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\begin{document}

\title{Relating Environmental Attitudes and Contingent Values:
How Robust Are Methods for Identifying Preference Heterogeneity?\footnote{This paper has benefited from 
comments provided by participants at the Canadian Resource and Environmental Economics Conference and workshops at Iowa State, NC State, UC Santa Barbara, and the University of Colorado.  We are grateful to 
James Stroud for assistance with Python.}}

\begin{comment}
\author{Gwendolyn A. Aldrich \\
{\small {University of New Mexico} }\\
%{\small {aldrichg@unm.edu}} 
\and Kristine Grimsrud \\
{\small {University of New Mexico}}
\\
%{\small {grimsrud@unm.edu}} 
\and Jennifer Thacher\footnote{Corresponding author: Department of Economics,
University of New Mexico, MSC05 3060, Albuquerque NM 87131; 
jthacher@unm.edu; 505-277-1965. 
}
\\
{\small {University of New Mexico}}\\
%{\small {jthacher@unm.edu}} 
\and Matthew J. Kotchen \\
{\small {University of California - Santa Barbara}}
\\
%{\small {kotchen@bren.ucsb.edu}}
}

\end{comment}
\maketitle

\newpage

\begin{abstract}
We assess the importance and robustness of cluster analysis and latent class analysis as methods to account for unobserved heterogeneity.  We provide a critique and comparison of both methods in the context of measuring environmental attitudes and a contingent valuation study involving endangered species.  We find strong evidence of robustness for these methods: group characterization and assignment of individuals to groups are similar between methods, and
willingness to pay estimates are
consistent. 
In addition, there are 
significant differences in willingness to pay across environmental attitudinal
groups, and we find that accounting for unobservable 
heterogeneity provides a significantly better fitting model.
\newline
\textbf{Keywords:} cluster analysis, contingent valuation, latent class 
analysis,  New Ecological Paradigm, unobservable heterogeneity, willingness to pay

\end{abstract}

\section{Introduction}

Economists are increasingly concerned with methods of identifying groups with 
homogeneous intra-group characteristics, as refining and targeting policy 
analysis often requires sorting individuals into different groups.  We use a 
unique survey data set containing attitudinal and willingness to pay (WTP) responses 
to investigate the comparative performance of two of the most commonly used 
techniques for identifying groups with heterogeneous inter-group 
characteristics and homogeneous intra-group characteristics: cluster 
analysis (CA) and latent class analysis (LCA).  Many behavioral sciences, such as 
marketing, psychology, and sociology, frequently rely on CA and LCA for data analyses.  
The methods are often applied to attitudinal or more 
general psychographic data.\footnote{Psychographic data includes information 
on personality traits, personal values, and lifestyle \citep{WedelKamakura2000}.}
In economics, applications of CA and LCA are few, especially applications 
to psychographic data.\footnote{CA applications include identification of 
market segments \citep{BakerBurnham2001}, examination of relationships between 
farmers' behavioral attitudes and their use of futures contracts 
\citep{PenningsLeuthold2000}, and assessment of the convergence of countries' 
per capita productivity levels \citep{HobijnFranses2000}. LCA has been used 
to identify motivations for wilderness recreation \citep{BoxallAdamowicz2002}, 
preference structure in rock climbers \citep{ScarpaThiene2005}, preferences 
regarding fishing characteristics \citep{MoreyEtAl2006}, and preferences 
for medical treatments \citep{ThacherEtAl2005}.}   This scarcity is surprising given the perceived importance of such data for understanding consumers in, for example, 
marketing.   The reason for the scarcity is unclear.  There may be the 
perception that heterogeneity is best accounted for by adding socioeconomic 
covariates in econometric models where the original economic model assumes 
all agents are identical. Furthermore, attitudinal and personal value data 
are sometimes criticized due to a concern that such data are 
unreliable predictors of behavior.\footnote{Interestingly, 
\cite{BakerBurnham2001} found that sociodemographic variables performed 
``little better than flipping a coin'' (p. 396), while a model with only 
cognitive variables performed well for predicting genetically modified 
organism (GMO) acceptance.}  Finally,
 CA and LCA may be regarded as prone to a great deal of subjectivity in the 
various steps of statistical analysis. In light of these concerns and criticisms 
we see a need to examine the potential of CA and LCA to contribute to our knowledge 
about the heterogeneity of economic agents' attitudes and preferences. 

In this paper we address several questions that are pertinent to the 
continued use and acceptance of CA and LCA techniques 
within economics.  First, are CA and LCA robust in the sense that they 
yield consistent results?  Second, does accounting for unobservable attitudinal 
heterogeneity add explanatory power?  Finally, what insights can be provided 
regarding a choice between the use of CA or LCA?  

We address these questions within the context of a contingent valuation 
(CV) study designed to estimate WTP for 
recovery for two endangered species, the peregrine
falcon and the shortnose sturgeon. 
A key feature of the survey is the
inclusion of attitudinal questions that comprise the New Ecological Paradigm
(NEP) Scale.  The NEP scale (originally proposed by 
\citet{DunlapVanLiere1978} and later revised by \citet{DunlapEtAl2000}) is 
designed to measure the strength of environmental attitudes. We apply both CA 
and LCA to the NEP data in order to identify heterogeneity of
environmental attitudes among survey respondents. We use the results
to partition respondents into different groups for the purpose of explaining
heterogeneity in WTP responses.\footnote{Environmental attitudes can arguably 
be modeled as either endogenous or exogenous. In this paper we choose to 
treat attitudes as exogenous; our treatment of environmental attitudes is 
similar to including environmental group membership as an independent variable.}

Our main findings are the following: We find strong evidence of convergence between 
CA and LCA.  Attitudinal groups identified using CA and LCA are similar in that there 
is consistency in the way the two methods assign individuals to attitudinal groups.  
Econometric models of WTP responses
are consistent for both methods.  Econometric 
models of WTP responses that include attitudinal group assignments have greater 
explanatory power than models that use only socioeconomic variables to capture 
preference heterogeneity.  Estimates of WTP for species recovery efforts derived 
using CA results are consistent with those derived using LCA results, and in 
both cases stronger pro-environmental attitudes increase the estimates of 
mean WTP.  Finally, CA may be the preferred method when groups are identified 
based upon a large number of variables or when the time available to learn a 
new technique is limited.  LCA may be the preferred method when the researcher 
desires more detailed output on predicted behavior 
and the ability to test the validity of results using a host of commonly used 
statistical tests.

The next section provides background information on the NEP and describes the 
data used for the analysis. Section 
\ref{sec:MeasuringAtt} provides an overview of CA and LCA in the context of 
identifying heterogeneity in environmental attitudes using the NEP. 
Section \ref{sec:Consistency} evaluates the consistency of results derived using 
the two methods for identifying attitudinal groups. Section \ref{sec:WTPnAtt} uses 
the CA and LCA results to investigate
the relationship between environmental attitudes and CV estimates. Section 
\ref{sec:Discuss} discusses the main results and concludes.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Data} \label{sec:Data}

The NEP is one of the instruments most commonly used by
social scientists to measure environmental attitudes, it has been used for several
decades, and the validity of its construction has been repeatedly confirmed.\footnote{The 
predictive, known-group, criterion, content, and construct validity of
the NEP have been demonstrated by numerous studies. Results obtained by \citet{EbreoEtAl1999}, 
\citet{BlakeEtAl1997}, and others suggest that the NEP
has predictive validity. Examples of studies that demonstrate known-group
validity include \citet{EdgellNowell1989} and \citet{Widegren1998}. Because
the NEP has been shown to have predictive and known-group validity, it also
has criterion validity. Content validity has been shown by \citet{KemptonEtAl1995}, 
and construct validity has been supported by most
studies involving the NEP, although especially strong evidence comes from 
\citet{PierceEtAl1987} and \citet{SternEtAl1995}.}  
The NEP scale consists of $15$ Likert-scale questions from which responses are typically combined
into a summated scale, with higher scores indicating stronger
pro-environmental attitudes. Table \ref{table:NEPfreq} lists the different statements, which are designed to probe
five facets of environmental attitudes. The
different facets and corresponding statements are the following: reality of
limits to growth (1,6,11), anti-anthropocentrism (2,7,12), the fragility of
nature's balance (3,8,13), rejection of the idea that humans are exempt from
the constraints of nature (4,9,14), and the possibility of an eco-crisis or
ecological catastrophe (5,10,15).

\begin{center}
[Table \ref{table:NEPfreq} here]
\end{center}

The data we use comes from a previously published study by
\citet{KotchenReiling2000}, who use the NEP data in the
traditional manner of constructing a summated scale in order to test
theoretical validity of CV responses. Consistent with attitude-behavior
theory, they find that respondents with stronger pro-environmental attitudes
are more likely to respond `yes' to a referendum CV question about protecting
an endangered species. Our analysis differs in that we apply CA and LCA
to the NEP data and use the results to compare the two methods. We estimate CV 
values for the purpose of using the NEP (an indicator of environmental attitudes) 
to compare CA and LCA as methods for identifying heterogeneity of latent attitudes. 

The mail survey, conducted in the spring of 1997, was sent to a
random sample of 1200 Maine residents. Mailing procedures
were conducted in accordance with the \citet{Dillman1978} Total Design Method.
After adjusting for undeliverable surveys, the survey response rate was 63
percent, which is relatively high for a survey of the general population.


The survey was designed to measure environmental attitudes and estimate
nonuse values for the protection of peregrine falcons and shortnose sturgeons,
both endangered species in Maine.\footnote{The survey was not designed to address
the issue that environmental preferences may change over time. See 
\cite{LeKamaSchubert2004}.} In order to avoid
potential bias resulting from asking respondents to value more than one
species, the sample was split such that one-half received questions
about peregrines and the other half received questions about sturgeons. 
WTP questions were asked in the context of a voter referendum for the
establishment of a state-wide fund designated for the purpose of protecting
the specified species. The proposed fund was to be instituted through a
one-time payment in the form of a tax increase. 
Of the 629 completed surveys, a useable sample of 563 surveys (272
sturgeon and 291 falcon) remains for the NEP analysis after deleting underage respondents and
observations with missing values for one or more NEP statements. 

We report descriptive statistics for responses to the NEP
statements in Table \ref{table:NEPfreq}. The response frequencies reflect substantial 
environmental attitude heterogeneity within
the sample. While there appears to be a general consensus about some
statements (e.g., 3,5,7,9,13), other statements (e.g., 4 and 10) elicit responses 
that are more evenly distributed across the various responses categories. For 
instance, the majority
of respondents strongly agree with statement 7, \textquotedblleft
Plants and animals have as much right as humans to exist,\textquotedblright\
but the responses differ widely regarding statement 10, \textquotedblleft The 
so-called `ecological crisis' facing human kind has
been greatly exaggerated.\textquotedblright\ The strongest pro-environmental attitudes are
associated with statement 9, \textquotedblleft Despite our special
abilities, humans are still subject to the laws of
nature.\textquotedblright\ The weakest pro-environmental attitudes are
associated with statement 6, \textquotedblleft The earth has plenty of
natural resources if we just learn how to develop them.\textquotedblright\
The general pattern of results reported in Table \ref{table:NEPfreq} is similar 
to that found in
other studies using the NEP (e.g., \citealt{DunlapEtAl2000, CooperEtAl2004}). 

Although a variety of views were expressed by survey
respondents (respondents both strongly agreed and strongly disagreed with
all NEP statements), the response frequencies and means indicate that in
general the survey respondents agreed with the pro-environmental NEP
statements and disagreed with the weak-environmental statements. The average
respondent's attitudes fall between undecided and strong pro-environmental.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Measuring environmental attitudes} \label{sec:MeasuringAtt}

In this section we introduce CA and LCA in the context of identifying heterogeneity in environmental attitudes using the NEP.  We discuss both methods as they relate to our objective of segmenting individuals into different groups based upon differences in environmental attitudes.\footnote{Factor analysis is another analytical technique sometimes used to segment individuals into groups.  Two applications of factor analysis in a CV context are \citet{Nunes2002} and \citet{NunesSchokkaert2003}.}  

\subsection{Cluster Analysis}
The application of CA techniques to NEP data segments survey respondents into environmental attitudinal groups such that respondents in the same group have similar environmental attitudes, but their attitudes differ from those of respondents in other groups.  Although a variety of clustering methods exist and the details of the
various methods differ, each method entails the same principal steps.  Clustering algorithms are applied either to raw data, standardized data, or proximity coefficients that measure the degree of similarity or dissimilarity between two observations.\footnote{Standardizing the data is an optional first step that removes arbitrary effects that can occur due to the variables' 
units of measure, and causes variables to contribute more equally to the proximity coefficients.  See 
\citet{Romesburg1984} for a description of available standardizing 
functions. Because responses to the NEP statements are measured using a 
Likert scale, and are thus measured in dimensionless units and contribute 
equally to the
calculation of proximity coefficients, standardization is an unnecessary
step for the present analysis.} In the present analysis we use proximity coefficients in the form of Euclidean distances, one of the most commonly used of several proximity coefficients appropriate for use with ordinal data 
\citep{Romesburg1984,AldenderferBlashfield1984}.\footnote{The 
Euclidean distance measure is $d_{ij}=\sqrt{\sum_{k=1}^{n}\left(
x_{ik}-x_{jk}\right) ^{2}}$, where $d_{ij}=$ the distance between individuals $i$ and $j$, and $k$ denotes the $k^{th}$ variable.} Proximity coefficients are
calculated for the $i^{th}$ and $j^{th}$ respondents for all $i=1,\ldots ,n$
and $j=1,\ldots ,n$, and are arranged in a symmetric $n\times n$ matrix referred to
as a resemblance matrix.  Because the Euclidean distances are calculated using responses to NEP statements, the proximity coefficients provide measures of the differences in respondents' expressed environmental attitudes.

We segment survey respondents into different environmental attitudinal groups
by applying Ward's minimum variance method to the resemblance
matrix.\footnote{Although the choice of a clustering method is relatively 
arbitrary, the decision depends in part upon the proximity coefficient used, 
as some algorithms and proximity coefficients are incompatible. Other 
considerations and strengths/weaknesses of various algorithms are detailed in 
\citet{AldenderferBlashfield1984}.}  Ward's method (one of the most commonly used clustering methods in the social sciences) is what is referred to as an agglomerative clustering method. This family of clustering algorithms iteratively merges $n$ observations (respondents) 
into a single cluster in a process
of $n-1$ steps. The various agglomerative algorithms use different criteria to determine which respondents or clusters of respondents are most similar and should thus be merged into a new cluster at each of the $n-1$ steps.   Ward's method uses an error sum of 
squares criterion to determine which
respondents to merge at each stage in the clustering procedure.  At each stage the
objective is to minimize the increase in the total within-cluster error sum
of squares: 
\begin{equation} \label{eq:ESS}
\text{min}\sum\limits_{m=1}^{g}ESS_{m}=\sum\limits_{m=1}^{g}
\sum_{i=1}^{n_{m}}\sum_{k=1}^{p}\left( x_{mi,k}-\frac{1}{n_{m}}
\sum_{i=1}^{n_{m}}x_{mi,k}\right)^2,
\end{equation}
where $ESS_{m}$ denotes the error sum of squares within the $m^{th}$
cluster, $x_{mi,k}$ is the value of the $k^{th}$ NEP variable for the $
i^{th} $ individual in the $m^{th}$ cluster, and $n_m$ is the number of
individuals in the $m^{th}$ cluster \citep{AldenderferBlashfield1984, EverittEtAl2001}.  The increase in the total within-cluster error sum of squares given in equation (\ref{eq:ESS}) is proportional to the squared Euclidean distance between the centroids of the merged clusters.\footnote{Although Ward's method is similar to the centroid clustering method, the two methods differ in that Ward's method weights the clusters' centroids.}    

Numerous texts, including \citet{Romesburg1984}, \citet{AldenderferBlashfield1984}, 
\citet{JohnsonWichern1992}, and \citet{EverittEtAl2001}, provide additional information and details relevant to the various steps and decisions involved in CA.  Applications of CA methods usually entail determining the appropriate number of clusters.  A variety of texts discuss the many heuristic procedures and more formal tests developed for determining the number of clusters; see for example \citet{EverittEtAl2001} and \citet{AldenderferBlashfield1984}.  Many software packages contain procedures for conducting CA. Examples include R \citep{R}, CLUSTAN \citep{CLUSTAN}, and STATA \citep{STATA}.  We conducted CA using SAS \citep{SAS}. 



\subsection{Latent Class Analysis}

The basic intuition of LCA in the context of the NEP is that response patterns of individuals who share similar environmental
attitudes will be highly correlated, but will differ from response patterns of those who have different environmental attitudes.
LCA assumes that each individual belongs to one and only one group; 
however, because class membership cannot be observed, it is treated as if it is probabilistic.  LCA typically assumes that answers to a series of questions are independent 
once class membership has been accounted for; in other words, it is only 
class membership that causes correlation between an individual's answers.

The estimation goals of LCA are two-fold.  The first is to determine the most likely
response probabilities given the
response pattern of all respondents.  We denote this $\pi _{qs|c}$, the probability that an individual in environmental attitudinal group $c$
gives answer $s$ to attitudinal question $q$; for example, it is the
probability that someone in group $c$ answers ``strongly agree'' to the
statement ``Humans have the right to modify the natural environment to 
suit their needs.''  The second goal is to find the unconditional class probabilities given the response pattern of all individuals. We denote this $\Pr \left( c\right)$, the probability
that any individual in the sample will belong to environmental group $c$.


The $\ln $\ likelihood function for a $C$-class model for the data in our
sample is \ 
\begin{equation}
\ln L=\sum_{i=1}^{N}\ln \left[ \underset{c=1}{\overset{C}{\sum }}\Pr
(c)\prod_{q=1}^{Q}\prod_{s=1}^{S}(\pi _{qs\left| c\right. })^{x_{iqs}}%
\right] \text{\label{eqlnl},}
\end{equation}
where $x_{iqs}$ is a dummy variable that reflects whether individual $i$
chose answer $s$ on question $q$. The objective is to find the 
values of $\Pr(c)$ and $\pi_{qs|c}$ that best explain the observed
response pattern.  

In the above maximum likelihood problem, class membership is unknown.
The E-M (expectation-maximization) algorithm is a technique that can be used
to perform maximum likelihood estimation in the case of incomplete information %
\citep{Dempster,Arcidiacono}. 
The basic idea of the E-M algorithm is that one replaces unobserved
information with its expected value and then conducts maximum likelihood
estimation as if these expectations were correct. The maximum likelihood
estimates can be used to update the original expectations, and the log-likelihood function re-estimated. 
This iterative process continues until the change in the log-likelihood
function becomes sufficiently small.
Using these methods, one can estimate the 
response and class membership probabilities
that maximize the log-likelihood function. 

Similar use of LCA can be found in \citet{MenzelScarpa2005}.  Standard
references to latent class models include \citet{TitteringtonEtAl1985}, \citet{BartholomewKnott1999},
and \citet{WedelKamakura2000}. 
A more detailed explanation of the derivation of this model and how it
can be estimated can be found in \cite{MoreyEtAl2006} and \cite{ThacherEtAl2005}.
\citet{Ben-AkivaEtAl2002} provide another example of the application of LCA to discrete choice models.  Typically an important part of any latent class modeling procedure is
to determine the number of classes and the fit of the model (see \cite{Forman2003} and
 \cite{EidEtAl2003} for information regarding model fit). 
A number of software
packages now exist that allow estimation of latent class models, including
Latent GOLD \citep{LatentGold} and Mplus \citep{Mplus}. The
results for this
study were estimated using LEM \citep{Vermunt1997}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Consistency between methods} \label{sec:Consistency}

In this section we assess the convergent and theoretical validity of the CA and 
LCA applications and the robustness of the results by comparing the CA and LCA 
results in two ways: the consistency of the assignment of individuals to 
environmental groups, and the response patterns across groups.\footnote{Convergent 
validity involves comparing results obtained using two different measures or 
approaches.  Theoretical 
validity can be addressed by testing whether relationships among the variables 
meet prior intuitive and theoretical expectations.}  Previous studies have used the NEP to identify 
three groups based on whether
respondents have \textquotedblleft strong,\textquotedblright\
\textquotedblleft moderate,\textquotedblright\ or \textquotedblleft
weak\textquotedblright\ pro-environmental attitudes (\citealt{KotchenReiling2000}, 
\citealt{CooperEtAl2004}). We follow the same
convention here and assume the existence of three latent groups.\footnote{For the purpose of 
the CA and LCA, we use combined data from the falcon and
sturgeon surveys. This does not pose any inconsistency in the analysis
because the surveys were identical except for the endangered species that
was valued.}  Thus, our comparison of CA and LCA is subject to the constraint of having the same number of groups (three in this case).  

In distributing respondents to the \emph{strong}, \emph{moderate}, and \emph{weak} attitudinal groups, 65 percent of respondents were assigned to the same attitudinal groups by CA and LCA.\footnote{
As discussed in the previous section LCA does not assign 
individuals to a particular group, but rather provides the probabilities that an 
individual belongs to each group.  In order to compare the CA and 
LCA results, individuals are assigned to the group for which 
they have the
highest conditional probability.}  The consistency is especially notable for the \emph{strong} group, but less so for the \emph{moderate} and \emph{weak} groups (see Table \ref{table:GrpAssignments}).   
As a point of
comparison, if group assignments had been random
only 33 percent of respondents would have been assigned to the same
group by the two methods.
Although 35 percent of respondents were assigned to different groups by CA and LCA, only one
individual was assigned to the \emph{strong} group by CA but the \emph{weak} group by
LCA, and no individuals were assigned to \emph{weak} by CA but \emph{strong} by LCA.  These results offer evidence of convergent validity, as the two methodologies demonstrate reasonable consistency in the assignment of individuals to environmental groups.  

\begin{center}
[Table \ref{table:GrpAssignments} here]
\end{center}

We further examine the consistency of the CA and LCA results by using the Mann-Whitney rank-sum test to assess whether mean respones to the NEP statements differ between methods (Table \ref{TableCAvsLCA_EthicsGrpMeans}).\footnote{Because the NEP data are not normally distributed, it is not appropriate to use the usual $t$-test to determine whether there are statistically significant differences between the groups' mean responses.  We therefore make use of the nonparametric Mann-Whitney rank-sum test.  \citet{BainEngelhardt1992} provide further information regarding this test.}  Results illustrate that for the \emph{strong} groups, mean responses differ for only three statements (\emph{modifyenv}, \emph{strongbalance}, and \emph{delicatebalance}).  However, the \emph{moderate} and \emph{weak} groups have statistically different means between methods for 11 and 12 of the NEP statements, respectively.  These results reiterate the fact that although the \emph{strong} CA and LCA groups are similar, the \emph{moderate} and \emph{weak} groups are somewhat dissimilar.  Results from comparisons of the methods' group assignments and mean NEP responses provide evidence (although not especially strong evidence) of convergent validity.

\begin{center}
[Table \ref{TableCAvsLCA_EthicsGrpMeans} here]
\end{center}

Tables \ref{TableEthicsGrpMeansCA} and \ref{TableEthicsGrpMeansLC} report mean responses to the 
NEP statements for each
attitudinal group for CA and LCA, respectively. This same data is illustrated in 
Figure \ref{figure:GrpChar}.
The results illustrate how mean responses differ between the 
\emph{strong}, \emph{moderate}, and \emph{weak}
environmental attitudinal groups.  As illustrated by the Mann-Whitney rank-sum 
test results (Tables \ref{TableEthicsGrpMeansCA} and \ref{TableEthicsGrpMeansLC}), the mean responses between the attitudinal groups are statistically 
different in almost all cases.  Exceptions occur for mean responses to the 
\emph{suffresources} and \emph{controlnature} statements for the \emph{moderate} 
and \emph{weak} pro-environmental groups  
derived using CA, and for the \emph{interfere}, \emph{lawsofnature}, and 
\emph{controlnature} statements for the \emph{moderate} and \emph{weak} groups 
derived using LCA.  The theoretical validity of the CA and LCA applications is supported 
by the fact that essentially all mean responses are statistically different.

\begin{center}
[Table \ref{TableEthicsGrpMeansCA} here]
\end{center}

\begin{center}
[Table \ref{TableEthicsGrpMeansLC} here]
\end{center}

\begin{center}
[Figure \ref{figure:GrpChar} here]
\end{center}

As expected, CA and LCA both yield attitudinal groups for which the \emph{strong} groups have the highest mean responses to each of the NEP statements, the \emph{weak} groups have the lowest mean responses to each NEP statement, and the \emph{moderate} groups have means that fall between those of the \emph{strong} and \emph{weak} groups.  This response pattern indicates that the \emph{strong} group has the most pro-environmental attitudes, whereas the \emph{weak} group has the least pro-environmental attitudes.  The across-group response pattern therefore provides further evidence of theoretical validity.  

There is similarity in the characterization of groups identified 
using CA and LCA, which indicates consistency in the results and provides further evidence of convergent validity.
The \emph{strong} pro-environmental groups have relatively strong
pro-environmental attitudes on all questions, although they consistently
demonstrate weaker attitudes regarding the sufficiency of the earth's
resources (\emph{suffresources}). The \emph{moderate} groups
are somewhat environmental on all questions, yet there are some statements for
which they are either undecided or appear to not hold a pro-environmental
attitude (e.g., \emph{suffresources}, \emph{ingenuity}). The \emph{weak} groups
consistently hold seemingly anti-environmental attitudes for statements
dealing with limits to growth, although responses on other statements are
more mixed. For example, responses to
statements associated with the rejection of exemptionalism suggest the presence of uncertainty or somewhat
pro-environmental attitudes.

A notable difference between the methods pertains to the size of
the different groups (Tables \ref{TableEthicsGrpMeansCA} and 
\ref{TableEthicsGrpMeansLC}). Although the \emph{moderate} 
pro-environmental group is
consistently the largest, there are substantial differences in group sizes
across the techniques. In particular, LCA yields a
much smaller \emph{weak} group. Consequently, the LCA \emph{weak} group has
weaker pro-environmental attitudes. This difference is most pronounced for
the \emph{ecolcrisis} statement and the questions associated with 
anti-anthropocentrism.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Relationship between attitudes and WTP} \label{sec:WTPnAtt}

We further examine the robustness
of CA and LCA
by testing
whether WTP for endangered species recovery varies across environmental attitudinal 
groups and between methods. If
the attitudinal groups are really different, we would expect WTP to vary by group. 
Moreover, consistency
between CA and LCA should imply that WTP estimates are similar for each
group under both analytical methods. In 
this section we also address the question of whether accounting for unobserved 
heterogeneity in environmental attitudes 
provides additional explanatory power in econometric models of WTP 
responses.\footnote{For the regression and WTP portions of our analysis we use 
data from the falcon and sturgeon surveys separately.}

We estimate logit models of dichotomous-choice CV responses, and include the same covariates 
used in \citet{KotchenReiling2000}---bid amount, previous knowledge about the good, 
household income, and environmental attitudes as
measured by the different methods.\footnote{While including income as a separate 
linear term 
is not utility
theoretic, 
it is common practice and acts as
a proxy variable for a number of other 
socioeconomic attributes \citep{Hanemann2001}.}
Table \ref{TableDescStats} provides the definitions and descriptive 
statistics for the variables we use.\footnote{Observations with missing values for any 
of the variables included in the
regression were deleted. In addition, following the approach taken
by \citet{KotchenReiling2000}, we exclude individuals
exhibiting protest behavior. Estimation was performed using non-linear maximization modules in  Python.}

\begin{center}
[Table \ref{TableDescStats} here]
\end{center}

Table \ref{TableLogit} reports coefficient estimates and
significance levels for the falcon and sturgeon surveys.
In both attitudinal models and for both data sets, the bid and income variables are significant 
and have the expected sign.
As expected, individuals are less likely to respond `yes' as the bid amount increases, 
but are more likely to respond `yes'
as income increases. 

\begin{center}
[Table \ref{TableLogit} here]
\end{center}

All of the attitudinal variables are significant in the sturgeon data set.
Under both the CA and LCA methods, the \emph{strong} environmental
groups are significantly more likely to agree to the referendum than
the \emph{weak} environmental groups. The same holds for the \emph{moderate}
environmental groups identified by each method. \emph{Knowledge} is a positive significant
explanatory factor at conventional levels for the LCA approach and at the
$6\%$ level for the CA approach.    

Attitudinal results for the falcon data set are less strong than those
of the sturgeon data set.
The \emph{strong} environmental group identified by the CA approach is
significantly more likely to agree to the referendum than the \emph{weak}
environmental group. The \emph{strong} environmental group identified by the
LCA approach and the \emph{moderate} group identified by the CA approach are
 significantly more likely to agree to the referendum than the \emph{weak} 
groups at only the $8\%$ and
$7\%$ levels, respectively. 
There is not a significant difference 
between \emph{moderate} and \emph{weak} groups in the LCA model. \emph{Knowledge}
is not a signficantly explanatory factor under either approach. 

We use the models in Table \ref{TableLogit}
to estimate WTP for 
each endangered species under each attitudinal measurement method.
Table \ref{TableGroupWTP} reports the estimated WTP for a representative individual
in each group,  
the $90$-percent confidence intervals around these estimates, and tests of whether
WTP differs between the groups and methods.

\begin{center}
[Table \ref{TableGroupWTP} here]
\end{center}

As expected, WTP is largest in 
magnitude for the \emph{strong} groups and smallest for the \emph{weak} 
groups. The $90$-percent confidence intervals show that mean WTP is significantly
greater than zero for the \emph{strong} and \emph{moderate} groups but is not
significantly greater than zero for the \emph{weak} groups.\footnote{Confidence 
intervals were calculated using the Krinsky-Robb
simulation method \citep{KrinskyRobb1986, ParkEtAl1991}. $1000$ random draws were taken from 
a standard normal distribution, weighted by a Cholesky decomposition of the variance-covariance
matrix, and added to the original parameter estimates. This process was replicated $100$ times.
These new parameter draws were used to calculate the distribution of WTP.} 

We use the method of convolutions to determine whether mean WTP differs significantly between attitudinal groups and between methods.\footnote{As
shown in \cite{PoeEtAl1994}, comparing confidence intervals between groups is
not an appropriate test because it relies on distribtional assumptions about WTP that
may not be satisfied.} To perform the test we performed $100$ replications of 
the Krinsky-Robb simulation method and calculated the mean WTP for each replication.
A distribution of the differences in means was then calculated. We report the $90$-percent
confidence intervals for this difference in means.  A confidence interval that excludes
$0$ signifies that the difference in mean WTP between the two
groups is significantly different from zero.
See \cite{PoeEtAl1994} and \cite{PoeEtAl2005} for additional explanation of this method.
The ``Test of Differences in Means across Groups'' section in Table \ref{TableGroupWTP}
shows that with one exception
(the \emph{strong} and \emph{moderate} LCA groups in the sturgeon
data set) the \emph{strong} environmental groups have a significantly higher mean
WTP than the \emph{moderate} and \emph{weak}  environmental groups, and the \emph{moderate}
environmental groups have a significantly higher mean WTP than the \emph{weak}
environmental groups. This provides strong evidence that groups with stronger
pro-environmental attitudes have significantly higher WTP, and additional evidence of CA and LCA theoretical validity.

Another question of interest is whether WTP significantly differs between
groups identified by CA and those identified by LCA. For example,
is there a significant difference in the mean WTP of the CA \emph{strong} group and 
the LCA \emph{strong} group?  In the sturgeon data set for each
attitudinal group, there are no significant differences in the mean WTP estimated using
the two methods (see section ``Test of Differences in Means between Methods'' of Table \ref{TableGroupWTP}). The results are not as strong in the falcon data set: only for the weak
group do the mean WTP not differ between the two methods. Thus, for the
sturgeon data set we find strong evidence that CA and LCA are identifying
groups with the same mean WTP (and thus evidence of convergent validity), but this result generally does not hold for the falcon data set.

\begin{comment}
As expected, the average
WTP for individuals in the \emph{strong}, \emph{moderate}, and \emph{weak} environmental groups 
are quite different in magnitude.  
The average WTP for those 
in the \emph{weak} environmental group is not significantly different from zero.
There are two important points to focus on in Table \ref{TableGroupWTP}: the 
significant differences
in estimated WTP between groups within a method and the insignificant
differences for a group between methods. Using 
the method of convolutions (\cite{PoeEtAl1994, PoeEtAl2005}), Table
\ref{TableGroupWTP} reports confidence intervals for the difference in means
over $100$ replications of $1000$ Krinsky-Robb simulations. 
The section entitled
``Test of Differences of Means across Group'' shows that with one exception
(the \emph{strong} and \emph{moderate} LCA groups in the sturgeon
data set), the \emph{strong} environmental group has a significantly higher mean
WTP than the \emph{moderate} and \emph{weak}  environmental groups and the \emph{moderate}
environmental groups have a significantly higher mean WTP than the \emph{weak}
environmental groups. The section entitled
``Test of Differences of Means across Methods'' shows that in the sturgeon data set for each
attitudinal group, there are not significant differences in the mean WTP estimated using
the two methods. The results are not as strong in the falcon data set: only for the \emph{weak}
group do the mean WTP not differ between the two methods.\footnote{We are grateful to a 
referee for suggesting the convolutions method.} 
Using mean WTP thus shows that both methods find significant differences in mean
WTP across groups and that the mean WTP measures obtained from the two methods
are fairly consistent.
\end{comment}

To test whether attitudinal data adds useful information beyond what
is already captured through demographic data, we run comparison models for both sets of data:
a full model that includes both demographics and attitudinal variables and a 
restricted model that excludes the attitudinal variables. 
Because we wish to assess the additional explanatory value provided by attitudinal 
data, we account for as much demographic heterogeneity as possible by adding
gender, age, and household size to the models presented in Table \ref{TableLogit}. The
likelihood ratio tests reported in  
Table \ref{TableAddedValue} 
show that the restricted models, which exclude  the attitudinal variables, consistently
provide a significantly  
worse fit. Demographics do not fully capture the same information as attitudinal
data. 

\begin{center}
[Table \ref{TableAddedValue} here]
\end{center}

Given that the models yield similar results, 
an obvious question of interest is whether the logit model using CA or LCA is preferred.
 Although information criteria such as the Akaike (AIC) or Bayesian (BIC) information criteria are commonly 
used for this purpose, these measures cannot test whether
the models are statistically different. For this reason we use 
the Vuong test \citep{Vuong1989} to test  
the null hypothesis that the
two models are equally close to the true specification.\footnote{The
Vuong test statistic is 
\begin{equation*}
\frac{1}{\sqrt{n}}\frac{LogL_{LCA}-LogL_{CA}}{\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n} \left[
\log L_{LCA}\left( i\right) -\log L_{CA}\left( i\right) \right] ^{2}-\left[ 
\frac{1}{n}\left( LogL_{LCA}-LogL_{CA}\right) \right] }} \sim N(0,1),
\end{equation*}
where $i$ is an individual in the sample.}
We do not find a significant difference between the two models, and thus cannot
conclude that there are statistical reasons for choosing one model over the other.  
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Discussion} \label{sec:Discuss}


We apply CA and LCA to NEP data in order to identify environmental 
attitude heterogeneity.  
Results stemming from CA
and LCA are similar; there is consistency in the assignment 
of individuals to attitudinal groups, as well as consistency in the 
characterization of the identified groups.  Moreover, we find that environmental
attitudes are strong and significant predictors of CV responses, regardless
of the method used to identify attitudes. Both methods
show that WTP increases with the strength of
pro-environmental attitudes. In all cases but one, we find
significant differences in WTP between the
attitudinal groups.  We also find that in four of the six cases there were
not significant differences in the mean WTP derived from the two 
grouping methods. 
Furthermore, we find that models that include attitudinal data have a significantly better 
fit than a model that includes solely sociodemographic variables.  
We thereby provide evidence that including unobservable attitudinal heterogeneity 
provides greater explanation of individuals' preferences and choices.

Given the consistency across the classification techniques, an
important question is the following: What insights might be provided regarding which method, CA or LCA, is most appropriate for
incorporating environmental attitudes into CV studies? Both methods have strengths
and weaknesses.  The primary limitation of LCA is that, depending on
the data, the number of parameters that must be estimated increases very
quickly. This is the case with the NEP data. Since there are 
five response categories for each of $15$ questions and three classes, one must
estimate four of the levels for each question and each class and two of 
the class probabilities,
translating into $182$ parameters. In some applications this may limit 
the number of groups that can be estimated. While techniques are available
for reducing the number of variables to be estimated, these techniques
require expertise on the part of researchers. 
Nevertheless, LCA has solid statistical tests for determining the fit of the model and indicative tests for determining the appropriate number of groups.\footnote{For example, the Pearson and Read-Cressie statistics, both measures of how well the observed and expected frequencies of responses compare, can be used to statistically determine whether the number of groups fits the data \citep{Forman2003}.  In the case of sparse data, these statistics can be bootstrapped \citep{EidEtAl2003}. Once the set of models that fit the data is determined, information criteria such as the AIC, CAIC, AIC$_C$, and BIC (which assess the fit of the model with a penalty imposed for the number of parameters) can be used to choose between the models \citep{Akaike1974,Bozdogan1987,HurvichTsai1989,Schwarz1978}.  The use of information criteria can be subjective if the criteria yield different results.}  Additionally, the outputs of predicted response probabilities, conditional
probabilities, and unconditional probabilities can be useful for economic analysis. 

CA has the advantages that it 
is available in numerous software packages, and it is not as complex as LCA and therefore does not require as much time to learn.
Additionally, CA is not limited in the number of groups that 
can be estimated.  A disadvantage of CA is the need to make 
somewhat arbitrary decisions between competing options and methods at various stages
of the clustering process. Although guidelines and considerations for the
various decision points exist, there are no definitive rules. Thus, although CA is relatively easy to implement in comparison with LCA,
its appropriate use requires in-depth knowledge and experience.  Another weakness of the CA procedure is that the Euclidean distance between two individuals is calculated using the differences between the values of their NEP responses rather than on the actual values of the responses.  Thus, the same Euclidean distance will result under the following two scenarios: 1) individual $i$ responds 1 and $j$ responds 3, and 2) individual $i$ responds 3 and individual $j$ responds 5.  For this reason LCA may be considered superior, as LCA segments individuals based upon an overall response pattern.  

Because 
CA and LCA have different strengths and weaknesses, 
the desired method for capturing attitudinal heterogeneity may depend on particular 
research goals and/or data availability. Fully addressing the question of whether CA or LCA is more appropriate for incorporating environmental attitudes into CV studies will require analysis of the techniques applied to other data sets.

In conclusion, this paper compares two methods for incorporating unobservable 
heterogeneity in valuation studies and finds the results are generally convergent---
the methods identified similar attitudinal groups, assigned individuals to  
groups with reasonable consistency, and WTP did not vary significantly between methods, but did vary 
significantly between groups. A model that excluded attitudinal variables 
illustrated that socioeconomic variables did not fully capture preference 
heterogeneity.  Future research should examine whether similar results are obtained when CA and LCA are applied to NEP responses within the context of other CV studies.  If similar results are obtained, this would corroborate that the NEP is a valuable tool for measuring environmental preference heterogeneity, and that CA and LCA are capable of identifying heterogeneous preference groups.  Future research might also strive to determine whether analyzing NEP responses using CA and LCA analysis provides superior results to the more common approach of capturing environmental attitudes.

\begingroup
\theendnotes
\endgroup

\bibliographystyle{greenbay}
\bibliography{bibfile20060911}

\newpage

\begin{figure}[tbp] 
\caption{Mean responses to NEP questions by method and class}\label{figure:GrpChar}
\begin{center}
%\fbox{\includegraphics[angle=90, width=1.0\textwidth]{GraphCompareLCCA}}
\fbox{\includegraphics[width=1.0\textwidth]{GraphCompareLCCA_20060823.eps}}
\end{center}
\end{figure}


% using include rather than input forces a pagebreak

\include{TableNEPAttitudes} 
\include{TableGrpAssignments} 
\include{TableCAvsLCA_EthicsGrpMeans_longtable}
\include{TableEthicsGrpMeansCA} 
\include{TableEthicsGrpMeansLC} 
\include{TableDescStats}
\include{TableLogit}
\include{TableGroupWTP}
\include{TableAddedValue_LRT}

\end{document}