Keywords: Contingent valuation, double-bounded dichotomous choice, Patient, proxy, anchoring, structural shift, influenza JEL Classification: I1, C2, H4

The small sample size of contingent valuation (CV) surveys conducted in patients may have limited the use of the single-bounded (SB) dichotomous choice format which is recommended in environmental economics.

In this paper, we explore two ways to increase the statistical efficiency of the SB format: 1) by the inclusion of proxies in addition to patients; 2) by the addition of a follow-up dichotomous question, i.e. the doublebounded (DB) dichotomous choice format. We found that patients (n=223) and spouses (n=64) answering on behalf of the patient had on average a similar willingness-to-pay for earlier alleviation of flu symptoms.

However, a patient was significantly more likely to anchor his/her answer on the first bid as compared to a spouse. Finally, our original DB model with shift effect and heterogeneous anchoring reconciled the discrepancies found in willingness-to-pay statistics between SB and DB models in keeping with increased statistical efficiency.

Introduction

Contingent valuation (CV) is used to assess the monetary value of non-market goods or services. This involves constructing a hypothetical market for those goods or services and directly asking individuals the maximum amount they are "willing-to-pay" (WTP) to get them. The CV method has been principally used in environmental and transport economics (Bateman and Willis, 1999;[START_REF] Jones-Lee | The economics of safety and physical risk[END_REF][START_REF] Beattie | On the contingent valuation of safety and the safety of contingent valuation: Part 1-caveat investigator[END_REF][START_REF] Carthy | On the contingent valuation of safety and the safety of contingent valuation: Part 2-the cv/sg "chained" approach[END_REF] accounting for 70% of the 2,071 studies referenced in the Environmental Valuation Reference Inventory.1 Over the last decade, CV has become increasingly popular in health economics to value health improvement from a patient's perspective [START_REF] Olsen | Theory versus practice: a review of 'willingness-to-pay' in health and health care[END_REF].

Several health economists recently acknowledged that transferring standards of practice from environmental economics to health economics may not be that straightforward [START_REF] Smith | The discrete-choice willingness-to-pay question format in health economics: should we adopt environmental guidelines?[END_REF][START_REF] Smith | Construction of the contingent valuation market in health care: a critical assessment[END_REF][START_REF] Hanley | Estimating the monetary value of health care: lessons from environmental economics[END_REF]. To begin, the construction and presentation of the CV market should take into account the specific features of health care services: 1) scenario descriptions should include important and relevant dimensions to the patients, and these should be elicited in pilot studies;

2) elements of risk or uncertainty should be provided according to the perspective of the surveyed population, i.e. either the uncertain benefits of a health intervention in patients or the risk of getting a disease in the general population; 3) the payment vehicle, e.g. out-of-pocket payments, as well as the time period over which WTP is assessed, should be in accordance with the country health care financing system [START_REF] Smith | Construction of the contingent valuation market in health care: a critical assessment[END_REF]. If these features of health care services are not properly defined, then the CV market may not be realistic enough and WTP answers may be altered by a strong "hypothetical bias" [START_REF] Liljas | On hypothetical bias and calibration in cost-benefit studies[END_REF].

Next, WTP can be elicited in CV surveys through various questionnaire formats [START_REF] Smith | The discrete-choice willingness-to-pay question format in health economics: should we adopt environmental guidelines?[END_REF][START_REF] Hanley | Estimating the monetary value of health care: lessons from environmental economics[END_REF][START_REF] Frew | Eliciting willingness to pay: comparing closedended with open-ended and payment scale formats[END_REF]. The single-bounded (SB) dichotomous choice format was introduced by [START_REF] Bishop | Measuring values of extra-market goods: are indirect measures biased?[END_REF]. The respondent is asked if he/she would be willing to pay or not a given amount (bid) for the provision of the non-market goods or services. The proposed bids are varied across respondents to trace out the WTP distribution and to estimate WTP statistics.

The SB dichotomous choice format has been recommended in environmental economics [START_REF] Arrow | Report of the NOAA panel on contingent valuation[END_REF]. It is consistent with referendum voting behavior which is an incentive compatible mechanism for provision of public goods. In addition, a "take-it-or-leave-it" question is easier for respondents to answer because it is a more market-like question as compared to an open-ended question. However, the SB dichotomous choice approach requires large sample sizes to reach a sufficient level of precision of WTP estimates [START_REF] Hanemann | Statistical Analysis of Discrete Response CV Data[END_REF]. In environmental economics surveys, the answers of the general population are of interest and the survey budget is the sole constraint to the sample size. In health economics, the answers of patients are usually of interest [START_REF] Olsen | Theory versus practice: a review of 'willingness-to-pay' in health and health care[END_REF]; therefore the type of respondent limits the sample size as compared to the general population. Accordingly, other questionnaire formats that require fewer respondents have been used, i.e. open-ended WTP questions and payment cards [START_REF] Olsen | Theory versus practice: a review of 'willingness-to-pay' in health and health care[END_REF].

If health economics CV surveys are to follow the SB dichotomous choice format recommended in environmental economics, then increasing statistical efficiency (lowering the standard error) of the mean WTP estimate may be achieved in several ways: 1) increasing the sample size by surveying other types of respondents than the patient; 2) increasing information about each respondent's WTP by adding a follow-up dichotomous choice question; and 3) optimizing the bid design.

The type of respondent is a crucial issue in health economics [START_REF] Dolan | Whose preferences count?[END_REF]. When a patient-based perspective is relevant for the CV survey, the inclusion of proxies as defined by relatives living with patients is an interesting way to increase the sample size. The question begged is: Will the proxies have the same WTP as the patients? The literature in health economics is scarce on the level of agreement between proxies and patients. In previous CV surveys, proxies were interviewed on behalf of the patient, either parents of sick children [START_REF] Sorum | Measuring patient preferences by willingness to pay to avoid: the case of acute otitis media[END_REF][START_REF] Liu | Mother's willingness to pay for her own and her child's health: a contingent valuation study in taiwan[END_REF][START_REF] Sach | Willingness-to-pay for pediatric cochlear implantation[END_REF] or spouses of cognitively impaired patients [START_REF] Chiu | Willingness of families caring for victims of dementia to pay for nursing home care: results of a pilot study in taiwan[END_REF][START_REF] Mulvaney-Day | Using willingness to pay to measure family members' preferences in mental health[END_REF]. Similarly, proxies were mostly interviewed on health status of patients when the need for proxies was the most salient.

The level of agreement between proxies and patients remains unclear from the few studies having compared both of their answers as illustrated by the following results: 1) agreement on functional status was better for observable behaviors such as "Activities of Daily Living" rather than for more subjective areas such as functional limitations, sensorial limitations or mental health [START_REF] Todorov | Bias in proxies' reports of disability: data from the national health interview survey on disability[END_REF]; 2) agreement was only moderate on health-related quality of life for patients with mild cognitive impairment [START_REF] Sneeuw | The use of significant others as proxy raters of the quality of life of patients with brain cancer[END_REF][START_REF] Tamim | Proxy reporting of quality of life using the eq-5d[END_REF]; 3) agreement was good on the utility of hypothetical disease vignettes [START_REF] Saigal | Differences in preferences for neonatal outcomes among health care professionals, parents, and adolescents[END_REF][START_REF] Lenert | Differences in health values among patients, family members, and providers for outcomes in schizophrenia[END_REF][START_REF] Dawson | Health state valuation in mild to moderate cognitive impairment: feasibility of computer-based, direct patient utility assessment[END_REF]; 4) agreement was poor on the utility of actual health states of 80-year patients who were willing to trade significantly less time for a healthy life than their proxies [START_REF] Tsevat | Health values of hospitalized patients 80 years or older. help investigators. hospitalized elderly longitudinal project[END_REF].

The statistical efficiency of health economics CV surveys may also be increased by using the double-bounded (DB) dichotomous choice method. Surprisingly, the DB dichotomous choice method has received little attention in health economics [START_REF] Bala | Valuing outcomes in health care: a comparison of willingness to pay and quality-adjusted life-years[END_REF][START_REF] Slothuus | Willingness to pay in arthritis: a danish contribution[END_REF][START_REF] Clarke | Valuing the benefits of mobile mammographic screening units using the contingent valuation method[END_REF][START_REF] Dalmau-Matarrodona | Alternative approaches to obtain optimal bid values in contingent valuation studies and to model protest zeros. estimating the determinants of individuals' willingness to pay for home care services in day case surgery[END_REF][START_REF] Kennedy | Revealed preference valuation compared to contingent valuation: radon-induced lung cancer prevention[END_REF][START_REF] Watson | Exploring preference anomalies in double bounded contingent valuation[END_REF]. The DB dichotomous choice method refines the information on each respondent's WTP by adding to the initial dichotomous choice a follow-up dichotomous question: a higher bid is proposed if the initial bid was accepted, and a lower bid otherwise [START_REF] Hanemann | Some issues in continuous and discrete response contingent valuation studies[END_REF]. [START_REF] Hanemann | Statistical efficiency of double-bounded dichotomous choice contingent valuation[END_REF] showed that the DB method was asymptotically more efficient than the SB method, whereas a third discrete question delivered relatively little efficiency gains compared to those achieved by the DB method [START_REF] Scarpa | Efficiency gains afforded by improved bid design versus follow-up valuation questions in discrete-choice cv studies[END_REF]. However, discrepancies in WTP statistics were commonly observed between the SB and DB methods. Several psychological and behavioral phenomenons were proposed to explain inconsistency between SB and DB methods. Starting point bias is a framing effect where respondents anchor their stated WTP value to the initial bid that represents a reasonable value [START_REF] Boyle | Starting point bias in contingent valuation bidding games[END_REF][START_REF] Herriges | Starting point bias in dichotomous choice valuation with follow-up questioning[END_REF][START_REF] Boyle | Anchoring and adjustment in single-bounded, contingent-valuation questions[END_REF][START_REF] Frykblom | An experimental testing of anchoring effects in discrete choice questions[END_REF][START_REF] Whitehead | Incentive incompatibility and starting-point bias in iterative valuation questions[END_REF]. Assuming a constant anchoring effect among respondents, [START_REF] Herriges | Starting point bias in dichotomous choice valuation with follow-up questioning[END_REF] showed that both mean and variance of the prior WTP could be biased. The follow-up question can also induce behaviors such as "yea-saying" or incentive incompatibility resulting in a "shift" in respondent's answers between the first and second dichotomous question. "Yea-saying" behavior occurs when a respondent is more likely to answer "yes" to the follow-up dichotomous choice question when he/she has previously answered "yes" to the first dichotomous choice question (see for instance [START_REF] Alberini | Modeling response incentive effects in dichotomous choice contingent valuation data[END_REF]. Incentive incompatibility occurs when a respondent who initially said "yes" is asked the same valuation question at a higher price, which would be equivalent to raising the price after the purchase has been made in a market setting, and the respondent could then respond with a "protest no" to the follow-up question even if his true WTP is higher than the increased price [START_REF] Alberini | Modeling response incentive effects in dichotomous choice contingent valuation data[END_REF][START_REF] Whitehead | Incentive incompatibility and starting-point bias in iterative valuation questions[END_REF][START_REF] Carson | Incentive and informational properties of preference questions[END_REF].

Finally, dichotomous choice-related WTP statistics are analytically a function of both the set of bids proposed to respondents and the unknown true WTP distribution. Changing the set of bids may result in significantly different WTP statistics [START_REF] Boyle | Bid design and yea saying in single-bounded, dichotomous-choice questions[END_REF][START_REF] Cameron | Referendum contingent valuation estimates: Sensitivity to the assignment of offered values[END_REF][START_REF] Cooper | Sensitivity of willingness-to-pay estimates to bid design in dichotomous choice contingent valuation models[END_REF]Kanninen, 1995a). While the mean WTP estimate converges asymptotically to the true parameter, standard deviation estimate still depends on the proposed bids [START_REF] Abdelbasit | Experimental design for binary data[END_REF][START_REF] Sitter | Robust designs for binary data[END_REF][START_REF] Hanemann | Statistical Analysis of Discrete Response CV Data[END_REF]. The set of bids is usually selected to cover WTP dispersion as found in a pilot study. Otherwise, optimal bid designs rely typically on the choice of one, two or three bids that depend on the underlying true WTP distribution and the number of observations [START_REF] Abdelbasit | Experimental design for binary data[END_REF][START_REF] Minkin | Optimal designs for binary data[END_REF][START_REF] Ford | The use of a canonical form in the construction of locally optimal designs for non-linear problems[END_REF][START_REF] Nyquist | Optimal designs of discrete response experiments in contingent valuation studies[END_REF][START_REF] Alberini | Optimal designs for discrete choice contingent valuation surveys: Singlebound, double-bound, and bivariate models[END_REF]. Paradoxically, an optimal bid design provides optimal efficiency when the underlying true distribution and parameters are known. Sequential bid design overcomes partially this limitation [START_REF] Abdelbasit | Experimental design for binary data[END_REF][START_REF] Minkin | Optimal designs for binary data[END_REF][START_REF] Nyquist | Optimal designs of discrete response experiments in contingent valuation studies[END_REF][START_REF] Kanninen | Design of sequential experiments for contingent valuation studies[END_REF]. In sequential CV sub-experiments, the WTP distribution is estimated using all available observations so that a new "optimal" bid design is derived for use with the subsequent iteration of the survey.

While successive updates improve the efficiency of the design for poor initial estimates, the sequential bid design is questionable in practice.

The previous literature review showed that each way to increase statistical efficiency of SB dichotomous choice data in health economics has the potential to bias WTP statistics. In this paper, we explore the statistical effects of increasing sample size by surveying patients as well as proxies and increasing information about each respondent's WTP by adding a follow-up dichotomous choice question. WTP for earlier alleviation of flu symptoms was elicited among adult patients or spouses by use of the DB dichotomous choice method. In each household surveyed, the respondent was either the patient or his/her spouse. We assumed that the SB dichotomous choice data of patients provided an unbiased WTP distribution of reference for earlier alleviation of flu symptoms. Three DB dichotomous choice models were compared: the "naïve" DB model; the DB model with shift effect and constant anchoring; and an original DB model with shift effect and heterogeneous anchoring, where anchoring depends on individual characteristics. In the four dichotomous choice models, a specific parameter accounted for potential discrepancies on mean WTP estimate between patients and spouses. Accordingly, we assessed the level of agreement on mean WTP between patients and spouses as well as potential bias on WTP statistics vs. statistical efficiency achieved by the follow-up dichotomous choice question.

The paper is organized as follows. Section 2 describes the DB dichotomous choice data used that relate to the WTP for earlier alleviation of flu symptoms; the econometric model; and the estimation procedure. Section 3 provides the comparative results between the four dichotomous choice models.

In Section 4, we discuss the results and the implications of the study.

Material and methods

Contingent valuation data

The CV survey was nested into a prospective household contact study on influenza burden of disease in France. Patients suffering from influenza-like illness were included when visiting their general practitioner during the influenza outbreak from January to March 2000. The general practitioner checked off flu symptoms (a flu symptom score was computed at entry with a score of 1 indicating the presence of all symptoms and a score of 0 indicating the absence of symptoms), and took nasal swabs for testing influenza A and B viruses. A household diary questionnaire had to be filled up over a 2-week period by the patient and each household member. The diary questionnaire included daily questions about flu symptoms and medical consumptions, as well as questions about individual past experience with flu (vaccination this year, influenza-like illness the previous year), the presence of chronic disease, risk factors for lung disease, and household socio-economic characteristics. More details on the prospective household contact study are provided elsewhere [START_REF] Carrat | Influenza burden of illness: estimates from a national prospective survey of household contacts in france[END_REF].

On day three after inclusion of the patient by the general practitioner, a CV survey was conducted by telephone. Due to budget constraints, we interviewed the first 435 recruited households (out of 946 -45%) [START_REF] Carrat | How much patients are willing to pay for an earlier alleviation of influenza ?[END_REF][START_REF] Carrat | Influenza burden of illness: estimates from a national prospective survey of household contacts in france[END_REF]. Respondents were asked about their WTP for earlier alleviation of flu symptoms. In each household, the respondent was either the adult patient or his/her spouse depending on who was answering the phone. Consequently, two WTP scenarios were prepared to reflect both a patient-based perspective where the respondent is the patient and a proxy-based perspective where the spouse is responding on behalf of the patient. Whatever the WTP scenario, respondents were randomly asked about their WTP for the alleviation of flu symptoms by 1 day or 3 days (see Appendix A).

A pilot CV study was performed by mail in autumn 1999 to test WTP scenarios for the alleviation of flu symptoms by 1 day and to select the bids. After visiting their general practitioner, patients with flu symptoms were sent a WTP question with a single bid randomly selected from [25; 50; 100; 200; 400] French Francs (FF). Two hundred seven patients of mean (sd.) age of 29 (21) years including 54% female, returned the CV questionnaire. The return rate differed significantly according to the bid proposed (25FF: 23%; 50FF: 15%; 100FF: 21%; 200FF: 34%; 400FF: 6%). In the standard probit model, the mean WTP for the alleviation of flu symptoms by 1 day was 137FF (deciles: 20th: 26FF; 40th: 103FF; 60th: 170FF; 80th: 247FF; 100th: 1216FF). Despite a possible bias on WTP distribution, we decided to keep the same set of bids for the main telephone CV survey as it traced out well the WTP distribution.

The elicitation mechanism in the telephone survey was based on the double-bounded dichotomous choice format. In other words, respondents were asked for two successive bids. The initial bid proposed to the respondent was randomly selected from [25;50;100;200;400] FF or [50;100;200;400;800] FF for the alleviation of flu symptoms by 1 day or 3 days, respectively. The follow-up bid was set at twice the initial bid if the respondent accepted the initial bid, and at half the initial bid otherwise. Out of 435 household respondents, 287 (66.0%) WTP interviews were used for the analysis, 10 (2.3%) were not willing to participate in the CV survey, and 138 were excluded from the analysis because of missing information -95 (21.8%) households did not send back the diary questionnaire including clinical and socio-economic variables, and 43 (9.9%) households did not complete the household income question. In the following, all bids, costs and WTP were converted to Euros (1 French Franc=0.152449A C).

Econometric Model

The true WTP distribution for earlier alleviation of flu symptoms is unknown. For comparison purposes among types of respondent and dichotomous choice models, we assumed that the SB model of patients' data provided an unbiased estimator of WTP statistics for earlier alleviation of flu symptoms. In all dichotomous choice models, we modeled the prior WTP of the proxy as a combination of the expected patient's prior WTP and a parameter θ. This parameter accounts for potential discrepancies on mean WTP estimate between patients and proxies. Potential heteroscedasticity between patients and proxies was also taken into account by an error term specific to each perspective.

We used standard interval-data modeling of double-bounded dichotomous choice data. Interval-data modeling assumes no changes in underlying WTP distribution between the first bid (the prior WTP according to the SB model) and the follow-up bid (the posterior WTP according to the DB model).

Interval-data modeling allows specification of the psychological effect (starting point bias) and behavioral effect (shift effect) induced by the follow-up discrete question. In addition to the naïve DB model without psychological/behavioral effects (where prior=posterior WTP) and the DB model with shift effect and constant anchoring (where prior differs from posterior WTP), we developed an original DB model with shift effect and heterogeneous anchoring-where anchoring depends on individual characteristics, e.g., the type of respondent. Originally, the econometrics developed in this paper allowed: 1) to assess the level of agreement on mean WTP estimates between patients and spouses in all dichotomous choice models; 2) to assess potential bias on WTP statistics vs. statistical efficiency induced by the follow-up dichotomous choice question.

Prior WTP of patients and proxies

First, let us consider the patient-based perspective where the patient answers the WTP question.

The prior WTP of the patient is defined as follows:

W * i1 = X i β + ǫ i , ǫ i ∼ N ID(0, σ 2 ) (1)
where X i is a set of explanatory variables which captures consumer preferences, and unknown parameters β and σ are respectively a vector and a scalar.

Second, let us consider the proxy-based perspective where the proxy answers the WTP question on behalf of the patient. The prior WTP of the proxy j answering on the behalf of the patient i is defined as follows:

W * j1 = E(W * i1 ) + θ + ν j , ν j ∼ N ID(0, ω 2 ) (2)
In our specification, the prior WTP of the proxy is a combination of the expected patient's prior WTP and a parameter θ. The parameter θ accounts for potential discrepancies between the prior WTP of the proxy and that of the patient. When θ is positive, the proxy overstates the patient's

prior WTP and understates it when it is negative. If θ is not significantly different from zero, the prior WTP of the proxy does not differ from that of the patient. The error term ν j is specific to the proxy-based perspective and it has a different variance ω2 than the error term ǫ i in the patientbased perspective. 2 We therefore take into account a potential heteroscedasticity induced by the two perspectives that could lead to misleading results otherwise (which is likely to occur in econometric models with discrete dependent variables, see [START_REF] Yatchew | Specification error in probit models[END_REF].

Equations ( 1) and ( 2) can be combined into one equation such that

W * i1 = X i β + θP i + u i , u i ∼ N ID(0, σ 2 i = σ 2 (1 -P i ) + ω 2 P i ) (3)
where P i is a dummy variable which is one when the proxy answers on behalf of the patient and zero otherwise. Equation (3) therefore reduces to (1) when P i = 0 and to (2) when P i = 1.

Dichotomous choice modeling

Let us now focus on the econometrics of the elicitation procedure. In the single-bounded dichotomous choice model (SB model), the respondent's answer to the first bid b i1 is observed, but not W * i1 . The answer of respondent i is then defined as follows:

W i1 = 1 if W * i1 ≥ b i1 , and W i1 = 0 otherwise. (4)
Assuming that the error term is identically and independently distributed according to a normal distribution with mean 0 and standard deviation σ i , W * i1 is estimated by means of the standard probit model with W i1 as the dependent variable.

In the double-bounded dichotomous choice model, a follow-up bid is proposed to the respondent.

When respondent i answers yes to the first bid, b i1 , the second bid b i2 is higher, and lower otherwise.

The "naïve" DB model assumes that WTP is independent of both first bid (b i1 ) and follow-up bid (b i2 ). W * i1 is estimated by means of standard multinomial model.

However, the follow-up discrete answer may be biased. First, the first bid may induce a starting point bias as respondents may anchor their WTP on this bid which may be seen as providing information about the "correct" price. Thus, respondents combine their prior WTP, W * i1 , with the first bid b i1 to form a revised or posterior WTP, W * i2 . To account for this phenomenon, [START_REF] Herriges | Starting point bias in dichotomous choice valuation with follow-up questioning[END_REF] proposed a DB model with constant anchoring in which the WTP W * i2 is defined as a convex combination of the prior WTP W * i1 and the first bid b i1 such that:

         W * i1 = X i β + θP i + u i W * i2 = (1 -γ)W * i1 + γb i1 (5)
where γ lies between 0 and 1.

Secondly, the follow-up bid may induce behaviors such as yea-saying or incentive incompatibility.

Yea-saying and incentive incompatibility are usually treated as a "shift" in respondents' answers between the first and second dichotomous question [START_REF] Alberini | Modeling response incentive effects in dichotomous choice contingent valuation data[END_REF][START_REF] Mcleod | Willingness-to-pay estimates using the double-bounded dichotomous-choice contingent valuation format: a test for validity and precision in a bayesian framework[END_REF][START_REF] Whitehead | Incentive incompatibility and starting-point bias in iterative valuation questions[END_REF]. Alberini et al. proposed a DB model with a shift parameter considering that the posterior WTP is based on the true prior WTP with a shift parameter [START_REF] Alberini | Modeling response incentive effects in dichotomous choice contingent valuation data[END_REF].

Although considered separately in previous papers, [START_REF] Whitehead | Incentive incompatibility and starting-point bias in iterative valuation questions[END_REF] proposed an econometric model which takes into account both constant anchoring and shift induced by the follow-up question (DB model with shift effect and constant anchoring). Formally, the econometric model is defined as follows:

         W * i1 = X i β + θP i + u i W * i2 = (1 -γ)W * i1 + γb i1 + δ (6)
where γ lies between 0 and 1, and δ is the "structural shift" parameter. The structural shift parameter could be positive due to yea-saying behavior (Kanninen, 1995a) or negative due to incentive incompatibility [START_REF] Alberini | Modeling response incentive effects in dichotomous choice contingent valuation data[END_REF].

In the previous DB model, the anchoring bias parameter γ was the same for all respondents.

However, in the presence of individual heterogeneity, results based on standard regression can be seriously misleading if this heterogeneity is not taken into account (see [START_REF] Aprahamian | Modeling starting point bias and unobserved heterogeneity in contingent valuation surveys: an application to air pollution[END_REF][START_REF] Aprahamian | Heterogeneous anchoring and the shift effect in iterative valuation questions[END_REF]. In this paper, we propose a DB model with shift effect and heterogeneous anchoring where the anchoring parameter is variable depending on individual characteristics. The model is now defined as:

         W * i1 = X i β + θP i + u i W * i2 = [1 -γ(Z i ; η)]W * i1 + γ(Z i ; η)b i1 + δ (7)
where γ(.) is a function of explanatory variables Z i and a parameter vector η. As in previous DB model with constant anchoring, γ(.) has to lie between 0 and 1. When one only includes the constant term in the explanatory variables Z i , the heterogeneous anchoring model 7 reduces to equation 5.

This model is of particular interest as it allows testing for different anchoring patterns depending on the perspective of the respondent, either patient-based or proxy-based. This is done by considering the dummy variable P i as an explanatory variable included in the function γ(.).

Estimation procedure

Both the prior WTP W * i1 and the posterior WTP W * i2 are not directly asked for. To estimate our econometric model, we have to define the probabilities associated to each pair of responses (no, no; no, yes; yes, no; yes, yes). The probability of a yes to the first bid depends on whether WTP is greater than the bid b i1 and is then defined as:

P (yes) = P (W i1 > b i1 ) = Φ b i1 -X i β -θP i σ i (8)
where Φ(.) is the normal c.d.f. since error terms are assumed to be normally distributed. Using equation 7, the probability of a yes to the first bid and a no to the second bid is given by:

P yn i = P (W i2 > b i2 ) -P (yes) (9) = Φ b i2 -γ(Z i ; η)b i1 -δ (1 -γ(Z i ; η))σ i - X i β + θP i σ i -Φ b i1 -X i β -θP i σ i ( 10
)
where the function γ(Z i ; η) is such that

γ(Z i ; η) = γ P a (1 -P i ) + γ P r P i + M i λ (11)
where γ P a is the anchoring parameter when the patient answers, γ P r is the anchoring parameter when the proxy answers on behalf of the patient and λ a vector of parameters associated with the variables M i . The variables M i are additional variables that may induce heterogeneity in anchoring.

It is possible to allow for some more flexibility such that:

γ(Z i ; η) = γ P a (1 -P i ) + γ P r P i + M i λ P a (1 -P i ) + M i λ P r P i (12)
In this specification, the effect of additional variables M i on anchoring can differ depending on the perspective of the respondent.3

The probabilities associated with other possible pair of response are calculated in the same way:

P nn i =Φ b i2 -γ(Z i ; η)b i1 -δ (1 -γ(Z i ; η))σ i - Xβ + θP i σ i (13) P ny i =Φ b i1 -X i β -θP i σ i -Φ b i2 -γ(Z i ; η)b i1 -δ (1 -γ(Z i ; λ))σ i - Xβ + θP i σ i (14)
P yy i =1 -Φ b i2 -γ(Z i ; η)b i1 -δ (1 -γ(Z i ; η))σ i - Xβ + θP i σ i (15)
Accordingly, the following log-likelihood function is defined as: We first tested for the significance of γ P a and γ P r in the γ(.) function. We then tested additional variables M i that may significantly induce heterogeneous anchoring. All explanatory variables recorded in the survey were tested one by one by means of standard likelihood ratio (LR) tests according to specification (11). We then introduced simultaneously all significant M i variables (p-value less than 0.10) in the γ(.) function by using specification (12). Hence, the remaining variables were included in the final DB model with shift effect and heterogeneous anchoring. Finally, we assessed the relevance of the DB model with shift effect and heterogeneous anchoring by testing the null hypothesis of no anchoring and no shift effect ("naïve" DB model) and the null hypothesis of constant anchoring and structural shift (DB model with shift effect and constant anchoring). This was done by means of LR tests.

ℓ(y; β, θ, δ, η,

Empirical results

Out of the 287 adult patients with flu symptoms included in the analysis, there were 51.9% male respondents and the mean (sd.) age was 41 ( 16) years (Appendix B). They had a mean flu symptom score at entry of 0.41 (0.16) with a higher score indicating higher severity, and 62.0% of cases were tested positive for influenza A or B virus, 14.3% had an influenza-like illness the previous year, and 10.8% had flu vaccination during the year of the study. Households had on average 3.1 (1.2) members, and a monthly income under A C1,524 in 31.0% and above A C3,049 in 21.3% of the cases. At the time of the CV survey, another member had flu symptoms in 24.4% of households in addition to the patient recruited by the general practitioner. Respondents provided their WTP for earlier alleviation of flu symptoms for themselves when they were the patient (77.7%) or for their symptomatic spouse (22.3%) when they were not the patient.

The discrete choice CV data is given in Tables 1.a and 1.b according to the size of clinical benefits (alleviation of flu symptoms by 1 or 3 days respectively) and the respondents' perspective. The columns separate respondents according to which initial bid was proposed and the rows provide the bid acceptance rate assuming that a respondent who accepts a bid accepts all lower bids. All columns show a decreasing probability of bid acceptance from lower to higher follow-up bids. In most instances when the respondent was the patient, the rows show a positive relationship between increasing initial bids and the probability of follow-up bid acceptance. For example, the probability of acceptance of the bid A C61 for the alleviation of flu symptoms by 3 days increased from 16% when the initial bid was A C30.5 to 68% with an initial bid of A C122 (p<0.001, Fisher's exact test).

Accordingly, the presence of a starting point bias was likely when the respondent was the patient.

When the respondent was the spouse answering on behalf of the patient, the trend was less salient, although sample size was smaller.

Table 2 presents the results of the four econometric models we used to describe the CV dichoto-mous choice data: the SB model (only the answer to the initial bid is considered), the "naïve" DB model (both answers to the initial and follow-up bids are considered), the DB model taking into account the starting point bias with constant anchoring (a unique parameter γ) and our original DB model with heterogeneous anchoring that depends on respondent's characteristics. A structural shift parameter was included in the two later models to control for possible behaviors associated with the follow-up question [START_REF] Whitehead | Incentive incompatibility and starting-point bias in iterative valuation questions[END_REF].

A large set of explanatory variables was considered to avoid a potential misspecification bias. All The "naïve" DB model provided efficiency gains when compared to the SB model, i.e. the standard error of mean WTP estimate decreased from 1.23 to 0.79 (Table 2). However, the "naïve" DB model significantly biased the mean WTP downward from A C58.8 (95% CI [56.4 to 61.3] in the SB model) to A C44.7 (95% CI [43.12 to 46.3] in the "naïve" DB model). In the DB model with shift effect and constant anchoring, we found a significant constant anchoring effect γ = .421 (p < .001), but no significant structural shift (δ = -.79; p = .391). The anchoring effect estimate seemed reasonable and was within the range of previous estimates in CV environmental surveys (from γ = .22 to γ = .38) [START_REF] Herriges | Starting point bias in dichotomous choice valuation with follow-up questioning[END_REF][START_REF] Green | Referendum contingent valuation, anchoring, and willingness to pay for public goods[END_REF][START_REF] Holmes | Dynamic learning and context-dependence in sequential, attributebased, stated-preference valuation questions[END_REF]. However, efficiency gains of the DB model with constant anchoring had to be balanced by the remaining bias downward of the mean WTP estimate (A C48.1, 95% CI [46.4 to 49.8])

In the DB model with shift effect and heterogeneous anchoring, the structural shift parameter was also negative (δ = -.67) and not significant (p = .483). Explanatory variables of the heterogeneous anchoring were selected in a three-stage procedure. We first considered whether anchoring was different across respondents' perspectives. The anchoring effect was significant when the respondent was the patient, but it was not the case when the respondent was the spouse answering on behalf of the patient (p < 0.001 and p = 0.601, respectively).

We then added explanatory variables one by one to test for further heterogeneity in anchoring anchoring showed a similar pattern: ω = 42.0 (se. 5.7) and σ = 55.1 (se. 7.2) with no significant difference (p = .127). Opposite results were found in other DB models where the standard deviation of the WTP of patients was smaller than that of proxies (ω > σ). This phenomenon was significantly marked in the DB model with constant anchoring (p = .147 in the "naïve" DB model and p = .008 in the DB model with constant anchoring).

Figure 1 illustrates the WTP for the earlier alleviation of flu symptoms by 1 day according to the level of anchoring of the respondent using the DB model with heterogeneous anchoring. In this example, the patient had the average characteristics provided in Appendix A. The spouse had no anchoring on the first bid and therefore provided a constant WTP of A C50.5 whatever the initial bid.

On the contrary, the patient anchored on the first bid. Accordingly, his WTP increased with the initial bid proposed, although to a lesser extent in case of another household member having the flu at the time of the CV survey. If the actual price of the drug related to the earlier alleviation of flu symptoms by 1 day had been proposed as a first bid to those later respondents (about A C30) [START_REF] Schwarzinger | Review of economic evaluations of neuraminidase inhibitors to control influenza[END_REF] their stated WTP would have been A C39.4 and A C44.3 respectively. If the initial bid had been 3-times higher (A C90), their stated WTP would have been A C73.2 and A C64.3, respectively.

Discussion

The context of health care services may partly limit the transfer of the standards of practice of contingent valuation (CV) surveys from environmental economics to health economics [START_REF] Smith | The discrete-choice willingness-to-pay question format in health economics: should we adopt environmental guidelines?[END_REF][START_REF] Smith | Construction of the contingent valuation market in health care: a critical assessment[END_REF][START_REF] Hanley | Estimating the monetary value of health care: lessons from environmental economics[END_REF]. In particular, the relatively small sample size of CV surveys conducted in patients as compared to the general population in environmental economics may have limited the use of the recommended single-bounded (SB) dichotomous choice format [START_REF] Arrow | Report of the NOAA panel on contingent valuation[END_REF]. Two ways to increase the statistical efficiency of health economics CV surveys based on the SB dichotomous choice format were explored in this paper: 1) the inclusion of proxies in addition to patients; 2) the addition of a follow-up discrete question as commonly used in environmental economics, i.e. the double-bounded (DB) dichotomous choice format.

In this paper, we analyzed DB dichotomous choice data for earlier alleviation of flu symptoms where patients or proxies were interviewed. We estimated a DB dichotomous choice model that accounted for respondent's perspective and heterogeneous anchoring. Our results are three-fold.

First, patients and proxies had on average a similar WTP. Second, our model showed that anchoring is a heterogeneous phenomenon. In particular, while a patient was likely to anchor his/her answer on the initial bid, a spouse answering on behalf of the patient was not. Finally, our original DB model with heterogeneous anchoring had the best statistics properties as compared to other DB dichotomous choice models.

In all dichotomous choice models, patients and proxies had on average a similar WTP for earlier alleviation of flu symptoms. Two arguments from the health state valuation literature support this original finding. Firstly, the knowledge of the health state plays a key role on health state valuation.

Both patient and proxy have an intimate knowledge about the subject's health state, whether the general public is usually in need for a description of the health state [START_REF] Brazier | Should patients have a greater role in valuing health states?[END_REF]. In case of influenza-like illness, the knowledge of the health state is even more realistic for proxies as they probably suffered the condition in the past or even may be infected during current epidemics, e.g., 24% of households had at least one secondary case of flu at time of interview. Secondly, patients provide on average a better health state valuation than other types of respondents due to adaptation, coping and adjustment of the patient to a chronic condition [START_REF] Cassileth | Psychosocial status in chronic illness. a comparative analysis of six diagnostic groups[END_REF][START_REF] Damschroder | The impact of considering adaptation in health state valuation[END_REF]. In case of influenza-like illness, i.e. an acute condition lasting about a week, these processes are unlikely to alter valuations between patients and proxies. Presumably, health benefits presented in the contingent valuation were valued by patients or proxies in reference to a disease of a similar perception. The economic interpretation of this finding is two-fold, though it is not possible to discriminate one interpretation from the other: either the proxy knows the preferences of his/her spouse and does not distort them during the interview, or the proxy has the same preferences as his/her spouse.5

Anchoring is a well-established phenomenon that has been recorded in many studies on healthy subjects [START_REF] Tversky | Judgments under uncertainty: heuristics and biases[END_REF][START_REF] Chapman | The limits of anchoring[END_REF][START_REF] Mussweiler | Comparison processes in social judgment: Mechanisms and consequences[END_REF]. Previous DB models assumed anchoring as a constant effect across the surveyed population. As shown by this study in health economics and recent studies in environmental economics, anchoring may depend on individual characteristics [START_REF] Flachaire | Heterogeneous anchoring in dichotomous choice valuation framework[END_REF][START_REF] Aprahamian | Modeling starting point bias and unobserved heterogeneity in contingent valuation surveys: an application to air pollution[END_REF]. In other words, framing effects may not be of the same magnitude for all respondents. To that extent, the DB model with heterogeneous anchoring provides an original means to measure the reliability of stated preferences by the level of anchoring of respondents.

Although patients and spouses had a similar WTP for earlier alleviation of flu symptoms, the DB model with heterogeneous anchoring showed that patients had a significant level of anchoring in their answers as compared to spouses. In addition, the level of anchoring of patients decreased from γ=0.563 to 0.333 when another household member had flu symptoms at time of the interview.

A possible explanation could involve notions of cognitive ability and sense of responsibility of the patient. On the one hand, former clinical experiments in adult volunteers showed that influenza viruses impaired cognitive performance to a level comparable to the deterioration seen in alcohol consumption [START_REF] Smith | Effects of influenza b virus infection on human performance[END_REF]. Accordingly, patients suffering from influenza-like illness may be cognitively impaired and more prone than asymptomatic proxies to framing effects in subjective valuations. On the other hand, the level of anchoring of patients decreased when another household member had flu. This trend may become critical when a dependent child becomes also ill as the adult patient's responsibility and reliability towards himself/herself and the child is more acute. It may lessen the impact of framing effects on valuation, but it does not cancel it.

Respondents in health economics may have salient characteristics, i.e. patient vs. asymptomatic proxy or disease-related characteristics such as cognitive impairment that could explain heterogeneous anchoring. It contrasts with environmental CV surveys on healthy subjects recording anchoring as a general phenomenon. However, recent environmental CV surveys have shown that some respondents were not likely to anchor their answer on the inititial bid. Flachaire et al. showed that respondents who had a more constructed opinion on the French natural park of Camargue were not subject to anchoring when asked for their WTP for a preservation program [START_REF] Flachaire | Heterogeneous anchoring in dichotomous choice valuation framework[END_REF].

Aprahamian et al. showed that 25% of respondents did not anchor when asked for their WTP for an air quality improvement [START_REF] Aprahamian | Modeling starting point bias and unobserved heterogeneity in contingent valuation surveys: an application to air pollution[END_REF]. The potential role on anchoring of explanatory variables related to the disease and recorded in health economics CV surveys should be subjected to further investigation.

Finally the DB model with heterogeneous anchoring was the best model among DB models to fit dichotomous choice WTP data for earlier alleviation of flu symptoms. It reconciled statistical discrepancies found previously between SB and DB models, i.e. the less biased mean estimate and similar standard deviation estimates. Similar results were found recently in a CV survey in environmental economics [START_REF] Flachaire | Heterogeneous anchoring in dichotomous choice valuation framework[END_REF]. Accordingly, the DB model with heterogeneous anchoring provides a means to both control for the anchoring effect on WTP estimation and optimize small sample sizes usually found in health economics CV surveys.

Several findings validated the DB model with heterogeneous anchoring presented in this paper.

Firstly, WTP increased with the level of household income, in keeping with economic theory. Secondly, WTP passed a between-subject "scope test" since WTP increased significantly with the size of clinical benefit, i.e. from 1 to 3 days of earlier alleviation of flu symptoms [START_REF] Hanley | Estimating the monetary value of health care: lessons from environmental economics[END_REF].

Other explanatory variables could be explained intuitively: a positive relationship between WTP and reassurance (smoker at risk of complications, antibiotic consumption); a negative relationship between WTP and mistrust in clinical benefits of the hypothetical drug (influenza-like illness the previous year, occurrence of another case at home).

This study has some limitations. Firstly, the sample may not be representative of the French patients suffering from influenza-like illness. One-third of respondents in the CV survey were excluded because they did not send back the household diary questionnaire containing explanatory variables.

However, there was no difference between excluded and included households according to the clinical variables recorded initially by the general practitioner. Secondly, the presentation of the CV market may not be realistic enough. Health benefits of the new treatment were limited to the description of an earlier alleviation of flu symptoms in accordance with the primary health outcome measured in related clinical trials [START_REF] Cooper | Effectiveness of neuraminidase inhibitors in treatment and prevention of influenza a and b: systematic review and meta-analyses of randomised controlled trials[END_REF]. However, elements of uncertainty such as the probability of success of the new treatment were not presented. Other features of the scenario descriptions such as the out-of-pocket payment were consistent with the lack of reimbursement of the drugs alleviating flu symptoms by 1 day and launched since then in the French health care system, i.e. zanamivir (RELENZA ) in 1999 and oseltamivir (TAMIFLU ) in 2002 [START_REF] Schwarzinger | Review of economic evaluations of neuraminidase inhibitors to control influenza[END_REF]. Further studies should validate the extent to which stated WTP from DB models relies to actual choices as it has been done with other WTP formats used in health economics [START_REF] Onwujekwe | Hypothetical and actual willingness to pay for insecticide-treated nets in five Nigerian communities[END_REF][START_REF] Blumenschein | Hypothetical versus real willingness to pay in the health care sector: results from a field experiment[END_REF][START_REF] Foreit | The reliability and validity of willingness to pay surveys for reproductive health pricing decisions in developing countries[END_REF].

Finally, we used standard interval-data modeling which relies on the assumption that the underlying WTP distribution is identical between the first bid (SB model) and the follow-up bid (naïve DB model). Alternatively, the bivariate probit model relaxes this assumption by allowing for the possibility of different means, dispersions, and non-perfect correlation across the two answers [START_REF] Cameron | Estimation using contingent valuation data from "dichotomous choice with follow-up" questionnaire[END_REF]. Several arguments support our modeling approach. Theoretically, Monte Carlo simulations showed that interval-data mean WTP estimates are robust from departures from a perfect correlation, and they are often superior to the bivariate probit model in terms of the mean square error of estimates (Alberini, 1995a;[START_REF] Haab | Estimation using contingent valuation data from a "dichotomous choice with follow-up" questionnaire: A comment[END_REF]. Other simulations showed that bivariate probit model mean WTP estimates are unbiased, but does not always perform well in terms of statistical efficiency as compared to the SB model [START_REF] Kang | Essays on methodologies in contingent valuation and the sustainable management of common pool resources[END_REF]. Actual CV data sets from environmental economics [START_REF] Kang | Essays on methodologies in contingent valuation and the sustainable management of common pool resources[END_REF] and health economics [START_REF] Clarke | Valuing the benefits of mobile mammographic screening units using the contingent valuation method[END_REF][START_REF] Kennedy | Revealed preference valuation compared to contingent valuation: radon-induced lung cancer prevention[END_REF] were inconclusive on the best modeling approach to fit double-bounded dichotomous choice data and relied eventually on the single-bounded model. In the present paper, we relied on interval-data modeling instead of the bivariate probit model to disentangle purposely the psychological effect (starting point bias) and behavioral effect (shift effect) induced by the follow-up discrete question.

Conclusion

The double-bounded (DB) dichotomous choice model has been less frequently used in contingent valuation surveys performed in health economics, relative to environmental economics. Recent studies in health economics using DB data did rely preferably on a single-bounded (SB) model due to "a starting point bias" [START_REF] Clarke | Valuing the benefits of mobile mammographic screening units using the contingent valuation method[END_REF][START_REF] Kennedy | Revealed preference valuation compared to contingent valuation: radon-induced lung cancer prevention[END_REF]. This paper proposed a DB model with heterogeneous anchoring that reconciled the discrepancies that were commonly found between the DB and SB related-WTP statistics. After correction for heterogeneous anchoring, the DB method still refines the information about a respondent's WTP, thereby allowing for smaller sample sizes than CV surveys relying on SB data. Furthermore, the model provides a means to test for the reliability of respondents' answers in dichotomous choice CV surveys.

Previous CV surveys were mainly restricted to one type of target-population only. That is, either patients were invited to disclose their own evaluation or the general population was asked to reveal the value they attached to health programs. The novelty of our study is that it included, in addition to the patient, a second target group, namely the respective spouse of the patient. This evaluation from different perspectives brings to light some interesting results. In particular, it shows that patients and proxies provided similar WTP for an earlier alleviation of flu symptoms, but the patient-self evaluation tends to be less reliable. This result suggests some immediate practical consequences for economic evaluation in health care. Health evaluation of relatives, partners and other people who are in close contact with the patient should receive more attention. This is an interesting and new aspect that needs to be further explored in future research. Previous CV surveys only relied on proxies when the need for proxies was the most salient, e.g. children or cognitively impaired patients. As a result of this study, patients' answers to WTP questions should be treated with more caution. However, patients still have more information about themselves than their relatives, thus their evaluation reveals some necessary information that should not be totally disregarded.

Acknowledgement

We would like to thank the patients, their families, and the 161 general practitioners of the French "Sentinelles" system who participated to the epidemiological study; Camille Sahler, MSc, Isabelle Goderel, BS, Audrey Lavenu, MSc, and Gregory Pannetier, BS, for technical assistance in data collection. We thank Frédéric Aprahamian, Susan Cleary, Cam Donaldson, Jean-Paul Moatti, Jason Shogren, Miriam Teschl, the Editor and two anonymous referees for helpful comments and remarks.

We also thank Dana Conley for her help with editing. The CV study was funded by GlaxoSmithKline, Marly-le-Roi, France. The sponsors of the study had no further role in the study design, data collection, data analysis, data interpretation, writing, or the decision to submit the report for publication. The views expressed in this paper are those of the authors and usual caveat applies.

A Description of the willingness-to-pay scenario with a patient-based perspective

We would like to measure the importance for you of earlier alleviation of influenza-like symptoms.

Please, assume that a new treatment could alleviate influenza-like symptoms by 1 ( 3

(n = 18) (n = 25) (n = 16) (n = 25) p-value (n = 4) (n = 5) (n = 6) (n = 5) (n =
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ • • • • • • • • • • • • • • • • • • • • • • • • • •

  CV dichotomous choice models showed construct validity: WTP increased significantly with the size of the clinical benefit from 1 to 3 days of alleviation of flu symptoms (INT3_DAYS significant in all four models) and WTP increased with income (although, not significantly). Among other explanatory variables of WTP, having had influenza-like illness the previous year significantly decreased WTP in the DB model with heterogeneous anchoring (p = .080). Other explanatory variables were not significant.The level of agreement between patients and spouses for the WTP for earlier alleviation of flu symptoms was assessed by means of a parameter θ (INT_PROXY) accounting for discrepancies between the expectation of the patient's prior WTP and that of the proxy. In all four econometric models, θ was not significantly different from zero (p = .246 in the SB model; p = .648 in the "naïve" DB model; p = .591 in the DB model with constant anchoring; p = .786 in the DB model with heterogeneous anchoring).

  of patients' answers (tests presented in Appendix C). The only significant explanatory variables were the occurrence of flu symptoms in another household member at the time of the CV survey (INT_OTHERFLU; p = .012) and the flu symptom score (FLU_SCORE; p = .058). Finally, we tested a heterogeneous anchoring model with 4 parameters whether INT_OTHERFLU and FLU_SCORE had a differential effect on the anchoring depending on the respondents' perspectives. Only one parameter was significant, i.e. INT_OTHERFLU on the anchoring effect of the patientbased perspective (p = .029). 4 Accordingly, the final DB model with heterogeneous anchoring presented in

  Figure

Figure 1 :

 1 Figure 1: WTP for the alleviation of flu symptoms by 1 day according to the level of anchoring of the respondent and the initial bid proposed in the double bounded model with heterogeneous anchoring.

Table 2

 2 Among the three DB econometric models, the DB model with heterogeneous anchoring presented the best fit for our CV data. The DB model with heterogeneous anchoring strongly rejected the null hypothesis of no anchoring and no shift effect (p < .001) and the null hypothesis of constant anchoring

	included an anchoring parameter when the respondent was the patient depending
	on the occurrence of new cases of influenza-like illness in the household at time of interview (γ = .563
	for INT_ONESELF only and decreased by 0.230 for INT_ONESELF × INT_OTHERFLU), and
	no anchoring parameter for proxies.

heterogeneous anchoring model were in accordance with those found in the SB model. In the SB model, the standard deviation of WTP of proxies ω (25.6, se. 7.8) was smaller than that of patients σ (63.8, se. 13.2), with no significant difference (p = .235). The DB model with heterogeneous

Table 1 .

 1 Characteristics of patients, households, and willingness-topay survey. Except where stated otherwise, values are expressed in percentage of subjects (n = 287) a: Bid acceptance rate for the earlier alleviation of flu symptoms by one day (n = 151) Bids were presented in FF at time of the survey and are converted to Euros in the Table (1FF=A C0.152449) Table 1.b: Bid acceptance rate for the earlier alleviation of flu symptoms by three day (n = 136)

	) day(s). The

B

  Bids were presented in FF at time of the survey and are converted to Euros in the Table(1FF=A C0.152449)

										6)	p-value
	3.8	100%						75%
	7.6	88%	94%				ns	75%	100%	ns
	15.2	65%	72%	92%			<.1	75%	100%	67%	ns
	30.5		56%	76%	81%		ns		60%	50%	80%	ns
	61.0			16%	50%	68%	<.001			17%	80%	17%	<.1
	122.0				6%	24%	ns			60%	0%	<.1
	243.9					0%				0%
	Note:

Table 2 :

 2 Econometric results (n = 287) ⋆⋆⋆ if p-value<0.01, ⋆⋆ if p-value<0.05, ⋆ if p-value<0.1

		Single-		Double-bounded
	Variables	Bounded reference	naïve	shift &Constant Anchoring	shift &Heterogeneous Anchoring
		Parameter	Parameter	Parameter	Parameter
		(p-value)	(p-value)	(p-value)	(p-value)
	CONSTANT	-7.986	16.626	9.421	9.473
		(0.767)	(0.328)	(0.685)	(0.678)
	FEMALE	5.917	1.063	1.445	3.534
		(0.520)	(0.846)	(0.845)	(0.656)
	AGE	0.424	0.301	0.496	0.408
		(0.223)	(0.184)	(0.115)	(0.176)
	FLU_SCORE	-4.995	0.097	-4.812	-8.185
		(0.864)	(0.995)	(0.836)	(0.748)
	FLU_DIAG	10.088	0.141	2.728	6.577
		(0.238)	(0.979)	(0.718)	(0.443)
	ATB	-10.400	-1.341	-2.228	-6.600
		(0.233)	(0.806)	(0.762)	(0.415)
	FLU_BEFORE	-8.408	-12.397	-15.467	-21.611 ⋆
		(0.468)	(0.101)	(0.134)	(0.080)
	FLU_VAC	-2.092	8.575	7.251	3.008
		(0.899)	(0.355)	(0.569)	(0.825)
	CHRONICDIS	-13.294	-11.376	-13.706	-11.254
		(0.264)	(0.109)	(0.152)	(0.294)
	SMOKER	17.649 ⋆	10.774 ⋆	13.698	9.547
		(0.099)	(0.082)	(0.103)	(0.310)
	HOUSE_SIZE	7.612 ⋆	1.579	3.022	4.915
		(0.089)	(0.536)	(0.385)	(0.234)
	HOUSE_INC2	15.199	2.325	3.669	4.463
		(0.157)	(0.711)	(0.666)	(0.648)
	HOUSE_INC3	19.244	12.756	14.765	10.562
		(0.143)	(0.114)	(0.175)	(0.365)
	HOUSE_MECOSTS	0.065	-0.009	0.025	0.074
		(0.715)	(0.920)	(0.838)	(0.576)
	INT_3DAYS	17.248 ⋆⋆	18.060 ⋆⋆⋆	13.407 ⋆	12.968 ⋆
		(0.047)	(0.000)	(0.071)	(0.098)
	INT_OTHERFLU	3.080	-2.514	-8.120	-6.234
		(0.767)	(0.680)	(0.334)	(0.504)
		Structural Shift
	Shift parameter δ	-	-	-0.791	-0.673
		-	-	(0.391)	(0.484)
		Anchoring
	Constant Anchoring γ	-	-	0.421 ⋆⋆⋆	-
				(<0.001)
	Heterogeneous anchoring
	INT_PATIENT	-	-	-	0.563 ⋆⋆⋆
		-	-	-	(<0.001)
	INT_OTHERFLU × INT_PATIENT	-	-	-	-.230 ⋆⋆⋆
		-	-	-	(<0.029)
		Proxy-perspective
	INT_PROXY (θ)	-11.741	3.569	6.583	-2.454
		(0.246)	(0.648)	(0.591)	(0.786)
		Prior predicted WTP ( c W i1 )
	Mean, A C	58.85	44.71	48.09	50.51
	(sd.)	(1.23)	(0.79)	(0.89)	(0.93)
	Dispersion, A C
	Patient (σ)	63.81	35.55	47.34	55.14
	(sd.)	(13.19)	(2.62)	(5.67)	(7.24)
	Proxy (ω)	25.578	45.807	76.086	41.962
	(sd.)	(7.769)	(7.430)	(15.869)	(5.768)
	Log-likelihood	-148.127	-363.588	-348.613	-342.059
	if

The Environmental Valuation Reference Inventory can be found at http://www.evri.ec.gc.ca/ -accessed on the 12th of July

We did not restrict the structure of the relationship between ω and σ so that ω can be either higher or lower than σ.

Note that it is useful to force γ to lie between 0 and 1 as commonly done with the double exponential function: γ(Z i ; η) = exp(-exp(Z i η)).

INT_OTHERFLU was not significant in the anchoring effect of the proxy perspective (p = .901). FLU_SCORE was not significant in both cases (p = .138 and p = .475)

Note that this interpretation is reminiscent of the distinction between pure altruism and pure paternalism (see Jones-Lee 1992b and Jones-Lee 1992a).

The initial bid was randomly selected in[25; 50; 100; 200; 400] FF or [50; 100; 200; 400; 800] FF for the alleviation of flu symptoms by one day or three days, respectively (1 Euro= 6.56 French Francs).

In the proxy-based perspective, the question was framed as follows: 'In your opinion, would you be willing to pay out-of-your pocket 100 FF* to alleviate by 1 (3) day(s) the symptoms of your partner if she(he) had flu again?'

The follow-up bid was doubled if respondent answered Yes to the initial bid, and halved otherwise.