Nathalie Havet
email: havet@gate.cnrs.fr

Magali Morelle
email: morelle@lyon.unicancer.fr

Alexis Penot
email: penot@gate.cnrs.fr

Raphaël Remonnay
email: raphael_remonnay@sfr.fr

The information content of the WTP-WTA gap: An empirical analysis among severely ill patients

Keywords: Economic evaluation, Contingent valuation, Willingness-to-pay and willingness-toaccept divergence, Blood transfusion, Home care services, Oncology JEL Classification: I1 (health), I3 (welfare), H41 (public good)

Large disparities between willingness to pay (WTP) and willingness-to-accept (WTA) are commonly encountered in empirical studies and raise some important controversies. Nevertheless, the relationships between WTA and WTP can help understand not only how a service is valued but also how it can be substituted or how its loss can be resisted. The purpose of this study was to examine cancer patients' preferences for blood transfusion setting from the perspective of WTA and WTP. A contingent valuation (CV) survey was administered to 139 patients receiving transfusions either at home or in the hospital. While few patients (6%) gave WTP protest responses, the WTA approach generated more protest responses (18%). The WTA-WTP discrepancy was confirmed. One in four of the patients reported that no amount was deemed sufficient to compensate for the renunciation of their home BT management. The main determinants of WTP were significantly different from WTA predictors. Our results suggest that individuals' strategies towards constructing WTP and WTA differ in terms of determinants, reasoning, use of information and economic rationality. They give empirical evidence on the usefulness to elicit both WTP and WTA responses in healthsurveys to help understand the economic evaluation of health technology assessment and care organization.

1.

Introduction

Valuing health care services the user receives in the national health system is quite complicated in the absence of any market. Contingent Valuation (CV), developed in the framework of cost-benefit analysis, is a particularly relevant method of determining benefits technology assessment when it comes to comparing two types of health care management which may differ in patients' well-being and satisfaction during the treatment process with no difference on health outcomes (Gerard & Mooney, 1993;[START_REF] Ryan | Assessing the benefits of health care: how far should we go?[END_REF]Ryan, 1996;[START_REF] Donaldson | Does 'process' utility exist? A case study of willingness to pay for laparoscopic cholecystectomy[END_REF]. CV values goods or services that cannot be exchanged in traditional markets by constructing a hypothetical but realistic scenario about the good or service in question. The value attributed by CV methodology can be studied from the perspective of willingness-to-pay (WTP) or by the willingness-to-accept compensation (WTA). In the WTP approach, potential beneficiaries receive descriptions of the expected consequence of the technology under appraisal, and are asked to elicit the maximum amount they would be willing to pay to ensure that this expected consequence occurred [START_REF] Olsen | Aiding priority setting in health care: Is there a rôle for the contingent valuation method[END_REF]. The WTA approach consists in asking actual users of the existing technology under appraisal to specify compensation for the hypothetical loss of their entitlement, for example if this technology is removed from the health care services. Individuals are also asked to elicit the minimum amount they would be willing to accept to make up for this loss. In most cases in health-related CV contexts, the WTP has been employed to assess the distribution of patients' preferences.

Nevertheless, the standard economic theory assumed that WTA could be inferred from WTP, since it stated that the marginal amount an individual is willing-to-pay to obtain a good is approximately equal to the marginal amount the individual is willing-to-accept to relinquish the same good [START_REF] Willig | Consumer's surplus without apology[END_REF].

However, empirical studies, where WTP and WTA have been simultaneously estimated, show substantial evidence of a disparity between the two CV's approaches [START_REF] O'brien | Assessing the value of a new pharmaceutical. A feasibility study of contingent valuation in managed care[END_REF][START_REF] Plott | The willingness to pay-willingness to accept gap, the « endowment effect », subject misconceptions, and experimental procedures for eliciting valuations[END_REF][START_REF] Sayman | Effects of study design characteristics on the WTA-WTP disparity: A meta analytical framework[END_REF][START_REF] Van Den Berg | The economic value of informal care: a study of informal caregivers' and patients'willingness to pay and willingness to accept for informal care[END_REF][START_REF] Whynes | WTP and WTA: Do people think differently?[END_REF][START_REF] Grutters | Willingness to Accept versus Willingness to Pay in a Discrete Choice Experiment[END_REF][START_REF] Martin-Fernandez | Differences between willingness to pay and willingness to accept for visits by a family physician: a contingent valuation study[END_REF]: the WTA value is often considerably larger than the WTP value for the same good or service. This disparity has been widely studied through both theory and experiments [START_REF] Horrowitz | A review of WTA/WTP studies[END_REF] for a nice review). Hicksian welfare theory has attributed the differences found in valuing a good or a service by WTA and WTP to income effects. But the implied income effects are implausibly high [START_REF] Horrowitz | A review of WTA/WTP studies[END_REF].

Consequently, beyond the standard microeconomic theory, the most studied theories consider loss aversion. For instance, the prospect theory [START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF][START_REF] Thaler | Toward a positive theory of consumer choice[END_REF][START_REF] Kahneman | Economic preferences or attitude expressions?[END_REF], based on the notion of reference-dependent preferences, also known as loss aversion or endowment effects, argues that there is no single value associated with a good and in particular that the prospect of losing a technology individuals use weights more heavily than the prospect of gaining one which they did not try. The explanation of this theory is essentially psychological. Economic theory tries to explain this resistance by information costs, the uncertainty about the value of the good and the impossibility of reversing the process of loss (Zhao & Kling, 2001;[START_REF] Guzman | Researching Preferences, Valuation and Hypothetical Bias[END_REF].

On the whole, economists have empirically investigated the effects on two main categories of factors on the WTA-WTP disparity: i) the CV's study design and elicitation techniques and ii) the nature of the good to be valued. [START_REF] Sayman | Effects of study design characteristics on the WTA-WTP disparity: A meta analytical framework[END_REF] present a meta analysis of design factors and show that on top of the payment mechanism described by the prospect theory, the WTA-WTP disparity is lower with iterative bidding than open-ended questions, and in a withinsubjects design -i.e. when WTA and WTP values are obtained from the same group of respondentsthan in a between-subjects design. About the nature of the good under appraisal, the WTA-WTP disparity depends on the availability of substitutes: since a WTA valuation simulates the need for a good about to be denied, goods with fewer substitutes are more difficult to replace and may require disproportionately more money as compensation. Therefore, the WTA-WTP disparity may be higher when there is no readily available substitute for the good [START_REF] Hanemann | Willingness to pay and willingness to accept: How much can they differ?[END_REF]. The market availability may also affect the WTA-WTP discrepancy: non-market goods can be characterized by a wider range of consumer valuations, and this ambiguity about the value of the good and the cost of gathering information lead individuals to overstate their WTA and understate their WTP value.

The WTA-WTP disparity tends to be much larger for non-market goods [START_REF] Brown | Why the WTA-WTP disparity matters?[END_REF][START_REF] Kolstad | Information and the divergence between willingness to accept and willingness to pay[END_REF] and in particular for public services. Finally, the WTA-WTP gap may result from emotional and ethical concerns, as well as from social responsibility considerations [START_REF] Biel | The willingness to pay-willingness to accept gap revisited: the role of emotions and moral satisfaction[END_REF]. Psychologists [START_REF] Peters | The role of affect in the WTA/WTP disparity[END_REF] provide evidence that WTA and WTP valuations are constructed by a process guided in part by affect and that emotions and feelings might play so disproportionately with respect to WTA and WTP. Moreover, the WTA-WTP gap may be higher for public goods because of a more ethical dimension since individual choices affect others. For example, [START_REF] Kahneman | Valuing public goods: The purchase of moral satisfaction[END_REF] suggested that the amount that is reported in survey-based hypothetical WTP studies may not primarily express the respondent's value of goods, but rather the moral satisfaction of behaving in an ethically admirable way, corresponding to the so-called warm glow hypothesis [START_REF] Andreoni | Impure altruism and donations to public goods: A theory of warm-glow giving[END_REF].

The purpose of the present study is to explore the relationship between WTA and WTP in order to improve the knowledge of the perception the cancer patients have of home blood transfusion, with respect to its economic value and the predisposition to the possible substitution of this service (in the hospital day-care unit). Actually, comparing WTP and WTA answers could be informative to identify individuals' behaviors, and particularly to evaluate how differences in emotions and/or moral perceptions could affect the contingent valuation.

Because of the important emotional, moral and ethical aspects we faced, the feasibility of our CV was by no means self-evident. Firstly, it was not guaranteed that the CV approach could be used in a particularly critical clinical situation, i.e. among cancer patients. Indeed, our study population consists in cancer patients who needed blood transfusion, which is often evidence of an advanced step of disease, and even of a pessimistic prognosis in the short term. Consequently, it was not a priori obvious that severely ill patients would be willing not only to participate in the survey, but also to give meaningful responses to WTA and WTP questionnaires because of the emotional and moral dimensions. Besides, to our knowledge, few CV studies have been used for eliciting preferences -use value -among patients with critical clinical prognosis (Iskedjan et al., 2009;Chuck et al., 2008). Secondly, our CV allows an obvious ethical dimension. In France, the costs corresponding to the blood transfusion (at home or in the hospital) are entirely covered by national health insurance. Now health insurance is often considered as vested interests, therefore the hypothetical scenarios of the WTA-WTP questioning process -the potential loss of health care services or possibility to pay to ensure them-could be unreliable and generate protest responses.

More precisely, in this paper, we tried to determine whether the two CV's approaches were feasible in a particularly critical situation from a clinical perception -i.e. among severely ill patients-, and with a strong emotional and ethical dimension, and to understand the process whereby individuals were making their CV decisions in such a context. Our analyses also focused on two main issues:

(i) the relationship between stated WTA and WTP values, and (ii) the identification of individual characteristics that account for the disparity between WTA and WTP values.

Material and methods

Context

The present study considers the empirical situation of blood transfusion (BT) for cancer patients.

Recourse to BT is frequent when patients have received aggressive treatment, with multiple chemotherapy and radiation courses. Blood transfusions have hitherto usually been administered in the hospital. However, with an ever-increasing demand for acute hospital services, alternative methods of delivering this health care need envisaging.

For most patients, BT can be delivered at home or in the hospital day-care unit with identical effectiveness and safety, and therefore identical health outcomes [START_REF] Idri | La transfusion à domicile ? Une alternative à l'hôpital de jour[END_REF][START_REF] Madgwick | A home blood transfusion programme for betathalassaemia patients[END_REF]. But, non-health benefits such as process utility may play a major role in the choice of care. The location of transfusion may be viewed differently from one patient to another.

Home blood transfusions may be psychologically beneficial, in that they keep the patient in familiar surroundings and avoid the complication of transportation. Particularly among older and more severely ill patients, transportation difficulties or waiting time at the medical facility may require additional effort, while remaining at home may decrease the patient's tiredness and thus be considerably more comfortable (Devin & Agnew, 2008). Nevertheless, transfusions at home are potentially stressful and anxiety-producing situations: it may increase the feeling of insecurity because of isolation and increased distance to emergency care [START_REF] Benson | Home is where the heart is: do blood transfusions belong there too?[END_REF][START_REF] Benson | Nationwide survey of home transfusion practices[END_REF].

Survey design

Our CV survey was realized at the Comprehensive Cancer Centre of the Rhone-Alps Region in Lyon (France). In addition to BT delivered in the hospital day-care unit, some patients also received BT at home in collaboration with the national blood service (EFS: Etablissement Français du Sang) within a regional cancer network stationed on the hospital. This prospective, non-randomised study was conducted during a 12-month period in [2003][2004]. All cancer patients needing a BT and who were more than 18 years of age were asked to participate. After obtaining informed consent, and less than 48 hours after BT administration, face-to-face interviews were conducted by a trained interviewer on the basis of a detailed guide. Although costly, this type of interview was chosen for minimizing hypothetical biases and improving the quality and responses rate (Mitchell & Carson, 1999;NOAA, 1993;[START_REF] Smith | A review of methodological issues in the conduct of willingness to pay studies in health care III : issues in the analysis and interprétation of WTP data[END_REF].

Patients were first asked whether they had already undergone, either BT in the hospital or at home, including the current procedure when it was administered at home, and whether they had already received home care other than BT. All patients were then given a detailed presentation of the BT management, either at home or at the hospital, and were told that effectiveness and safety were identical in both cases.

WTP/WTA questioning process

All patients were given a general presentation of the CV method. We carefully explained the benefits using this method in order to provide policy-makers with valuable information about patients' preferences. It was indeed feared that this approach might be poorly accepted by French patients, particularly when seriously ill, who are strongly attached to their national health insurance system.

Then they were asked to imagine two hypothetical scenarios where they would need another BT and where the only freely available BT management would be hospital BT. The first one evaluated WTP for home BT by asking patients how much they would be willing to pay for this service rather than the free alternative and, immediately after that, what level of compensation they would be willing to accept for renouncing it. More precisely, patients were asked to imagine that the government decided to reduce the number of BT outside the hospital. Their WTP value would be the maximum amount of money they were willing to contribute to still be transfused at home rather than at the hospital which is fully covered by Health Insurance. Their WTA value would correspond to the minimum compensation they would accept for undergoing hospital BT rather than home BT.

Considering the payment vehicle, we chose out-of-pocket expenditure, which is recommended when respondents are personally involved [START_REF] Smith | Construction of the contigent valuation market in health care: A critical assessment[END_REF][START_REF] Mitchell | Using Surveys to Value Public Goods: the Contingent Valuation Method[END_REF][START_REF] O'brien | When do the 'dollars' make sense? Toward a conceptual framework for contingent valuation studies in health care[END_REF]. A bidding process was chosen rather than closed-ended questions because of our small sample size [START_REF] Donaldson | Limited dependent variables in willingness to pay studies: applications in health care[END_REF] and the lower expected WTA-WTP disparity [START_REF] Sayman | Effects of study design characteristics on the WTA-WTP disparity: A meta analytical framework[END_REF]. The process involved three steps. First, an initial bid was proposed to the patients.

In the case of WTP (WTA), if they agreed to pay (refused to accept compensation), we proposed a higher bid. Conversely, if they refused to pay (agreed to receive compensation), we proposed a lower one. This step was repeated twice. In the WTP case, if the patients refused to answer (a zero WTP value was attributed to this non-response) or when the bidding process yielded a zero value, the interviewers were instructed to ask follow-up questions to identify the reasons for this choice. In the WTA case, follow-up questions were introduced again if the bidding process yielded a zero value or if patients reported that no amount was deemed sufficient to compensate for the renunciation of the home management (we referred these responses as "infinite WTA values").

Patients were randomly assigned to two initial bids (40€ and 80€) to test for a possible anchoring bias (Herriges & Shorgen, 1996;[START_REF] Flachaire | Starting point bias and respondent's uncertainty[END_REF]. The range from 40€ to 80€ was chosen because it was close to the range of fees normally covered by social insurance in France for a home visit by a specialist (including call-out charges). Respondents rated their certainty about their WTP and WTA on a 10 cm Visual Analogue Scale (VAS) (from 0='not at all certain' to 10='absolutely certain') 1 .

Socioeconomic characteristics (age, employment, monthly family income before tax, marital status, children, distance from hospital, etc.) were documented by the patients themselves. Information on stage of disease (curative, palliative or terminal) was reported by the oncologist in charge of the Home Care Unit at the Cancer Centre. We also constructed a health-related quality of life index including tiredness as measured by the Functional Assessment of Cancer Treatment General scale (FACT-G) (Cella, 1993), which varied from 0 (worst possible situation) to 108 (best possible estimation).

1

The WTP/WTA questionnaire is available from the authors upon request.

Statistical analysis and econometric specifications

Summary statistics for patients' characteristics and WTP and WTA amounts, including mean and standard errors, were calculated. Using nonparametric statistics, as appropriate, we first examined differences between stated WTP and WTA values, and their respective level of certainty.

Subsequently, multivariate analyses were carried out in order to identify and compare the determinants of WTP and WTA. The choice of appropriate WTP and WTA modeling was also driven by the obtained distribution.

Zero values should receive particular attention in the econometric analysis of WTP determinants, as is usually the case in the CV literature. Not only the patients going zero responses represented an important proportion of the whole sample, but also there was a range of possible explanations for zero values, from true zero responses consistent with economic and rational decisions (i.e. nil or very small preference, inability to pay) to protest responses with no economic significance, which indicate respondents' disapproval [START_REF] Strazzera | Modelling zéro values and protest responses in contingent valuation survey[END_REF]. Indeed, protest responses can result from informational constraints, social responsibility considerations, objections to the support payment and/or emotional concerns. For example, they could reflect an ideological position against health care valuing. According to the empirical literature, the double-hurdle is the most appropriate econometric approach for zero and protest responses [START_REF] Dalmau-Matarrodona | Alternative approaches to obtain optimal bid values in contingent valuation studies and to model protest zeros. Estimating the determinants of individual's willingness to pay for home care services in day case surgery[END_REF]). Nevertheless, it cannot be used when the number of protest responses is too small like in our study (see section 3).

In [START_REF] Havet | Cancer patients'willingness to pay for blood transfusions at home: results from a contingent valuation study in a French cancer network[END_REF], a first study of our WTP values, it was showed that the truncated regression model could be an appropriate econometric methodology for modeling them 2 : the estimation was based on strict positive WTP only and took into account the elimination of all zero values to obtain valid results for all patients. More precisely, this model specified the probability to observe the amount of WTP, conditional on the expression of positive willingness-to-pay:

(1) where is the value of the i th observation on the dependent variable WTP and its corresponding latent value, described by the relation with X i the independent variables that influence the WTP amount, β the unknown parameters and u i the error term such as ~.

As regard the distribution of the WTA values (left-censoring at zero with protest responses and a significant part of « infinite » values, which constituted a right censoring, see section 3), we estimated a Tobit model on truncated sample. Indeed, for the same reasons as the WTP analysis, our WTA estimation was based on strict positive WTA values only and so on a truncated sample. In order to allow « infinite » responses in the regression, we replaced these missing values by an arbitrary threshold, noted c (representing a very high WTA value). The Tobit model was also the most appropriate to the treatment of the right censoring at the c value. Therefore, the standard Tobit was modified to obtain valid results for all patients despite the elimination of all zero values3 :

(2)

where is the value of the i th observation on the dependent variable WTA and its corresponding latent value, Z i the independent variables that influence the WTA amount, the unknown parameters and v i the error term such as ~. We reported the results of the WTA model obtained with c equal to 1500 euros. We also tried other c values for robustness purposes:

they confirm the independence of our results to the chosen threshold. In our estimations, the dependent variables and were expressed in logarithm.

3.

Results

Sample Characteristics

Over the study period, all 153 solicited patients consented to participate. However, 14 people could not be interviewed within the 48 hours following their BT either because of lack of availability or because they felt too tired. As a result, 139 patients (90.9%) were enrolled.

The mean age of the patients (69 males and 70 females) was 57.5 years (±12.8). Most lived in a marital relationship (73%) and had no children at home (71.2%). Less than half (43.2%) had a professional occupation. Pre-tax net monthly household income distribution was as follows: less than 800€ -12.2%, between 800 € and 1 499 € -28.2%, between 1 500 and 2 299 € -25.9%, between 2 300 and 2 999 € -16.8%, 3 000 € or more -16.8%. The average distance from home to hospital was large (nearly 40 km) because of the over-representation of rare tumors. Almost one in two respondents had previous experiences of home BT on the one hand and of home care (except home BT) on the other (43.2% and 45.3% respectively). Regarding stage of disease, patients were almost equally distributed between curative (50.4%) and palliative or terminal stages (49.6%). Quality of life scores, assessed according to the FACT-G scale ranging from 0 to 108 (worst to best quality of life), were rather poor, with an average index of 61.7. For the 132 patients without comprehension problems and non-responses, WTP and WTA values are displayed in Figure 1.

Descriptive analysis of WTP and WTA responses

Fig. 1. Survey distribution of WTP and WTA for home blood transfusion

The WTP distribution contains a high proportion of zeros (47.0%) and otherwise is positively skewed. The highest WTP value is 300€ for home BT. The graph presents typical pattern of WTP studies from open-ended questions or a bidding process [START_REF] Jorgensen | Protest Responses in Contingent Valuation[END_REF][START_REF] Lindsey | Market models, protest bids, and outliers in contingent valuation[END_REF][START_REF] Donaldson | Limited dependent variables in willingness to pay studies: applications in health care[END_REF]. Follow-up questions allowed to distinguish between "genuine" zero values and "protest" responses and therefore to better understand the individual preferences and to assess the ability to implement cost-benefit analysis. We were able to determine that 8 (12.9%) of the 62 zero WTP values were "protest" zeros, on the basis of statements such as "I have paid health insurance premiums all my life and I should not have to pay anything more", "the goal of this kind of study is that we pay more and more for care" or "I do not want to pay for blood when the donors are unpaid volunteers". Clearly, these 8 patients reacted negatively to the payment vehicle. The remaining 54 zero WTP values were considered as "real" zeros, as opposed to the protest responses:

among them, 12 patients participated in the bidding process which resulted in zero values, either because of low income or without any justification, and the other 42 patients clearly expressed that they would prefer to receive BT in the hospital rather than at home and also were not willing to pay for home BT.

The WTA distribution shows a U-shaped pattern: in addition to the large number of zero responses (54.5%), 35 patients (26.5%) reported that no amount was deemed sufficient to compensate for the renunciation of their home BT management and were also classified as "infinite WTA values". In the follow-up questions, most of these 35 patients justified their WTA answers by either utilitarian argument such as "maintaining the quality of life is priceless" or by ethical consideration of the kind: "it is out of question to renounce health rights". In the particular context of our study, the expression of an infinite WTA could correspond to a rational choice: patients were seriously ill, and even likely to receive palliative care in the short or long term. It is reasonable to expect that any deterioration in quality of life would have a significant impact on these patients and that they found it impossible to consider a substitute for their preferred BT management. These patients' responses correspond to an extreme preference, which is not quantifiable.

Among the 72 WTA zero values, 24 (33.3%) could be considered as protest responses and 48 (66.7%) as genuine zero. Indeed, in the follow-up questions, 11 patients clearly stated a moral position or expressed a refusal on principle to accept monetary compensation for failure to obtain their home BT. It also seemed especially immoral for patients to be paid for receiving blood when donors are volunteers, even if the payment covered the administration of blood and not the blood itself; there would be a transfer of moral taboos associated with the merchandising of blood. These 11 responses did not seem to reflect the actual preferences of the patients. The remaining 13 protest bidders indicated a zero WTA without giving any reason while they indicated a WTP greater than 0.

It was also legitimate to assume that these 13 WTA values did not correspond to actual preferences: by expressing a non-zero WTP, they showed that they were not indifferent between hospital and home BT. Among the 48 "genuine" zero values, there were again the 42 patients preferring hospital BT and 6 patients who expressed both zero WTA and WTP values with no justification. We considered these latter responses as true zero preferences for lack of additional information.

Quantitative analysis of WTP-WTA discrepancy

Data from the patients who had problems understanding WTP and/or WTA questioning were eliminated from empirical analyses.

Univariate analyses

Since "infinite WTA" responses represent a very high but unquantifiable preference, the WTA-WTP gap is incalculable. That is why we examined the difference related WTP and WTA values using the Sign test, which compares the number of positive or negative differences between the two measures but ignores the size of these differences. We obtained that WTA was significantly greater than WTP, whether excluding protest zeros (p<0.0001) or including protest responses as zero WTP (or WTA) (p=0.0028).

However, the WTA approach generated more protest responses than the WTP approach (18.2% of the complete responses versus 6.1%). Consequently, we wanted to know how patients felt about the two CV methods and particularly if they had a preference for one approach. To do this, we used the responses obtained about the WTA and WTP certainty by the Visual Analogue Scale. The Wilcoxon signed-rank test allowed to identify possible differences in the degrees of certainty between WTP (mean: 8.0; SD: 2.6) and WTA (mean: 8.5; SD: 2.3) for patients choosing home BT.

Results indicated no significant differences between the two scales (p=0.1540).

Regression models on WTP and WTA data

We tried to identify the individual characteristics that could account for the observed difference between WTA and WTP values with regression models. Since the WTA-WTP gap was incalculable because of the "infinite WTA" values, our strategy was to find the determinants of the WTA and WTP values separately and to compare them. First, considering the truncated regression results, the sign and significance of several covariates behaved as expected. The health economics literature suggests that income positively influences WTP values [START_REF] Donaldson | Valuing the benefits of publicly-provided health care: does 'ability to pay' preclude the use of 'wilingness to pay[END_REF][START_REF] Smith | A review of methodological issues in the conduct of willingness to pay studies in health care III : issues in the analysis and interprétation of WTP data[END_REF][START_REF] Drummond | Methods for the economic evaluation of health care programmes[END_REF] and that choosing a higher initial bid value could increase WTP responses, but would in no way decrease them [START_REF] Smith | A review of methodological issues in the conduct of willingness to pay studies in health care III : issues in the analysis and interprétation of WTP data[END_REF]Herriges & Shorgen 1994;[START_REF] Flachaire | Starting point bias and respondent's uncertainty[END_REF]. This income effect and the anchoring bias hypothesis, often considered as a guarantee of the construct and content validity of a survey, were confirmed in our study: all other things being equal, the WTP value for home BT increased with higher income and with higher initial bid. Several patient characteristics were also significantly associated with WTP for home BT: experience of home care, distance from home to hospital, stage of disease, professional and familial statuses and age, as well as health-related quality of life. However, neither previous experience of home BT nor gender and the presence of children seemed to influence WTP values. More precisely, elderly patients or patients living far from the hospital stated a higher WTP for home BT than other patients. Those with a higher healthrelated quality of life stated a lower WTP for home BT, which means that home BT was all the more appreciated as patient quality of life (including tiredness) was low. This is in agreement with the medical literature [START_REF] Devlin | An evaluation of domicillary blood transfusion service for palliative care patients in Northern Ireland[END_REF][START_REF] Ademokun | Home blood transfusion: a necessary service development[END_REF][START_REF] Benson | Home is where the heart is: do blood transfusions belong there too?[END_REF] which suggests that sparing painful transportation and waiting time are strong reasons for preferring home BT. Conversely, the WTP for home BT was lower by 40% for advanced-stage (palliative or terminal) than early-stage (curative) patients. We could assume that patients would be all the more sensitive to safety concerns that there lives would be threatened, and that hospital could be generally considered as safer by patients. Previous experience of home care proportionally increased the WTP for home BT. Lower WTP values for home BT were obtained for patients living with a partner. All other things being equal, WTP for home BT was lower for patients with a professional occupation.

By contrast to the WTP equation, few explanatory variables emerged as a significant influence on WTA. Only three patients characteristics were significantly associated with WTA for home BT:

experience of home care, distance from home to hospital and to a lesser extent familial status. The WTA analysis confirmed that patients living far from the hospital and patients with a previous experience of home care would have higher preferences for home BT. However, the statistical insignificance of the income and initial bid variables could raise doubt about the construct and content validity of the WTA survey. Therefore, the WTP-WTA gap should not be a surprise since the responses to both CV methods do not seem to share the same determinants.

Discussion

Although the WTA-WTP disparity has been commented on at considerable length in behavioral economics, only a small number of studies have conducted WTA analysis in health care context [START_REF] O'brien | Assessing the value of a new pharmaceutical. A feasibility study of contingent valuation in managed care[END_REF][START_REF] Borisova | Measuring the value of time for méthadone maintenance clients: willingness to pay, willingness to accept, and the wage rate[END_REF][START_REF] Van Den Berg | The economic value of informal care: a study of informal caregivers' and patients'willingness to pay and willingness to accept for informal care[END_REF][START_REF] Whynes | WTP and WTA: Do people think differently?[END_REF][START_REF] Martin-Fernandez | Differences between willingness to pay and willingness to accept for visits by a family physician: a contingent valuation study[END_REF]. And yet, the differences between WTA and WTP can help to understand not only the value attributed to a health service but also the capacity to substitute it or the resistance of its loss. For example, a positive WTA-WTP discrepancy would mean that patients require a larger compensation for the withdrawal of an existing service than they would have been willing to pay for its introduction, which has an important implications for economic evaluation of health technology assessment and care organization. Here, it seems to us that the comparison of the WTP and WTA to our very particular situation -blood transfusion setting in advanced stage cancer patients -would provide some additional understanding of the behavior of respondents. More precisely, our analyses focused whether cancers patients would accept the two elicitation formats, and if yes, they would behave differently with the two approaches proposed, While administration of a CV survey to patients receiving intensive chemotherapy or near the end of life may appear somewhat incongruous, we found that patients have massively responded. More than 90% of the solicited patients consented to participate. For instance, in their literature review about WTP in health and health care, [START_REF] Smith | A review of methodological issues in the conduct of willingness to pay studies in health care III : issues in the analysis and interprétation of WTP data[END_REF] found a mean participation rate of 74% among the studies. Moreover, very few participants had comprehension problems of the CV method: approximatively 2% and 5% for the WTP and WTA approaches respectively.

These results are particularly interesting because they show that the criticality of a health situation is not an a priori obstacle to the achievement of such a study. The ability of respondents to accept a CV questioning in a context of high health benefits had already been partially investigated in the literature. A dozen studies had conducted WTP questioning on health programs with major consequences either on psychical or physical well-being of respondents' quality of life or on their life expectancy (e.g., recently [START_REF] Geelhoed | Parental perspective of the benefits of genetic testing in children with congenital deafness[END_REF][START_REF] Radtke | Willingness-to-pay and quality of life in patients with vitiligo[END_REF][START_REF] Iskedjian | Willingness to pay for a treatment for pain in multiple sclerosis[END_REF][START_REF] Saulo | Willingness and ability to pay for artemisinin-based combination therapy in rural Tanzania[END_REF]Chuck et al., 2008;[START_REF] Lacour | What women with ovarian cancer think and know about genetic testing[END_REF][START_REF] Khanna | Willingness to pay for a cure in patients with chronic gout[END_REF][START_REF] Bishai | Willingness to pay for drug rehabilitation: implications for cost recovery[END_REF].

Nevertheless, none of these studies have dealt with so severely ill patients, with so limited life expectancy and only two of them had simultaneously conducted a WTA analysis [START_REF] Whynes | WTP and WTA: Do people think differently?[END_REF][START_REF] Borisova | Measuring the value of time for méthadone maintenance clients: willingness to pay, willingness to accept, and the wage rate[END_REF].

Several reasons can explain why so many patients have accepted to answer the valuation questions.

First we put a great emphasis on the information we gave them through face-to-face interviews and we introduced ourselves as independent researchers with no ties to hospitals and welfare-financing agencies in order to avoid any suspicion of instrumentation results. Second, we did explain to transfused patients that contingent valuation aimed at gathering their preferences, which could be used to define health policy. These patients appeared to be concerned about their preferences being taken into account as users and contributors to the health-insurance system. Third, we wanted to elicit preferences for a health program with which patients were familiar enough to easily picture its outcomes. Using respondents with a relatively realistic contingent market, all patients would have experienced blood transfusions at the hospital, and all would have information on close substitutes.

This might reduce the risk of hypothetical biases associated with hypothetical questions and would produce a more realistic valuation.

This intimate knowledge of the good under appraisal probably helped patients to internalize its costs and benefits and to give meaningful responses, at least for the WTP approach. Any far-fetched or exaggeratedly high WTP value had been expressed and only 5.8% of participants gave protest responses. Moreover, the econometric results on the WTP determinants were consistent with what was expected. For instance, the classical anchoring effect was confirmed. We also obtained, all other things being equal, a significant and positive relationship between income and WTP. This positive income elasticity is one criterion generally required for the construct WTP validity [START_REF] Donaldson | Valuing the benefits of publicly-provided health care: does 'ability to pay' preclude the use of 'wilingness to pay[END_REF][START_REF] Smith | A review of methodological issues in the conduct of willingness to pay studies in health care III : issues in the analysis and interprétation of WTP data[END_REF][START_REF] Drummond | Methods for the economic evaluation of health care programmes[END_REF]: it is the evidence that patients do internalize their budget constraint when revealing their preferences and that our WTP results are in accordance with theoretical explanations.

Our results with the WTA approach are a bit less convincing. First, the proportion of protest responses for home BT was higher (18%). Even if patients can justify their protest responses, the criterion validity of CV will suffer from too high a number of them and cost-benefit analysis will be hard to implement. Second, 26% of the patients reported that no amount was deemed sufficient to compensate for the renunciation of their home BT management and were considered to have "infinite" WTA values. In our particular critical situation, such arguments for infinite WTAs are understandable: patients at the end of life may refuse to renounce their quality of life whatever the amount of financial compensation. However, even if these patients answer in accordance with their preferences, they provide no usable values in a cost-benefit analysis by decision-makers. Third, any socioeconomic variable significantly influence the patients' WTA values. In theory, we expected that the same explanatory variables play on WTA and WTP values. For example, we could expect a positive impact of income on WTA since the marginal utility of a given amount is larger for the poorest. It is also obvious from our econometric results that individuals' strategies towards constructing their WTP and WTA values differ in terms of determinants, reasoning, use of information and economic rationality.

The direct consequence is the significant WTA-WTP disparity. Yet, we had purposely chosen an iterative bidding process and a within-subjects design, in order to limit the gap between the two CV approaches. Similarly, we have evaluated the patients' preferences for home BT, which has a very close substitute a priori since home BT and hospital BT do not differ in safety and health outcomes.

Unfortunately, we can not exactly measure the WTA-WTP gap because of infinite WTA values. As an approximation, we have computed the ratio of mean WTA to mean WTP but ruled out the patients with protest responses to at least one of the two CV tasks and those who did not give a precise value for their WTA. For this sub-sample of 71 patients, the WTA/WTP ratio was 2.3 (20.57 euros/47.58 euros). However, since it overlooks patients with infinite WTA, it widely underestimates the difference for all patients. Therefore, we can conjecture that the true WTA-WTP divergence in our case is very much higher than values found in other studies when referring to goods or services in the field of health [START_REF] O'brien | Assessing the value of a new pharmaceutical. A feasibility study of contingent valuation in managed care[END_REF][START_REF] Borisova | Measuring the value of time for méthadone maintenance clients: willingness to pay, willingness to accept, and the wage rate[END_REF][START_REF] Van Den Berg | The economic value of informal care: a study of informal caregivers' and patients'willingness to pay and willingness to accept for informal care[END_REF][START_REF] Whynes | WTP and WTA: Do people think differently?[END_REF][START_REF] Martin-Fernandez | Differences between willingness to pay and willingness to accept for visits by a family physician: a contingent valuation study[END_REF]. Higher rate of protest responses, important proportion of unquantifiable values and the nonsignificance of the major socioeconomic determinants suggest the existence of strong ethical and moral considerations in the individual WTA decisions. They show that the patients' preferences can only be imperfectly elicited by a monetary equivalent and then show the limits of CV methods.

Others have suggested that the influence of a moral perspective may be particularly strong under WTA and may also explain a large part of the WTA-WTP divergence [START_REF] Boyce | An experimental examination of intrinsic values as a source for the WTA-WTP disparity[END_REF][START_REF] Irwin | Buying/selling price preference reversals: Preference for environmental changes in buying versus selling modes[END_REF][START_REF] Nyborg | Homo economicus and homo politicus: Interpretation and aggregation of environmental values[END_REF][START_REF] Peters | The role of affect in the WTA/WTP disparity[END_REF]Sayman & Öncüler 2005). The ethical and emotional dimensions are very likely to have the dominant effect in our empirical situation. Indeed, our contingent valuation focused on a public good which was far from an « ordinary private good available in the market » [START_REF] Kolstad | Information and the divergence between willingness to accept and willingness to pay[END_REF][START_REF] Brown | Why the WTA-WTP disparity matters?[END_REF]Horowitz & McConnel, 2002) and was health related [START_REF] Chapman | Preferences reversals in monetary and life expectancy evaluations[END_REF][START_REF] Brown | Why the WTA-WTP disparity matters?[END_REF]. In the WTA approach, people were asked to imagine a situation where, for a hypothetical reason (government decision to promote hospital, organizational difficulties, saturation of healthcares), a compensation would be granted to renounce blood transfusion at home. People might have considered that, if home health care was socially difficult to be maintained, it would not be very ethic to ask for a compensation of their loss of life quality. Thus, a zero WTA would not mean only a refusal of this questioning mode; it would also illustrate that the patients' utility to behave as citizens and to take part to the preservation of the health care system is greater than the disutility to not benefit from their favorite BT management. Many patients also gave an infinite WTA on the basis of moral principles. They indicated in their free comments that they refused to renounce an universal right of care access, considered as an endowment. Indeed, the French health care system relies on universal access and on the solidarity between all members through their contributions.

For long-lasting diseases such as cancer, it guarantees a full coverage of costs. In the context of a growing private sector, some patients expressed their will to preserve this system based on the participation of all. Therefore, with a zero or infinite WTA, individuals might have maximized their moral satisfaction instead of their purely economic utility. Our results seem consistent with the "warm-glow" hypothesis, shown for other public goods [START_REF] Kahneman | Valuing public goods: The purchase of moral satisfaction[END_REF]. They could also be interpreted at the light of Mc Graw & Tetlock (2005)'s arguments who defined the «tabootrade-off» notion. The WTP approach is based on scenario where patients hypothetically buy a good or a service but the WTA approach (in our paper but also in other health-related cases) offers patients the possibility to trade for money some potentially sacred aspects such as life quality or maintaining a relationship with relatives, which is a transgression of the forbidden. Given the emotional impact on patients of the deterioration of their well-being, a trade-off in this direction becomes abhorrent and taboo [START_REF] Whynes | WTP and WTA: Do people think differently?[END_REF].

Conclusion

As [START_REF] Whynes | WTP and WTA: Do people think differently?[END_REF], we found that the evaluation of public health services made from the perspective of gain or loss is different, and this should be taken into account if patient preferences are to be incorporated to health planning. Our results indicate that we should have higher confidence with WTP than WTA to build a cost-benefit analysis. In the health care context, the numerical style of WTP elicitation can be difficult to transpose for WTA. The WTP scenario refers to a structure market revealing preferences while the WTA one appears to be a measure on emotions and the respect of social norms. In all cases, ethical and moral concerns must be taken into account for the health policy definitions and health care strategies. Since the patients' preferences seem to be only imperfectly elicited by a monetary equivalent, a qualitative analysis of the responses is necessary. If warm glow and taboo trade-off effects are substantial and underline the limits of CV methods, an alternative method could consist in resorting to experimental economics to reveal preferences as suggested by [START_REF] Robin | Mesurer les préférences du consommateur pour orienter les décisions des pouvoirs publics[END_REF].

  In a classic evaluation, O' Brien et al. (1998) obtained a WTA/WTP relationship of approximately 2 for a new drug to treat cancer. In another paper on the treatment of drug-dependent persons, the WTA/WTP ratio was situated at about 1.3 (Borisova &Goodman, 2003). Martin-Fernandez et al. (2010) reported aWTA/WTP ratio of about 3.3 when valuing the visit to family physician. The valuation of surgical techniques, a cochlear implant, in a population of children, produces values for this ratio approaching 4[START_REF] Whynes | WTP and WTA: Do people think differently?[END_REF]. When valuing providing care to the chronically ill the relationship between WTA and WTP closely approaches the unit (van den[START_REF] Van Den Berg | The economic value of informal care: a study of informal caregivers' and patients'willingness to pay and willingness to accept for informal care[END_REF].

Table 1 . Cross-tabulation of WTA and WTP values for home blood transfusion

 1 Table1exhibits a cross-tabulation of WTP and WTA values. Three patients (2.2%) expressed comprehension problems with the two CV tasks and offered so neither WTA nor WTP values for home BT. Three others did not understand the WTA method and one patient refused to answer WTA questions.

	WTP	Comprehension	Zero	Positive	Total
	WTA	Problem	value	value
	Comprehension problem	3	0	3	6
	Non-response	0	0	1	1
	Zero value	0	54 (a)	18	72
	Positive value	0	4	21	25
	Infinite value	0	4	31	35
	Total	3	62	74	139

(a) Of whom 42 patients preferred hospital BT.

Table 2 . Results of the truncated regression model of WTP (log) and WTA (log) values WTP (log)

 2 Table 2 reports the results of the WTP truncated regression and the WTA Tobit model on truncated sample.

		WTA (log)
	Truncated regression	Tobit model on
		truncated sample

[START_REF] Havet | Cancer patients'willingness to pay for blood transfusions at home: results from a contingent valuation study in a French cancer network[END_REF] showed that similar results were found with a type II Tobit. However, the truncated regression model was preferred here for an easier comparison with the WTA modelling.

The estimation procedure for the Tobit on truncated sample was programmed using the software Stata 10.1.


Malo Cadoret

Estefania De Mirandés

Pierre Cladé

Saïda Guellati-Khélifa

Franc ¸ois Nez

Franc ¸ois Biraben

Measurement of the ratio h/m Rb and determination of the fine structure constant

Keywords:

We present a review of the most precise determinations of the fine structure constant α which are obtained in different domains of physics. We describe the measurement of the ratio h/m Rb between the Plank constant and the mass of Rubidium atom which leads to a precise value of α which is very few dependent of the QED. Finally, we present a review of the different determinations of the von Kitzling constant R K .

Nous présentons une revue des déterminations les plus précises de la constante de structure fine α qui sont obtenues dans différents domaines de la physique. Nous décrivons la mesure du rapport h/m Rb entre la constante de Planck et la masse de l'atome de rubidium qui conduit à une valeur de α très précise et très peu dépendante de l'électrodynamique quantique. Finalement nous présentons une revue des différentes déterminations de la constante de von Kitzling R K .

Introduction

The fine structure constant α characterizes the strength of the electromagnetic interaction. It is defined as:

α = e 2 4πǫ 0 c , (1)
where ǫ 0 is the permittivity of vacuum, c the speed of light, e the electron charge and the reduced Planck constant ( = h/2π). It was introduced by Arnold Sommerfeld in 1916 to explain the fine structure of the levels of hydrogen atom. The fine structure constant is a dimensionless quantity, i.e. it is independent of the system of units used. The determinations of α are obtained in different domains of physics, from the quantum Hall effect and Josephson effect in solid state physics, from the combination between the precise measurements of atomic physics and quantum electrodynamics (QED) calculations, and from the measurements of the ratio h/m between the Planck constant and the mass m of a particle. In 2006, the recommended value in the last report of the CODATA (Committee on Data for Science and Technology) [1] was:

α -1 = 137.035999679(94).

(

) 2
This value was mainly deduced from the combination of the measurement of the electron anomaly a e made in 2006 by Gabrielse at Harvard University [2] by Kinoshita [3]. Since 2006, several progress have been made in the determination of the fine structure constant: an error in the QED calculation of a e has been detected, Gabrielse has improved his measurement of a e , and, in our group, we have made a new measurement of the ratio h/m Rb and obtained a new value of α. Moreover, the calculation of the fine structure in helium has been improved by Pachucki and, now, the value of α deduced from this fine structure is in agreement with the other determinations. The aim of this paper is to relate these different progress. In section 2 we make a review of the different methods used to determine α. Section 3 presents in detail our measurement of the ratio h/m Rb and, in section 4 we analyze the α measurements for the determination of the von Kitzling constant R K .

Determination of the fine structure constant

Figure 1 shows the most precise determinations up to date of the fine structure constant with a relative uncertainty smaller than 10 -7 . The first determination uses the precise measurement of the ground-state hyperfine splitting of the muonium (an atom formed by a positive muon and an electron). This splitting varies mainly as α 2 R ∞ , and it can be calculated very accurately. Indeed the muonium is a purely leptonic system, therefore there is not the difficulty due to the structure of the proton as in the calculation of the hydrogen hyperfine structure. The CODATA report [1] presents the analysis of the measurements carried out in 1982 and 1999 at LAMPF (Los Alamos Meson Physics Facility) [4]. The obtained value is:

α -1 = 137.036 0017(80), (3)
with a relative uncertainty of 5.8 × 10 -8 which is mainly due to the muon-to-electron mass ratio.

The atomic fine structures vary also as α 2 R ∞ . Unfortunately, in hydrogen, the measurement of the 2P 1/2 -2P 3/2 splitting (about 11 GHz) is limited by the natural width of the 2P level (100 MHz). Then the uncertainty of the best 2P 1/2 -2P 3/2 measurement [5,6] is only 15 kHz, corresponding to a relative uncertainty of about 1.4 × 10 -6 , and providing an α value at the level of 7 × 10 -7 . On the other hand, the fine structure of the 2 3 P J states in helium is a more promising case with a 30 GHz splitting and a 1.6 MHz natural width. The corresponding intervals have been accurately measured by radiofrequency spectroscopy by the group of Hessels [7], and by laser spectroscopy of the 2 3 S 1 -2 3 P J line by the groups of Shiner, Inguscio and Gabrielse [8,9,10]. The best result has a relative uncertainty of 2.4 × 10 -8 . On the theoretical side, the QED calculations of the two electrons system is difficult and, up to recently, there was a disagreement (12 standard deviations) between the theory and the measurements [11]. Recently, Pachucki and Yerokhin have overcome this problem by calculating all the QED terms up to the order α 5 R ∞ [12]. They obtain the value:

α -1 = 137.036 0011(39)( 16),

where the first uncertainty is due to the theory and the second to the experiment. With a relative uncertainty of 3.1 × 10 -8 , this value is in perfect agreement with the other α determinations (see figure 1). The most precise determinations of α are deduced from the measurement of the electron anomaly a e . During twenty years, the best a e value was the one obtained at the University of Washington with a relative uncertainty of 3.7 × 10 -9 [13]. In 2006, this result was superseded by the one of Gabrielse at Harvard University with an uncertainty reduced by a factor of about 6 [2]. Finally, in 2008, Gabrielse has improved again the a e measurement up to a relative uncertainty of 2.4 × 10 -10 [14]. During this period, the QED calculation of the electron anomaly has been continuously improved by Kinoshita who has calculated all the α 4 contributions [3]. In 2006, the combination of these experimental and theoretical results provided the value of the fine structure constant α -1 = 137.035 999 710(96) with a relative uncertainty of 7 × 10 -10 . This result determined the value given by the last report of the CODATA (see equation ( 1) [1]). Nevertheless, in 2007, Kinoshita and collaborators found an error in the calculation of the term in α 4 . With this correction, the value of α is shifted to α -1 = 137.035 999 070(98) [15]. This value is labeled "Harvard 2007" on figure 1. Finally, the value of the fine structure constant deduced from the last measurement of a e is:

α -1 = 137.035 999 084(51).

With a relative uncertainty of 3.7 × 10 -10 , this result, labeled "Harvard 2008" on figure 1, is the present most accurate determination of α.

Another way to obtain the fine structure constant is the measurement of the ratio h/m X between the Planck constant and the mass of a particle X. Indeed, from the ionization energy of the hydrogen atom (hcR ∞ = α 2 m e c 2 /2), one can deduce an expression of α:

α 2 = 2R ∞ c h m e , (6)
where m e is the electron mass. In this equation, c is exactly known and the relative uncertainty of R ∞ is smaller than 10 -11 . On the contrary there is no precise determination of the ratio h/m e .

To circumvent this limit, one introduces the mass ratio m X /m e to write the equation:

α 2 = 2R ∞ c m X m e h m X , (7)
where the limiting factor is the ratio h/m X . This method provides values of α which are only slightly dependant on the QED calculations. Actually, it is possible to extract the Rydberg constant from the combination of the radio frequency measurement of the 2S Lamb shift [5] and from the optical frequency measurement of the 2S 1/2 -8D 5/2 made in our group [16,17]. One obtains the frequency interval 2P 1/2 -8D 5/2 (about 771 THz) where the first QED contribution is the Lamb shift of the 2P 1/2 level (about 13 MHz [18]), i.e. 1.7 × 10 -8 of the total frequency interval. By contrast, the value of α deduced from the measurements of a e depends wholly on the QED calculations. Consequently, the values of α obtained from the h/m ratio can be used to test the QED calculations of the electron anomaly. This method has been implemented for the first time on a neutron beam to determine the ratio h/m n between the Planck constant and the neutron mass [19]. The principle is to measure together the velocity υ of the neutron beam with a time of flight technique and the de Broglie's wavelength λ DB by reflection on a silicon crystal (h/m n = υλ DB ). Consequently this α determination depends also of the measurement of the lattice spacing of silicon. The value obtained on this experiment is [1]:

α -1 = 137.036 0077 (28).

This result, labeled "h/m(neutron)" on figure 1, has a relative uncertainty of 2.1 × 10 -8 . The possibility to deduce the ratio h/m A (m A is the mass of the atom A) from the measurement of the recoil velocity was pointed out in 1976 by Hall and coworkers [20]. Indeed an atom at rest which absorbs a photon acquires the velocity υ r = k/m A , where k is the wave vector of the photon. Nevertheless a precise measurement of the ratio h/m A requires cold atom techniques and atom interferometry [21]. In a pioneering atom interferometry experiment in the nineties at Stanford University, Chu and colleagues succeeded in doing the first accurate measurement of the recoil velocity of 133 Cs atom [22]. As the m Cs /m e was precisely measured at MIT [23], the deduced value of α is [1]:

α -1 = 137.036 0000 (11), (9) with a relative uncertainty of 7.7 × 10 -9 (labeled "h/m(Cs)" on figure 1). In Paris we have followed the same ideas to measure the ratio h/m Rb between the Planck constant and the mass of 87 Rb atom. This experiment is described in the next section. Its originality is the use of Bloch oscillations to transfer to the atom a very large number of photon momenta. In 2006 we obtained a first result without using atom interferometry [24,25]:

α -1 = 137.035 998 84(91), (10)
with a relative uncertainty of 7.7 × 10 -9 . In 2008, this result was slightly improved by combining Bloch oscillations with atom interferometry and we obtain the value of α [26]:

α -1 = 137.035 999 45(62). ( 11
)
With a relative uncertainty of 4.5 × 10 -9 , this is the most accurate result after the values deduced from the electron anomaly measurements. Finally solid state physics allows two precise determinations of α. In the quantum Hall effect, the von Kitzling constant R K = h/e 2 is directly linked to the fine structure constant by the relation R K = µ 0 c/2α where µ 0 and c are exactly known in the Système International of units. The measurement of R K has been made in several NMIs (National Metrology Institute) by using the calculable capacitor of Thomson and Lampard. The CODATA report gives a complete review of these results. From the average of these measurements, one obtains the value α -1 = 137.036 0030 (25), (12) with a relative uncertainty of 1.8 × 10 -8 [1]. The details of these measurements will be described in section 4.

T R ¹/2 ¹/2 N Bloch oscillations T R ¹/2 ¹/2 v 0 v 0 + 2 N v r T R ¹/2 ¹/2 T R ¹/2 ¹/2 N Bloch oscillations N Bloch oscillations T R ¹/2 ¹/2 T R ¹/2 ¹/2 v 0 v 0 + 2 N v r
The last point of figure 1 is a combination of the low field gyromagnetic ratio measurements and the Josephson and quantum Hall effects. It corresponds to the mean value of the two results deduced from the measurements of the gyromagnetic ratio of the proton and helion ( 3 He nucleus). The principle is to replace in equation ( 6) the ratio h/m e by the product (h/e) × (e/m e ) where the first factor is given by the Josephson constant K J = 2e/h. The detail of this derivation will be described in section 4. Following the CODATA report [1], the value of the fine structure constant, labeled "Γ ′ p,h-90 " on figure 1, is:

α -1 = 137.035 9875(43), (13)
with a relative uncertainty of 3.1 × 10 -8 . This value is in slight disagreement (2.7 standard deviations) with the one deduced from the measurement of the electron anomaly.

Measurement of the ratio h/m Rb

We present in this section the Paris experiment in which the ratio h/m Rb is measured. The details are given in references [24,25,26]. The principle of the experiment is to coherently transfer as many recoils as possible to the atoms at rest (i.e. to accelerate them) and to measure the final velocity distribution. In our experiment, the atoms are efficiently accelerated by means of N Bloch oscillations (BO). In our latest measurement, Bloch oscillations are combined with a Ramsey-Bordé interferometer to precisely measure the induced atomic velocity variation (see figure 2).

The experiment develops in three steps. i) Firstly, a pair of π/2 pulses of a Raman transition transfers the 87 Rb atoms from the F = 2 hyperfine sublevel to the F = 1 one and produces a fringe pattern in the velocity distribution of these atoms. The width of the envelope of this velocity distribution varies inversely with the π/2 pulse duration τ, while the fringe width varies as 1/T R , where T R is the delay between the two π/2 pulses. ii) Secondly, we transfer to the selected atoms as many recoils as possible by means of Bloch oscillations. Bloch oscillations have been first observed in atomic physics by the groups of Salomon in Paris and Raizen in Austin [27,28,29]. Bloch oscillations can be interpreted as Raman transitions in which the atom begins and ends in the same energy level, so that its internal state (F = 1) is unchanged while its velocity has increased by 2υ r per Bloch oscillation (see figure 3). Bloch oscillations are produced in a one dimension vertical optical lattice which is accelerated by linearly sweeping the relative frequencies of the two counter propagating laser beams (frequencies ν 1 and ν 2 ). This leads to a succession of rapid adiabatic passages between momentum states differing by 2 k. iii) Finally, we measure the final velocity of the atoms by a second pair of π/2 pulses which transfers the atoms from the F = 1 to the F = 2 hyperfine level. The frequency difference between the two pairs of π/2 pulses is scanned to obtain a fringe pattern from which we can deduce the velocity variation between the two pairs of π/2 pulses.

In the vertical direction, an accurate determination of the recoil velocity would require an accurate measurement of the gravitational acceleration g. In order to circumvent this difficulty, we make a differential measurement by accelerating the atoms in opposite directions (upward and downward trajectories) keeping the same delay between the two pairs of π/2-pulses. Thus the photon-recoil measurement is a determination of the frequency difference between the central fringes of two opposite interferometers (upward and downward). Figure 4 shows two records obtained with 800 Bloch oscillations between the pairs of π/2 pulses. The frequency difference between the two spectra corresponds to 3200 recoil velocities. Moreover, the contribution of some systematic effects (energy level shifts) changes sign when the direction of the Raman beams is exchanged: for each up or down trajectory, the Raman beams directions are reversed and we record two velocity spectra. Finally, each determination of h/m Rb and α is obtained from 4 velocity spectra.

Our determinations of h/m Rb and α have been derived from 221 experimental data points. Each point corresponds to a 20 minute measurement. The total number of Bloch oscillations N up + N down has been varied from 200 to 1600. The dispersion of these n = 221 measurements is χ 2 /(n -1) = 1.85 and the resulting relative statistical uncertainty on α is 3 × 10 -9 . The systematic effects have been analyzed in detail in reference [25]. They are listed in Table 1 with the corresponding contributions and uncertainties. The two main effects are due to the geometry of the laser beams and to the second order Zeeman effect. To evaluate these effects, we have measured the wave front curvatures with a Shack-Hartmann wave front analyzer and mapped the frequency shift due to the magnetic field following the procedure described in reference [30]. Finally the relative uncertainty due to the systematic effects is 3.4 × 10 -9 and we obtain for α the value given by equation (11).

The most precise determinations of the fine structure constant are reported on figure 5. There is a very good agreement between our measurements (labeled "h/m(Rb)") and the value deduced from the electron anomaly measurement. Even if the uncertainty on this value is 10 times smaller than our result, the comparison of these two results provides the most stringent test of the QED calculations or, assuming these calculations exact, it gives a limit to test a possible internal structure of the electron, or the existence of low-mass dark-matter particles [31].

Determination of von Kitzling constant

The two determinations of the fine structure constant derived by solid state physics assume the exactness of the relations R K = h/e 2 and K J = 2e/h. To test this validity we introduce two small deviations characterized by ǫ K and ǫ J [32]:

R K = h e 2 (1 + ǫ K ) K J = 2e h (1 + ǫ J ). (14)
In this section we present the different determinations of R K which correspond to the values of the fine structure constant presented in section 2. This analysis is illustrated in figure 6. The first group of R K values corresponds to the direct measurements of the von Kitzling constant with the Lampard capacitor. The values obtained in the different NMIs are detailed on this figure [33,34,35,36,37]. These determinations of R K are in good agreement. The weighted mean value of these results is: R K = 25 812.808 18 (47) Ω.

The value of α given in equation ( 12) is directly obtained from this value of R K by the relation α = µ 0 c/2R K .

A second method to obtain the von Kitzling constant R K is to use the relation:

R K = (1 + ǫ K )µ 0 c 2α ( 16
)
with the values of α deduced from the h/m X ratio or from the electron anomaly. Assuming that ǫ K = 0, we have reported in figure 6 and the last measurement of the electron anomaly. These results are in good agreement with the direct measurements of R K .

) 1 /( K K R e + Value of R K deduced from G' p-90 (lo) Value of R K deduced from G' h-90 (lo) ) 1 ( J K R e + 25812,
) 1 /( K K R e + Values of R K deduced from a g-2 LKB-08 LKB-06 g-2 LKB-08 LKB-06 ) 1 /( K K R e + Value of R K deduced from G' p-90 (lo) Value of R K deduced from G' h-90 (lo) ) 1 ( J K R e + Value of R K deduced from G' p-90 (lo) Value of R K deduced from G' h-90 (lo) ) 1 ( J K R e +
Finally, it is also possible to deduce R K from the gyromagnetic ratio measurements. From the proton gyromagnetic ratio, we deduce the fine structure constant:

α -2 = c 4R ∞ × µ ′ p µ B × 2e/h γ ′ p ( 17
)
where µ ′ p and γ ′ p are the magnetic moment and the gyromagnetic ratio of the shielded proton. Taking into account the equation ( 14), one deduces:

α -2 = c 4R ∞ × µ ′ p µ e × g e 2 × K J /(1 + ǫ J ) γ ′ p ( 18
)
where the electron g-factor and the ratio µ ′ p /µ e are precisely well known [14,38]. The gyromagnetic ratio of the shielded proton has been measured in low field in terms of the conventional electrical units V 90 and Ω 90 with a relative uncertainty of 1.1 × 10 -7 [39]. The values of the gyromagnetic ratio of the shielded proton in terms of SI units (γ ′ p ) and in terms of the conventional units (Γ ′ p-90 (lo)) are linked by the relation [1]:

γ ′ p = Γ ′ p-90 (lo) × K J R K K J-90 R K-90 ( 19
)
where K J-90 and R K-90 are the conventional values of the Josephson and von Kitzling constants expressed in Hz/V 90 and Ω 90 . Then one obtains:

α -2 = c 4R ∞ × µ ′ p µ e × g e 2 × K J-90 R K-90 R K Γ ′ p-90 (lo) × 1 1 + ǫ J ( 20
)
and deduces R K :

R K = c 4R ∞ × µ ′ p µ e × g e 2 × K J-90 R K-90 Γ ′ p-90 (lo) × α 2 × 1 1 + ǫ J ( 21
)
where the limiting factor is the measurement of Γ ′ p-90 (lo). We can obtain a similar expression from the measurement of the gyromagnetic ratio of the shielded helion Γ ′ h-90 (lo):

R K = c 4R ∞ × µ ′ h µ ′ p × µ ′ p µ e × g e 2 × K J-90 R K-90 Γ ′ h-90 (lo) × α 2 × 1 1 + ǫ J ( 22
)
where µ ′ h is the magnetic moment of the shielded helion. The two values of R K (1 + ǫ J ) deduced from the equations ( 21) and ( 22) are reported in figure 6. If we suppose ǫ J = 0, these values are in slight disagreement with the other determinations of R K . This result suggests a non null value of ǫ J as already mentioned in reference [32].

Conclusion

We have presented a review of the most accurate determinations of the fine structure constant and described in detail the measurements of the h/m Rb ratio. This last experiments leads to a value of α which is the most precise after the determinations of α deduced from the electron anomaly. The agreement between these two results is the more stringent test of the QED. We have built an improved experimental setup and we expect to reduce the uncertainty of α to 1 ppb. A review of several determinations of the von Kitzling constant R K is also presented. There is a slight disagreement for the values deduced from the gyromagnetic ratios of the proton and the helion (relative difference of 2.5 × 10 -7 and 3 × 10 -7 respectively). This suggests a non exactness of the relation between the Josephson constant K J and 2e/h. It would be advisable to understand the origin of this discrepancy before the change of the definition of the units of the Système International.

Figure 1 :

 1 Figure1: Determination of the fine structure constant in different domains of physics. The CODATA value is shifted with respect to the most precise recent results because of an error in the QED calculations of the electron anomaly a e . Détermination de la constante de structure fine dans différents domaines de la physique. La valeur du CODATA est déplacée par rapport au résultat récent le plus précis à cause d'une erreur dans les calculs QED de l'anomalie de l'électron.

Figure 2 :

 2 Figure2: Scheme of the interferometer used for the measurement of h/m Rb . The first pair of π/2 pulses produces a fringe pattern in the atomic velocity distribution which is measured by the second pair of π/2 pulses. Between these two pairs of pulses, the atoms are accelerated upwards or downwards. The solid line corresponds to the atom in the F = 2 state, and the dashed line to the F = 1 state. Schéma de l'interféromètre utilisé pour la mesure de h/m Rb . La première paire d'impulsions π/2 produit une structure de franges dans la distribution de vitesse des atomes qui est mesurée par la seconde paire d'impulsions π/2. Entre ces deux paires d'impulsions, les atomes sont accélérés vers le haut où vers le bas. La ligne en trait plein correspond aux atomes dans l'état F = 2, et celle en pointillé à ceux dans l'état F = 1.

Figure 3 :

 3 Figure 3: Acceleration of cold atoms in a frequency chirped standing wave. The variation of the kinetic energy versus the atomic momentum is given by a parabola. This momentum increases by the quantity 2 k at each cycle. The Ramsey fringe patterns represent the momentum distribution of the atoms in the F = 1 hyperfine level. Accélération des atomes dans une onde stationnaire accélérée. La variation de l'énergie cinétique en fonction de l'impulsion de l'atome est représentée par une parabole. Cette impulsion augmente de 2 k à chaque cycle. Les franges de Ramsey représentent la distribution des impulsions des atomes dans l'état hyperfin F = 1.

Figure 4 :

 4 Figure4: Velocity spectra obtained when the atoms are accelerated downwards and upwards. The spectrum on the left corresponds to the downwards acceleration (800 Bloch oscillations) and on the right to the upwards acceleration (800 Bloch oscillations). The frequency difference between these spectra corresponds to 3200 recoil velocities. Franges d'interférence obtenues quand les atomes sont accélérés vers le bas et vers le haut. Le spectre de gauche correspond à une accélération vers le bas (800 oscillations de Bloch) et celui de gauche à une accélération vers le haut (800 oscillations de Bloch). La différence de fréquence entre ces deux spectres correspond à 3200 vitesses de recul.

Figure 5 :

 5 Figure 5: The five most accurate determinations of the fine structure constant. Les cinq déterminations les plus précises de la constante de structure fine.

Figure 6 :

 6 Figure 6: Determination of the von Kitzling constant R K . Détermination de la constante de von Kitzling R K . LNE: Laboratoire National d'Essais, France; NIM: National Institute of Metrologie, People's Republic of China; NPL: National Physical Laboratory, United-Kingdom; NMI: National Metrology Institute, Australia; NIST: National Institute of Standards and Technology, USA.

  and strenuous QED calculations of a e

	CODATA 2006 CODATA 2006 CODATA 2006
	hfs muonium hfs muonium hfs muonium hfs muonium
			Helium fine structure Helium fine structure Helium fine structure
	QED QED QED	g -2 of the electron (UW) g -2 of the electron (UW) g -2 of the electron (UW)
		g -2 of the electron (Harvard 2007) g -2 of the electron (Harvard 2007) g -2 of the electron (Harvard 2007)
		g -2 of the electron (Harvard 2008) g -2 of the electron (Harvard 2008) g -2 of the electron (Harvard 2008)
		h/m (neutron) h/m (neutron) h/m (neutron)
	h / m h / m h / m	h/m (Cs) h/m (Rb) 2006 2008 h/m (Cs) h/m (Rb) 2006 2008 h/m (Cs) 2006 h/m (Rb) 2008
	G' p,h-90 G' p,h-90 G' p,h-90 G' p,h-90	quantum Hall effect quantum Hall effect quantum Hall effect	Solid state physics Solid state physics Solid state physics
	137.035 990 137.035 990	137.036 000 137.036 000	137.036 010 137.036 010	a -1 a -1

Table 1 :

 1 Error budget on the determination of 1/α.

	Source			Correction Relative uncertainty (part in 10 9 )
	Laser frequencies						0.4
	Beams alignment			-2			2
	Wavefront curvature and Gouy phase	-11.9			2.5
	2nd order Zeeman effect			4.9			1
	Quadratic magnetic force			-0.59			0.2
	Gravity gradient			-0.07			0.02
	Light shift (one photon transition)					0.1
	Light shift (two photon transition)					0.01
	Light shift (Bloch oscillation)		0.48			0.2
	Index of refraction atomic cloud					0.3
	Index of refraction background vapor	-0.36			0.3
	Rydberg constant and mass ratio [1]					0.23
	Global systematic effects			-9.54			3.4
		h/m(Cs) h/m(Cs)
						2005 2008 2005 2008 2005 2008	h/m(Rb) h/m(Rb) h/m(Rb)
			a e (UW) a e (UW)
				a e (Harvard) a e (Harvard)
	599,8 599,8	599,85 599,85	599,9 599,9	599,95 599,95	600 600	600,05 600,05	600,1 600,1	600,15 600,15
			(a -1 -137.03) × 10 5 (a -1 -137.03) × 10 5

  the values of R K obtained from our measurements of h/m Rb 8

					LNE-01
					NIM-95
		Measurements of R K with the		NPL-88
			Lampard capacitor		NMI-97
					NIST-97
		Values of R K deduced from a		LKB-06
					LKB-08
					g-2
	25812,79	25812,795	25812,8	25812,805	25812,81	R K (W)