Abstract
In a site-selection choice experiment various hypotheses are tested related to spatial heterogeneity in willingness to pay (WTP) for environmental improvements. Spatial heterogeneity is measured through distance-decay effects, substitute sites inside and outside the experiment’s choice set, and spatial trend variables. We demonstrate that distance-decay functions differ between users and nonusers and across study sites. Also the distance to substitutes outside the choice set explains spatial variation in WTP. We show that further extending the model with spatial trend variables reveals additional spatial heterogeneity in choices. Accounting for spatial heterogeneity patterns results in significantly different WTP estimates for environmental improvements. (JEL Q25, Q51)
I. Introduction
Many stated preference (SP) studies essentially involve spatial choices among environmental improvements at different locations in a confined geographical area. The characteristics of the spatial context of the ecosystem service under valuation, notably the location providing the service, the availability of possible substitute sites, and distance from the services to the population of beneficiaries, are all likely to affect the value attached to the ecosystem service. Distance and substitutes may considerably influence willingness to pay (WTP), especially for water quality improvements in areas encompassing a large number of water bodies, such as rivers, creeks, or lakes (Johnston, Swallow, and Bauer 2002).
Despite the importance of the geographical context, many valuation studies using SP techniques can be characterized as aspatial single-site studies. Only about 25 SP studies in the immense environmental valuation literature over recent decades were found to report the effect of distance from the respondent’s home to the asset under valuation on WTP values (see Bateman et al. 2006 for an overview). WTP often declines as the distance increases between a respondent’s place of residence and the site providing ecosystem services. This effect is referred to as distance- decay in the SP literature (Loomis 2000). Hanley, Schlapfer, and Spurgeon (2003) hypothesize that distance-decay relationships vary between different types of environmental goods, but also among similar goods due to differences in the availability of substitutes in the surrounding area. However, this hypothesis has not been tested empirically in the SP literature.
For many environmental goods, the number of available substitutes increases with distance from the site, thereby contributing to distance-decay effects. The distance from respondents in an SP survey to the locations providing ecosystem services affects the substitutability of these services. Therefore, distance and substitution effects are interdependent in spatial choice studies. Surprisingly, there appear to be no published SP studies in which distance to substitute sites is included in the same WTP function as distance to the site under valuation. In general, existing SP studies are limited in their consideration of spatial factors, which may affect the reliability and validity of their results. An important critique is that "simply adding a distance variable does not overcome the peculiarities of spatial choice" (Pellegrini and Fotheringham 2002, 506). Among the main limitations of existing studies is the isotropic approach, assuming that distance-decay effects for an ecosystem service are the same in all directions from the site, which leaves the distances to substitutes unaccounted for in survey and model design.
In the study presented here, we develop a choice experiment that explicitly accounts for the effect on the estimated WTP welfare measure of spatial heterogeneity in public preferences for water recreation sites and the ecosystem services they provide. We test three specific hypotheses related to spatial heterogeneity. The first hypothesis is based on work by Hanley, Schlapfer, and Spurgeon (2003) that distance-decay effects differ across water bodies providing different ecosystem services in the same confined geographical area. The second hypothesis tested here is that distance- decay effects differ across users and nonusers (Bateman et al. 2006). Our third and final hypothesis is that distance-decay is not a simple unidirectional phenomenon, but displays significant multidirectional variation due to the spatial distribution of substitutes and heterogeneity in the spatial perceptions of respondents.
By including distance and substitution effects in our choice experiment (CE) and model design, we aim to obtain more valid and reliable WTP results. To test the first and second hypotheses, we specify different distance-decay functions for each of the study sites and control for differences between users and nonusers of these sites in the utility model. We estimate two models to test the third hypothesis. First, we expand the model and include the effect of distance to substitute sites, other than the sites in the CE choice set, to evaluate possible variation in distance-decay effects due to the nonrandom distribution of alternatives. A disadvantage of this approach is that it requires the identification of the full set of relevant substitutes affecting WTP, either based on researcher or respondent perception of relevant substitutes. Secondly, we include spatial trend variables in the model based on a so-called spatial expansion model (Cassetti 1972; Fotheringham, Brunsdon, and Charlton 2002) to identify any remaining spatial heterogeneity. The latter would relate to more complex multidirectional spatial preferences and unidentified associated substitution effects. This geographical model allows for a two-dimensional analysis and accounts for spatial heterogeneity measured through the direction of distance-decay effects.
II. Substitution And Distance In Environmental Valuation
Two characteristics of the spatial context of an environmental good or service that are most likely to affect WTP, yet rarely addressed in SP studies, are (1) the distance between the respondent and a site providing the ecosystem service under valuation and (2) the distance between the respondent and substitute sites. The former has been studied more thoroughly in the environmental valuation literature than the latter, but primarily in revealed preference studies. One of the main reasons to include distance-decay effects in SP research is to determine the geographical boundaries of the market for the environmental good in question (Loomis 2000). This market represents the population within a geographical area over which the WTP values can be aggregated to calculate the total welfare change of a proposed scenario of environmental change. The distance-decay function reflects the trade-off between site benefits and the cost of increasing distance (Smith 1975). The distance-decay rate can be used to estimate at what distance from the site under valuation the population mean WTP equals or approaches zero.1 In travel cost and recreational choice studies (e.g., Adamowicz, Louviere, and Williams 1994), travel costs, and sometimes also travel time, are used as a price proxy to calculate the WTP for a site. Distance can also serve as an indicator for the effect of the frequency of visitation, information, and knowledge about the good under valuation, as it is often inversely correlated with these commonly used explanatory variables of WTP in SP studies (e.g., Pate and Loomis 1997; Bateman et al. 2006; Concu 2005, 2007).
While distance-decay is expected to be found in the WTP for use values, there is no explanation within standard economic theory for nonuse values to decrease with distance. Results on this issue in the SP literature are mixed and complicated by the problem of separating pure nonuse from use values. For the protection of charismatic species that people rarely see (Loomis 1996, 2000; Bulte et al. 2005), and for environmental policies generating mainly nonuse values (Payne et al. 2000), no distance-decay effects are found. Distance-decay effects are also less likely to affect the WTP for goods that have importance at a large scale, such as national parks (Barrick and Beazley 1990). These goods may be widely known, iconic, and have fewer substitutes, while sometimes their uniqueness has given rise to a protected status. However, Sutherland and Walsh (1985) find distance- decay for nonuse values for a locally important lake, for which respondents living at greater distance may have considered conservation to be less important. Some studies compare the distance-decay rates of users and nonusers, where the latter are expected to show lower distance-decay rates. Also in this case, the empirical results are mixed. For example, Bateman, Langford, and Nishikawa (1999) and Hanley, Schlapfer, and Spurgeon (2003) find significant distance-decay for WTP values held by nonusers. Current nonusers may show distance-decay if the WTP scenario describes a future improvement for which they hold option values.
Another explanation for distance-decay effects is the availability and accessibility of substitutes (Sutherland and Walsh 1985; Brouwer and Slangen 1998). Often, the availability of substitutes for nonunique sites will increase with distance from the site. The greater the supply at a given distance, the lower will be the demand and WTP for the study site. Brown and Duffield (1995) and Pate and Loomis (1997) appear to present the only two studies that include both the distance to the site and the presence of substitutes in the regression analysis. Many contingent valuation studies in the literature focus on a single site and do not provide information about the effect on choices of changes in the availability or characteristics of alternatives. These single-site studies do not provide insight into the prioritization of sites by the general public when a policy program will involve changes at multiple sites simultaneously (Carson, Flores, and Meade 2001). The exclusion of substitute characteristics from the WTP-model may cause omitted variable bias. Parsons (2003) argues that single-site travel cost models should therefore include a variable reflecting the price of substitutes. Kling (1989) shows that the level of bias depends on the substitutability of the goods under study.
Two practical constraints may explain the limited attention that has been paid to estimating substitution effects in spatial choice settings in SP studies. First, the selection of alternatives that constitute the relevant set of substitutes, also known as the consideration set or opportunity set (Haab and Hicks 1997), is one of the most difficult tasks in stated choice studies. The definition of the relevant choice set is particularly complicated in spatial choice studies, because the number of potential alternative locations may be very large (Cummings, Ganderton, and McGuckin 1994). In water valuation studies, for example, the case study areas are often not only abundant in water locations, but also involve a diverse range of recreational activities and opportunities, such as fishing, swimming, boating, walking, surfing, watching wildlife, and so on. The choice set definition is important, because it will affect parameter estimates of models and hence WTP estimates (Parsons and Hauber 1998; DeShazo and Fermo 2002). To determine the relevant alternatives in the choice set, its spatial boundaries and relevant selection criteria affecting site substitutability, such as site characteristics, have to be defined. Either researcher-defined criteria (for instance, all available nature sites in the study area) or respondent-defined criteria (such as the sites known or visited previously by the respondent) can be applied (Parsons 2003; Freeman 2003). In random utility travel cost studies based on revealed preference data, researcher-defined choice sets are most common (Parsons and Hauber 1998). Here, the spatial boundaries of the choice set are often defined using distance criteria. In recreation studies, for example, only those sites within a certain range are expected to be chosen, reflecting respondents’ propensity to travel and their limited leisure time. In SP studies aiming to elicit nonuse values, distance may be a less relevant criterion, because markets for nonuse values could theoretically approach infinite geographical horizons. In general, the relevant set of substitutes may vary across goods and sampled populations (Bateman and Willis 2002). Analysts may not reliably observe the relevant choice set unless respondents are asked to list their subjectively relevant substitutes in surveys (e.g., Peters, Adamowicz, and Boxall 1995).
A second complicating factor is that the design of SP studies can include only a limited number of questions and variables. SP studies depend on the use of questionnaires, and therefore a balance must be found between the completeness of information provision in a survey and the patience and cognitive capabilities of respondents. As a result of this trade-off, it is often difficult to devote an equal amount of attention to substitutes, as well as a proper description of the study site and good being valued (Rolfe, Bennett, and Louviere 2002). This may explain why, in practice, most contingent valuation studies focus on single sites. One of the main concerns of such studies is that parameters will be biased and WTP values overestimated when respondents state their WTP focusing on one particular location only, without consideration of the full set of alternatives (Hoehn and Loomis 1993; Carson, Flores, and Hanemann 1998; Johnston, Swallow, and Bauer 2002; Freeman 2003). Similarly, CEs are limited with respect to the number of alternatives they can include. If all available alternatives are included as options in a choice task, the realism of the CE may increase, but at the same time the cognitive burden for the respondent may become insurmountable. Hence, large choice sets may affect the decision process, resulting in simplifying heuristics, and may complicate the experimental design of the CE (DeShazo, Cameron, and Saenz 2009).
Variation in substitute availability across space may cause heterogeneity in distance-decay and WTP. In many regions, substitute sites will not be randomly distributed over space in terms of quantity, quality, or both. In the econometric analysis, spatial heterogeneity may lead to variation in the model coefficients or error terms of the model, where this variation follows a systematic spatial pattern. The location of the observations is crucial in explaining the variation (Anselin 1999). This may produce anisotropic distance-decay effects, that is, nonuniform distance-decay effects across different directions from the site.
The effect of heterogeneity in the spatial distribution of alternatives on decision outcomes has been addressed in other research fields, for instance, in migration studies (e.g., Fotheringham and Pitts 1995; Pellegrini and Fotheringham 2002). In the hedonic pricing literature, there are a few studies that correct for directional heterogeneity in distance-decay (Herriges, Secchi, and Babcock 2005; Cameron 2006; Agee and Crocker 2010). These studies address the impact of wind direction on distance-decay, which in turn determines the distribution of airborne pollution affecting house prices downwind from the pollution source. Herriges, Secchi and Babcock (2005) and Agee and Crocker (2010) include dummy variables for different directions, whereas Cameron (2006) uses continuous spatial trend variables. Cameron (2006) shows that including only a distance variable in the model and ignoring directional effects can lead to insignificant distance-decay effects or produce biased distance estimators due to omitted variables.
There are a number of studies in the revealed preference literature that include the distance to other locations representing local (dis)amenities, substitutes, or complementary goods and services to control for different sources of spatial heterogeneity. Hedonic pricing models typically include the distance to a number of local facilities besides the environmental good of interest, such as the city or retail center (e.g., Tyrvainen and Miettinen 2000; Cavailhes et al. 2009). Cameron and McConnaha (2006) include the distance to relevant geographical features in a migration model. Similarly, some travel demand studies have included the distance to additional recreational amenities, such as Jones et al. (2010), who show that the number of visitors to forests depends, among other things, on the travel time to the nearest inland waterway, heathland, and the coast.
Existing distance-decay studies in the SP literature all apply a one-dimensional distance indicator from the respondent’s place of residence to the study site and ignore the effect of the spatial distribution of alternatives, so the obvious question is how reliable are previous distance-decay results and associated market delineation procedures. In the next section, we address this issue by comparing different modeling approaches. By employing spatial trend variables, we test for site-specific distance-decay effects, as well as the effects of distance to substitutes and any remaining spatial heterogeneity.
III. Modeling Approach
We present three models to assess different sources of spatial heterogeneity in choices for environmental quality improvements. Heterogeneity is reflected across respondents in their spatial preferences, including the effects of distance to the study sites and to substitutes. The more restricted model (Model I) is used to test the first hypothesis of uniform distance- decay rates across sites as well as users and nonusers. This model is compared with two models (Models II and III) that account for the effect on WTP of spatial heterogeneity related to substitutes to test hypotheses 2 and 3.Before presenting the models, we will outline the general modeling framework and specification of the utility function.
To properly capture spatial heterogeneity and substitution among sites in a CE, models that allow for flexible substitution patterns between alternatives are needed. The standard multinomial model represents the most common structure of choice models (McFadden 1974), but the substitution patterns in this model are restricted by the "independent of irrelevant alternatives" assumption (Kanninen 2007). In our analysis, we therefore employ mixed logit models estimated through simulation methods.2 Mixed logit models are able to account for respondent differences (taste heterogeneity) and repeated choices (Train 2003), that is, the panel structure of the data (Hensher and Greene 2002). Mixed logit models have been regularly used in environmental valuation studies, for example, to estimate the WTP for erosion control (Colombo, Cala- trava-Requena, and Hanley 2006), water quality (Hanley, Wright, and Alvarez-Farizo 2006), and animal welfare (Carlsson, Frykblom, and Lagerkvist 2007).
The general utility specification U of alternative j for individual i at choice moment t is presented in equation [1]:
[1]In this function, the parameter a refers to the alternative specific constant of alternative j;3 Xijt denotes the set of attributes and, Pijt the price of alternative j described at time t.
and βp are the choice set parameters. For each attribute, it is tested if the beta-coefficient βj is significantly different from the coefficient of the same attribute for other sites. The set of respondent characteristics Yi includes theoretically expected variables, such as household income, which enter the model in interaction with dummy-variable Hj that takes the value 1 for each of the sites, and zero for the opt-out. The associated parameter estimate βy reflects the probability effect on choosing an alternative for which respondent i is willing to pay compared with the opt-out. The term λij denotes the zero-mean parameter with a standard normal distribution 
for the error component related to dummy variable Hj The term λij is an individual random effect common to all the hypothetical responses by individual i. We include this error component to allow for correlation across the hypothetical alternatives compared with the opt-out. In this way, the model controls for status quo bias that may arise if the hypothetical alternatives are perceived differently from the actual situation represented by the opt-out (Scarpa, Ferrini, and Willis 2005). The remaining term, εijt is the standard extreme value type I error.
In its most general form, utility is influenced by the site-specific distance-decay function f(Distij,Zij,Userij)Generically, the function f depends on Distij reflecting the effect of road distance from respondent i to site j,4 a set of explanatory variables Zij that can be related to the sites or respondents, and a dummy variable taking the value 1 if respondent i has recreated at site j under the current site conditions prior to the survey, denoted Userij The latter variable is included in the models to control for differences in distance- decay between users and nonusers.5
The three models differ in their specification of the distance-decay function f. Model I, based on equation [2], is the basic variant of this function. It accounts for differences in the effect of distance across the study sites in the CE and across user and nonuser groups, providing a test of the hypotheses suggested by Hanley, Schlapfer, and Spurgeon (2003). Differences in distance-decay between users and nonusers are examined by specifying the distance-decay function f(Distij,Zij,Userij) as a function of distance Distij with a coefficient of 
for nonusers arid a user-related coefficient differential of 
for users.
[2]The term μi is introduced to accommodate respondent heterogeneity and is multiplied by the user dummy variable Userij This random term is assumed to be constant over choice occasions t and distributed normally (
. This model specification assumes that the error term of this random parameter is now uncorrelated with the model residuals εijt The standard deviation of the random parameter μi is common for all three sites, but site-specific mean effects for users are not restricted to equal zero or to be equal across sites in the models, because Userij is a component of the distance-decay function f. This specification gives us flexibility in specifying the relationship between distance and the user dummy in the model and allows testing of our hypotheses regarding differences in distance- decay between users and nonusers and between sites.
In empirical SP studies for environmental valuation, the four most commonly applied functional forms for the effect of distance on WTP are the linear, quadratic, lognormal, and exponential forms. From these, the nonlinear distance-decay functions are used most often. Sometimes, different specifications are applied within the same study. For example, Concu (2005) shows that distance-decay effects can differ across different characteristics of a good in a CE survey focusing on the WTP for environmental changes in different types of bushland in Australia. Economic theory provides no reason for the existence of a similar effect of distance on the WTP for different goods or a compelling argument for the use of a particular functional form. The use of similar distance-decay specifications across goods may be preferred for reasons of model consistency but can lead to biased or insignificant distance coefficients and inaccurate aggregate WTP values. Given the importance of the estimation of reliable distance-decay parameters for the purpose of reliable WTP value aggregation, we test and compare the significance and impact on model fit of different functional forms using likelihood ratio tests. In the estimation procedures, we compare the four most common specifications listed above and present only the functional specification of the distance-decay effects that results in the best model fit, allowing for differences in functional forms across sites.
Our first null hypothesis 
is that distance-decay effects are similar across sites:
[3]Rejection of H will follow if the distance coefficients or distance-related user effects of at least two alternative sites j and k in the choice set of the CE are significantly different from each other. The second related null hypothesis H2 is that distance-decay is similar across users and nonusers, in other words, there is no significant effect of presurvey visitation behavior on choices, as formulated in equation [4]:
[4]This null hypothesis is rejected if we find that users and nonusers display different choice behavior in terms of distance-decay.
The third objective is to evaluate possible spatial variation in distance-decay effects, for which we develop two models, with more-inclusive specifications of the distance-decay function and related hypotheses (
, 
and 
). In addition to the effect of distance to the study sites in the CE, our second model in equation [5] (Model II) also accounts for the effect of road distance from the individual i to substitute sites outside the CE-set for alternative j(DistSubij) on WTP, where 
is the site-specific coefficient of substitute sites for nonusers and 
is the differential coefficient activated for users:
[5]Different substitute sets will be defined and tested for their significance to explore if "respondent-reported" or "researcher-assigned" sets best reflect respondents’ opportunity sets and capture the effect of substitutes outside the choice set on WTP. The third hypothesis to evaluate possible spatial variation in distance-decay effects due to distance to substitute sites is tested as follows:
[6]where the null hypothesis is that there exists no significant effect of distance to substitute sites outside the choice set for either users or nonusers.
In the third model (Model III), we use the so-called spatial expansion method to assess any remaining spatial heterogeneity, by testing for directional differences in distance-decay effects among users and nonusers (Casetti 1972; Brown and Jones 1985). The spatial expansion method allows for two-dimensional spatial variation in the parameters of the model by including spatial trend variables. Underlying this method is the assumption that functional relationships are not constant over space. To this end, the model coefficients are allowed to vary with their spatial context, that is, as functions of the coordinates (x, y) of the locations:
[7]In this paper, we employ the spatial expansion method in a similar way to the hedonic pricing study presented in Cameron (2006) and model distance-decay parameters as a function of their location relative to any point of reference, such as the study sites in the CE.6The objective of our application, however, differs from this study in a number of ways. First, we apply the method in an SP context for environmental valuation and add the directional effects to a discrete choice model. Second, we modify the model to control for differences between users and nonusers. Third, the spatial expansion method is used to test the spatial heterogeneity that remains after controlling for the distance to substitutes.
It should be noted that the different elements of the spatial expansion function are not drivers of preferences per se. However, where a respondent lives determines which substitutes are available to him or her and hence the relative scarcity of the good under valuation. The availability and proximity to substitute sites are considered by economic theory to be among the main spatial factors affecting WTP besides distance, and these vary across directions from the site. The elements of the spatial expansion function capture the location of the respondent and hence show how this location, as a determinant of the direction to the site being valued and distance and direction to other relevant sites, affects WTP once road distance to the site under valuation is accounted for.
To apply the spatial expansion method, we can rewrite equation [8] in terms of differences in longitudinal and latitudinal location between the CE study sites and respondent i:
[8]First, all longitudinal and latitudinal Euclidean distances from the homes of the respondents to the study sites have to be calculated.7 These are added to the model as linear spatial trend variables, with their coefficients γ1j and γ2j reflecting the continuous relationship between individual choices and the longitudinal (Longij) and latitudinal (Latij) distance from the respondents to the sites, respectively. In other words, they reflect a linear effect from the east to the west, or the north to the south, of the site.
Next, following Cameron (2006), the site- specific constants αj of equation [1] are allowed to differ by direction and vary systematically, with θij reflecting the angle (measured in radians) between the study site and the respondent using sine and cosine variables. The resulting site-specific distance-decay function is presented in the following equation:
[9]The sine and cosine intercept shifters 
enable marginal WTP to differ between east and west (and north and south) from the site following the angle (direction) between the study and respondent locations, independent of distance.
Finally, Model III includes the variables reflecting distance to substitutes DistSubij similar to Model II. We are also interested in differences in spatial preferences between users and nonusers, so we again interact the six components of the multidirectional distance effect with the user dummy Userij The distance-decay function in Model III is specified as follows:
[10]where 
is the parameter for distance Distij, δ1jand δ2j are the parameters for the intercept shifters cosθij and sinθij for nonusers, and δ3j and δ4j are corresponding increments to these coefficients that only apply to users. Likewise, γ1j and γ2j are the parameters for the longitudinal and latitudinal distances for nonusers and γ3j and γ4j are the related increments for users.
Our null hypothesis 
is that there are no significant multidirectional distance effects among users and nonusers:
[11]Furthermore, we hypothesize that the variables for road distance to the substitute sites DistSubij as specified in Model II will become insignificant in Model III, because the spatial variation in distance-decay caused by distance to substitute sites is now captured by the spatial trend variables:
[12]IV. Case Study Area and CE Design
A novel site-selection CE was developed to analyze the spatial distribution of WTP values for the study sites and their substitutes. Three sites in the Dutch Scheldt estuary, located in the southwestern part of the Netherlands, were chosen and included as labeled alternatives in the CE: Breskens, Braakman, and Saeftinghe.8 These three sites represent three typical water bodies in the area. They provide different types of water-based recreation opportunities and are well known among local residents. A map of the area is presented in Figure 1. Breskens is a popular beach site attracting local, national, and international visitors. Braakman is the mouth of a small river, suitable for family recreation. Saeftinghe is a tidal mudflat with restricted visitor access to preserve its natural conditions and without any swimming possibilities, unlike the other two sites. There are a number of other water bodies in the area providing similar opportunities to recreate and enjoy nature.
Map of the Study Area
The general objective of the CE was to estimate public WTP for changes in ecosystem services provided by the study sites. The quality improvement scenarios in the CE are based on the ecological standards that must be achieved by the year 2015 under the European Water Framework Directive (WFD). The ecological quality at the study sites currently does not meet WFD standards due to high concentrations of nutrients and heavy metals and geo-morphological changes in the natural conditions of the sites. Achieving the WFD standards is expected to generate considerable public use and nonuse values for ecosystem services (Brouwer 2008).
In the CE, the WFD ecological objectives were translated into three easily understandable attributes, reflecting the main use and nonuse components that were expected to benefit respondents: recreational walking, recreational swimming, and nature appreciation. Each attribute was specified at three ecological quality levels: the current level, termed bad, a moderate level, and the good level. The last level corresponds with the WFD objective of good ecological status. Table 1 gives an overview of the attributes and their levels. The biophysical outcomes of achieving improved quality depend on the local characteristics of the water bodies and will differ across the three study sites. Therefore, site-specific photographs and texts were used in the survey to explain the attributes. For instance, different bird and fish species were used across the sites. The fourth attribute in the CE is a monetary attribute, expressed as an increase in the local water agency’s annual rates that are paid by all local households. This payment vehicle was chosen after careful pretesting. In previous SP studies focusing on water quality changes in the Netherlands, the use of the water agency’s rates as a payment vehicle was preferred and did not result in increased status quo bias (Brouwer 2006). The six levels of the price attribute ranged from Ĩ5 toĨ80 per year. The final experimental design consisted of 24 choice set versions with five choice tasks each, based on a D-efficient design that minimized the correlation among attributes and alternatives.
Overview of Attributes and Levels
The CE started with an explanation of the choice task, including an overview of all attributes and levels, the current situation at each of the three sites, and an example of a choice task. Next, five different choice cards were presented to every respondent. Figure 2 presents an example of a choice card. Each choice card presented four alternatives: the three sites improving in at least one attribute against a certain payment, and the "opt-out." The opt-out was defined in terms of the current quality levels. These were equated with "bad" for all attributes at all sites at zero price, except for the current swimming quality conditions at Breskens, which were equated with "moderate" quality. The survey text emphasized that the money paid for the chosen alternative would be used only for improvements at the chosen site and not for improvements at the other sites.
Example Choice Card
Respondents were asked if they had visited the three sites in the CE before and, after each choice task, whether they would visit the chosen site under the improved scenario. The follow-up question allows us to examine whether option values also drive the decision-making process. If future use considerations are negligible, distance-decay effects are expected to be lower for nonusers than for users.
Besides possible substitution between the three sites in the CE, detailed questions about other sites were included prior to the start of the CE. These questions were designed to encourage people to think actively about the alternatives in their relevant choice set before expressing their preferences for the three sites in the CE. Respondents were asked to list other frequently visited sites (other than those in the CE), the activities they undertake at these locations, and what they would do if their most frequently visited site among the three CE sites would face a decrease in water quality such that it could no longer provide the same recreational amenities.
In addition to the set of alternative sites identified by respondents, two researcher-assigned selection strategies were used to identify relevant sites for inclusion in the WTP models to test for possible substitution or complementarity effects. These site-specific sets were restricted to locations within the sampling area. These two strategies resulted in different sets for each study site:
A strict selection of sites, using as selection criteria the nature types identified in the province’s nature policies and the water types identified in the WFD. These sets included sites that could provide the same environmental goods and services as the sites included in the CE if all attributes were at "good" WFD objectives levels. This approach resulted in three substitute sites for Breskens, three for Braakman, and five for Saeftinghe.
A broad selection of sites that are either expected to meet the 2006 Bathing Water Directive or protected under Natura 2000 legislation. The sets under this selection strategy included all official coastal swimming water locations for Breskens, all inland swimming locations for Braakman, and all Natura 2000 sites for Saef- tinghe.
The identification and acknowledgment in policy documents of the sites included in these sets is considered to be indicative of their high quality, and they are therefore expected to provide relevant substitute amenities for the alternatives in the choice set of the CE. The researcher-assigned and respondent-reported sets of alternative locations do not entirely overlap. One of the reasons for the differences between the sets of alternative sites is that respondents often visit smaller sites that are not listed in official directives or policy documents on which the researcher-assigned sets are based.
It should be noted that the relationship between the study sites and the possible alternative locations depends on the preferences of the respondents. Both complementarity and substitutability effects between sites are possible (Hailu, Adamowicz, and Boxall 2000). In Model II, we include the distance to the nearest alternative providing similar ecosystem services in the utility specification for each of the three study sites, based on the two different strategies for selecting substitutes as specified above. Based on economic theory, we expect that respondents living closer to an alternative site that provides amenities similar to a study site will assign a lower value for that study site as the relative abundance of these amenities increases. This substitution effect would be reflected in a positive effect on WTP of distance to substitutes. A negative relationship between the distance to the nearest alternative and the WTP for a study site indicates complementarity effects. The inclusion of distance to the nearest option of the full set of respondent-selected alternatives in the model provides a test of the importance of a wider range of alternative sites relevant to the respondents, compared with the researcher-assigned substitute sets.
Whereas the researcher-assigned alternatives for Braakman and Saeftinghe are distributed over the entire sampling area, alternative beach locations for Breskens are available only along the coastline. Given this spatial pattern of beach availability, we expect that the directional effects of distance-decay in Model III will be especially significant for Breskens and indicate lower WTP-values along the coastline. Distance-decay rates among respondents living along the coastline are expected to be different from those living farther inland in the eastern part of the study area.
To maximize the variance in distance across the sample, we employed a spatial sampling strategy, which reflects the population distribution in the sampling area and at the same time ensures that there exists sufficient variation in distance and substitute availability across the sample. The sample included respondents from 46 towns and villages located at different distances around the three sites. Based on respondents’ home addresses, we specified the shortest travel distance to the three sites and other substitutes using geographical information systems (GIS). Travel distance was calculated based on the existing road network, as Euclidean distances were not expected to accurately reflect real travel distances because natural barriers, such as waterways, limit straight-line travel routes. The most important physical barrier is formed by the estuary, which divides the study area into a northern and southern part and can be crossed only by a tunnel (see Figure 1). The final paper-and-pencil survey was implemented via face-to-face interviews between July and September 2007.
V. Results
Sample Characteristics and Perception
Almost 800 households participated in the survey, and the response rate was 34%.9 After data cleaning, the analysis was based on 780 completed questionnaires. There are more female (61%) than male respondents. The average age of the respondents is 51. The average household size is 2.7, which is slightly higher than the Dutch average of 2.3. Disposable household income in the sample is, on average, Ĩ2,143 per month, which is close to the provincial average level. Most respondents have completed secondary education (comparable to senior high school). Table 2 presents the descriptive statistics for the sample.
Descriptive Statistics of the Sample Related to Sociodemographic Characteristics,Recreation, and Distances
As expected, given the abundance of water in the study area, a majority of respondents (96%) recreate at open water sites.10 The three sites in the CE have never been visited by 146 respondents (19%), while all three sites have been visited at least once by 156 respondents (20%). Less than 4% of respondents had never heard of any of the locations, and 85% had heard of all three locations prior to the survey. Breskens is the most popular site: 60% of the respondents have visited this location at least once, compared with 44% for Braakman and 40% for Saeftinghe.
A number of results stress the importance of including substitution effects. Respondents reported substitution behavior in terms of sites and activities. If the quality at their most frequented site decreased to such low levels that their most preferred activity would no longer be possible, 40% would go to another location. Another 20% would not engage in any water-based recreation at all, and 30% would continue going to their preferred site; more than half of those would switch to another activity. Three-quarters of the respondents visit other sites besides the three in the CE and travel on average 23 km to get to their preferred alternative site.
CE Results
Estimation results for our three models are presented in Table 3, where the results for different subgroups of variables are labeled AG. In Model I, we specify three site-specific distance-decay functions for the three study sites in set J, in Model II, we include the distance to substitute sites not included in the CE, and Model III controls for any remaining additional directional heterogeneity in distance-decay by applying the spatial expansion method. In all models, we test for differences in spatial heterogeneity in WTP between users and nonusers. Starting from the full model specifications in equations [2], [5], and [10], we produce and present models including only statistically significant nonzero explanatory factors by excluding variables with persistently statistically insignificant coefficients until all remaining variables are significant at the 5% significance level. Models including only significant nonzero variable coefficients can be used more safely for function transfer over space to aggregate WTP estimates over the relevant nonsurveyed population. The results in Table 3 are based on a broad exploration of alternative forms of the distance-decay function and represent the reduced best-fitting specification for each site.
Model Results
Part A in Table 3 presents the estimates of the site-specific constants for Breskens (Bres), Braakman (Braa), and Saeftinghe (Saef). As expected, Price (Pijt) presented in part B of Table 3, has a negative effect on choice probabilities, and income (Hj Yi) has a positive effect. Higher household income increases the probability of choosing one of the improvement alternatives for which respondents are willing to pay extra. In general, all site attributes (Xijt), presented in part C of Table 3, bear coefficients that are significant at the 1% level. The attribute effects all have the expected signs: walking, swimming, and nature quality improvements have a positive effect. We find site-specific values for swimming quality being improved to a good level: improved swimming quality at Braak- man has a significantly higher value than at Breskens 
. Similarly, improving nature quality to a good level at Saeftinghe is valued significantly higher than at Breskens and Braakman, for which the estimated values are not significantly different 
.
As can be seen at the bottom of Table 3, in part G, the significant standard deviation of the error component (λij Hj) confirms that the three hypothetical alternatives were perceived to be significantly different from the status quo. Moreover, for the user dummy variable, equal to one for those respondents who have visited the sites in the past, the standard deviation of the random parameter with zero mean (μi), is significant in all three models. The significant effect implies that there is significant heterogeneity in the sample with respect to the effect of previous visits to the three sites.
Part D in Table 3 presents the results of the distance-decay and related user effects. In Model I, we specified three site-specific distance-decay functions for the study sites, interacted with the user dummy. We estimated different distance-decay effects based on linear, quadratic, loglinear, and exponential functional forms and present the statistically best-fitting results. We find significantly different best-fit distance-decay effects for all sites, hence rejecting the first hypothesis. A model with a generic instead of site-specific specification of the distance-decay effect also resulted in significant coefficients for distance and the user dummy, but the model fit was much lower. The variation in distance-decay and related user effects can be explained by the differences across the amenities provided by the sites under the hypothetical scenarios.
A quadratic distance-decay function for Breskens gives the best model fit, implying a distance-decay pattern where WTP decreases at an increasing rate the further away the respondent lives 
. Previous visits to the site have a positive effect on the probability of choosing Breskens 
. However, there is no significant difference in the distance effect between users and nonusers, and we fail to reject our second hypothesis for this site. We expect that this result is because nonusers of Breskens hold a high share of option val- ues,11 which are also expected to decrease as distance increases for site-specific improvement scenarios. Also for Braakman, previous visits increase the probability of choosing this alternative 
. Distance-decay is found to be significant only for users, as reflected by the interaction term for distance and the dummy representing previous visits to the site 
, rejecting in this case the second hypothesis. Here, the distance-decay effect is linear. The distance-decay function for Saef- tinghe yields the best fit when specified as a logarithmic function 
. This indicates that WTP values decline quickly within a short range of the site and stabilize at larger distances for all respondents. We furthermore find a significant positive effect of the interaction between distance and the user dummy 
, implying that the positive effect of previous visits increases WTP at greater distances from Saeftinghe, also rejecting the second hypothesis for this site. In other words, WTP of users declines more slowly with distance compared with that of nonusers. The higher WTP among current users is likely to reflect a positive WTP for use values related to the ecological quality improvements, as reflected in the attributes.
Model II accounts for additional spatial heterogeneity due to differences in substitute accessibility, measured as the road distance from the respondent’s place of residence to the nearest researcher-assigned or respondent-reported substitute. The results are presented in part E of Table 3. The third hypothesis 
can be rejected for Breskens only where the researcher-assigned broad selection of alternative beach locations for Breskens has a significant effect on choice probabilities, namely, the distance to the nearest beach under the Bathing Water Directive. The positive coefficient of this variable 
indicates that as distance to other beaches increases, so does the probability of choosing Breskens. This result implies substitution between Breskens and other beaches under the Bathing Water Directive, and there is no difference between users and nonusers. The significant substitution effect is an additional indicator of the theoretical validity of our results.
For the other researcher-assigned and respondent-reported substitution sites, measured and tested through the minimum and average distances to these sites, no significant effects can be found. One of the reasons for the insignificance of the respondent-reported substitute sets may be the low variation in distances to the substitute sites reported by respondents. Due to the many water bodies in the study area, most respondents have an alternative site available in close proximity to their homes. The likelihood ratio test,12Bayesian information criterion (BIC), and Akaike information criterion (AIC) indicate that Model II has a better fit than Model I. Accounting for these substitutes improves the predictive ability of the model.
Model III includes longitude and latitude variables and intercept shifters, thereby controlling for additional directional heterogeneity in distance-decay, as specified in equation [10]. The results presented in part F of Table 3 for Model III are based on the reduced model that was estimated after stepwise exclusion of the variables that were found to be insignificant in the full model (the full model is included in the Appendix). Four variables of the spatial expansion method are found to be significant in the reduced Model III. Moreover, accounting for directional heterogeneity significantly affects the distance-decay to substitute sites outside the CE. The impact of distance to substitutes for Breskens remains significant, but only for users, partly rejecting hypothesis 
.
In Model III, significant directional heterogeneity in the choice probabilities of two of the three sites is found, most prominently for Breskens. These directional effects are additional to the direct effects on WTP of travel distance to the CE sites and distance to substitute sites outside the CE. The two significant intercept shifters in the function for Breskens reflect how WTP varies with the angle between the site and respondent locations. The cosine intercept shifter 
in the function for Breskens has the strongest positive effect east of the site, where cos(0°) = 1. This means that respondents living east of the site are more likely to choose Breskens, compared with respondents located in other directions from the site. Similarly, the negative sine intercept shifter 
in the function for Breskens implies lower WTP values north of the site, where sin(90°) = 1, and higher WTP values south of the site,where sin(270° = — 1). The longitudinal distance effect 
is significant among users of Breskens and implies that the farther away these users live, east of Breskens, the less likely they are to choose the site, which partially offsets the effect of the cosine intercept shifter for users. The overall effect is that people living northwest of Breskens have a lower probability of choosing Breskens, whereas respondents, especially nonusers, living southeast of the site are more likely to choose the Breskens alternative and derive higher utility from improvements at this site. This result can be explained by the presence of the many beaches in the northwestern part of the sampling area along the coastline and the long travel distances faced by respondents living in the north.13
There is also some directional heterogeneity in the distance-decay of Saeftinghe. Only the cosine intercept shifter 
for users is significant and negative. The intercept shifter shows that users living west of the site, where the cosine equals — 1, have a higher probability of choosing Saeftinghe. No significant directional effects were found for Braakman.14
The results of Model II in part E show that the alternative beaches in the northwestern area included in the researcher-assigned substitute set cause systematic spatial variation in the WTP for Breskens, which is controlled for by including the distance to the nearest beach from this researcher-assigned substitute set in the model. However, the significance of the directional effects in Model III suggests that there are other variables besides distance to substitutes causing spatial heterogeneity in distance-decay. The results furthermore show that the effect of the distance to substitutes remains significant among users of Breskens 
. The positive coefficient implies that as users of Breskens live closer to another beach site, they are less likely to choose Bres- kens. The coefficients of the attributes are similar across the three models. Therefore, compared with Models I and II, we have no reason to believe that the variables for the directional variability in choice probabilities of Model III are picking up any other nonrandom spatial distribution of explanatory variables. Despite the fact that Model III has a lower log- likelihood and the spatial trend variables are significant according to their z-statistics, the overall fit of Model III is not necessarily better than of Model II according to the AIC and BIC statistics. We therefore also conducted the Ben-Akiva and Swait (1986) test for nonnested choice models. This test indicates that Model III outperforms Model II. The probability that the goodness of fit measure of Model II is greater than that of Model III is virtually zero with 
, Φ() being the standard normal cumulative distribution function.
WTP Results
An important question is whether accounting for the additional spatial heterogeneity and distance to substitutes outside the CE choice set in Models II and III leads to significant differences in WTP results, compared with the use of site-specific distance-decay functions of Model I. We use the model results to estimate mean WTP for achieving good ecological quality (i.e., all attribute levels at "good") at Breskens for user and nonuser groups in the four-digit postal code areas in the study area. The centroids of the postal code areas are used to calculate the road distance to Breskens and substitute sites.
Comparison of the WTP results for Bres- kens of Model II with those of Model I shows that accounting for road distance to the nearest possible substitutes results in differences between Models I and II ranging from — 25% to +25% for users and — 35% to +40% for nonusers. When we also account for directional heterogeneity by including spatial trends as in Model III, the differences in WTP compared with those of Model I vary from -22% to + 30% for users and — 42% to + 31% for nonusers. Nonparametric Kolmo- gorov-Smirnov tests show that, compared with Model I, Models II and III lead to significantly differently distributed WTP values for both users and nonusers at the 5% level.15Additional tests as described by Poe, Giraud, and Loomis (2005) show that these differences are significant for some, but not all, postal code areas.
Overall, the results indicate that accounting for directional heterogeneity by including spatial trends and distance to substitutes in the deterministic part of the WTP function, as in Models II and III, results in significantly different mean WTP estimates. This provides a clear rationale for accounting for distance to substitutes and spatial heterogeneity in the model specification.
Finally, there are differences in WTP per postal code area between Models II and III, varying from — 19% to + 20% for users and 18%to +29% for nonusers. Figure 3 depicts the mean individual WTP estimates based on Models II (left) and III (right) for achieving good ecological quality at Breskens held by users. The spatial heterogeneity displayed in Figure 3 is caused by distance-decay and substitution effects and (only in Model III) by spatial trend variables. From the differences in the grayscales in the maps, it is immediately clear that the two models do not have the same predictions and result in a different spatial pattern of mean WTP. The maps show that Model III results in more spatial variation in WTP estimates. Compared with Model III, failure to account for directional heterogeneity in Model II overestimates the mean WTP in the northeast and northwest, whereas mean WTP is underestimated in the central southern part.
Maps of Individual WTP (in 
VI. Conclusions and Recommendations
Substitution and distance-decay effects in SP studies have long been ignored or addressed in a relatively simple manner, either by including a dummy variable for the presence of substitutes or by including a single uniform distance variable in the WTP model. Existing studies capture none or only part of the spatial variation in WTP values caused by the nonrandom spatial distribution of substitutes, for instance, across the different directions from the study site. This is expected to diminish the reliability of existing WTP results from SP studies involving spatially defined environmental improvements and their use in welfare aggregation procedures.
In the study presented in this paper, spatial heterogeneity in SPs and WTP for environmental quality improvements at three different sites in a confined geographical area has been tested based on the functional form of site-specific distance-decay functions, controlling for user, substitution, and directional effects. Hence this paper presents the most inclusive set of tests for spatial heterogeneity in the literature on CEs for environmental valuation so far. The results of the CE show that distance-decay effects differ among sites in magnitude and functional specification, and between users and nonusers. These differences are attributed to site characteristics on the one hand, such as the function of sites and the type of ecosystem goods and services they provide, and respondent characteristics on the other, such as recreational use, familiarity with the sites, and option values held by nonusers. An option for future research is to use a latent class model for further analysis of differences between subgroups of respondents, which may also reveal classes of future, rather than just past, users and nonusers.
To reveal additional spatial variation in distance-decay patterns, we included the distance to the nearest potential substitute sites as an additional explanatory variable in the choice model. To our knowledge this is the first CE study in the environmental valuation literature that controls for distance to substitutes in the WTP function. We used a selection of both respondent-reported and researcher-assigned sites. Only the latter appear to have a significant effect, and only on the choice probability of one of the three CE sites, namely, the beach site, which is affected by the presence of other beach sites. Disregarding these substitution effects leads to under- and overestimations of individual WTP values ranging between - 35% and +40%.
In addition to the distance to substitutes, for which the selection process of relevant substitute sites may require a considerable amount of research time, we proposed to use the spatial expansion method. When extending a model that already contains variables reflecting the distance to substitutes with spatial trend variables, the results show that there is some remaining spatial heterogeneity. We find significant coefficients for the directional effects in the choice probabilities, thereby reducing imprecision in distance-decay effects, and suggesting that including distance to the nearest potential substitutes is not sufficient to capture all the spatial variation in distance- decay. On the other hand, applying the spatial expansion method is also insufficient to capture all spatial heterogeneity in WTP resulting from the effect of substitute availability. The inclusion of spatial trend variables and distance to substitutes leads to a significantly different spatial distribution of WTP estimates.
Finally, it should be noted that we accounted only for the effect of alternative sites providing similar environmental goods and services on choice behavior. Further research is needed to determine whether the availability of other recreational amenities, including sites of historical or cultural interest, or locations outside the study area, also affect WTP for water quality changes. The main conclusion is that distance-decay effects are not uniform as generally assumed and modeled in the limited number of studies found in the literature. Instead, they are shown to vary across sites, respondents, and directions and to depend on the accessibility of substitute sites. We recommend that future valuation studies of spatially defined environmental goods test for and, where necessary, take this spatial variation into account in the specification of distance-decay functions to produce more reliable WTP results and market delineation procedures.
Acknowledgments
This study was carried out and financed under the EU DG Research project AquaMoney (SSPI- 022723). See www.aquamoney.org. The authors wishto thank Silvia Ferrini and two anonymous reviewers for their constructive comments.
Appendix
Table A1 presents the results of a full specification of Model III. The four variables of the spatial expansion method that are significant in the reduced Model III version are denoted with a superscript "a" in the table. As can be seen from the comparison of these results with those of the reduced model presented in the paper, the parameter estimates of the attribute levels remain the same. Only one(γ3j) of the four significant variables in the reduced Model III is also significant in the full model, but the remaining three others have higher t-statistics than most of the other variables. These become significant at the 5% level when the variables with the lowestt-statistics are excluded. Without the random coefficient on the user dummy variable, the four variables of the spatial expansion method that are significant in the reduced model are also significant in the full model. In general, one would not expect all spatial trend variables to be statistically significant, but only those that mark a particular spatial pattern in choices and WTP.
Results of the Full Specification of Model III
Footnotes
All authors are at the Department of Environmental Economics, Institute for Environmental Studies, VU University, Amsterdam; the first, third, and fifth authors are senior researcher, the second and fourth full professor; the first author is currently senior research associate at CSERGE, School of Environmental Sciences, University of East Anglia, U.K.; the fourth author is also affiliated with Faculty of Economics and Business Administration, VU University, Amsterdam, ICREA, Barcelona, and Institute of Environmental Science and Technology, Universitat Autònoma de Barcelona.
↵1 As suggested by an anonymous reviewer, it should be noted that individual heterogeneity in WTP will create different market sizes for different respondent profiles, also if this heterogeneity is not spatially dependent. These other respondent-specific factors in WTP, which cause an upward or downward shift of the WTP-function, could be used to determine the markets for different respondent profiles.
↵2 Hensher and Greene (2002) and Train (2003) describe of the specification of mixed logit models and the maximum simulated likelihood procedure used to estimate these models.
↵3 No constant was specified in the model for the status quo (the fourth choice option), because the model included three site-specific alternative specific constants. It is beyond the scope of this paper to further specify and test the status quo option as a function of respondents’ perceptions and socio-demographic characteristics.
↵4 We use road distance to the site (rather than Euclidean "as the crow flies" distance) measured from the home location, assuming that the home location is the most relevant point of reference for the respondents. The respondents’ self-reported estimates of travel distance were highly correlated with the GIS-based travel distances based on home outset locations. We cannot control for the distance from other outset points, such as the work location, because we do not have the necessary information. Considerations about different outset zones are expected to be less of a concern in this CE where the price attribute is used to estimate the WTP for the quality attributes and non-use values are expected to play an important role too, compared with a travel cost study where the exact computation of travel distance is more crucial as it is used as a proxy for price.
↵5 Anonymous reviewers have suggested that this dummy-variable may lead to endogeneity due to correlation in the error term between the user dummy and the choice outcome related to unobserved preferences. We argue that the user-dummy, which does not reflect the number of visits, may be related to utility derived from the site under its current quality levels, but not to hypothetical future conditions. However, the WTP models reflect utility derived from future hypothetical quality improvements and WTP estimates therefore reflect the utility associated with a departure from the current state, which may include non-use values as well. We therefore argue that there is no circular relationship between utility for future site changes and past visitation behavior. We also refer to previous WTP studies on environmental changes which include user-variables (for recent examples, see Kataria et al. 2012; Moore, Provencher, and Bishop 2011; Carlsson and Kataria 2008). Endogeneity is a topic that has received relatively little attention in the discrete choice literature (Louviere et al. 2005), let alone in CE studies for environmental valuation (Hoyos 2010). Excluding the user-variable from the models would lead to omitted variable bias, including in the parameters of the spatial trend variables in Model III. We include a random coefficient on the user-variable, following Train (2003). It is beyond the scope of the paper to explore instrumental variables to control for endogeneity.
↵6 An extended practical description of the application of the spatial expansion method in hedonic pricing studies as proposed by Cameron (2006) can be found in Section 3 in the working paper version (Cameron 2003).
↵7 Longitudinal and latitudinal distance differences are based on (x,y) coordinates in decimal degrees, and a conversion factor is used to express these distances in kilometers (see Cameron 2003). Unlike Cameron (2006), we keep distance based on the road network for the main distance- decay effect of the variable Dist, and only the longitudinal and latitudinal distance differences are expressed in Euclidean (straight line) distances. We also do not convert these distance differences into trigonometric terms.
↵8 The number of alternatives in the CE was, after pretesting, set to three to limit the cognitive burden of having to choose between too many different sites.
↵9 This is a typical response rate for Dutch valuation studies based on face-to-face interviews (Brouwer 2006). The main reasons for non-participation were lack of time and unwillingness to participate in surveys in general. There was no opportunity for further investigation of nonresponse bias. The introductory text of the survey was neutral and unrelated to environmental issues and the sample was representative, and therefore we have no reason to believe that any particular subgroup in society is under or over-represented in our sample.
↵10 Open water was explicitly defined in the questionnaire to respondents as outdoor water bodies, including the sea, lakes, streams, rivers, canals, ponds, etc.
↵11 We asked respondents, in a follow-up question included after each choice task, whether they would visit the site in the future under the proposed improved quality level. The results indicate that almost half of the current non-users would consider visiting the site if it improved.
↵12 LR-test statistic = 2 X (3,342 — 3,331) = 22 >v(p = 0.05, df = 1) = 3.84.
↵13 We have no reason to suspect that the latitudinal and longitudinal variables reflect travelling constraints. We are using road distance instead of simple straight-line Euclidean distance, so travel barriers such as waterways and tunnels are reflected in the distance variable. The tunnel between the northern and southern part of the study area (see Figure 1) is not known for congestion. Including travel time instead of distance reduced the model fit and was therefore not used here.
↵14 It is interesting to note that in a model, specified with a generic linear distance-decay for the three study sites and the same user X distance interactions, leaves more spatial heterogeneity unaccounted for than a model specification with different functional forms for distance-decay. This can be shown by applying the spatial expansion method. In a model with a generic distance-decay specification, more spatial expansion parameters become significant when added to the model compared to the site-specific specification. As argued by Cameron (2006), ignoring spatial heterogeneity will result in biased distance-decay estimates. These results hence argue for defining site-specific distance-decay functions. The generic distance-decay model results are available from the authors upon request.
↵15 Test results are available from the authors upon request.
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