Mathilde Touvier
email: m.touvier@uren.smbh.univ-paris13.fr

Caroline Méjean

Emmanuelle Kesse-Guyot

Clothilde Pollet

Aurélie Malon

Katia Castetbon

Serge Hercberg

Caroline Me

• Aure ´lie Malon

Á C Me

Á E Kesse-Guyot

Á C Pollet

Á A Malon

Á S Hercberg Uren

Á S Hercberg

Comparison between web-based and paper versions of a self-administered anthropometric questionnaire

Keywords: Anthropometry, Comparative study, Internet, Self-administered, Web-based questionnaire

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

An increasing body of literature suggests causal relationships between diet, physical activity, overweight and adiposity, on the one hand, and several chronic pathologies such as cancers and cardiovascular diseases on the other [START_REF]Food, nutrition, physical activity and the prevention of cancer: a global perspective[END_REF][START_REF]Diet, nutrition and the prevention of chronic diseases[END_REF][START_REF]The challenge of obesity in the WHO European Region and the strategies for response[END_REF]. However, there is still a need for carefully conducted, large prospective cohort studies involving hundreds of thousands of subjects and collecting numerous data on various potential confounding factors, in order to more thoroughly investigate and confirm such relationships. Traditional performance of these studies is complex and costly, requiring heavy logistic resources related to printing and mailing of paper questionnaires, as well as data entry, checking and cleaning of collected information. Studies based on telephone interviews may improve several logistic aspects, but remain very costly. Therefore, data collection via the internet might considerably facilitate the entire process and advantageously replace traditional methods in the near future [START_REF] Ekman | New times, new needs; e-epidemiology[END_REF].

The quality of collected data is also of utmost importance. Traditional paper-based questionnaires may generate data entry mistakes along with missing, inconsistent or abnormal data. The remaining inconsistencies in the nutritional epidemiology of overweight and obesity could be partly due to methodological difficulties that prevented generation of the necessary evidence. Web-based questionnaires can minimize these errors by adding automating controls and skip or branching logic.

Literature is lacking on controlled empirical studies investigating the potential for differential effects of administrative methods [START_REF] Hardre | Testing differential effects of computer-based, web-based and paper-based administration of questionnaire research instruments[END_REF][START_REF] Reips | Standards for internet-based experimenting[END_REF][START_REF] Schatzkin | Observational epidemiologic studies of nutrition and cancer: the next generation (with better observation)[END_REF]. Therefore, studies showing how new instruments compare to traditional ones are of interest.

In May 2009 in France, our group launched the NutriNet-Sante ´Study. This is the first web-based prospective cohort study worldwide on such a broad scale, as it is scheduled to include 500,000 volunteers (aged C18 years) over the next 5 years and will investigate the relationship between intake of nutrients, food group dietary patterns, nutritional status and health (especially the risk of chronic diseases), as well as examining determinants of dietary behavior. Participants will be followed for at least 5 years via a web site specifically created for that purpose (www.etude-nutrinet-sante ´.fr ). All questionnaires and forms are directly filled out online (internet) using a secure HTML interface. Before launching the NutriNet-Sante ´cohort, we conducted test studies to compare the web-based version of baseline questionnaires to corresponding usual methods.

The objective of the present study was to compare the web-based version of the NutriNet-Sante ´self-administered anthropometric questionnaire with the paper-based version, by administering both versions to a sample of about 150 subjects of mature age, accustomed to filling out paper questionnaires, but who were not specifically trained web users. Our hypotheses were that (1) there would be a high agreement between the answers to the two versions and (2) the paper version would generate a substantial number of errors that could be avoided in the web-based version.

Methods

Study population

In January 2009, a call for participation was launched among subjects of the SU.VI.MAX. cohort (''Supple ´mentation en VItamines Mine ´raux et AntioXydants'') both verbally (during the annual meeting of subjects) and by e-mail (collected by paper questionnaire during the SU.VI.MAX follow-up for 5,949 subjects out of 13,017). Persons who were interested in participating had to register their candidacy on a specific web-site and to provide an actualised e-mail address. A valid e-mail address and a potential access to the internet were required to participate. Before the end of January, already 1,090 persons were volunteers and were available for participating in NutriNet-Sante ´validation studies; thus we stopped the call for participation. Among them, 170 were randomly selected to test the anthropometric questionnaire. They were between 35 and 60 years of age at the beginning of the SU.VI.MAX. study in 1994. The design, methods and rationale of this cohort have been described elsewhere [START_REF] Hercberg | A primary prevention trial using nutritional doses of antioxidant vitamins and minerals in cardiovascular diseases and cancers in a general population: the SU.VI.MAX study-design, methods, and participant characteristics. SUpplementation en VItamines et Mineraux AntioXydants[END_REF]. All procedures were approved by the Ethical Committee for Studies with Human Subjects and the Commission Nationale Informatique et Liberte ´(CNIL N°908450). Electronic informed consent was obtained from all subjects.

The NutriNet-Sante ´anthropometric questionnaire (web-based and paper versions)

The self-administered NutriNet-Sante ´anthropometric questionnaire [9] includes 17 questions divided into subquestions (55 variables in all), dealing with height, weight, hip and waist circumferences, weight history, practice of restrictive diets, weight self-perception and Sorensen-Stunkard's silhouettes [START_REF] Sorensen | The accuracy of reports of weight: children's recall of their parents' weights 15 years earlier[END_REF]. For the present test study, two versions of this questionnaire were developed: a paper version and a web-based version, used in the NutriNet-Sante ´cohort study. The 170 subjects were asked to fill in both versions of the questionnaire. They were randomly assigned to two groups (N = 85 each) according to the order in which the two versions were given to them. For group 1, the paper version was given to them first (the login and password were provided to the subjects for access to the web-based version only when we had received their completed paper questionnaire by regular mail); group 2 included those who received the web-based version first (the paper version was mailed to participants when they had completed the web-based version).

In the paper version, the conditional branch was indicated by written instructions (e.g. ''if yes, then go to question No. 5, and if no, go to No. 6''). In the web-based version, subjects were automatically directed to the appropriate question according to their previous answers. Besides, the web-based version did not allow missing values (all items had to be completed in order to access the next screen) and boundary logic controls were included (e.g. height had to range from 90 to 220 cm). If these conditions were not respected, the subject was alerted by an alarm message indicating the nature of the problem and the need to correct it before going onto complete the questionnaire.

In both versions, identical written instructions, including photographs of self-measurement of weight and hip/waist circumferences, were provided to participants. In addition, in the web-based version, subjects had access to videos explaining body self-measurements and general explanations on filling out the questionnaire via the internet (e.g. how to navigate between screens and answer questions).

Data from the web-based questionnaire were directly included in our database. The answers to the paper version were captured by two independent trained keyboard operators. When an inconsistency between the two operators was detected, data were checked on the paper questionnaire and the correct answer was retained.

''Satisfaction'' questionnaire After completing the two versions of the anthropometric questionnaire, participants were requested to fill in a webbased ''satisfaction'' questionnaire describing their attitude toward the internet (self-evaluated computer and web knowledge, web access, type of connexion), how long it took to fill out each version, their opinion of each version (attractiveness of layout, fluency of use, etc.) and their preferred version.

Statistical analyses

Agreement between the two versions of the anthropometric questionnaire was quantified by intraclass correlation coefficients (ICCs) for the eight continuous variables [START_REF] Shrout | Intraclass correlations: uses in assessing rater reliability[END_REF] and kappas for 18 categorical variables [START_REF] Feinstein | High agreement but low kappa: I. The problems of two paradoxes[END_REF]. For the remaining 29 categorical variables corresponding to details on restrictive diet motivation (e.g. vegetarianism, food allergies), methods used for weight reduction (e.g. use of laxatives or meal substitutes) and the ability to carry out waist and hip measurements (yes/no), kappa was not a reliable estimate for agreement because of a substantial imbalance in the table's marginal totals (for each modality, only a few participants were concerned) [START_REF] Feinstein | High agreement but low kappa: I. The problems of two paradoxes[END_REF]. Thus, kappas were not presented in this paper for these variables. ICCs and kappas were calculated overall and stratified by handout order (which version was given first), age (\ or C60 years), gender and web knowledge (novice/inexperienced or experienced/expert). For the self-perceived Sorensen silhouette variable, which comprised a high number of categories, weighted kappas were calculated in order to assign different weights for the closest and the most distant disparities.

Next, we quantified the number of data entry mistakes and missing, inconsistent or abnormal data inherent in the paper version (non-existent in the web-based version due to integrated controls), and we compared them between the two handout groups using binomial tests.

Third, all answers to the ''satisfaction'' questionnaire (time needed for filling out each version, along with the evaluation) were compared between the web-based and the paper version of the anthropometric questionnaire using Mac-Nemar Chi 2 tests.

Finally, we evaluated the cost related to the paper version of the anthropometric questionnaire (printing, mailing and staff needed for data entry, non-existent in the webbased version by definition) and the cost related to the development and support of the web tool.

For all analyses, the significance level was two-sided and set at 0.05. All statistical analyses were performed using SAS software (version 9.1, SAS Institute Inc., Cary, NC, USA).

Results

Characteristics of the study population

We excluded 23 subjects from the 170 initially selected: 15 subjects did not respond despite several reminders, 5 had misunderstood the modalities of the study (e.g. necessity for web connection and for filling out two versions of the same questionnaire) and withdrew their participation, 2 had internet connection problems, and one participant stopped because he felt the web system was too complex for him. Thus, 147 subjects remained for analyses (76 completed the paper version first and 71 the web-based version first); among them, 128 subjects filled in the satisfaction questionnaire. The median number of days between filling out the two versions was 9 (minimum 2 days, maximum 61 days). Table 1 presents sociodemographic and web characteristics of the study population. We checked that randomized handout order groups were well balanced for all characteristics (P [ 0.1, data not tabulated). Participants were mature in age (mean ± SD: 62.7 ± 5.8 years; age range: 49-75 years) and about 25% considered themselves to be novices/inexperienced in computer and web use.

Agreement between the web-based and paper versions

Table 2 provides ICCs for agreement between the two versions (for quantitative variables). ICCs ranged from 0.86 to 1.00 for the eight continuous variables in the questionnaire. Table 3 provides agreement and kappas between the two versions for the main qualitative variables. Kappas calculated for 18 categorical variables ranged between 0.69 and 1.00, except for the four variables corresponding to the questions ''Did you carry out this body measurement specifically for this study?'' (for height, weight, and hip and waist circumferences). For the four latter variables, kappas were low (e.g. 0.18 for height measurement), since many subjects answered ''yes'' only in the first version they had completed (and ''no'' in the second version). For each of the remaining 29 categorical variables, very few subjects had differing responses in the two versions (maximum 6). For each variable, kappas and ICCs were similar in all stratification groups except for weight evolution during the last 3 months and current practice of a restrictive diet, for which lower kappas were observed in subjects below 60 years of age and in women.

Data entry mistakes and missing, inconsistent or abnormal data inherent in the paper version

Corresponding numbers are presented in Table 4. Most of the missing data (i.e. no information for the variable; no numeric value provided, nor ''I do not remember'' box checked) consisted of age at maximal and minimal weight (missing for eight subjects, i.e. 5.4% of the study population) and these maximal and minimal weights themselves (missing for 6 subjects, i.e. 4.1% of the study population) (data not tabulated). Similarly, most of the inconsistent data involved the age at minimal (17) and maximal [START_REF] Marsden | Validation of Web-based questionnaires regarding osteoporosis prevention in young British women[END_REF] weights, and the minimal (8) and maximal (5) weights themselves (the ''I do not remember'' box was checked and a numeric value was also provided).

Duration, satisfaction and cost

Within the 128 subjects who completed the ''satisfaction questionnaire'', although the time needed to fill out the questionnaires was comparable for the two versions, the web-based version was considered more satisfactory than the paper version (Table 5). The web-based version was preferred by 92.2% of the subjects (data not tabulated). About 87.5% of participants filled out the web-based questionnaire in one sitting. About 74.2% consulted the ''recap form'', which provides a concise summary of all answers to the questionnaire. Among them, 18.8% declared that it helped them to detect a mistake and correct it. About 94.5% of the participants felt that the web-based interface was user-friendly, while 88.3% appreciated the ''mid-questionnaire support message''. Only 29 subjects (22.7%) viewed the help video; all of them were entirely or fairly satisfied. Once conceptually designed, the elaboration of the paper version of the questionnaire (writing and editing) took only 1 day. Cost specific to the paper version involved: printing of the questionnaire (0.025 €/printed page; hence 0.18 €/subject), mailing it to the participants (1.67 €/subject), payment of mailing costs for returning the completed questionnaire (1.71 €/subject ? a package cost, for the entire cohort, for postal service enabling free mailing by the subjects: 460 € the 1st year, and then 445 €/y) and cost for double data entry (6.38 €/subject, calculated on the basis of the cost for one full-time equivalent: 1913 €/month, including charges); this corresponded to a total of 9.94 € (14.1 US$) for one subject and 4,965,833 € (7,052,383 US$) for a cohort of 500,000 subjects (i.e. the targeted effective for the NutriNet-Sante ´study). Cost specific to the web-based version was much lower, as it involved only developmental and support cost for the study web-site. Such cost depends on technical skills and resources of implicated research teams and on standards of the product (in terms of user-friendliness, security level, quality of graphic design, etc.). Cost for our web-based tool, developed by professional web-designers, amounted to 150,000 € (213,000 US$; 9 months of development) for the entire NutriNet-Sante ´study (and not only for the baseline anthropometric questionnaire). It comprised the whole study process: secure inscription system, development and administration of baseline questionnaires, monitoring of the cohort (e-mails to the subjects, newsletter, etc.), equipment and hosting. Cost for an additional questionnaire equivalent to the anthropometric questionnaire would cost about 5,000 € (7100 US$; 4 days of development).

Discussion

Our baseline hypothesis was confirmed: agreement between the two versions of the anthropometric questionnaire was high. Moreover, even with an overall short and simple questionnaire such as that evaluated in the present study, errors (data entry mistakes and missing, inconsistent or abnormal values) observed in the paper version were avoided in the web-based version due to automatic controls. Finally, the web-based version was highly appreciated and led to considerable logistic cost savings.

Earlier studies had compared web-based versus paper questionnaires in diverse fields such as diet [START_REF] Beasley | Evaluation of a web-based, pictorial diet history questionnaire[END_REF], 166.9 8.2 166.7 8.3 0.99 (0.99-0.99) 0.99 (0.98-0.99) 0.99 (0.98-0.99) 0.97 (0.95-0.98) 0.99 (0.99-1.00) 0.98 (0.97-0.99) 0.98 (0.97-0.99) 0.98 (0.95-0.99) 0.99 (0.99-0.99) Weight (kg, N = 146) 69.8 14.3 69.8 14.2 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) Lifetime maximal weight a (kg, N = 134) 74.0 15.1 73.9 15.0 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00 (1.00-1.00) 1.00( .9 (0.8-1.0) 0.9 (0.9-1.0) 0.8 (0.7-1.0) 0.9 (0.8-1.0) 0.9 (0.8-1.0) 0.9 (0.7-1.0) 0.9 (0.8-1.0) 0.9 (0.8-0.9) 0.8 (0.7-1.0) 0.9 (0.8-1.0) 0.9 (0.8-1.0) 0.8 (0.7-0.9) 0.8 (0.7-0.9) 0.9 (0.8-1.0) 0.9 (0.7-1.0) 0.9 (0. 2 0.7 (0.6-0.8) 0.7 (0.6-0.9) 0.7 (0.6-0.9) 0.8 (0.6-1.0) 0.7 (0.6-0.9) 0.7 (0.6-0.9) 0.7 (0.6-0.9) 0.7 (0.4-0.9) 0.8 (0.6-0.9) No. 1 2 1.4 3 2.1 0.9 (0.8-0.9) 0.8 (0.8-0.9) 0.9 (0.9-1.0) 0.9 (0.8-1.0) 0.9 (0.8-0.9) 0.8 (0.8-0.9) 0.9 (0.8-1.0) 0.9 (0.7-1.0) 0.9 (0.9-1.0)

No. 2 No. 9 (men only) 0 0 0 0 Percentage of subjects with identical responses in the two versions. For each variable, statistics were calculated among subjects who gave an answer in both versions of the questionnaire. Number of participants is specified in parenthesis. For the analyses stratified on web experience, 19 subjects were excluded due to missing data for this information a Weighted kappas were calculated in order to assign different weights for the closest and the most distant disparities (different levels of agreement contribute to the value of kappa)

osteoporosis prevention [START_REF] Marsden | Validation of Web-based questionnaires regarding osteoporosis prevention in young British women[END_REF] and acute mountain sickness [START_REF] Beidleman | Validation of a shortened electronic version of the environmental symptoms questionnaire[END_REF]. In such pilot/validation studies involving small sample sizes, even a few misclassifications could lead to substantial variations in kappa values, thus making it meaningless to precisely compare the levels of agreement across studies. In general, however, the authors of those investigations [START_REF] Beasley | Evaluation of a web-based, pictorial diet history questionnaire[END_REF][START_REF] Marsden | Validation of Web-based questionnaires regarding osteoporosis prevention in young British women[END_REF][START_REF] Beidleman | Validation of a shortened electronic version of the environmental symptoms questionnaire[END_REF] consistently confirmed strong agreement between the two versions. The research team of A Schatzkin (National Cancer Institute, USA) is currently undertaking detailed evaluation studies so as to compare their new web-based dietary and physical activity questionnaires to traditional methods and biomarker data [START_REF] Schatzkin | Observational epidemiologic studies of nutrition and cancer: the next generation (with better observation)[END_REF].

A key question in large-scale internet-based studies concerns their capacity to reach a sufficiently broad and diversified population. Although use of computer-based questionnaires may exclude segments of the population without access to, or the capacity to use, computers (notably the elderly [START_REF] Klovning | Web-based survey attracted age-biased sample with more severe illness than paper-based survey[END_REF]), internet access is constantly increasing throughout the world and within most sociodemographic groups. France had 33 million internet users in February 2009 [17], and ranks first in the European classification for internet use according to the European Interactive Advertising Association [18]. Around 62% of the population over the age of 11 connected to the net in a 1-month period [18]. 29% of internet users belong to low social classes [19] and one internet user in four is over 55 years of age [18]. In Europe, 178 million persons go online each week and over half (55%) of internet users go online every day [18]. In the United States, internet use rose to 73.9% by June of 2009 [20]. More than half of persons aged 64 or older go online at least once a month [START_REF] Golvin | The state of consumers and technology: benchmark[END_REF]. This increasing internet use provides an attractive option for large-scale prospective cohort studies in developed countries.

Although online data collection may not solve traditional problems faced for reaching the population sample [START_REF] Lefever | Online data collection in academic research: advantages and limitations[END_REF], several studies argue in favor of high acceptability and feasibility of web-based questionnaires [START_REF] Bexelius | Evaluation of an internet-based hearing testcomparison with established methods for detection of hearing loss[END_REF][START_REF] Ekman | Feasibility of using web-based questionnaires in large populationbased epidemiological studies[END_REF][START_REF] Kendler | A web-based study of personality, psychopathology and substance in twin, other relative and relationship pairs[END_REF]. Moreover, the latter, particularly usable for analytical studies that focus on associations between variables [START_REF] Etter | A comparison of cigarette smokers recruited through the internet or by mail[END_REF], have major advantages. First, data quality is positively impacted, since missing, inconsistent or abnormal data can be minimized by adding alerts to users and automated skip or branching logic. These improvements are even more effective for questions dealing with the past, such as those concerning lifetime maximal and minimal weight. Overall, percentages of errors observed in our study were low, but absolute numbers of mistakes and percentages of subjects c When information provided was inconsistent with skip or branching logic, e.g., when a subject indicated a numeric value for minimal weight reached during his/her lifetime and also checked the ''I do not remember'' box d When information fell outside of accepted boundaries: 1 subject declared a body weight outside of the 35-200 kg range; 1 subject declared a height outside of the 90-220 cm range; and 1 subject declared an age for minimal weight outside of the 18-120 year range with at least one error were not negligible. Besides, this questionnaire is very simple in terms of conditional branch and type of information requested. Thus, one could expect a much higher percentage of mistakes in a more complicated questionnaire. Second, the study burden is lowered for participants, who can complete their questionnaires via a user-friendly interface that provides interactive help tools (such as videos and animated screens). Indeed, in our study, we observed that time needed for completion was less frequently viewed as a constraint with the web-based version than with the paper version. This could be partly explained by the convivial easy-to-use nature of the webbased questionnaire. Third, even in non-anonymous surveys, internet introduces a distance between the investigator and the subject, thereby encouraging the latter to deliver uncensored personal information [START_REF] Joinson | Social desirability, anonymity, and Internet-based questionnaires[END_REF]. This is particularly important for sensitive questions related to weight status, body shape and self-perception, and to other topics such as use of alcohol, tobacco or other substances [START_REF] Wang | Survey of substance use among high school students in Taipei: web-based questionnaire versus paper-and-pencil questionnaire[END_REF]. However, a recent study observed differences in underreporting of dietary intake by gender and weight status using a web-based survey, similar to observations made using paper-based 24 h recalls and dieticians' interviews [START_REF] Vance | Self-reported dietary energy intake of normal weight, overweight and obese adolescents[END_REF].

Fourth, data processing is greatly facilitated, and reliability is increased by eliminating data entries or scanning of paper forms. Fifth, there is a considerable reduction in expenses and paper waste in web-based studies, since printing, mailing and treatment of paper questionnaires are avoided. Eventually, web-based studies also enable further reduction in cost when modifying the questionnaires over time, along with a high flexibility.

Whether web-or paper-based, our anthropometric questionnaire remains self-administered, which provides potentially biased information on body measurements compared to those taken by a trained interviewer [START_REF] Rowland | Self-reported weight and height[END_REF]. However, in large cohorts such as the NutriNet-Sante Śtudy (500,000 subjects), body measurements by interviewers would be extremely complex and costly. Nevertheless, validation of self-reported anthropometric data by objective measurements will be possible on subsamples of the cohort, providing an evaluation of the bias associated with self-reported data.

Our study had several limitations. First, paper-based questionnaires cannot represent the gold standard for their internet-based counterparts [START_REF] Van Der Lei | Of new times, new opportunities, and old problems[END_REF]. Therefore, our study was comparative rather than being a validation study using paper as the gold standard. Second, in this type of study, for certain variables, comparisons between the two versions are impossible or very difficult, since questions do not refer to the same period of time (e.g. weight evolution during the last 3 months). Moreover, lack of interest on the part of participants when completing the second version may affect their answers. Changes can also occur between the periods of completion of the two versions. For instance, lower kappas observed in weight evolution and current restrictive diet among subjects under 60 and among women may be explained by a new restrictive diet begun by some participants between completion of the two versions of the questionnaire. Finally, our participants belonged to middle/ high education and occupational categories; thus, caution is required when extrapolating our results to the general population, and especially to people of a low socioeconomic status. However, one-fourth of the subjects were internet novices or inexperienced, mean age was relatively high (62.7 years) and participants were accustomed to filling out paper but not web-based questionnaires (they had filled in more than 25 different paper questionnaires during their participation in the SU.VI.MAX cohort), which would tend to penalize the web-based version.

In conclusion, our study supports the idea that webbased questionnaires provide information of equal or superior quality when compared to the paper-based method, with substantial logistic and cost advantages. More studies are needed in order to find out about acceptance and validity of web-based questionnaires in not welleducated subjects, manual workers, and never-employed individuals, and in subjects that are not trained in filling out questionnaires. Online surveys are able to access wide and diversified populations and achieve quick returns. Their use should increase worldwide, with appropriate adaptation of web-based tools to country-specific cultural aspects [START_REF] Schatzkin | Observational epidemiologic studies of nutrition and cancer: the next generation (with better observation)[END_REF]. This will advance our understanding of the relationship between anthropometry and health outcome. The internet also promises a new generation of tools for public health improvement [START_REF] Underbakke | Web-based resources for medical nutrition education[END_REF], such as e-learning [START_REF] Tse | E-health for older people: the use of technology in health promotion[END_REF], nutritional education [START_REF] Brug | Past, present, and future of computer-tailored nutrition education[END_REF] and lifestyle counselling [START_REF] Van Wier | Phone and e-mail counselling are effective for weight management in an overweight working population: a randomized controlled trial[END_REF] in the field of weight management and health in general.

Table 1

 1 Characteristics of the study population (N = 147)

		All (N = 147)		Paper version		Web-based version
				first (N = 76)		first (N = 71)
		Mean	SD	Mean	SD	Mean	SD
	Age (years)	62.7	5.8	62.4	5.5	63.0	6.1
		n	%	n	%	n	%
	Female	79	53.7	40	52.6	39	54.9
	Education level
	Elementary school	18	12.2	11	14.5	7	9.9
	Secondary school	62	42.2	34	44.7	28	39.4
	University or equivalent	67	45.6	31	40.8	36	50.7
	Occupational category a
	Managerial staff	55	37.4	25	32.9	30	42.3
	Self-employed, farmers	6	4.1	5	6.6	1	1.4
	Intermediate professions, employees	84	57.1	45	59.2	39	54.9
	Manual workers	1	0.7	1	1.3	0	0.0
	Never employed	1	0.7	0	0.0	1	1.4
	Self-evaluated computer knowledge b
	Novice or inexperienced	31	24.2	17	25.8	14	22.6
	Experienced or expert	97	75.8	49	74.2	48	77.4
	Self-evaluated web knowledge b
	Novice or inexperienced	33	25.8	15	22.7	18	29.0
	Experienced or expert	95	74.2	51	77.3	44	71.0
	Type of connexion b
	\512 k	36	28.1	15	22.7	21	33.9
	C512 and \1,024 k	47	36.7	30	45.5	17	27.4
	C1,024 k	36	28.1	18	27.3	18	29.0
	Do not know	9	7.0	3	4.5	6	9.7

a Current occupation or most recent job if they were retired or unemployed b 128 subjects (66 for the paper version first and 62 for the web-based version first) returned the satisfaction questionnaire, thereby providing this information

Table 2

 2 Intraclass correlation coefficient (for quantitative variables) between the web-based and paper versions, overall and according to handout group, age, gender and web knowledge

Table 3

 3 For each variable, statistics were calculated among subjects who gave an answer in both versions of the questionnaire. Number of participants is specified in parenthesis. For the analyses stratified on web experience, 19 subjects were excluded due to missing data for this information Agreement and kappa statistic (for qualitative variables) between the web-based and paper versions, overall and according to handout group, age, gender and web knowledge

	1.00-1.00)

a From the age of 18 Comparison between web-based and paper versions 291

Table 4

 4 Number of data entry mistakes and missing, inconsistent or abnormal data inherent in the paper version (for the 55 variables of the NutriNet-Sante ´anthropometric questionnaire) P value for the difference between the two handout groups, from binomial tests b Percentages of errors out of the number of total entries overall = 5,269 entries (2,656 in the group ''paper version 1st'' and 2,613 in the group ''web-based version 1st''), with a mean of 35.8 entries per subject (the number of entries varies across subjects, depending on the conditional branch)

		All (N = 147)		Paper version		Web-based version		P a
				1st (N = 76)		1st (N = 71)
		Number	% b	Number	%	Number	%
	Number of data entry mistakes
	Keyboard operator No. 1	102	1.9	56	2.1	46	1.8	0.4
	Keyboard operator No. 2	61	1.2	30	1.1	31	1.2	1.0
	Mean	81.5	1.5	43	1.6	38.5	1.5	0.5
	Mean number of subjects with at least	52 (35.4)		27.5 (36.2)		24.5 (34.5)		0.6
	one data entry mistake (%)
	Number of missing values	60	1.1	24	0.9	36	1.4	0.2
	Number of subjects with at least	16 (10.9)		7 (9.2)		9 (12.7)		0.8
	one missing value (%)
	Number of inconsistent data c	57	1.1	28	1.1	29	1.1	1.0
	Number of subjects with at least	31 (21.1)		16 (21.1)		15 (21.1)		1.0
	one inconsistent data (%)
	Number of abnormal data d	3	0.1	2	0.1	1	0.0	-
	Number of subjects with at least	3 (2.0)		2 (2.6)		1 (1.4)		-
	one abnormal data (%)

a

Table 5

 5

	Comparison of completion time and satisfaction		Paper version	Web-based version	P a
	between the web-based and		N	%	N	%
	paper versions of the NutriNet-
	Sante ´anthropometric	Completion time needed					0.07
	questionnaire	\5 min	19	14.8	22	17.2
		5-9 min	32	25.0	41	32.0
		10-14 min	38	29.7	35	27.3
		C15 min	39	30.5	30	23.4
		Completion time was acceptable					0.02
		Agree	114	89.1	124	96.9
		Disagree	14	10.9	4	3.1
		Completion time is not a barrier for participation					0.003
		Agree	114	89.1	126	98.4
		Disagree	14	10.9	2	1.6
		Filling out this questionnaire was easy b					0.046
		Agree	122	96.1	126	99.2
		Disagree	5	3.9	1	0.8
		The questionnaire is nicely presented b					0.05
	Among the 128 subjects who	Agree	116	91.3	123	96.9
	returned the satisfaction questionnaire a P value for the difference between the two versions, from	Disagree I liked this version b Agree	11 105	8.7 82.7	4 123	3.1 96.9	0.0002
	Bowker's test of symmetry for	Disagree	22	17.3	4	3.1
	completion time and from Mac-Nemar Chi 2 tests for other variables b Data missing for 1 subject out of 128	Consultation of the ''Help'' section b Yes (n or %, as appropriate, of subjects entirely or fairly satisfied) No	58 (54) 45.7 (93.1) 55 (54) 43.3 (98.2) 69 54.3 72 56.7	0.7

Acknowledgments We thank the scientists, dieticians, technicians and assistants who helped to carry out the NutriNet-Sante ´test studies, and all dedicated and conscientious volunteers. We especially thank Soizic Gueho (data manager) for performing statistical analysis and Gwenael Monot (computer scientist), who is coordinating the computer aspects of the NutriNet-Sante ´Study. We thank Voluntis (a healthcare software company) for developing the NutriNet-Sante ẃeb-based interface according to our guidelines. We are grateful to Jerri Bram for English editing of the manuscript. This work was supported by the Ministe `re de la Sante ´(DGS, French Ministry of Health); the Institut de Veille Sanitaire (InVS); the Institut National de Pre ´vention et d'Education pour la Sante ´(INPES); the Fondation pour la Recherche Me ´dicale (FRM); the Institut National de la Sante ´et de la Recherche Me ´dicale (Inserm); the Institut de Recherche en Sante ´Publique (IRESP); the Institut National de la Recherche Agronomique (Inra); the Conservatoire National des Arts et Me ´tiers (CNAM); and Paris 13 University.


Xavier Pautrel

Abatement technology and the environment-growth nexus with education

research documents, whether they are published or not. The documents may come


Marc Baudry
email: marc.baudry@univ-nantes.fr

Precautionary principle and the cost benefit analysis of innovative projects

Keywords: JEL classification :. Q55, D81, C61 Precautionary principle, Real options, Bayesian learning, Environmental preservation, Irreversibility, Uncertainty

Public authorities often invoke the precautionary principle to ban or postpone the development of innovative projects with uncertain but potentially harmful irreversible impacts on the environment or health. As stated in the Rio declaration, the precautionary principle suggests balancing costs and benefits associated with irreversible decisions while taking account of the perspective to acquire better but costly information in the future. Though the real option theory seems to be an appropriate tool to deal with the precautionary principle it has two important limits with this respect. First it focuses on Markovian processes rather than on Bayesian learning. Second, it disregards the role of preferences whereas preferences are at the core of the seemingly linked concept of precautionary saving. The article is an attempt to circumvent these two limits. A canonical model of Bayesian real option expressed in terms of intertemporal utility maximisation is presented and solved. The optimal decision rule is discussed in the light of the precautionary principle. It is then shown how to switch consistently to an equivalent problem expressed in terms of costs and benefits.

INTRODUCTION

Implications from the use of new technologies are often ambivalent. On the one hand new technologies are by essence intended to satisfy human needs and, as such, they generate benefits. On the other hand, unforeseen effects on the environment and health may appear.

Impacts on the environment associated to the use of Genetically Modified Organism in agriculture or consequences on health of the use of cellular phones are among the most often quoted and analysed examples. Facing the decision whether to allow or ban new technologies in the presence of suspicions of harmful effects on the environment and/or health, public authorities in Europe generally invoke the precautionary principle to argue in favour of a, at least temporary, ban. The precautionary principle they referred to is that defined at the earth summit that took place at Rio de Janeiro in 1992. Indeed, principle fifteen of Rio declaration states that "Where there are threats of serious or irreversible damage, lack of full scientific certainty shall not be used as a reason for postponing cost-effective measures to prevent environmental degradation"1 . Similar definitions have been enacted in other national or international laws (see [START_REF] Gollier | Should we beware of the Precautionary Principle?[END_REF][START_REF] Immordino | Looking for a guide to protect the environment : the development of the precautionary principle[END_REF]. From an economist point of view there are three key concepts embedded in the Rio conference definition. The first key concept is that of irreversibility. If damages were not irreversible, inappropriate decisions could be corrected and associated costs could be avoided thanks to a feedback effect. Irreversibility makes such a correction ineffective and thus strengthens the effects of an inappropriate decision. The second key concept is that of a lack of full scientific certainty. Scientific knowledge is not exogenous but cumulates through time as the result of a costly research activity. Thus the first two key concepts call for a dynamic approach to the precautionary principle. The third key concept is that of cost-effectiveness. It implicitly means public authorities have to balance costs and benefits. Surprisingly, how to balance costs and benefits in the presence of potential irreversible harmful effects and scientific evolving knowledge is not much detailed in the economic literature.

Since the seminal articles by [START_REF] Arrow | Environmental Preservation, Uncertainty, and Irreversibility[END_REF] and [START_REF] Henry | Investment Decisions Under Uncertainty: The Irreversibility Effect[END_REF] and the clarification made by [START_REF] Hanemann | Information and the Concept of Option Value[END_REF], economists generally invoke the "irreversibility effect" to explain that the more uncertain we are about future returns from an irreversible project, the more the postponement of the project is relevant. The "irreversibility effect" has received much attention due to the ability to implement it to irreversible investment choices thanks to the real option theory synthesized in the now well known articles and textbook by [START_REF] Pindyck | Irreversibility, Uncertainty and Investment[END_REF], [START_REF] Dixit | Investment and Hysteresis[END_REF], and [START_REF] Dixit | Investment Under Uncertainty[END_REF]. Although the real option theory is originally build on a comparison with financial options (see, for instance, the pioneering works of [START_REF] Mcdonald | The Value of Waiting to Invest[END_REF]Siegel 1986 or Brennan and[START_REF] Brennan | Evaluating Natural Resource Investments[END_REF], its relationship with the concept of options and the "irreversibility effect" originally developed and clarified by Arrow Fisher Henry and Hanemann (AFHH) has been outlined, among others, by [START_REF] Lund | Financial and Non Financial Option Valuation[END_REF] and [START_REF] Fisher | Investment under Uncertainty and Option Value in Environmental Economics[END_REF]. The real option theory is commonly quoted in works dealing with the precautionary principle but it has never been applied, to our knowledge, to explicitly model it.

We suggest two ways to explain this paradox.

First, as noted by [START_REF] Ulph | Global Warming, Irreversibility and Learning[END_REF], assumptions about stochastic variables made in the real option theory are not appropriate for a wide range of economic problems implying both irreversibility and uncertainty. The reason for this is that the real option theory focuses on problems where independent exogenous and repeated shocks affect the evolution of a key variable in the decision of whether to develop or postpone an irreversible project. This approach, thereafter called Markovian approach, is relevant to deal with uncertainty affecting future prices but it is not suitable to deal with uncertainty about an unknown parameter, the true value of which is invariant with time. The latter kind of problems entails an analysis in terms of Bayesian uncertainty. The matter is that, in its primary version, the concept of option introduced by [START_REF] Arrow | Environmental Preservation, Uncertainty, and Irreversibility[END_REF] is sufficiently large to be thought of as a problem with either Markovian uncertainty or Bayesian uncertainty while the reference to a learning process and Bayesian uncertainty is made more explicit in the subsequent literature. [START_REF] Fisher | Quasi-option Value: Some Misconceptions Dispelled[END_REF] and [START_REF] Hanemann | Information and the Concept of Option Value[END_REF], for instance, explicitly refer to Bayesian uncertainty when focusing on uncertainty about biological and engineering parameters or uncertainty as to whether the offshore structures contain oil in commercial quantities. By contrast, the analysis of the optimal timing of environmental policies proposed by [START_REF] Pindyck | Irreversibilities and the timing of environmental policy[END_REF] still refers to Markovian uncertainty.

Second, economists often have other theoretical references than the real option theory in mind when dealing with the precautionary principle. Indeed, precautionary saving is a well known and extensively concept in macroeconomic dynamics. For many economists, adapting the concept of precautionary saving to environmental policies thus seems to be a natural way to analyse the precautionary principle. One of the most noticeable articles that obey to this logic is that by [START_REF] Gollier | Scientific progress and Irreversibility: an Economic Interpretation of the Precautionary Principle[END_REF]. As a result, preferences, and more specifically attitude toward risk, play a crucial role in determining whether the precautionary principle is theoretically founded or not. This contrasts with the real option theory that may be thought of as an advanced Costs Benefits Analysis method specified in terms of monetary units without explicit reference to the concept of utility. The fact that models adapted from the concept of precautionary saving focuses on continuous choices in a macroeconomic context rather than discrete choices in a microeconomic context adds to the gap with the real option theory.

Nevertheless, citations of [START_REF] Dixit | Investment Under Uncertainty[END_REF], considered as the core of the real options literature, are still made in works building on the concept of precautionary saving (and more specifically in the article by [START_REF] Gollier | Scientific progress and Irreversibility: an Economic Interpretation of the Precautionary Principle[END_REF].

This article is an attempt to unify the different ways suggested to analyse the precautionary principle. It starts with a Bayesian approach to the dynamics of scientific knowledge and highlights the role of prior beliefs in the debate about the precautionary principle and the importance to accord to the arrival of new information (section 2). We then turn to the presentation of a canonical model of Bayesian real option expressed in terms of intertemporal utility maximisation (section 3). The optimal exercise rule of the option helps identifying rules to balance advantages and disadvantages when facing a discrete choice with irreversible but uncertain impacts. These rules are used as a guideline to characterise cost effectiveness in the precautionary principle (section 4). For this purpose, a reformulation of the canonical Bayesian real option model is proposed in the specific case of a project with marginal impact. We more specifically focus on how to switch consistently from a specification in terms of utility to a specification in terms of monetary values.

A DYNAMIC APPROACH TO SCIENTIFIC UNCERTAINTY

This section introduces a Bayesian representation of the dynamics of scientific knowledge. How to model in a simple setting the arrival of informative but noisy messages about potential harmful effects of a project is first examined (2.1). We then turn to a discussion of decision errors and illustrate it with the case of a risk of dissemination of Genetically Modified Organisms (GMOs) in agriculture (2.2). We also examine in what respect the basic model proposed by [START_REF] Arrow | Environmental Preservation, Uncertainty, and Irreversibility[END_REF] introduces some confusion between Markovian and Bayesian approaches to uncertainty.

Bayesian learning and scientific progress

Since the seminal article of [START_REF] Arrow | Environmental Preservation, Uncertainty, and Irreversibility[END_REF], the prospect of an arrival of new information is the core of the analysis of irreversible economic decisions. New information takes either the form of the observation of exogenous random shocks (described as Markov processes) affecting the evolution of some key variables or the form of messages used to revise the beliefs of economic agents. The real option theory typically focuses on the first representation of information while the work of AFHH and, more especially, the subsequent literature is merely influenced by the second representation. This second representation constitutes the basis for modelling scientific uncertainty as regards the irreversible consequences of developing an innovative project. It is generally assumed that there is a finite set of possible values of the unknown parameter and that there exists an a priori, or subjective, discrete probability distribution on this set representing the beliefs about the true value of the unknown parameter. Messages arriving with time are used to revise the probability distribution. Some basic or intuitive properties of the revision process are expected. First of all, uncertainty decreases in the long term. This means that the probability for one of the possible values tends to increase with time while the probability assigned to other values tend to decrease. Some authors rule out the eventuality of contradictory messages, which allows representing the evolution of knowledge as an information structure which becomes finest as time goes2 . As a result, the evolution of probabilities assigned to the different possible values of the unknown parameter is monotonic. If the eventuality of contradictory or noisy messages arriving at different dates is not ruled out, the evolution of probabilities is not monotonic: it may be the case that a probability increases and then decreases on two successive periods of time. Kolstad (1996a) for instance takes account of such noisy messages. Then, the evolution of probabilities may be viewed as a stochastic process with an attractor corresponding to a vector made of zeros except one of its components (that associated to the correct value of the unknown parameter) equal to unity.

Another important property of the evolution of beliefs is that the revision of probabilities may be consistent with the probability theory, more especially with Bayes' theorem. This point has not always received attention in the literature dealing with irreversible decision when facing uncertainty. A noticeable exception is the article by [START_REF] Kelly | Bayesian Learning, Growth and Pollution[END_REF]. Though their model is not formalised in terms of real options, it has set the intuition for our canonical model.

Consider an innovative project intended to generate an additional flow of revenues for all future dates. Nevertheless, there are threats that developing the project generates irreversible though uncertain damages. Uncertainty affects damages in the sense that two scenarios are envisaged. The optimistic scenario refers to the case where the development of the project does not generate damages and, conversely, the pessimistic scenario refers to the case where the development of the project generates damages. Whether damages are generated by the project or not is known with certainty only once the project is developed.

Before the project is developed, economic agents have beliefs about what is the correct scenario. Beliefs are based on common and public but costly information. C denotes the deterministic flow of investigation costs that has to be incurred on one unit of time to eventually receive additional information with probability 1 where is the probability that the investigation yields no new result. Investigation costs are expressed in terms of consumption units. Probability of receiving additional information on one period of time for a given investigation cost C may be thought of as a rough measure of the speed of knowledge acquisition. Due to the arrival of information through time, beliefs are nor static but dynamic. Beliefs are represented by subjective probabilities associated to each scenario. Let X t (respectively X t 1

) denote the subjective probability at time t that the true scenario is the optimistic (respectively pessimistic) scenario. In order to make the dynamics of X t more explicit, we consider that they are two types of informative messages that may be received on one period of time. The first type is a message M a which is more likely when the correct scenario is the optimistic scenario. M a is received with probability (conditional on the arrival of an informative message) 2 1 p a if the correct scenario is the optimistic scenario and with probability (conditional on the arrival of an informative message) 2 1 1 p b if the correct scenario is the pessimistic scenario. The second type of message is a message M b which is more likely when the correct scenario is the pessimistic scenario. M b is received with probability (conditional on the arrival of an informative message) 2 1 p b if the correct scenario is the pessimistic scenario and with probability (conditional on the arrival of an informative message) 2 1 1 p a if the correct scenario is the optimistic scenario.

Accordingly, given that an informative message is received, economic agents consider that there is a probability

p X p X b t a t a t 1 1 Pr that it is a message M a and a probability p X p X b t a t b t 1 1 Pr that it is a message M b .
Probabilities p a and p b are objective probabilities known by all economic agents. As a result, economic agents are assumed to revise their beliefs on the correct scenario by applying Bayes' theorem. This yields the dynamics of beliefs illustrated by Figure 1. A key feature of the stochastic process X t representing these beliefs is that it admits two absorbing points which are respectively 0 X t and 1 X t

. This is consistent with the fact that X t is a probability and does not take values outside the interval 1 , 0 when starting from the interior of this interval. It also reflects the fact that when economic agents have no doubts about what is the correct scenario they will never change their beliefs whatever the type and number of new messages received.

Insert Figure 1

Bayesian learning and decision errors: an example

In the context of the use of GMO for agricultural purposes, investigation takes the form of scientific tests to determine whether a GMO plant may disseminate easily or not. The optimistic scenario corresponds to the absence of dissemination. In this case, producers adopting the GMO plant do it in their own interest so that there are exclusively net benefits. The pessimistic scenario corresponds to a significant dissemination of the GMO plant, more specifically dissemination on fields devoted to organic agriculture. As a result products harvested on these fields no longer conform to the rules of organic agriculture and cannot be sold under the organic label, thus implying a loss for organic producers. For the problem to make sense, the loss incurred by organic producers is supposed to exceed the benefit of farmers adopting the GMO plant so that there is a net loss. Scientific tests may invalidate some factors alleged to induce GMO dissemination. Such a conclusion is logically more likely to occur under the optimistic scenario and is assimilated to a message of type M a .

Conversely, scientific tests may confirm factors alleged to induce GMO dissemination. Such a conclusion is assimilated to message of type M b because it is more likely to occur under the pessimistic scenario. Probability p b 1 reflects the risk that no dissemination is proven by scientific tests in spite of its existence, due to measurement errors and/or inappropriate protocols. Probability p a 1 captures the risk that a conclusion in favour of dissemination is obtained whereas the correct scenario is that without dissemination. Such a risk originates in errors in the protocol or manipulations and is clearly expected to be far lower than the previous risk. These two types of risk are referred to as the risk of first type and the risk of second type in statistics. Table 1 adapted from [START_REF] Dorman | Evolving knowledge and the precautionary principle[END_REF] helps illustrating the nature of the risk conditionally on the correct scenario and the decision. It also helps understanding the debate between the partisans of GMOs and their opponents. On the one hand, partisans of GMOs stress the risk of a ban of GMOs whereas they may constitute a major innovation for economic and human progress. What they have in mind is the risk of first type. By sometime disregarding the risk of second type they implicitly reveal that they believe without doubt that the correct scenario is the optimistic scenario (i.e. 1 0

X

). On the other hand, opponents to GMOs point out the threat of an irreversible dissemination of GMO plants and thus have in mind the risk of second type. They sometime do not mention the risk of banning a harmless and profitable innovation and thus implicitly reveal that they believe without doubt that the correct scenario is the pessimistic scenario (i.e. 0 0 X ). For this two polar positions, scientific investigation aiming at obtaining additional information is worthless because both 1 X t and 0 X t are absorbing points of the stochastic process describing the dynamics of beliefs and thus both extreme partisans and extreme opponents are locked in their initial beliefs. Scientific investigation makes sense only in the presence of some initial doubts. A reasonable position could consist for instance in affecting a same probability (i.e. 2 1 0 X ) to each scenario to reflect the absence of a priori about the problem at stake. Beliefs then evolve through time as new information is received and how the risks of first and second types should be balanced is a dynamic matter.

Insert Table 1

What distinguishes our modelling assumptions form the standard model of [START_REF] Arrow | Environmental Preservation, Uncertainty, and Irreversibility[END_REF] and the subsequent works applying the real option theory to the analysis of optimal choice when facing risk and irreversibility is the explicit Bayesian nature of the underlying stochastic process. Indeed, the dynamics of beliefs used in [START_REF] Arrow | Environmental Preservation, Uncertainty, and Irreversibility[END_REF] is obtained as the peculiar case of the stochastic process described in Figure 1 with 0 , 0 p a and 0 p b . Said less formally, [START_REF] Arrow | Environmental Preservation, Uncertainty, and Irreversibility[END_REF] consider an informative message is received for sure and, moreover, it is not a noisy message. As a result, waiting one period of time enables switching from incomplete to complete information so that the dynamics of the problem is fully handle by a two period model. From a technical point of view, these additional assumptions imply that the stochastic process does not depart from a standard Markovian process. This is less obvious if errors of first type and/or errors of second type are introduced.

A BAYSIAN REAL OPTION APPROACH TO THE OPTIMAL DEVELOPMENT DECISION

This section starts with the decision problem of a public authority having to maximise the intertemporal utility of a representative agent. Reference to the agent's preferences is thus explicit but the problem departs from a standard model of precautionary saving by the discrete nature of the decision (3.1). It is shown that the optimal decision rule may be determined as the optimal exercise rule of a real option problem (3.2). A numerical example is provided to highlight how the perspective of acquiring costly information may drastically reduce the set of state of beliefs for which either definitive abandonment or immediate development is optimal, thus calling for postponement of project and further scientific investigation about its consequences (3.3).

The decision problem

The public nature of the decision to develop or not the innovative project in the canonical problem introduced in the previous section arises from two elements. First, as they generally correspond to previously unobserved phenomena, no mechanism exists to internalise the potential harmful impacts of the development of the innovative project.

Second, scientific knowledge is a non rival good the production of which is suboptimal if let to private economic agents. With these two elements in mind, we consider a public authority facing the problem to either abandon the innovative project or develop it immediately or postpone the development while financing scientific investigations about its consequences.

There is a representative agent or equivalently numerous identical agents with an additive ~ where B ~ is the random additional benefit expressed in terms of consumption units that the project generates for all future dates. In the optimistic scenario, environmental quality (or health) is preserved when the project is developed so that Q remains equal to its initial level q . The instantaneous utility level is then given by q B y u , ~. Conversely, the pessimistic scenario implies that Q

falls to D q
~ where D ~ is the measure of random damages in physical terms and the resulting instantaneous utility level for all future dates is D q B y u , ~. The exact values of additional consumption and damages are unknown until the project is developed. Risk as regards these values is introduced in order to analyse the impact of risk aversion and link our results to that of [START_REF] Gollier | Scientific progress and Irreversibility: an Economic Interpretation of the Precautionary Principle[END_REF]. B

~ and D ~ have known probability distributions. For the problem to make sense we also assume that q y u D q B y u , , ~ whatever the realisations of B ~ and D ~. Finally, the consumption level is given by C y as long as postponement of the development and further scientific investigation is decided, which yields the instantaneous utility level q C y u , . The public authority seeks to maximise the intertemporal expected utility of the representative agent. In case of abandonment, the intertemporal utility level is deterministic and defined by

W A 0 1 , t t q y u q y u , 1 (1)
In case of an immediate development without damages, the intertemporal expected utility level is given by the same expression with q B y u E , ~ in place of q y u , . The corresponding expression in case of damages deserves more attention. Indeed it is not standard to use the expected utility model with a utility function having two random arguments. Therefore, we first express the instantaneous utility level obtained with the pessimistic scenario in terms of the willingness to pay W ~ for preserving environmental quality or health. By definition we have

D q B y u q W B y u , , ~
so that only the first argument of the instantaneous utility function is random. Accordingly, the intertemporal expected utility level in case of damages is given by the same expression than (1) with

q L y u E , ~
in place of q y u , where B W L ~ stands for random net equivalent losses expressed in consumption units and associated with damages to the environment or health.

Note that the second argument in u remains unchanged whatever the situation considered. In order to make notations more concise we thus replace u by v with consumption as the sole argument thereafter. Function v may be interpreted as an indirect utility function. Therefore, the expected intertemporal utility level, conditional on information received so far, when immediate development is decided but just before development reveals whether there are damages or not is defined as

L y v E X B y v E X X W t t t D 1 1 1 (2) 10
Postponing the development while financing scientific investigation about its consequences yields the instantaneous utility level q C y u C y v , at the current date plus the expected discounted intertemporal utility level corresponding to the optimal choice between definitive abandonment, postponement with scientific investigation and immediate development at the next date. Therefore, the optimal expected intertemporal utility level at the current date is given by

X W p X W p X W X W C y v W Max X W t D b t a t b t a t a t a t t A t 1 Pr 1 * Pr Pr * Pr 1 * * (3)
The expected optimal intertemporal utility level at the next date in case of postponement is defined on the basis of the dynamics of beliefs described in Figure 1. With probability no additional informative message is received in spite of scientific investigation. The optimal intertemporal utility level then remains unchanged. With probability Pr 1 a t it is believed that a message of type M a will be received and thus that an upward revision to , even if informative messages are received. According to the discussion made in the previous section about the respective positions of extreme opponents and extreme partisans of the development of innovative projects susceptible to have harmful impacts on the environment and /or health, this explains why both of them reject the idea of further scientific investigations. Whatever the reason, if the perspective of acquiring more information is disregarded, the optimal choice consists in developing immediately the project when

W X W A t D
and definitely abandon it otherwise. Using (1) and ( 2), it is easily checked that immediate development is preferred if and only if X t exceeds a threshold value defined as a ratio of variations for the instantaneous utility level:

X X W X W t A t D (4.a) with L y v E B y v E L y v E y v X ~ (4.b)
If the perspective of acquiring more information is taken into account, a definitive abandonment of the project as soon as X X t could reveal to be suboptimal at the next date.

It more specifically happens if messages of type M a (leading to an increase of X t ) are received. Conversely, an immediate development of the project as soon as X X t could be regretted if the pessimistic scenario reveals to be the correct one. Indded, waiting just one period of time more could induce the receipt of a message M b leading to a drop of X t and thus justifying not to develop the project. Therefore, taking account of the arrival of additional information, it is optimal postponing the decision to abandon or develop the project as long as beliefs stay inside an interval X X D A , where X X A is the lower bound behind which definitive abandonment is optimal and X X D is the upper bound above which immediate development is optimal. In mathematical terminology, is the optimal waiting region of the optimal stopping time problem defined by (3). The optimality of this form of waiting region is checked ex post by verifying that the resulting intertemporal utility level

X W t * exceeds both W A and X W t D
for values of X t in .

The option value of additional information

The optimal thresholds X A and X D are find by solving the optimal stopping time problem (3). For this purpose, note that as long as X t lies inside the optimal waiting region the value function

X W t * satisfies the following definition X W t * C y v 1 * X W t X X X p X W p X W D A t b t a t b
p p p p p p b b a b b a 1 1 1 1 1 (7)
For a strictly positive discount rate the left hand side of equation ( 7) is a positive, higher than unity, constant. Let h denote the right hand side of equation ( 7 . Then, solutions of the homogeneous part of equation ( 6) are linear combinations of the two independent solutions

X X X X X f t t t t t 1 1 1 1 and X X X X X f t t t t t 1 1 2 2
. Solutions to the complete equation ( 6) are obtained by adding the particular solution

C y v W P 1 (8)
A parallel with (1) shows that this particular solution defines the intertemporal utility level in case of permanent postponement and investigation. Of course, it is expected that postponement will not be permanent and that at some date in the future a sufficient amount of information will be acquired so that it will be optimal to either abandon the project or develop it. This perspective justifies incurring the investigation cost for acquiring more information and generates an option value, expressed in terms of utility, that adds to W P in order to obtain X W t * . The expression of this option value X OV t is given by the general solution to the homogeneous part of equation ( 6). Therefore

X OV X f K X f K W C y v X W t t t P t 2 2 1 1 * 1 (9)
Where the two constants K1 and K 2 remain to be determined on the basis of the behaviour of with respect to X t . It seems that it is not possible to obtain an analytical solution for

X

A and X D . Therefore we turn to a numerical solution to illustrate the optimal decision rule and visualise the importance of the additional intertemporal expected utility generated by the opportunity to postpone realisation of the project in order to acquire more information. . The optimal waiting region is not centred on this latter value but extends more on its left than on its right.

The perspective to acquire additional information thus plays more in favour of a reduction of the range of beliefs that justifies definitive abandonment of the project than in favour of a reduction of the range of beliefs in favour of an immediate realisation of the project. In spite of a wide optimal waiting region, the supplement of intertemporal expected utility generated by the opportunity to postpone the project and pay for additional information remains rather limited. It amounts at most to % 26 . 1

of W A at X X .

Insert Figure 3 4

. IMPLICATIONS FOR THE COST BENEFIT ANALYSIS OF INNOVATIVE PROJECTS

When facing projects without macroeconomic incidence, public authorities generally implement a Costs Benefits Analysis. This section examines how to make such a Costs Benefits Analysis consistent with the approach to the precautionary principle developed so far in the article. Rewriting the problem in terms of monetary units requires adding corrections to gross measures of costs and benefits in order to correctly take account of preferences (4.1).

The real option model with Bayesian learning presented in the previous section may then be converted from a problem in terms of utility to a problem in terms of monetary units (4.2).

Switching from utility to monetary units

Implementing Cost Benefit Analysis to the problem of developing, abandoning or postponing the innovative project presumes that net benefits and net losses incurred in case of development but also investigation costs associated to postponement are small compared to the initial consumption level y . The indirect utility function may then be replaced by its second order approximation in the neighbourhood of the initial consumption level. In the case of postponement, the instantaneous utility function is thus replaced by

C y v C y v y v C y v 2 2 1 (12)
Moreover, utility terms have to be converted in monetary units by dividing by the marginal utility level y v

. We then obtain the following decomposition of ( 12)

m C y v y v C v y v y v v y v C y v p a p 2 2 1 ( 13
)
where v p is the instantaneous utility level in terms of monetary units obtained in case of postponement and investigation whereas va is the instantaneous utility level in terms of monetary units obtained in case of abandonment. Postponing the project to benefit from scientific investigations not only induces the additional investigation cost C compared to abandonment but also an additional monetary value referred to as expression mp . Given that the marginal utility of consumption is positive but decreasing, mp is a cost. Imposing constant marginal utility of consumption is the only way to obtain a value of zero for mp . Therefore, mp is interpreted as the equivalent monetary penalty to be added to the investigation cost C to take account of the decreasing marginal utility of consumption. Substituting L ~ to C in (13), taking expectation and noting that the variance of net losses is given by

L L L L a L t y v y v m y v y v v y v y v v y v L y v E L 2 2 2 1 2 1 ~ (14)
In addition to the monetary penalty mL due to the decreasing marginal utility of consumption, the expected loss L is also reduced by the Arrow Pratt approximation of the risk premium. The fact that both the decreasing marginal utility of consumption and risk aversion are captured by the concavity of the instantaneous utility function implies that mL and have close expressions. The decomposition is almost identical except the negative sign of the second term of the right hand side in case of development and net benefits:

B B B B a B y v y v m y v y v v y v y v v y v B y v E B 2 2 2 1 2 1 ~ ( 15
)
where B and 2 B respectively denote the expected value and variance of net benefits. Note that whether losses or benefits are considered, taking account of the marginal utility of consumption always yields a penalty.

Precautionary principle in terms of costs and benefits

The next step to switch from the utility maximisation problem to a Cost Benefit Analysis problem consists in rewriting program (3) in terms of variations of the expected intertemporal utility level compared to the level W A obtained with abandonment and dividing all expressions by the marginal indirect utility y v to convert all terms in monetary units.

Using approximations (13) ( 14) and ( 15) and proceeding along the same lines than in section 3, the option value is now defined in monetary units by two constants y

v K K 1 1 and y v K K 2 2
that solve the following two "value matching" boundary conditions replacing boundary conditions (10.a) and (10.b)

0 1 2 2 1 1 X f K X f K m C A A p (16.a) X f K X f K m C D D p 2 2 1 1 1 L L L D B B B D m X m X 1 1 1 (16.b)
The optimal thresholds X A and X D of beliefs solve the following "smooth pasting" conditions that replace boundary conditions (11.a) and (11.b)

0 2 2 1 1 X f K X f K A X A X (17.a) X f K X f K D X D X 2 2 1 1 L L L B B B m m 1 (17.b)
where the subscript X is used to denote the derivative of functions with respect to the state of beliefs. of losses is generated if the pessimistic scenario is correct. Both flows are discounted at rate and corrected to take account of decreasing marginal utility of consumption and risk aversion. By this way, preferences and attitude toward risk affect the optimal decision rule. Given the current state of beliefs X t , development of the project yields the expected discounted monetary value that appears in the third line of (18). According to (18), the project is valued by the expected discounted sum of benefits and losses, respectively weighted by the current beliefs that the correct scenario is the optimistic scenario or the pessimistic one, if and only if immediate development is optimal.

L L L t t B B B t b t a t b t a t a t a t t p t m X m X p X V p X V X V m C Max X V
This occurs when X X D t

. The project is worthless if the current beliefs in favour of the optimistic scenario fall below the threshold X A . Finally, the project is valued at the option value

X f K X f K m C X V t t p t 2 2 1 1 1 (19)
for all states of beliefs such that X t is in the range X X D A ,

. This value may be used, for instance, to define the maximal initial investment cost that justifies not abandoning the project. The optimal exercise rule of the real option highlights how to balance costs and benefits in the presence of evolving scientific uncertainty about potential irreversible impacts.

Costs and benefits are not only weighted by current beliefs, they also play a key role in defining the optimal thresholds of the state of beliefs above which immediate realisation of the project is optimal and below which definitive abandonment has to be chosen. As regard this second effect, their influence is highly non linear if the perspective of acquiring costly additional information is properly taken into account.

CONCLUSION

The canonical model developed in this article is intended to provide a guideline for implementing the precautionary principle in Cost Benefit Analysis. First, it builds on the real option theory but extends it to Bayesian learning processes instead of the standard Markovian processes generally postulated in the literature. This extension makes the real option approach more consistent with the logic of evolving scientific knowledge that is at the core of the precautionary principle and results from a costly search for additional information. Second it conciliates the real approach to precautionary principle and the strand of literature that analyses the precautionary principle by adapting the concept of precautionary saving to environmental preservation. This suggests applying two kinds of correction to flows of costs and benefits. The first correction is intended to take account of the marginal utility of consumption. The second correction follows on from risk aversion and takes the form of a risk premium. Using standard method of resolution of real options models, it is then shown

  instantaneous utility function and denotes the utility discount rate. The instantaneous utility function depends on the consumption level Y and on the level Q of environmental quality or health. Consumption amounts to y units in the absence of development and scientific investigation. In case of an immediate development of the project, the consumption level raises to B y

  to the optimal stopping problem By definition, at the lower bound X A of the waiting region the public decider is just indifferent between postponing the development of the innovative project and definitely abandoning it. As a result the following condition holds the upper bound X D of the waiting region the public decider is just indifferent between postponing the development of the innovative project and immediately realising it. of optimal stopping time problems, (10.a) and (10.b) are the "value matching" conditions. Solving simultaneously (10.a) and (10.b) with respect to K1 and K

  value, we obtain a somewhat similar decomposition of the monetary equivalent vL of the instantaneous utility level in case of development and net losses

Finally, boundary

  conditions (16.a), (16.b), (17.a) and (17.b) are those obtained when directly solving the optimal stopping time problem defined by

  t

  corresponds to (3) expressed in terms of monetary costs and benefits. X V t is the monetary option value of postponing the project to acquire additional information at the cost m C p , to be thought of as the exercise price of the option. Abandonment yields a monetary value of zero corresponding to the first line in (18). Development of the project generates the flow B B B mof benefits for all future dates if the optimistic scenario is the

  Figure 1 Dynamics of beliefs

  Figure 2 Solutions to the characteristic equation

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at http://www.unep.org/Documents.multilingual/Default.asp?DocumentID=78&ArticleID=1163

This is typically the case in[START_REF] Freixas | On the Irreversibility Effect[END_REF] orKolstad (1996b) who merely follow the primary model of[START_REF] Henry | Investment Decisions Under Uncertainty: The Irreversibility Effect[END_REF].

See Brekke and Oksendal (1991),[START_REF] Dumas | Super Contact and related Optimality Conditions[END_REF] or[START_REF] Dixit | The art of smooth pasting[END_REF] for technical details on the link between the optimality of the thresholds values and the smooth pasting conditions. Sodal (1998),[START_REF] Dixit | A markup interpretation of optimal investment rules[END_REF] or[START_REF] Shackleton | Smooth pasting as rate of return equalization[END_REF] provide an alternative, equivalent but more intuitive for economists, explanation of optimality conditions for basic option problems.

how corrected costs and benefits influence the choice between abandoning, realising or postponing innovative projects with uncertain but irreversible potential harmful effects.