Aggregation Bias in Recreation Site Choice
Models: Resolving the Resolution Problem
M. K. Haener, P. C. Boxall, W. L. Adamowicz,
and D. H. Kuhnke
ABSTRACT. This paper examines the effect of
differing levels of spatial resolution on recreation
site choice models and welfare resulting from
changes in site attributes. These issues are important where the spatial scale at which recreationists
make choices is unknown, but information exists
on choice attributes at larger spatial scales. We
estimate choice models at various scales of spatial
resolution and incorporate the size of the aggregate
sites and heterogeneity parameters in the model.
Accounting for the size of the aggregates in estimation improved model fit and alleviated aggregate
parameter bias. We provide advice for applied
modeling based on these results. (JEL Q26)
I. INTRODUCTION
Geographic information systems (GIS)
technology and improvements in computer power have allowed land managers
to maintain large amounts of digital spatial
information at various levels of resolution.
Since this information is increasingly available, scientists have been developing ecological models that utilize data at fine spatial scales that more appropriately correlate
with the scale of ecological processes. These
advancements can be a double-edged sword
for economists who wish to develop ecological-economic models. Improvements in spatial databases mean that there are richer data
available for economic models that utilize
landscape characteristics to explain behavior. However, available socio-economic data
are often only available or feasible to collect
at much higher levels of spatial resolution.
The result is that the level of resolution or
scale in ecological and economic models may
be quite different, which makes integrated
Land Economics • November 2004 • 80 (4): 561–574
ISSN 0023-7639; E-ISSN 1543-8325
 2004 by the Board of Regents of the
University of Wisconsin System
analysis challenging. Resulting interrelationships between ecological and economic processes are becoming difficult to understand.
We explore the possibility of using results
from aggregate economic models in applications with finer resolutions of spatial scale
by examining the relationship between models at various scales.
Understanding the impact of aggregation is also important for understanding
spatial choice behavior. If researchers only
observe choices from the aggregate of spatial alternatives (coarse scale), when in fact
individuals are making choices based on
fine scales, they may produce biased measures of choice and economic welfare. In
this paper, we examine models at various
spatial scales and evaluate approaches to
deal with aggregation issues in practice.
An option that may be available to researchers is to use a very fine resolution for
the economic analysis. However, such an approach is often impractical. Researchers
must usually aggregate fine resolution alternatives due to data or computational limitations. Researchers who have investigated the
implications of this practice (Parsons and
Needelman 1992) found that aggregation
can lead to biased coefficients and welfare
measures. They found that “aggregation
bias” can be partially alleviated by accounting for the degree of aggregation in
The authors are, respectively, policy analyst with the
Ministry of Aboriginal Affairs, Government of the
Northwest Territories, Canada; professor in the Department of Rural Economy, University of Alberta, Canada;
Canada Research Chair and professor in the Department of Rural Economy, University of Alberta, Canada;
and forest analyst with the Canadian Forest Service
in Alberta, Canada. The authors thank Weldwood of
Canada Limited (Hinton Division), the Alberta Department of Sustainable Resource Development, and the
Canadian Forest Service for assisting with this study.
562
Land Economics
the estimation of parameters. However,
these previous studies have not fully investigated the issue of aggregation and heterogeneity and particularly the role of error
variance in modeling aggregation.
Given these findings, a relevant question is whether scale or resolution really
matters or whether models across scales
can be modified appropriately. Perhaps it
is possible to use recreation site choice
models estimated at a coarse level of resolution to represent choice behavior at finer
scales if one can account for aggregation
in estimation. If this is the case, is this type
of calibration practical, and what are the
limitations of calibrating economic information to predict behavior at these finer
scales? While we examine these questions
in a recreation demand framework, it is
important to recognize that these issues
affect all economic models that have a spatial component.
In this paper, we investigate the influence of the scale of aggregation on recreation site choice models, specifically the
logit specification of the random utility
model (RUM). Using a common set of recreation site choice data, we compare models estimated at several levels of aggregation. We extend previous work by formally
testing whether model coefficients become
biased when alternatives in the choice set
of the model are aggregated. To do this, we
test whether coefficient vectors are jointly
equal within a factor of proportionality using
procedures described by Swait and Louviere
(1993). We also consider whether the degree
and means of aggregation influence the resulting welfare measures for various hypothetical environmental changes. We compare several approaches to mitigating aggregation bias and offer a practical means
of calibration. The empirical application
of these procedures involves recreational
hunting, which offers some unique challenges regarding the aggregation process.
We also examine the effect of aggregation
on welfare measures and the variation in
welfare measures for key attributes of the
situation we examine.
November 2004
II. BACKGROUND AND THEORY
For a recreation site choice model to be
consistent with economic theory, the set
of alternative recreation sites should represent those elements that individuals consider when making their choice decision.
For a number of reasons the data used to
estimate choice models may not be at the
same aggregation level as the choice process. Collection of economic data at a fine
scale may be infeasible or there may be a
very large number of alternatives in an
individual’s choice set. The latter makes
model estimation computationally intractable (Kanaroglou and Ferguson 1996).
For these reasons, researchers often aggregate sites and model behavior at a coarser
scale. Several papers have shown that unless very restrictive conditions hold, the
practice of aggregating sites leads to “aggregation bias” which influences model coefficients and corresponding welfare measures (Kanarogolou and Ferguson 1996;
Parsons and Needelman 1992).
The conditions under which aggregation
bias is likely to be significant and the appropriate means for correcting this bias
have been reported in the literature. Initial
theoretical work dates back to McFadden
(1978). More recent work includes research by Ferguson and Kanaroglou (1998,
1997, 1995). These studies show that incorporating additional terms in the utility
function can circumvent aggregation bias,
at least theoretically. Ferguson and Kanaroglou (1997) refer to the corrected model
as the aggregated spatial choice model. We
refer to the logit specification as the aggregated spatial logit (ASL) model. Ferguson
and Kanaroglou (1995) derive the generalized form of the aggregated spatial choice
model in the context of inter-regional migration. A similar, but less general, derivation is given in Ben-Akiva and Lerman
(1985).
Following Ferguson and Kanaroglou
(1998) and Kanaroglou and Ferguson (1996)
assume that the landscape area of interest
is divided into elemental alternatives among
which N choice makers with known origins
are selecting alternative destinations. The
80(4)
Haener et al.: Aggregation Bias in Recreation Site Choice Models
Aggregate J of L=6 zones.
�
Elementj of M'=4 elements in Aggregate J.
FIGURE 1
Illustration of Indexing Variables in the
Aggregated Spatial Logit Model
the perspective of the researcher and can be
expressed as: UjJ ⫽ VjJ ⫹ εjJ ⫹ εj, where VjJ
represents the systematic portion of utility
determined by attributes of aggregate J and
element j; εjJ is the unobserved variation relating to the elemental unit j 僆 {1, 2, …, MJ}
and εJ represents the unobserved variation
related to the aggregate unit J. εjJ is assumed
to be a Gumbel distributed variate with parameters (0, ␮J) and [εjJ ⫹ εJ ] is also assumed
to be Gumbel distributed with parameters
(0, ␮). Thus, Var[εjJ ] ⫽ ␲2/6 (␮J)2 and Var
[εjJ ⫹ εJ ] ⫽ Var[ε jJ] ⫹ Var[εJ ] ⫽ ␲2/6 ␮2
(Kanaroglou and Ferguson 1996).
Kanaroglou and Ferguson (1996, 275)
utilize this information to show that the
utilities of any elemental alternatives that
are members of the same aggregate have
the same covariance (and are correlated)
but that elements belonging to different
aggregates are uncorrelated. The correlation of any two elements in an aggregate is
related to the ratio ␮/␮J which Kanaroglou
and Ferguson show can be represented by
␮
⫽
␮J
elemental alternatives are combined into
L aggregate zones. Each aggregate, J 僆 {1,
2, . . . , L}, contains MJ elemental alternaL
tives (J indexing aggregates) with 兺 MJ ⫽
J⫽1
M, the total number of elemental alternatives in the landscape. The set of elemental
alternatives defines the finest resolution
on which choice-makers base their selection, but the researcher can only observe
which of the aggregates a choice maker
selects. In other words, the researcher can
only observe the aggregate, J, to which
the selected elemental alternative belongs.
Figure 1 provides an illustration of this situation.
The utility that a representative individual choice maker receives when visiting
element j within aggregate J is denoted
UjJ. Following conventional economic theory, the individual is assumed to select the
elemental alternative that provides maximum utility, and as a result, also selects the
aggregate zone to which that chosen elemental alternative belongs. UjJ is random from
563
Var(ε ) .
冪Var(ε
⫹ ε)
J
j
J
J
j
[1]
This ratio is represented by parameter ␭J
which can be aggregate specific and has
a theoretical range of 0 ⱕ ␭J ⱕ 1. Thus,
when the elemental error occupies all possible variance Var[εjJ ] → ␲2/6, Var[εJ ] → 0,
and ␭J → 1. This case would suggest that the
systematic utilities of the elemental alternatives within aggregate J are uncorrelated.
Conversely when Var[εjJ ] → 0, Var[εJ ] →
␲2/6, then ␭J → 0. suggesting that the elemental utilities are highly positively correlated.
Individuals are assumed to be utility
maximizers and compare the aggregate alternatives on the basis of the utilities of
their elemental members. Thus, when selecting a destination, choice makers consider the elements with highest utility in
each aggregate and choose the aggregate
on the basis of the maximum elemental
utility it can offer. Since these utilities are
random variables and are conditional on
being chosen, they can only be compared
on the basis of their expected value VJ ⫽
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Land Economics
E关maxj僆{1,2, . . . ,MJ}UjJ兴. Following Kanaroglou
and Ferguson (1996) and Parsons and
Needelman (1992) the probability the choice
maker selects aggregate J is
P(J) ⫽
exp ␮VJ
, J 僆 [1,2, . . . , L} ,
L
兺 exp[␮V
[2]
K
]
K⫽1
where ␮ is a scale parameter. If one assumes that ␮ ⫽ 1, and that the systematic
utilities are assumed to represent the average characteristics of all the elements in J,
then this formulation resembles the typical
conditional logit model merely applied to
the choice of aggregate spatial units. This
model would represent the typical case in
much of the empirical literature on recreational hunting site choice (Adamowicz et
al. 1997; Boxall et al. 1996).
However, since the aggegate zone consists of elemental alternatives which may
be the “true” object of the choice makers
attention, the systematic component of the
utility function is actually the maximum
of the expected utility of the elemental
alternatives in the aggregate or
J
1 M
ln 兺 exp(␮JVJj ) ,
␮J j⫽1
[3]
where ␮J is related to Var[εjJ]. Kitamura,
Kostyniuk, and Ting (1979) showed that
the degree of heterogeneity in the elemental attributes in aggregates was also required in the utility function, along with
measures of average attributes and the effect of aggregate size. Kanaroglou and Ferguson (1996) and Parsons and Needelman
(1992) suggested that the systematic utility
can also be represented by the following
expression:
VJ ⫽ VJ ⫹
⫹
冤
冢
冣冥
1 M
␮J
␮
ln J 兺 exp 共VJj ⫺ VJ兲
J
␮
M j⫽1
␮
␮
lnMJ .
␮J
J
[4]
This equation reveals that the utility associated with choosing an aggregate consists
of three components. The first term, Vj is
the mean utility effect that is a function of
November 2004
the mean elemental attribute levels in the
aggregate. The second term is the heterogeneity effect that captures the variability
of the systematic utility in the aggregate.
␮
The final term, J lnMJ, is a function of the
␮
number of elemental alternatives in the
aggregate zone and is called the size effect.
In the case of the conditional logit
model ␮ or the scale of the utilities is not
identifiable and for convenience can be set
to 1. This allows the ratio of the strictly
elemental unobserved variance and total
unobserved variance, ␭J J 僆 {1, 2, . . . , L},
to be estimated.
Formally examining the degree of heterogeneity of sites within aggregate alternatives requires estimating the ASL model,
which can be represented by substituting
␭J for ␮/␮J in [4]:
VJ ⫽ VJ ⫹ ␭J ln
⫹ ␭J lnMJ .
冤
J
冢
冣冥
1 M
1
exp J共VJj ⫺ VJ兲
J兺
M j⫽1
␭
[5]
Estimation of the ␭J parameter will provide
an indication of the relative importance
of within aggregate heterogeneity. As ␭J
approaches zero, the heterogeneity term
in [5] attains its highest value and the size
effect diminishes. The unobserved variation unique to elements within the aggregate also approaches zero. Therefore, the
element with maximum utility in the aggregate can be identified with near certainty.
As ␭J approaches one the size effect will
tend to dominate the heterogeneity effect
with elemental systematic utilities not being well distinguished. In this latter case,
random error makes identification of the
element with the highest utility difficult.
To our knowledge, only Ferguson and
Kanaroglou (1998) estimated the ASL
model with the heterogeneity term. Parsons and Needelman (1992) incorporated
information about aggregation (number of
elemental sites and variability of the sites)
as parameter estimates in a nested logit
framework formed over the elemental alternatives. They do not estimate heterogeneity terms as described above. Estimation
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Haener et al.: Aggregation Bias in Recreation Site Choice Models
of the ASL model is computationally difficult and requires knowledge of attributes
at a fine scale of resolution. Advances in
software and computing power, however,
make estimation of the ASL model an option for cases where choice data are only
available at an aggregate level, but landscape data may be available at a finer scale.
The research focus in this paper is the
comparison of choice models using unaggregated or elemental alternatives and aggregated alternatives. In the empirical application examined here, recreational big
game hunting, this is an important issue.
In most cases, the site choices of hunters
are collected at very large aggregate spatial
levels for administrative convenience. For
example, in Alberta the Wildlife Management Unit (WMU) is used to regulate
hunting activity and frequently represents
the finest level of choice resolution available to researchers interested in hunting
site choice. Past studies on the choice behavior of hunters have used this scale of
resolution (Adamowicz et al. 1997; Boxall
et al. 1996). However, WMUs in Alberta
can be greater than 8,000 km2 in size. It is
difficult to believe that hunters consider
attributes of WMUs when selecting hunting sites, but more likely there are much
smaller areas within WMUs that are being
chosen on the basis of their attributes.
These smaller areas are influenced by industrial activity such as forestry, which affects wildlife habitat and human access levels. The forest industry typically collects
spatial information on forest and access
characteristics at finer levels of resolution
than the WMU. This information could
be more useful in developing hunting site
choice models because of its degree of resolution and accuracy.
In this paper, the ASL model will be used
to determine the importance of accounting
for heterogeneity in aggregate models of recreation site choice. In practice one would
not be required to use a model like the
ASL model if one knew they had the elemental alternatives. However, in most
cases researchers do not know if they have
the elemental alternatives, or they know
that they are using aggregated alternatives
565
and wonder what bias the aggregation is
introducing. Our approach of using the elemental alternatives as a test of performance of the ASL model versus aggregate
models provides insight into the degree of
bias and the options for reducing bias.
We will also assess the influence of aggregation by comparing the parameter
vectors of elemental and aggregate models
that do not account for heterogeneity, but
do account for the size effect. Parsons and
Needelman (1992) examined aggregation
bias in recreation site choice models, but
parameter vectors were only compared
qualitatively for directional or magnitude
differences. This qualitative comparison
was paired with the comparison of welfare
measures to determine whether the aggregation of choice alternatives (elements) led
to biased coefficients. Although directional
differences between significant coefficients
are a concern, differences in magnitude
could be attributable to scale (error variance) differences between the models. In
comparing the models in this paper we use
formal tests of whether model coefficients
become biased when alternatives in the
choice set are aggregated. We utilize the
test developed by Swait and Louviere (1993)
to determine whether coefficient vectors are
jointly equal within a factor of scale.
Finally, we also compare the marginal values of attribute changes calculated using the
parameter vectors from elemental and aggregate models. Using these methods, we
consider whether the degree and means of
aggregation influence aggregation bias.
III. DATA
The Empirical Case Study
This study examines recreational hunting in the Weldwood Forest Management
Agreement (FMA) Area in Alberta, Canada. This area consists of one million hectares, and is divided into 147 compartments
(Figure 2). Each compartment within this
industrial forest leasehold contains a high
proportion of similar forest stand types,
based on uniformity in either age or species composition. The compartment is the
566
Land Economics
o
November 2004
Compartment boundaries
WMU boundaries
10
0
--
10 Kilometers
FIGURE 2
A Map of the Forest Management Agreement Area in West Central Alberta
Used to Examine Recreational Hunting in a Spatial Context
smallest administrative unit used by the
forest company in its landscape planning.
Compartments vary widely in size, ranging
from 75 ha to 23,400 ha, with a mean size
of 7,291 ha. This level of resolution is the
finest digitally available and seemed to researchers to be reasonable as the base or
elemental level at which to accurately model
the choice of hunting sites. Recreational
hunting is permitted in 139 of the 147 compartments. Some of the compartments are
provincial parks, coal mines, and other lands
not open to hunting.
Other administrative boundaries can also
be used to segment the FMA into larger
regions.1 For example, the land base is also
intersected by the boundaries of 9 different
WMUs which are used by provincial wildlife
managers to set hunting regulations. Previous studies of hunting in this area used surveys of licensed big game hunters (McFarlane, Boxall, and Adamowicz 1998), which
elicited trip information from sampled hunters using the WMU boundaries. The results
from these trip logs could be used in estimating models of recreation site choice; how1
The forest company, for example, divides the landbase
into five large regions called working circles (WC).
These WCs consist of groupings of the compartments.
80(4)
Haener et al.: Aggregation Bias in Recreation Site Choice Models
567
ever, there is concern that this spatial scheme
may not be appropriate for depicting the
actual choice behavior of hunters. In order
to test this concern another survey was
administered to a convenience sample to
collect trip data on a finer spatial scale.
The compartment level formed the basis
for the collection of this additional data.
A survey was developed which asked respondents to mark the location of all hunting
trips taken between 1996 and 1998 in the
FMA on a pair of customized 1:405 000 scale
maps.2 Information concerning each trip was
requested including: the year the trip was
taken, the species hunted, the number of
other trips taken in the same general location during the survey period, and the number of years that the respondent hunted in
the area prior to 1996. Respondents were
also asked the total number of days spent
in the same general location across all trips
during the survey period (all day trips were
rounded up to one day). Socioeconomic information was also collected from respondents including age, nature of their rural or
urban area of residence, whether their primary livelihood came from forestry, and
their household income level.
Following two focus groups in which the
survey was improved, the final version was
mailed to 400 individuals who were members of fish and game clubs located in two
Alberta towns located near or in the FMA
(Edson and Hinton), and additional hunters whom we knew visited the area previously, but did not reside in the FMA.
About 72% of the sample resided in the
Hinton and Edson, and the remainder came
from other areas of the province.
The overall survey response rate was
64%, of which 45% (117 respondents) fully
completed the survey.3 Respondents who
hunted took an average of 3.6 trips each.
This resulted in a sample of 425 trips. Ninetysix of the 139 compartments were visited at
least once and at most 19 times. Elk was
most commonly indicated as the target species followed by deer and moose. For 26 of
the 425 trips the target species was either
bear or bighorn sheep. These trips were not
included in the analysis since bear and sheep
hunting differs considerably from elk, deer,
and moose hunting. Therefore, information
on a total of 399 trips was available for
analysis.
Trip locations were transferred to digital maps of the FMA and their location
with respect to a compartment recorded.
Distances from respondents’ places of residence to the center of each compartment
were determined with the aid of a geographic information system (GIS). These
distances were converted to travel costs
using $0.48/km as the out of pocket vehicle
expenses, and the value of time calculated
at an average speed of 100 km/h and one
third of the wage rate calculated using the
respondent’s reported income (the same
approach used by Parsons and Needelman 1992).
The forest company provided spatial data
consisting of roads, seismic lines, clear-cuts
and other man-made patches (polygons),
and spatially explicit habitat suitability indices for several ungulate game species based
on relevant characteristics of forest stands
and man-made patches in each compartment. Spatial information on forest fires and
additional road spatial information were
provided by the provincial forest management agency. From these data we were able
to generate a variety of site attribute variables.
2
Because the study was based on a mail-out survey,
the number of maps that could accompany each survey
had to be pared to a minimum to present the respondent
with a survey that does not appear too daunting. Concerns over the confidentiality of the location of preferred
or favorite hunting sites were also clearly voiced during
focus groups. Maps that accompanied the survey therefore had to be of a relatively small scale.
3
A high number of responses were unusable because
the marked location of hunting trips covered an area too
large to be usable in this study (these were usually day
trips) or because the marked locations of hunting trips
could not be matched with the requested trip details
(they were to have been marked with accompanying trip
numbers). In addition, 87 respondents did not hunt at all
during the survey period or hunted outside the FMF.
568
Land Economics
November 2004
TABLE 1
Description of Choice Sets and Aggregation Schemes Used to Examine
Recreational Big Game Hunting in Alberta
Aggregation
Scheme
ELEM
20AG
10AG/WMU
10AG/EVEN
Choice Set Description
96 compartments
96 compartments aggregated into 20 zones
96 compartments aggregated into 10 zones using Wildlife
Management Unit boundaries
96 compartments aggregated into 10 zones of relatively
equal area
IV. METHODS
We used the various regulatory and
planning boundaries to develop different
aggregation schemes. The “base case”
used to compare aggregation schemes is
the compartment level model in which a
choice set of the 96 units was used. We
believe this model is as close to the “true”
elemental model in terms of representing
site choice attributes and refer to this
model as the elemental model (ELEM, Table 1).
The set of models compared to the
ELEM model involved combining compartments into various aggregate units.
These cases all result in smaller choice sets.
The first was the aggregation of compartments into 20 different spatial units. To
do this, WMU and forest planning borders
were used to define the boundaries for the
aggregate units. This model is labeled 20AG
(Table 1). These 20 units were then further
aggregated into 10 units using the WMU
borders as a guide (the largest WMU was
divided into 2 units). This model is labeled
10AG/WMU. Since this latter scheme led
to aggregates with large variation in the
number of compartments in each aggregate (see Table 1), another 10-unit aggregation scheme (called 10AG/EVEN) was
created that grouped compartments more
equally.4
4
An additional aggregation scheme was also attempted in which the Working Circle boundaries were
used to group compartments into five units. The results
of the models estimated using this scheme are not reported here but are available from the authors upon
request.
Choice
Set Size
Range of Compartments per
Aggregate Unit (min.–max.)
96
20
10
2–7
1–21
10
7–13
The base case or ELEM model involved
a choice set of all the 96 compartments
visited by the hunters in the sample. The
specification with the largest value of the
log likelihood at convergence and ␳2 was
chosen as the preferred model. This specification was also used to estimate all of the
aggregate models. Thus, the choice models
included travel costs (COST), a quadratic
travel cost term (QCOST), road density
(ROAD), presence or absence of a forest
fire in the last 2 years (HBURN), and thousands of elk habitat units (ELKHU) in the
indirect utility function. Many other site
attribute variables thought to be important
determinants of site choice were correlated with these variables and could not
be included in the utility function (i.e. the
density of seismic access disturbances).
This model specification is based on the
elemental model; however, we feel that
this is consistent with our assumption that
the elemental model is the “true” model. In
addition, as we shall see in the results, even
with this specification choice, some aggregate models preformed well, strengthening
the case for their use.
First, conditional logit models with no
consideration of aggregation bias were estimated for each aggregation scheme. Formal
tests were then conducted of whether the
coefficients of these simple aggregate models were significantly different from those of
the ELEM model when scale differences are
taken into account. This test involved joint
estimation of the ELEM data and the other
data from the aggregate of interest. The
estimation restricted the coefficient vector
80(4)
Haener et al.: Aggregation Bias in Recreation Site Choice Models
of the ELEM model and the respective
aggregate model to be equal within a factor
of proportionality (see Swait and Louviere
1993). If the parameter vectors were found
to be significantly different, there is indication of aggregation bias.
Kaoru, Smith, and Liu (1995) and Parsons and Needelman (1992) included a size
variable for the aggregate units in the utility function and found that it could be used
to mitigate aggregation bias. However,
choice of the variable used to represent
the effect of size can be an issue. In the case
of fishing site choice examined by Parsons
and Needelman (1992) for example, the
number of lakes in each aggregate was
used as the size variable. In the hunting
case examined in this paper the number
of compartments in an aggregate would be
a similar representation of size. In theory,
however, the larger the spatial size of the
choice alternative the higher the chance it
would be selected. Thus, for hunting site
choice the natural logarithm of the actual
area of each aggregate (LNAREA) in each
aggregation scheme was used to represent
the size effect.5 Including the size effect in
the choice models represents a simplified
version of the ASL model that sets the
heterogeneity term equal to zero. We formally test whether the coefficients of the aggregate model specifications with LNAREA
are significantly different from two ELEM
specifications (one with and one without
LNAREA) when scale differences are accounted for.
As a final comparison, ASL models
(with the heterogeneity term) were estimated for each aggregation scheme. As
discussed above, the ASL model includes
an additional piece in the likelihood function that utilizes elemental characteristics
to account for the influence of spatial het5
Models were also estimated where the number of
compartments in each aggregate represented the size
effect. While the results are not substantially different
from using the area, it seems to make more sense to
use the area in this empirical example rather than the
number of elemental units in an aggregate. The parameter estimates and other results from using the number
of elements in each aggregate are available from the
authors upon request.
569
erogeneity on choice. This term includes
␭J, but the ␮ term in the numerator of this
parameter is not identifiable whereas the
denominator ␮J is. Thus, ␮ was set equal
to 1.0 and the ␮J terms were free to be
estimated. We attempted to estimate models that allowed for ␭J to be different for
each aggregate, or to be equal across subsets of aggregates. However, these models
were difficult to estimate and in many
cases did not converge. As a pratical solution for comparative purposes, ␭J was constrained to be equal across all aggregates
in each aggregation scheme. Since this coefficient (labeled ␭ hereafter) represents
the positive correlation among elemental
characteristics in each aggregate, exp(␪) ⫽
␮J ⫽ (1/␭) was estimated in the ASL models
in the maximum likelihood function. Note
that the results presented in Table 2 include estimates of the parameter actually
estimated (exp(␪), estimated as such to
guarantee a positive variance), the translation of this estimated parameter into an estimate of ␮J, and the inversion of ␮J into ␭.
V. RESULTS AND DISCUSSION
Comparison of Model Parameters
Table 2 presents parameter estimates
for the various choice models estimated
with the elemental and three aggregate datasets. With few exceptions, the parameters are statistically significant and consistent with a priori expectations. The first
sets of models in each spatial scheme (the
model 1s in Table 2) do not include any
spatial aggregation variables. The parameter estimates suggest that the probability
of a hunter visiting a site increases with
decreasing travel costs, decreasing road
density, and increasing elk habitat. Although not significant in all models, the
likelihood of visiting a site also increases
if the site has not burned in the last two
years as evidenced by the negative sign
on HBURN.
Model 2 in the elemental and aggregation schemes in Table 2 includes the
LNAREA parameter. This parameter is
significant and positive in all the models.
⫺1.8657
(⫺8.329)
0.1580
(3.337)
⫺1.9133
(⫺5.130)
⫺0.2490
(⫺2.128)
0.5277
(6.528)
0.6753
(7.437)
Model 2
⫺1.4014
(⫺6.223)
0.0829
(1.516)
⫺0.0929
(⫺0.108)
⫺0.2710
(⫺1.363)
0.7131
(5.628)
Model 1
⫺1.9477
(⫺7.556)
0.1213
(2.357)
⫺2.2313
(⫺3.218)
⫺0.6558
(⫺3.003)
0.7112
(5.711)
0.8919
(5.593)
Model 2
⫺1744.05
⫺1712.46
⫺1117.94
⫺1096.06
0.0423
0.0567
0.0647
0.0830
⫺1.6891
(⫺7.752)
0.1271
(2.722)
⫺1.4981
(⫺4.358)
⫺0.0751
(⫺0.6052)
0.4929
(0.6443)
Model 1
20AG
0.8639
(2.261)
2.3724
0.4215
⫺1106.97
⫺1.9571
(⫺7.705)
0.1166
(2.302)
⫺1.0236
(⫺1.816)
⫺0.3086
(⫺1.020)
0.4224
(3.707)
ASL
⫺760.62
0.1721
⫺1.8744
(⫺6.459)
0.1085
(1.890)
⫺1.3076
(⫺1.524)
⫺0.7689
(⫺3.410)
1.8114
(8.72)
Model 1
⫺743.66
0.1906
⫺2.2693
(⫺7.052)
0.1308
(2.239)
⫺3.2171
(⫺3.499)
⫺0.5601
(⫺2.279)
0.9690
(3.969)
0.9777
(5.535)
Model 2
10AG/WMU
0.2020
(1.392)
1.2238
0.8171
⫺744.23
⫺2.2834
(⫺7.106)
0.1219
(2.169)
⫺3.3450
(⫺3.417)
0.0316
(0.093)
0.6246
(3.755)
ASL
⫺854.07
0.0704
⫺1.8197
(⫺6.228)
0.0875
(1.591)
⫺4.0589
(⫺4.369)
0.1666
(0.690)
0.9099
(5.967)
Model 1
⫺838.86
0.0870
⫺2.5881
(⫺7.569)
0.1590
(2.868)
⫺3.9567
(⫺4.211)
⫺1.0172
(⫺3.077)
0.6936
(4.155)
1.1894
(5.328)
Model 2
10AG/EVEN
⫺0.2561
(⫺1.261)
0.7741
1.2919
⫺850.87
⫺1.8888
(⫺6.645)
0.0759
(1.390)
⫺2.0649
(⫺2.621)
0.1332
(0.503)
0.3024
(2.107)
ASL
Note: In the ASL models the size and heterogeneity terms generate one additional parameter. This parameter is estimated by exp(␪) to constrain it to be positive, and is then
transformed into the corresponding values of ␮J and ␭. See the text for details regarding the relationship between ␮J , ␭ and exp(␪).
␮J
␭
Log L.
␳2
Exp(␪)
LnAREA
ELKHU
HBURN
ROADD
COST 2
COST
Variable
ELEM
TABLE 2
Maximum Likelihood Estimates of Parameters (t-statistics) for Choice Models Using Various Spatial Configurations
570
Land Economics
November 2004
80(4)
Haener et al.: Aggregation Bias in Recreation Site Choice Models
571
TABLE 3
Results of Log Likelihood Ratio Tests for Equality of Parameter Vectors Between
the ELEM Models and the Aggregation Models
Aggregation
Scheme
⌺ LogL
Joint
LogL
␹2
Scale (p-Value)
Relative to ELEM
ELEM (model 1) vs.
Aggregation schemes
(model 1)
20AG
10AG/WMU
10AG/EVEN
⫺2861.99
⫺2504.67
⫺2598.12
⫺2869.35
⫺2520.10
⫺2601.37
14.72
30.86
6.48
1.1774 (0.0952)
2.3788 (0.0000)
1.2587 (0.0482)
ELEM (model 1) vs.
Aggregation schemes
(model 2)
20AG
10AG/WMU
10AG/EVEN
⫺2840.11
⫺2487.71
⫺2582.91
⫺2842.81
⫺2490.09
⫺2588.43
5.38
4.75
11.04
1.2753 (0.0234)
1.5653 (0.0007)
1.4260 (0.0043)
ELEM (model 2) vs.
Aggregation schemes
(model 2)
20AG
10AG/WMU
10AG/EVEN
⫺2808.52
⫺2456.12
⫺2551.32
⫺2810.14
⫺2457.46
⫺2554.80
3.24
2.68
6.96
1.2164 (0.0385)
1.4861 (0.0001)
1.3856 (0.0044)
Model Comparisons
Note: Gray shading indicates results suggesting that the parameter vectors are equal at the 5% level of significance (critical ␹2 ⫽
9.49 (4 df) and 11.07 (5 df)).
The addition of this variable appears to
adjust the cost and site quality coefficients.
The fact that LNAREA is significant for
the ELEM model suggests that compartments may not represent the unit of choice
for the respondents in this sample and that
the LNAREA variable may be correcting
this bias. Thus, even in the ELEM model
the inclusion of LNAREA may reduce aggregation bias. The 20AG and 10AG/
EVEN models with LNAREA show considerable improvement in the value of the
log likelihood at convergence and in ␳2.
One would expect model fit to improve
as the degree of aggregation increases and
choice set size decreases because a smaller
choice set increases the chances of predicting choices correctly. However, based
on ␳2 the best model is 10AG/WMU; otherwise, the pattern of improved fit with increasing aggregation is displayed. One possible explanation for this observation is
that compared to the other aggregation
schemes, the WMU aggregation scheme
captures some of the influence of the
WMU administrative units used to establish provincial hunting regulations on the
individual choice process.
Table 3 reports the results of testing
whether the parameter vectors of the each
ELEM models are equivalent to those of
the relevant aggregate models when scale
differences are taken into account. The first
comparison was among model 1 for each aggregation scheme. The 10AG/EVEN model
is the only aggregate model that passes the
scale test, suggesting that its parameter vector is not significantly different from the corresponding ELEM parameter vector. It is
noteworthy that the model with the best fit,
10AG/WMU, failed this test. This result suggests that for the 10AG/WMU model, aggregation influenced model coefficients and can
be interpreted as evidence of aggregation
bias.
The second set of parameter vector comparisons involves the aggregation scheme
in which LNAREA has been added to correct for possible aggregation bias. In this
comparison, both the 20AG and 10AG/
WMU parameter vectors are not significantly different than the ELEM specification in which LNAREA was not included
(ELEM model 1). It would appear that
adding LNAREA to these models had an
influence on the bias resulting from aggregating compartments. Even for the 10AG/
EVEN-aggregation scheme, the ␹2 value
of the likelihood ratio test is not far from
the level of insignificance.
The last set of parameter vector comparisons involves ELEM model 2 in which
LNAREA was included. In these cases all
three of the aggregate models are not significantly different than the corresponding
ELEM model. This information suggests
572
Land Economics
that when LNAREA is included in the estimation of the aggregate model the results
conform to those of the ELEM model. If
this particular ELEM model represents
the “truth,” then the inclusion of the area
effect appears to mitigate bias caused by
aggregating elemental alternatives.
For the comparisons in Table 3 where
the hypothesis of parameter equality was
accepted, the scale parameters can be compared among the models with confidence
(Swait and Louviere 1993). One would expect the scale (error variance) to increase
(decrease) steadily as the level of aggregation increases. This is generally seen in Table 3 where the estimated scale parameter
is larger for larger aggregation schemes.
To consider whether the aggregate models can be improved further by incorporating
the influence of elemental heterogeneity,
ASL models were estimated at each level
of aggregation. These results are reported
in the columns labeled ASL in Table 2.
Although one would expect these ASL
models to perform better than their less
complicated counterparts, the ASL models
display little or no improvement in the fit
in each aggregation scheme (Table 2). This
suggests that the size effect dominates the
heterogeneity effect in these data. The ␭
parameters are all positive and it is interesting to note that they suggest a pattern
of correlation among elemental attributes
that seems consistent with the degree of
aggregation. For example, in the 20AG
model ␭ is 0.4215 suggesting that the correlation among the characteristics of the elements is not strong. In the 10AG/WMU
model, however, this correlation increases
to 0.8171, which is expected given the
greater degree of aggregation (from 20 to
10 units). Finally in the 10AG/EVEN
model ␭ is 1.2919 and this value is not
significantly different than 1.0. This increase in the correlation of elemental characteristics in the aggregates should be expected given that the aggregation units are
more equal in area than those of the 10AG/
WMU scheme (see Table 1).
These results are promising for researchers since it appears that including a variable
that accounts for aggregate size can in part
November 2004
mitigate aggregation bias. It also suggests
that a model estimated at a coarse level of
resolution could be calibrated for use at finer
spatial scales. In this empirical example,
however, it was assumed that the highest
level of resolution for which we have data
(compartments) is representative of the
choice process of the respondents in our
sample. The choice of compartments provided the units used in determination of the
size variable (LNAREA). This assumption
may not be valid because it is unknown if
compartments actually represent the elements of choice considered by the individual
hunters. This problem is common for researchers where choice data at the aggregate
level are available and there is uncertainty
about the resolution at which the choice process occurs. This issue is pertinent for activities like hunting where single trips are taken
to areas that are large and may contain many
kinds of spatial configurations. In cases
where this issue arises, it appears that including the area of the aggregate rather than the
number of “elements” is the proper approach to account for size bias in the aggregation exercise.
Implications of Aggregation on
Welfare Measures
It is also important to consider the influence of aggregation on estimates of welfare measures. The marginal values of attribute changes were calculated using each
models’ coefficients and the corresponding
variance-covariance matrix. To attain a distribution of marginal values, 1,000 draws
from the variance-covariance matrix were
taken. Figure 3 displays the mean and 95%
confidence intervals (displayed as error
bars) of the marginal values of a unit change
in road density, forest burns in the last two
years, and elk habitat units. The figures show
that the confidence intervals of marginal values generated from the 20AG, 10AG/
WMU, 10AG/EVEN aggregate models
overlap with the ELEM model with
LNAREA. The marginal values are most
similar for road density and elk habitat
units. The confidence intervals for recent
burns still overlap, but the mean marginal
value varies from model to model.
80(4)
Haener et al.: Aggregation Bias in Recreation Site Choice Models
573
Forest Area Burned
Road Density
0.0
0.0
-0.1
-0.5
-1.0
-0.2
4--j---
-0.3
+---t---
-0.4
-1.5
-0.5
t---------t-----t---�F___;
-0.6
t---
-t----
----
-t-___;
-----
-0.7
-2.0
-0.8
-0.9
-2.5
t-------�"'--___;
t-------___;
---'
-1.0 "20AG
ELEM
10AG/wmu
10AG/even
-3.0
B.EM
20AG
1OAGIw rnu
10AGIeven
Elk Habitat Units
1.0
,------,
0.9
t---------
0.8
+----
--------
--
----
1 ----
----\
---
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
ELEM
20AG
10AG/wmu
10AG/even
FIGURE 3
Comparisons of the Marginal Values of Changes in Three Attributes across the Four Choice
Models. Shaded Bars Represent the Mean Value and the Error Bars the 95%
Confidence Interval from 1,000 Draws from the Coefficient Distributions
VI. CONCLUSION
Although there may be exceptions, our
findings suggest that for developing site
choice models of recreational hunting, the
size of the choice alternative in spatial context matters. If researchers are able to account for the size of aggregates in estimation, model parameters will be equivalent
across scales. The inclusion of a size variable improves model fit and appears to
alleviate aggregation bias allowing model
parameters to be equivalent (within a factor of proportionality) across different spa-
tial resolutions. These results are consistent with previous studies that also find
that aggregation bias can largely be eliminated by including a size term in the utility
function (Parsons and Needelman 1992).
However, what comprises this size term
has not heretofore been a topic of consideration. Other recreation researchers incorporate size by including the number of
elemental alternatives aggregated in each
aggregate choice alternative.
Our results are supported by more rigorous tests than previous studies. We formally test for differences in parameter vec-
574
Land Economics
tors between elemental level and aggregate
models. This test is more reliable than the
qualitative comparisons used in previous
studies since it accounts for possible scale
differences. As recommended by Parsons
and Needelman (1992), we also consider
the influence of heterogeneity between
sites (as opposed to assuming homogeneity) in aggregate models. The inclusion of a
heterogeneity term did not provide significant improvement over models which just
include a size term. Welfare measures are
also compared in a more comprehensive
manner. What remains to be examined,
however, is how well these corrected aggregate models predict behavior out of
sample (Haener, Boxall, and Adamowicz
2001). This is an important topic for future
research. Furthermore, we did not model
the frequency of trips or the correlation of
trip choices over each individual. These
provide additional avenues for extension
of our approach.
These findings are of importance to researchers who wish to integrate recreation
site choice models with ecological models
developed at finer spatial scales. It appears
that recreation site choice models can be
‘calibrated’ to finer spatial scales simply
by including a size or area term in their
estimation. For our data, site heterogeneity did not play a significant role. This may
not hold true in other regions or for other
activities. In cases where heterogeneity is
likely to be important, it may be necessary
to account for this in estimation through
use of the ASL model. An important topic
for future research will be the identification of empirical cases where this heterogeneity effect is important.
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