion g1 . g1 (bh ) − g1 (a) ≤ 0 a

g1 (bh ) − g1 (a) ≥ 0 b

bh

q1 (g1 (bh ))

p1 (g1 (bh ))

q1 (g1 (bh ))

p1 (g1 (bh ))

b4 b3 b2 b1

5 3 0 0

6 4 1 0

3 1 0 0

4 3 0 0

a b

The above interpretation seems difficult to understand without considering carefully the core objective of the decision aiding sorting model as well as the ART medical treatment context. Fig. 1 aims to clarify such an interpretation. This figure shows that small differences on the performances are more relevant for older women, where at least for a 40 years

Table 5 – Discriminating thresholds on the criterion g2 .

a b

g2 (bh ) − g2 (a) ≤ 0 a

g6 (bh ) − g6 (a) ≥ 0 a

g6 (bh ) − g6 (a) ≤ 0 b

q6 (g6 (bh ))

p6 (g6 (bh ))

q6 (g6 (bh ))

p6 (g6 (bh ))

1 1 0 0

2 2 0 0

0 0 0 0

1 1 0 0

Inverse thresholds. Direct thresholds.

old woman, any difference is significant of strict preference according to the criterion g1 . We have to stress that this is a new way of modeling the thresholds, which was not published before. The thresholds functions must be consistent to the core objective of the main analysis provided in the decision aiding sorting context. Table 5 presents the variable discriminating thresholds on the criterion g2 . The longer the duration of infertility is, the more urgent the ART treatment is. Table 6 presents the variable discriminating thresholds on the criterion g6 . It was considered that under a classification of 16/20, any difference between the classifications of two sets of embryos is important. In the next section, the intrinsic weights of the criteria and the veto thresholds are provided for achieving the modeling process of this ART application.

2.6.

Modeling the role of the criteria

g2 (bh ) − g2 (a) ≥ 0 b

q2 (g2 (bh ))

p2 (g2 (bh ))

q2 (g2 (bh ))

p2 (g2 (bh ))

1 1 0 0

2 2 1 1

1 1 0 0

2 2 1 1

Inverse thresholds. Direct thresholds.

b4 b3 b2 b1 b

– Any woman is strictly preferred to the older 25 years old reference woman if the difference of their ages is strictly greater than four years, p1 (g1 (b4 )) = 4. On the contrary, any woman is strictly preferred by the younger 25 years old reference woman if the difference of their ages is strictly greater than six years, p1 (g1 (b4 )) = 6.

b4 b3 b2 b1

bh

a

Inverse thresholds. Direct thresholds.

bh

Table 6 – Discriminating thresholds on the criterion g6 .

Within the Electre family of decision aiding methods several techniques can be used to assign the weights to the criteria, wj , j = 1, . . ., n [20–22]. These intrinsic weights are interpreted as the voting power of such criteria on a pairwise comparison between two different couples according to their performances. One of the techniques which are used to determine appropriate values for the weights of the criteria is the revised Simos’ procedure [20]. This procedure is usually

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Table 7 – The weights according to the embryologist. Code g5

Rank a

White cards b

Non-normalized weights

Normalized weights c

1

2

2.67

6

4.33

10

6

15

7.67

19

8.78

21

11

27

1 2

g3

2 2

g2

3 2

g4

4

267

As for the criterion g1 , variable veto thresholds were also defined (see Table 8). Such thresholds were considered to be more important for older women. When a woman is compared to a younger 38 years old reference woman, there is a veto (no transfer of three embryos!) if the difference of ages is at least ten years, v1 (g1 (bh )) = 10. On the contrary, when a woman is compared to an older 38 years old reference woman, there is a veto if the difference of their ages is at least six years, v1 (g1 (bh )) = 6.

2 g7

5 1

g6

6 3

g1

7

Total a b c

100

1 = worst rank and 7 = best rank; Z value = 11; Format = zero decimal places.

well accepted by the experts who are not familiarized with this kind of decision adding methods. In what follows, only the weights obtained in interaction with the embryologist are taken into account. When using the revised Simos’ procedure, the names of the criteria, written in “white cards”, are ranked from the least important criterion to the most important one according to the expert perspective (some criteria may have the same importance). Then, the difference of importance between successive levels of the rank obtained previously is expressed through a number of “white cards” introduced between those levels, and it is also recorded how many times the most important criterion is considered to be more important than the least important one in the ranking (Z value). These inputs are introduced in the SRF (Simos-Roy-Figueira) software, which was developed with an algorithm for assigning a numerical value to the weights of the criteria by determining the nonnormalized and the normalized weights (see Table 7). The power of the criteria can also be reinforced by introducing veto thresholds. A veto threshold, vj , is the minimal difference of performance of a couple over another couple to exceed on the criterion gj , for being incompatible with an overall outranking statement. In this ART application, for the criteria gj , j = 2, . . ., 7, constant veto thresholds were considered, such as v2 = 20, v3 = 7, v4 = 7, v5 = 3, v6 = 12, and v7 = 5. There are no desirable veto effects on the criterion g2 , which means that it is enough to consider a veto value greater than any possible differences between any two couples [15].

Table 8 – Veto thresholds on the criterion g1 . bh

g1 (bh ) − g1 (a) ≤ 0 a

g1 (bh ) − g1 (a) ≥ 0 b

b4 b3 b2 b1

15 12 10 10

15 10 6 6

a b

Inverse thresholds. Direct thresholds.

3.

Results

The set of characteristic reference couples defined for this ART application (see Table 2) achieves a minimum credibility level equal to 0.007 (the best value is 0 and the worst value is 1). This means that the characteristic couples are distinct enough to increase the interpretation of the assignment results provided by the Electre Tri-C method [16]. A set of 51 couples were evaluated according to the set of seven criteria to be assigned to the set of four embryo-transfer categories (see Table B.1). Then, the credibility indices of the comprehensive outranking of the couples over the characteristic reference couples, and vice-versa, are computed. These indices are then compared to the chosen credibility level in order to obtain the final assignment results. The chosen credibility level validated by the medical expert was  = 0.80 (see Table 9). This is the sum of the weights of the criteria which are favorable with each one of the assignment results. According to the weights provided in Table 7, this means that the “age of the woman” (g1 ) and the “morphological quality of the four best embryos” (g6 ) play the effective role of the most important criteria because at least one of these two criteria must “agree” with each assignment result. The Electre Tri-C assignment results, the chosen medical treatment, and the success of the ART treatment (pregnancy) are presented in Table B.3. According to these results, 44 (86%) of the 51 studied couples are assigned to only one transferembryo category. A brief overall comparison to the medical expert choice is presented in Table 10. These results show that the “recommendations” provided by both the method and the medical expert go in the same direction. The medical experts’ team recognized that they could have taken a different medical decision if the sorting results of Electre TriC were available when such decisions were taken since the results of the model were performed after such decisions. According to the medical experts the results presented in Table B.3, the success of the ART methodology in terms of pregnancy is 61%. This means that 31 pregnancies were obtained from the 51 couples which were subjected to an ART treatment. Within these results, 24 single pregnancies (47%) and 7 twin pregnancies (14%) were obtained. Such a methodology has achieved 30 live births (79%) of a total of 38 possible pregnancies taking into account the level of success of the ART methodology. According to the assignment results presented in Table B.3, the Electre Tri-C method provides a unique category in 44 (86%) of the 51 studied couples. The sorting accuracy of the type I (when Electre Tri-C provides a unique category which is the same as the one provided by the medical experts) is 61%

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Table 9 – Choice of the credibility level, . Version

Unique Ch 1 2 3 4 5 a b

0.55–0.70 0.75 0.80 0.85 0.90

ACR I a

UC



51 (100%) 47 (92%) 44 (86%) 43 (84%) 40 (78%)

(m = 51) 71% 65% 61% 59% 59%

ACR II b (m = UC)

(m = 51)

71% 70% 70% 70% 75%

71% 73% 75% 75% 80%

Sorting accuracy of the type I. Sorting accuracy of the type II.

Table 10 – Summary of the assignment results. Ch

Nature of the category

C1 [C1 , C2 ] C2 [C2 , C3 ] C3 C4

Transfer of four embryos Transfer of three or four embryos Transfer of three embryos Transfer of two or three embryos Transfer of two embryos Transfer of one embryo

Total

which is exactly the same as the success of the ART methodology, given by the overall pregnancy rate. The sorting accuracy of the type II (when Electre Tri-C provides a range of categories which contains the one provided by the medical experts) is 75% (see also Table 9).

4.

Discussion

Our approach is an informatics tools to deal with a medical application. It is based on a theoretical sound basis [23]. It is co-constructed through an interactive process between the analyst and the medical experts. It is also an open tool to incorporate other tools, in particular those related to statistics. The key findings of this work were particularly important, with a special emphasis on the “confront” between the “decisions-suggestions” provided by the tool and the embryologist current decisions. The fact that the embryologist agreed with the revision of some of his judgments represents the major finding to add to the body of pertinent knowledge. Some mechanisms related to this finding should be pinpointed. As an example, consider the couple a50 (see also Tables B.2). Taking into account the set of parameters, this couple a50 strongly dominates the reference couples b1 and b2 . This means that (a50 ,b1 ) = (a50 ,b2 ) = 1.00 and (b1 ,a50 ) = (b2 ,a50 ) = 0.00. The assignment of a50 to C1 or C2 cannot be justified! The reference couple b4 does not dominate a50 only because of the criteria g3 and g5 , but such a reference couple is strictly preferred to a50 . Once again the assignment of a50 to C4 is very difficult to be justified taking into account the power of the criteria g3 and g5 . The criteria g3 and g4 are used for justifying a small difference in the judgment of the assignment of a50 in C3 . But, since the power of each one of these criteria is at most 0.15, then the assignment of a50 to

Electre Tri-C ( = 0.80)

Medical expert

0 (0%) 2 (4%) 9 (18%) 5 (10%) 31 (61%) 4 (7%)

1 (2%)

40 (78%) 1 (2%)

51 (100%)

51 (100%)

9 (18%)

C3 is well supported by 85% of the criteria, including the two most important criteria, g1 and g6 . Despite the obtained results, we need to apply this model in a prospective way and step-by-step to allow fine tuning of the different parameters of the model. This work allowed defining a clear perspective of this research. As shown in Table 7, the subset of criteria {g4 , g6 , g7 } has a relevant influence on the results, since the cumulative sum of the weights is 55%. Therefore, further research is needed to design at least two specific decision aiding assignment models: one of them must provide a “recommendation” concerning the evaluation of the women according to several criteria; and the other one must provide “recommendations” concerning the choice of the four best embryos as well as the evaluation of their morphological and developmental quality. This work suggests a way to decreasing multiple pregnancies while maintaining the overall pregnancy rates. A relevant next step will be to confirm these findings provided by the model in a future prospective study. In addition, extending this type of analysis by incorporating detailed views of the clinicians will be another future avenue of research.

Author contributions The idea for developing this informatics tool was firstly proposed by J.R.F. and C.E.P. Then S.M. has modeled a first prototype of this decision aiding tool, based on a first version of the Electre Tri-C method proposed by J.A.D., J.R.F., and B.R. Gathering data and structuring the problem through the set of criteria was done interactively by the decision analysts (in this case J.R.F. and S.M.) and the decision makers or medical experts (M.J.C. and C.E.P.). Some aspects related to the modeling process were done by all the authors. J.A.D., J.R.F., and B.R. applied a new version of Electre Tri-C. J.A.D. and S.M. made a draft of a first version of the paper,

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Summary points What was known before the study • The successful application of multiple criteria decision analysis tools to health care problems and medicine, but not related to ART. • A set of elementary consequences in ART (medical exams, couple characteristics and attributes, etc.) that have been further grouped into homogeneous consequences leading to the construction of the set of criteria. • The basic concepts of the decision aiding informatics tool applied.

ences increase when the criterion performances increase too. Consider two couples, denoted a and a . When using the discriminating thresholds defined in Section 2, the following binary relations can be derived for each criterion [16]: (1) If | gj (a) − gj (a ) |≤qj , then a is indifferent to a according to gj , denoted a Ij a . Let C(a I a ) be the subset of criteria such that a Ij a . (2) If gj (a) − gj (a ) > pj , then a is strictly preferred to a according to gj , denoted a Pj a . Let C(a P a ) be the subset of criteria such that a Pj a . (3) If qj < gj (a) − gj (a ) ≤ pj , then a is weakly preferred to a (an hesitation between indifference and strict preference), denoted a Qj a . Let C(a Q a ) be the subset of criteria such that a Qj a .

What the study has added to the knowledge • Despite the variety of studies applied to health care problems and medicine to our best knowledge no multiple criteria informatics tool was applied to deal with ART problems. • The way of taking into account the preferences from the ART medical experts is new, in particular, the one related to the definition of some thresholds to take into account the imperfect knowledge of the data regarding ART. • The fact that the informatics tool was well-accepted by the ART medical experts. After observing the results, they agreed that the tool could provide improved decisions and they are looking forward to implement the referred tool, with additional options for statistics, in future ART decision processes.

which was improved mainly by J.A.D. with the comments by J.R.F., C.E.P., and B.R. All authors have approved this revised final version.

Conflicts of interest statement

The credibility of the comprehensive outranking of a over a , denoted (a,a ), which reflects the strength of the statement “a outranks a ” (a is at least as good as a ) when taking all the criteria from F into account, is defined as follows (see also Table B.2):

(a, a ) = c(a, a )

Let us assume, without loss of generality, that all the criteria gj ∈ F are to be maximized which means that the prefer-

(A.1)

where

Tj (a, a ) =

 1 − d (a, a ) j 1 − c(a, a ) 1

if

dj (a, a ) > c(a, a ),

(A.2)

otherwise.

⎧ 1 if gj (a) − gj (a ) < −vj , ⎪ ⎨ g (a) − g (a ) + p j j j dj (a, a ) = if − vj ≤ gj (a) − gj (a ) < −pj , pj − vj ⎪ ⎩ 0

if

gj (a) − gj (a ) ≥ −pj .

(A.3)

c(a, a ) =



wj +

j ∈ C(aPa )

 j ∈ C(aQa )

wj +

 j ∈ C(aIa )

wj +



wj ϕj ,

j ∈ C(a Qa)

(A.4)

Acknowledgements

Appendix A. The outranking credibility index

Tj (a, a ),

j=1

None declared.

Juscelino Almeida Dias acknowledges the financial support from the Fundac¸ão para a Ciência e a Tecnologia, Portugal (Grant SFRH/BD/22985/2005), the COST Action Number IC0602, and the Fundac¸ão Calouste Gulbenkian, Portugal (Grant 109475). The first two authors also acknowledge the financial support from the Luso-French bilateral cooperation agreement between LAMSADE and CEG-IST (FCT/CNRS 2009).

n 

ϕj =

gj (a) − gj (a ) + pj pj − qj

∈ [0, 1[.

(A.5)

where (a,a ) is based on the aggregation of the partial concordance indices, expressed by formulas (A.4) and (A.5), with the partial discordance indices, expressed by formula (A.3). For this aggregation issue, we assume without loss of generality n that = 1 and wj > 0, j = 1, . . ., n. All the thresholds have j=1 non-negative values such that vj ≥ pj ≥ qj ≥ 0.

Appendix B. The comprehensive results Tables B.1–B.3.

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Table B.1 – Performances of the couples according to the set of criteria. Couples

Criteria g1

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26

36 28 38 28 44 42 30 33 36 27 39 36 37 27 30 33 38 30 32 30 33 37 40 34 28 33

g2 1 1 1 6 3 4 1 2 3 2 3 1 2 2 3 4 6 2 3 3 12 2 3 3 2 3

Couples

g3

g4

g5

g6

g7

6 2 6 2 2 4 6 4 5 2 4 4 4 3 3 5 3 3 1 3 5 6 1 1 3 5

4 5 1 6 3 3 4 4 2 4 3 3 5 3 2 4 4 4 3 3 3 2 2 3 3 3

3 2 3 3 1 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

16 20 14 8 11 14 7 17 9 16 5 14 15 18 16 16 13 17 14 20 14 14 12 10 12 12

3 5 4 3 3 5 3 5 2 5 1 2 3 4 5 3 4 5 4 5 4 1 3 3 4 2

Criteria g1

a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51

37 34 36 33 34 33 39 37 34 38 33 34 35 34 38 26 34 28 30 31 33 41 40 32 40

g2 6 3 6 5 13 5 3 3 3 4 3 2 2 3 7 5 2 3 1 2 2 2 2 3 4

g3

g4

g5

g6

1 3 3 5 1 4 5 5 3 4 4 4 3 2 6 3 3 4 3 3 3 3 5 1 5

4 2 3 4 2 3 2 4 2 5 5 3 5 4 4 2 3 3 6 2 4 3 2 3 3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3

17 13 15 9 15 14 8 16 14 13 16 16 19 13 10 15 10 14 10 16 15 19 12 16 12

g7 5 5 4 4 5 5 2 4 5 5 5 4 5 5 1 5 3 4 1 5 4 5 2 4 4

Table B.2 – Comprehensive outranking credibility indices. Couples

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30

(a,bh )

(bh , a)

b1

b2

b3

b4

b1

1.00 1.00 1.00 1.00 0.73 1.00 1.00 1.00 1.00 1.00 0.79 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.94 1.00 0.79 0.79 0.71 0.73 0.73 1.00 0.79 1.00 0.29 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90 0.75 0.73 1.00 1.00 1.00 1.00 1.00 1.00 0.79

0.60 0.98 0.22 0.36 0.01 0.22 0.16 0.85 0.29 0.85 0.01 0.45 0.33 0.85 0.85 0.60 0.27 0.85 0.64 0.85 0.48 0.04 0.10 0.45 0.64 0.38 0.48 0.64 0.54 0.42

0.01 0.77 0.00 0.07 0.00 0.00 0.00 0.30 0.00 0.58 0.00 0.01 0.02 0.39 0.46 0.00 0.00 0.58 0.03 0.69 0.00 0.00 0.00 0.01 0.29 0.00 0.12 0.08 0.00 0.00

0.00 0.00 0.01 0.00 0.09 0.01 0.00 0.00 0.00 0.00 0.22 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

b2 0.01 0.00 0.48 0.00 0.27 0.12 0.09 0.00 0.61 0.00 0.67 0.13 0.00 0.00 0.01 0.01 0.35 0.00 0.00 0.00 0.05 0.40 0.36 0.09 0.00 0.16 0.01 0.04 0.03 0.15

b3 0.90 0.14 0.90 0.52 0.94 0.81 0.90 0.60 1.00 0.48 1.00 0.90 1.00 0.46 0.75 1.00 0.94 0.54 0.94 0.54 1.00 1.00 0.94 0.94 0.67 1.00 0.54 0.75 0.94 1.00

b4 1.00 0.79 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.79 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

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– Table B.2 (Continued) Couples

a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51

(a,bh )

(bh , a)

b1

b2

b3

b4

b1

0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.90 1.00 0.52 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.65 0.98 1.00 1.00 0.81 1.00 1.00 0.73 0.73 1.00 0.73

0.54 0.54 0.02 0.52 0.64 0.52 1.00 0.85 1.00 0.64 0.00 0.36 0.45 0.64 0.60 0.85 0.64 0.58 0.04 0.85 0.28

0.10 0.12 0.00 0.00 0.09 0.03 0.17 0.04 0.52 0.12 0.00 0.26 0.01 0.23 0.14 0.22 0.07 0.00 0.00 0.04 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

b2 0.05 0.00 0.88 0.01 0.03 0.12 0.00 0.00 0.00 0.00 0.83 0.00 0.15 0.00 0.12 0.01 0.00 0.05 0.67 0.00 0.29

b3 0.75 0.81 1.00 1.00 0.75 0.81 0.81 1.00 0.54 0.75 1.00 0.48 0.94 0.73 0.69 0.75 0.94 0.54 1.00 0.94 1.00

b4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Table B.3 – Electre Tri-Cversus medical expert’s assignment results. Electre Tri-C results a

ART current results

Couples

Lowest category

Highest category

Medical expert a

Pregnancy

Nr. of live birth b

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35

C3 (2) C4 (1) C2 (3) C2 (3) C1 (4) C2 (3) C2 (3) C3 (2) C2 (3) C4 (1) C1 (4) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C4 (1) C3 (2) C4 (1) C3 (2) C2 (3) C2 (3) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C3 (2) C3 (2) C2 (3) C3 (2) C3 (2)

C3 (2) C4 (1) C2 (3) C3 (2) C2 (3) C3 (2) C3 (2) C3 (2) C2 (3) C4 (1) C2 (3) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C4 (1) C3 (2) C4 (1) C3 (2) C2 (3) C2 (3) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C3 (2) C3 (2)

C2 (3) C3 (2) C3 (2) C3 (2) C2 (3) C2 (3) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C1 (4) C3 (2) C3 (2) C3 (2) C3 (2) C4 (1) C3 (2) C2 (3) C3 (2) C3 (2) C2 (3) C3 (2) C3 (2)

Single No No No No No No No Twin No Single Twin Single Single No No No No No No No No Twin Twin No Single No No No Single Single Single Single Single Twin

1 (1) 1 (2) 1 (1) 2 (2) 1 (1) 0 (1) 1 (2) 2 (2) 1 (1) 0 (1) 1 (1) 1 (1) 1 (1) 0 (1) 2 (2)

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– Table B.3 (Continued) Electre Tri-C results a

ART current results

Couples

Lowest category

Highest category

Medical expert a

Pregnancy

Nr. of live birth b

a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51

C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C2 (3) C3 (2) C2 (3)

C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3) C3 (2) C2 (3)

C2 (3) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C3 (2) C2 (3)

Single Twin Single Single Single Single Single Single Single Single Single Single Twin Single Single Single

1 (1) 2 (2) 1 (1) 1 (1) 1 (1) 0 (1) 1 (1) 0 (1) 1 (1) 1 (1) 1 (1) 1 (1) 2 (2) 0 (1) 1 (1) 1 (1)

a b

( ) the corresponding number of embryos. ( ) the possible number of live birth.

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