Attribute Based Consideration Sets
by
Timothy J. Gilbride1 and Greg M. Allenby
Fisher College of Business
2100 Neil Avenue
Ohio State University
Columbus, OH 43210
Phone: (614) 688-8163
Fax: (614) 292-0879
gilbride.1@osu.edu
allenby.1@osu.edu
October, 2002
Acknowledgements
The authors thank Mike Lotti and Eastman Kodak Company for providing the data used in this
study.
1
Corresponding author.
Attribute Based Consideration Sets
Abstract
Consumers choose offerings with attributes that provide benefits. In many product categories
there are numerous offerings, each with multiple attributes, resulting in a complex consumer
decision problem. Consideration sets have been proposed as one mechanism for simplifying the
choice process. However, the simplification that consideration sets provide to consumers leads
to complexities in modeling choice, whenever the consideration set is not revealed. We propose
a consideration set choice model in which consumers evaluate products and attributes using a
conjunctive, disjunctive, or compensatory screening process. Bayesian data augmentation and
Markov chain Monte Carlo methods are used to navigate over the unobserved consideration sets
in a discrete choice model. The method is illustrated using a conjoint study of cameras. The
results indicate that 92% of respondents form consideration sets by screening alternatives on one
or more attributes.
Keywords: Conjoint Analysis, Decision Process, Hierarchical Bayes, Revealed Choice
1
Attribute Based Consideration Sets
1. Introduction
As consumers pursue the tasks and interests of their lives, they consider the attributes and
benefits of marketplace offerings, judging if they are worth purchasing. Consider, for example,
an individual purchasing a single-use (i.e., disposable) camera on his/her way to a family
reunion. Attributes of the camera may include: an automatic film advance/rewind; a flip-up flash
to reduce red-eye; look like a real camera; produce pictures that have a date stamp; develop
pictures instantly; let you know that the film has advanced properly; good for group shots. The
evaluation of all single-use cameras on every attribute is an involved process. The individual
may therefore form a subset of offerings with one or more of these attributes that are critical to
them, which are then evaluated in more detail.
Consideration sets have been proposed as a mechanism for simplifying the choice
process. However, the simplification that consideration sets provide to the consumer results in
complexities in studying consumer choice whenever the set is not revealed. For a product
category with J brands, there are 2J-1 different possible sets, among which the choice probability
for an offering varies. If the consideration set is not revealed to the researcher, it must be
integrated out of the expression for the choice probability when specifying the likelihood
function and predicting consumer choice. The difficulty in executing this operation is greatly
reduced when the number of possible sets is small, and, as a result, most empirical studies have
been limited to choice among a small number of choice alternatives.
In this paper we propose a new approach to modeling and estimating consideration sets
that can accommodate alternative models of set formation. Our approach employs the Bayesian
technique of data augmentation, and uses Markov chain Monte Carlo methods to integrate over
2
the unobserved consideration sets. The approach can handle a large number of potential sets,
and facilitates various set formation rules that include overall utility as well as the levels of
specific attributes, employing both conjunctive and disjunctive screening mechanisms.
The organization of the paper is as follows. In section 2 we describe our modeling
approach and describe how it deals with the complexities of estimating consideration set choice
models.
Alternative models of set formation are then discussed in section 3.
Section 4
introduces an empirical example in the context of a choice-based conjoint study for a new
camera type.
Results are presented in section 5 where we find that 92% of the survey
respondents use some form of non-compensatory screening when choosing among the choice
options. Implications and directions for future research are discussed in Section 6.
2. Estimating Consideration Set Models
Consumers use various decision rules or heuristics to simplify complicated decision
tasks. Theoretical and empirical support for the use of consideration sets by consumers can be
found in Payne and Easton (1978) who elicit decision protocols of consumers in supermarkets,
Bettman (1979) who argues that a phased decision process reduces the cognitive demands of the
decision maker, and Bettman and Park (1980) who provide evidence that the formation of the
consideration set is linked to specific attributes and that the final selection is more holistic.
Consideration set models have been applied in a variety of domains, including the analysis of
survey data (Gensch 1987) scanner panel data (Manrai and Andrews 1998), and data collected in
laboratory settings (Johnson and Meyer 1984). Johnson, Meyer, and Ghose (1989) and Andrews
and Manrai (1998) demonstrate that parameter bias and incorrect inferences result when the
decision process is not correctly modeled.
3
Consideration sets induce a complicated structure in the likelihood function. Consider,
for example, a simple discrete choice model without consideration sets. The choice probability
is:
Pr(i) = Pr(Vi + εi > Vj + εj for all j)
= Pr( εj < Vi – Vj + εi for all j)
+∞
=
∫
−∞
Vi − V j + ε i Vi − Vm + ε i

 ∫ L ∫ f (ε j ) L f (ε m ) dε j L dε m  f (ε i ) dε i
 −∞

−∞
(1)
where Vi denotes the deterministic portion of utility of choice alternative i, εi is the stochastic
portion, and it is assumed that the stochastic terms are distributed independently. The probit
model is obtained by assuming that the stochastic terms are distributed normal, and the logit
model is obtained when the stochastic terms are distributed extreme value. The discrete choice
probabilities are derived by partitioning the domain of the error space corresponding to a choice
outcome, and integrating the error densities over the corresponding region. The regions for a
simple discrete choice model are indicated it the upper and lower limits of the integrals.
The likelihood for an observed sequence of choices is the product of the choice
probabilities:
l(V | data ) ≡ π (data | V ) = ∏ Pr(i )
(2)
i
The likelihood surface for simple discrete choice models is continuous and globally concave
(Maddala 1983). Small changes in the deterministic portion of utility (V) results in small
4
changes in the limits of integration, and small changes in the resulting choice probabilities.
Moreover, unique values of the deterministic portion exist that maximize the probability of the
observed choices. The regularity of the likelihood surface allows use of gradient-based methods
(e.g., Newton-Raphson) to solve for values of the deterministic portion (V) that maximize the
likelihood.
The presence of consideration sets destroys the regularity of the likelihood function. As
choice alternatives enter and leave the consideration set, the choice probabilities are obtained by
integrating over an error space of changing support.
Adding an alternative (k) to the
consideration set results in an additional factor in equation (1), with an additional stochastic error
(εk) and appropriate limits of integration. The likelihood surface for a consideration set model is
therefore discontinuous, and not necessarily globally concave.
Researchers have historically dealt with the non-regularity of the likelihood surface by
writing the choice probability as the product of a (conditional) second stage choice probability
and a marginal consideration set probability:
Pr(i,C) = Pr(i|C) × Pr(C)
(3)
where C denotes the consideration set that reduces the number of choice alternatives in equation
(1).
This approach, coupled with the assumption that that the values of the deterministic
components {Vi} are constant across the consideration set, results in a smooth and globally
concave likelihood surface.
Andrews and Srinivasan (1995) use this approach to study
consideration sets in consumer choice using scanner panel data, where the probability that an
5
alternative is considered is modeled with a binary probit model. Ben-Akiva and Boccara (1995)
employ a logit model for the consideration set probability.
The estimation of consideration set models using equation (3) has historically been
approached sequentially, sometimes with independent data. Fader and McAlister (1990) use
display and feature data to identify sets of promoted offerings and assume the consideration set is
either the set of previously purchased brands currently on promotion or all previously purchased
brands. Jedidi, Kohli and DeSarbo (1996) and Roberts and Lattin (1991) have respondents
indicate the acceptability of alternatives in a survey. When independent data are available to
identify the acceptability of an offering, the evaluation of consideration set models can employ
standard maximum likelihood techniques.
If independent data are not available to identify the consideration set, it becomes a latent
construct that must be simultaneously estimated with the observed choices. Complications arise
in the evaluation of the choice probabilities because the partition of the error space described in
equation (1) must also be partitioned with respect to the consideration set (see Mehta, Rafiv and
Srinivasan, 2002).
Only one realization of the error vector, ε, is associated with each
observation, and it is not appropriate to estimate the two components in equation (3) by assuming
that each are associated with a different realization. Such an assumption leads to a non-identified
model specification unless additional assumptions are made to avoid associating each
observation with two errors. The complexity of the evaluation is aggravated by the 2J-1 different
sets that are possible when J offerings are studied. Researchers have therefore tended to focus on
a small number of choice alternatives (J < 4) where the effects of the consideration sets are not as
pronounced.
6
An alternative to partitioning the error space is to directly estimate the probability of all
consideration sets. Chiang, Chib and Narasimhan (1999) consider equation (3) as a hierarchical
Bayes model, where the 2J-1 sets are assigned a prior distribution and posterior estimates of set
composition are obtained.
While this approach avoids the complications involved with
identifying suitable partitions of the error space to derive choice probabilities, the consideration
sets are not linked to the characteristics of the offerings other than through the fact that the
chosen brand is an element of the set. Consideration sets with high posterior probabilities
comprise offerings that have been chosen and exclude offerings that have not been chosen.
Our model structure allows for consideration set formation based on the attributes of the
choice alternatives and/or the overall value of the offering, and can employ either conjunctive
(i.e., and) or disjunctive (i.e., or) rules for combining these components. We navigate the
complicated partitioning of the error space by employing data augmentation and Markov chain
Monte Carlo methods. We next describe the use of data augmentation to evaluate the standard
probit model in equation (1) above. We then show how this method can be used to estimate
choice models with consideration sets.
Data Augmentation
Direct evaluation of the probability in equation (1) can be avoided in a Bayesian
approach to estimation if the parameters {V} are augmented with a vector of latent variables, z
(see Tanner and Wong 1987), where
zi = Vi + εi ,
εi ~ Normal(0,1)
(4)
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and the model is written hierarchically:
y |z
(5a)
z|V
(5b)
where y denotes the observed choice outcomes. Equation (5a) reflects the censoring of the
continuous variables z, where only the offering with the maximum value of z is revealed to the
researcher by the choice data. Equation (5b) reflects the assumption that the latent variables are
normally distributed, with the mean of each element equal to Vi and variance one. In marketing
applications of the probit model, the mean is typically parameterized Vi = xi'β. Markov chain
Monte Carlo estimation of the probit model proceeds by drawing iteratively from the two
conditional distributions:
π(z|y,V) = Truncated Normal (V,I)
(6a)
π(V|z)
(6b)
= Normal( z , I/n)
where "I" is the identity matrix, and the truncation in equation (6a) corresponds to the alternative
selected being the maximum draw. If Vi is parameterized as xi'β then the mean and variance in
equation (6b) would correspond to the OLS regression estimate and covariance matrix. Detailed
treatments of estimating the probit model with data augmentation can be found in Albert and
Chib (1993), McCulloch and Rossi (1994), Allenby and Rossi (1999) and Edwards and Allenby
(2002).
8
The upper limits of the integral in equation (1) that complicate the evaluation of the
choice probability are replaced with a much simpler procedure that involves drawing from a
truncated normal distribution. The truncation points correspond to the limits of integration, and
the Markov chain is used to navigate the domain of the error space consistent with the observed
data to arrive at the posterior distribution of model parameters.
Thus, data augmentation
provides a clever way of avoiding the estimation of the integral.
Application to Consideration Set Models
The consideration set restricts the set of alternatives that are evaluated for final selection.
The considered choice alternatives are identified with an indicator function, I(x,γ), that equals
one if the decision rule is satisfied and equals zero otherwise. The choice model can then be
written in general form as:
Pr(i) = Pr(Vi + εi > Vj + εj for all j such that I(xj,γ) = 1)
(7)
where xj denotes a generic argument of a function that reflects the decision rule applied to the jth
offering. For example, if price is used to screen the choice alternatives, so that only offerings
above a lower threshold are considered, then I(xj,γ) = I(pricej > γ). If an upper threshold for price
is present, then negative price can be used as an argument, I(xj,γ) = I(-pricej > γ).
This
formulation allows for dynamic consideration sets when an attribute (e.g., price) changes over
the choice history. Moreover, as described below, the indicator function can be specified with
multiple attributes and alternative decision rules.
9
If an alternative is in the consideration set, then its choice probability is determined
relative to the other offerings in the set. If an alternative is not in the consideration set, then its
choice probability is zero. The choice model can be written hierarchically as:
y|z,I(x,γ)
(8a)
z|V
(8b)
where zi = Vi + εi, and estimation proceeds by drawing iteratively from the conditional
distributions:
z|y,V,I(x,γ)
(9a)
γ|y,z,x
(9b)
V|z
(9c)
The conditional distribution in equation (9a) is a truncated normal distribution for those
items in the consideration set. That is, the element of z corresponding to the chosen alternative is
greater than the elements of z that correspond to the alternatives not chosen, given that they are
in the consideration set. If an alternative is not in the consideration set, which is determined by
the indicator function I(x,γ), then z is drawn from a non-truncated distribution. Here we assume
I(x,γ) does not involve the deterministic component of utility, V; changes to the hierarchy to
accommodate this decision rule are discussed below.
10
The conditional distribution for the cutoff parameters, γ, in equation (9b) is dependent on
the observed data, y, and the augmented parameter, z. The conditional distribution takes the
form of an indicator function because equation (8a) is a deterministic function that censors the
continuous latent variables z, identifying the considered offering with the highest value.
Variation in the value of the cutoff parameter changes the set of considered alternatives.
Permissible values of the cutoff parameters are those that lead to a consideration set where the
maximum of the augmented variable, z, corresponds to the observed choice. If a value of the
cutoff leads to a consideration set where the alternative with the maximum z is not the alternative
chosen, then the cutoff parameter is not permissible. This can happen because some values of
the augmented parameter in equation (9a) are drawn from a truncated distribution while others
are not. Hence, variation in z results in different consideration sets by defining the permissible
values of the cutoffs.
Finally, the conditional distribution in equation (9c) is identical to equation (6b).
Standard normal distribution theory can be used to generate the draw of V. If V is parameterized
as a regression function, V=X'β, then draws of β are generated from a normal distribution with
mean and covariance matrix corresponding to the standard OLS model. Details of the estimation
routine for our empirical study, reported below, are provided in the appendix.
The Markov chain defined by equations (9a) – (9c) generates draws from the full
conditional distribution of model parameters despite the irregularity of the likelihood surface.
Given the cutoffs, γ, the model (equation 7) becomes a standard discrete choice model. Given
the set of augmented variables, {z}, some values of the cutoff are acceptable and others are not.
Across multiple draws of the augmented variables, however, the acceptable range of γ varies, and
the Markov chain converges in distribution to the full conditional distribution of all model
11
parameters. The augmented variable z reflects a single error term that gives rise to the joint
probability of the consideration set and the resulting choice.
3. Alternative Decision Rules
Alternative models of consideration set formation can be investigated with the indicator
function relationship, I(x,γ) = 1. Flexibility exists in the specification of this relationship because
its complexity does not increase the complexity of estimating the consideration set choice model.
For each of the non-compensatory models described below, the conditional distributions
described in equations (9a) – (9c) are the same, except for the screening rules leading to different
alternatives for consideration in equation (9a). Changes required by the use of a compensatory
model are discussed below.
Consideration sets formed by a conjunctive decision rule requires that an alternative be
acceptable on all relevant attributes for it to be included in the consideration set. A conjunctive
rule is formed by multiplying indicator functions across the attributes (m) of an offering:
Conjunctive Rule:
Πm I(xjm > γm) = 1
(10)
where xjm is the level of attribute m for choice alternative j. The cutoff value, γm, is the smallest
level of the attribute that needs to be present for the decision maker to consider the offering. If
the cutoff value is smaller than all levels of the attribute, then the attribute is not used to screen.
For nominally scaled attributes (i.e., 0,1 attributes), a cutoff value greater than zero indicates that
the attribute is required.
Use of the multiplicative function, Πm, results in a conjunctive
12
combination of the factors where all components must be satisfied for the combination to be
satisfied.
A disjunctive decision rule is one where at least one of the attribute-levels is acceptable:
Disjunctive Rule:
Σm I(xjm > γm) ≥ 1
(11)
and a compensatory decision rule is one where the deterministic portion of the utility of the
offering must exceed a threshold value to be acceptable.
Compensatory Rule: I(Vj > γ) = 1
(12)
The compensatory decision rule requires a change in the hierarchical model. Since the
deterministic portion of utility, V, is used to determine both the consideration set and the final
choice, the hierarchy is:
z|y,V,I(V,γ)
(9a')
γ|y,z,V
(9b')
V|z ,y
(9c')
The exact form of these conditional distributions and the estimation algorithm is contained in the
appendix.
In addition to the form of the decision rule, researchers can specify that the cutoff
parameters (γ) take on a limited range of values. If an attribute, x, is continuous (e.g., price),
13
then γ can also be continuous. However, if the attribute takes on a restricted number of values,
then the cutoff is not well identified over an unrestricted range. In a conjoint analysis, for
example, there are rarely more than four or five levels of an attribute. In these cases one can
assume that the cutoff parameter takes on a small number of finite values by a prior assumption
that γ is distributed multinomial over a grid of values, and estimating the posterior distribution of
the multinomial mass probabilities. We explain this approach in the context of our empirical
application next.
4. Data and Models
The consideration set model is illustrated in the context of a discrete choice conjoint
study of consumer preferences for the features of a new camera format, the Advanced Photo
System (APS). This photographic system offers several potential advantages over standard
35mm film systems, including mid-roll film change, camera operation feedback, and a magnetic
layer on the film that offers the ability to annotate the picture with titles and dates. A more
complete description of the data is available in Allenby et. al. (2002).
Respondents were screened and recruited by telephone, and interviewed at field locations
in nine metropolitan areas. A total of 302 respondents participated. The screen was designed to
exclude individuals who had little or no interest in photography, and retained approximately 55%
of the population contacted. This compares to approximately 67% of the population that owns
still cameras (Simmons, 1997). Individuals participating in the study were therefore assumed to
be good prospects for cameras in the price range of the new APS offerings, and representative of
the APS camera market.
14
Respondents were asked to evaluate 14 buying scenarios. Each scenario contained full
profile descriptions of three compact 35mm cameras, three new APS cameras, and a seventh no
buy option. Table 1 lists the attributes that describe each of the cameras, and identifies the
attributes that are new and specific to APS cameras. Table 1 also displays restrictions on the
availability of certain attributes. For example, the 4× zoom lens was available only on the "high"
body style. Prices ranged from $41.00 to $499.00, the average price for APS cameras was $247,
and the average price of the compact 35mm cameras was $185.
== Table 1 ==
Each buying scenario requires the respondent to engage in a complex decision. With six
offerings described on eight attributes, plus the no-buy option, we consider this to be a good
dataset to investigate the formation of consideration sets. The no buy option was selected 14.7%
of the time or an average of 2.1 times in the 14 choice tasks. One of the APS camera offerings
was chosen 39.9% of the time, and one of the 35mm cameras was selected 45.4% of the time. In
the analysis that follows, 12 choice tasks per respondent were used to calibrate the model and the
final two retained for predictive testing. This results in 3,624 observations for model estimation
and 604 for hold-out validation.
Five models are fit to the data. In each model we make the assumption that the no buy
option is always in the consideration set. The first model is a heterogeneous probit model that
assumes all choice options are considered by the respondent. For identification, effects-level
coding is used for the non-price attributes, with the lowest level of each attribute set to zero,
except for body style. For the no buy option, all attributes are set to zero. Price is parameterized
as a natural logarithm. The effects-level coding results in 18 parameters, 17 corresponding to
binary indicator variables and the 18th corresponding to ln(price). In order to fix the scale in the
15
probit model, the error term is assumed to be independent and identically distributed across
choice tasks with unit variance. The latent continuous variable in our model setup is therefore:
Pr(j)hi = Pr(zhij > zhik for all k)
(13)
zhi = Xhiβh+ εhi,
(14)
εhi ~ N(0,I)
βh ~ Normal( β , Σβ)
(15)
where h indexes the respondent, i is the index for the buying scenario, and j and k indicate the
choice alternatives within the buying scenario.
The second model assumes that the consideration set comprises choice alternatives with
the deterministic value of utility (Vhi = Xhiβh) greater than a threshold value, γh. That is, the
selected alternative has greatest utility of the options for which Vhik > γh:
Pr(j) hi = Pr(zhij > zhik for all k such that I(Vhik > γh) = 1)
(16)
We allow for heterogeneity in the cutoff value, γh through the normal distribution:
γh ~ Normal( γ , σ γ2 )
(17)
The third model assumes that the consideration set is formed using a conjunctive decision
rule over the m attributes for each alternative:
Pr(j) hi = Pr(zhij > zhik for all k such that Πm I(xhijm > γhm) = 1)
16
(18)
where xhijm is the level of the attribute for respondent 'h' in buying scenario 'i' for alternative 'j'
and attribute 'm'. γhm is a respondent-level parameter representing the threshold, or acceptable
level, of attribute 'm' for respondent 'h'. We employ two alternative heterogeneity distributions
for the threshold parameters. When attribute m is continuously distributed (e.g., price), then we
employ a normal distribution to allow for heterogeneity:
γhm ~ Normal( γ m , σ m2 )
(19)
When an attribute takes on a small number of discrete levels we assume that the threshold
parameter is distributed multinomial. For example, consider the zoom lens attribute in Table 1
taking on values of none, 2x and 4x. We recode the attribute levels (e.g., 0,1,2) and specify a grid
of possible cutoffs (e.g., -0.5, 0.5, 1.5, 2.5) where the lowest cutoff value indicates that all
attribute-levels are acceptable, and the highest level indicates that none of the levels are
acceptable. For nominally scaled attribute-levels, where a prior ordering is not obvious (e.g.,
color), the grid has three points (-0.5, 0.5, 1.5). The full set of attributes, levels, and cutoffs is
contained in Table 5. The distribution of heterogeneity of the cutoff parameters with discrete
levels is:
γhm ~ Multinomial(θm)
(20)
17
where θm is the vector of multinomial probabilities associated with the grid for attribute m.
The fourth model assumes that the consideration set is formed using a disjunctive
decision rule:
Pr(j) hi = Pr(zhij > zhik for all k such that Σm I(xhijm > γhm) ≥ 1)
(21)
The upper and lower grid points specify the role of the attribute in determining the
consideration set, depending on the decision rule. An attribute determines membership in the
consideration set only if it changes the value of the alternative indicator function. For the
conjunctive model, if the threshold for attribute k is equal to –0.5, then I(xhijk > γhk) always
equals one and the value of Πm I(xhijm > γhm) does not change. Since different levels of the
attribute do not affect consideration sets, we conclude that attribute k is not being used to screen
alternatives. For the disjunctive model, if γk = 2.5, then I(xhijk > γhk) always equals zero and the
value of Σm I(xhijm > γhm) does not change across different levels of attribute k.
For the
disjunctive model, the upper value of the threshold implies no screening based on that attribute.
Figure 1 illustrates the procedure and the interpretation.
== Figure 1 ==
The fifth model assumes that consumers are either using a conjunctive (equation 18) or a
disjunctive (equation 21) decision rule.
That is, we allow for the presence of structural
heterogeneity (Yang and Allenby, 2000):
Pr(j) hi = φPr(zhij > zhik for all k such that Πm I(xhijm > γhm) = 1) +
(1-φ)Pr(zhij > zhik for all k such that Σm I(xhijm > γhm) ≥ 1)
18
(22)
where φ is the portion of the sample using the conjunctive decision rule and (1-φ) is the portion
using the disjunctive decision rule.
Inference is conducted via Bayesian MCMC methods. The chain converged quickly in
all models. Convergence was assessed by inspecting time series plot from multiple starting
points. The chain was run for 10,000 iterations with the final 5,000 iterations used to estimate the
moments of the posterior distributions. Details of the estimation algorithms for all models are
provided in the appendix.
5. Results
The results indicate that the conjunctive model fits the data best.
In sample and
predictive fits of the five models are presented in table 2. The log marginal density is computed
using the GHK estimator (Keane, 1994, Hajivassilious, McFadden and Rudd, 1996) and the
importance sampling method of Newton and Raftery (1994). Hit probability is defined as the
average probability associated with the observed choices. Hit frequency is the number of times
the predicted choice matches the actual choice.
The parameters reported below for the
conjunctive model result in 30% of the alternatives screened out of the final consideration set,
which averaged five (out of seven) alternatives including the no buy option.
Fit results for the other models are as follows. The fit for the disjunctive model is slightly
worse than the standard probit model, with parameters that lead to only 2% of the alternatives
screened from final consideration. The slight decrease in the log marginal density for the
disjunctive model is from the increase in the dimension of the model due to the cutoff
parameters. Although the compensatory model screened out 23% of the alternatives to form the
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consideration set and has a slightly better in-sample fit than the probit model, its predictive
accuracy is nearly identical.
The structural heterogeneity model assigned nearly all the
respondents (99%) to the conjunctive consideration set model as opposed to the disjunctive
model, so the fit statistics closely match those of the conjunctive model. The results provide
evidence that the conjunctive model is superior to both the standard probit and non-conjunctive
models.
Tables 3 and 4 report part-worth estimates for the probit and conjunctive models. Table 3
displays the mean and covariance matrix of the part-worths for the probit model, and table 4
reports the same statistics for the conjunctive model. Substantive differences exist in the partworth estimates for body style, zoom, camera setting feedback and price. The conjunctive model
estimates are much smaller for these attributes, with the average price coefficient moving to zero.
The reason for this difference is discussed below.
== Tables 3 and 4 ==
The cutoff parameters are reported in table 5. For discrete attributes, the cutoffs are
reported in terms of multinomial point mass probabilities associated with the grid described in
figure 1. The probabilities indicate the fraction of respondents with a cutoff parameter (γ) that
screens out choice alternatives. For the conjunctive model, θ1 = .598 indicates that 59.8% of the
respondents are not screening alternatives based on "Body Style," θ2 = 0.324 implies that 32.4%
of the respondents screen out offerings with the low body style, θ3 =0.059 implies that 5.9% of
the respondents screen out offerings with low and medium body style, and θ4=0.015 implies that
1.5% of the respondents screen on all alternative levels. Estimates of θ equal to approximately
2% reflect the influence of the prior distribution and can be disregarded for practical purposes.
== Table 5 ==
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The use of screening rules to form consideration sets is pervasive among the respondents
in this research. Analysis of the individual-level cutoff values indicates that 92% of respondents
form consideration sets using the conjunctive screening rule; 58% are forming consideration sets
based on a single attribute, 33% are using two attributes, and 2% are using 3 attributes. A
strength of the Bayesian approach outlined in this paper is the ability to make individual level
inferences and estimate not only what attributes are important in forming consideration sets but
also who is forming consideration sets.
Consider, for example, an aggregate analysis of the effect of the price threshold,
estimated to be equal to –6.269. This corresponds to an average price threshold of $527. Since
the maximum price of an APS camera is $499, this implies that, on average, respondents are not
screening on price. However, such analysis masks the tail behavior of the distribution of cutoff
values. As displayed in figure 2, 10% of the respondents will only consider cameras under $276,
20% will only consider cameras under $356, etc. In all, 41% of the respondents are screening
alternatives based on price.
== Figure 2 ==
Figure 3 displays the proportion of respondents screening on each of the attributes. Body
style, zoom, camera settings feedback and price are the attributes most often used to form the
consideration set. These attributes are available on both 35mm and APS cameras, and as a result
respondents are more fully aware of their benefits. In contrast, none of the new APS attributes
(e.g., mid-roll film change) are being used to form consideration sets. In addition, the attributes
used to form consideration sets have substantively different part-worths in tables 3 and 4. Our
results indicate that once the consideration set is formed, the price and body style do not play a
role in the final decision. The role of the zoom and camera feedback features however, continue
21
to play a role in the final selection of a camera beyond just forming consideration sets. Previous
authors have hypothesized that an attribute plays either a role in the screening phase or the
choice phase (Bronnenberg and VanHonacker, 1996) or that once an attribute is deemed
acceptable it will play no role in the final selection process (see Bettman, 1979); these results
provide evidence on this issue.
== Figure 3 ==
The participants in the survey were recruited and screened so that they were a prospect
for cameras in the price range of the APS offering. Despite this pre-qualification, some 41% of
respondents used price to form consideration sets and excluded those cameras above a particular
threshold. Exploratory analysis revealed no relationship between the posterior estimate of the
price threshold value and demographic variables including income. However, replies to the
question "Typically, when you attend informal social gatherings away from your home, how
often do you take a camera along with you?" were associated with the price threshold (p-value =
.0362). Those respondents that indicated they "almost always" take their camera (44% of
respondents) have an average price threshold $68 higher than respondents who take their camera
less frequently. Although exploratory in nature, this analysis shows that other variables correlate
with the price threshold and may be used in conjunction with the model to better identify
prospects in the market place.
The empirical results support the conjunctive consideration set model. Better in-sample
and out of sample fit despite the large increase in the number of parameters argue for the use of
the consideration set model over the more parsimonious probit. Consumers tend to screen
alternatives using attributes they are already familiar with and an exploratory analysis was
conducted to demonstrate how the price threshold can be associated with other variables in a
22
posterior analysis. After controlling for their effect in forming consideration sets, the importance
of the screening attributes in the final choice among acceptable alternatives is smaller than that
indicated by a standard probit model.
6. Discussion
In this paper we introduce a model of attribute based consideration sets and estimate it
with data from a discrete choice conjoint study.
We find support for the formation of
consideration sets and for the conjunctive model in particular. The use of consideration sets is
pervasive with 92% of respondents using this heuristic to manage the complexity of the choice
problem. The empirical results also show that consumers screen alternatives using attributes that
are well known as opposed to the new and novel attributes also included in this study. Some
attributes appear to be only used in forming consideration sets, while others are used in both
stages of the model: screening and final product choice.
Although previous research has
documented the improved fit of consideration set models and statistical biases that result from
ignoring them, our method can apply a variety of screening rules and handle realistic sized
problems. Our ability to capture a non-compensatory screening process and make attribute level
inferences is due to the unique model formulation and estimation technique.
A key element in our model formulation is accurately navigating the likelihood of the
data where only the final choice is observed. Past researchers have attempted to isolate the
consideration set probability Pr(C) in equation (3) to create a smooth likelihood surface. This
approach requires a complex partition of the error space corresponding to the 2J-1 consideration
sets and the choice probability within each set (see Mehta, Rajiv and Srinivasan 2002).
Alternatively, the consideration set probability can be estimated by treating each of the 2J-1
23
partitions as a model parameter (see Chiang, Chib and Narasimhan 1999). Our approach is to
not marginalize the consideration set probability, but to specify the joint event of consideration
set formation and choice, Pr(i,C). The advantage of this approach is that it allows us to model
consideration set formation based on various compensatory and non-compensatory decision
rules.
This research highlights the benefits of and the need for more research on extended
models of choice (Ben-Akiva, et. al. 1999). In this research we have demonstrated that the data
is consistent with a two-stage decision process utilizing decision heuristics to form consideration
sets. In an exploratory analysis we show that the level of the price threshold was related to a
camera usage question thought to encompass varying concerns and interests related to pursuing
photography. It may very well be that the use of any decision heuristic, which decision heuristic
(conjunctive or disjunctive), and/or which attributes to screen are also related to individual
concerns and interests that may vary within an individual across usage occasions or choice
contexts. For example, choosing a camera as a gift for a loved one may result in more effortful
decision making and less reliance on decision heuristics; choosing a single use camera to take to
the beach may be entirely characterized by a lexographic decision process (e.g. the cheapest).
The model proposed in this paper permits testing of various decision-making processes using
revealed choice data.
The prevalence and improved fit of the conjunctive consideration set model illustrates the
important role that individual attributes play in consumer choice processes. This paper and
previous research demonstrate the importance of two-stage decision making and consideration
sets in consumer choice processes (see Shocker, Ben-Akiva, Boccara, and Nedungandi, 1991).
Our study extends this research and examines competing theories of compensatory, conjunctive,
24
and disjunctive consideration set formation within a single dataset and provides evidence in
favor of the conjunctive model.
The conjunctive model eliminates an alternative that is
unacceptable on any single attribute regardless of the other elements in the marketing mix. The
behavioral decision making literature hypothesizes that consumers make either brand based
decisions or attribute based decisions (Bettman, Luce, and Payne, 1998).
Brand based or
alternative based decisions involve an evaluation of all the attributes of an alternative to
determine its acceptability. Attribute based decisions permit the elimination of an alternative
based on a single attribute. The support for the conjunctive consideration set model indicates
that, at the consideration stage, consumers are making screening decisions based on the levels of
the product attributes.
Several improvements on the existing model are possible. First, attribute level thresholds
or the propensity to use one decision process over another in the structural heterogeneity model
can be parameterized as functions of explanatory variables. The Bayesian hierarchical model can
then be expanded by another level. In this data, there was very little evidence that consumers
used a disjunctive decision rule to form consideration sets. It may be that more than twelve
observations per individual are necessary to identify the parameters in this very flexible model.
The methodology presented in this paper can be extended to other choice contexts. The
conjoint exercise used in this paper involved six alternatives described on eight attributes plus a
"none" option. Conjoint researchers may be interested to know if the number of alternatives or
attributes influences respondents' use of consideration sets; this question can be studied using the
proposed model and an experimental design that varies the number of attributes and alternatives.
The model can also be applied to scanner panel datasets or any other panel dataset that captures
25
repeated purchase behaivor. Such a study would reveal the importance of individual attributes
and/or the use of consideration sets in an actual market setting.
Managers frequently define markets based on particular attributes (e.g. body style) or
price points, recognizing that certain consumers will only consider offerings with specific
attribute-levels, regardless of other elements of the product offering. Past methods of identifying
these attributes and levels have tended to rely on compensatory models that may not be a good
representation of the decision process. While compensatory models can often predict as well as
non-linear models, inferences for strategic decisions are often biased. Our model overcomes the
statistical and practical problems of estimating models of attribute based consideration sets.
26
Appendix: Estimation Algorithm
Equations (9a) – (9c) list the conditional distributions for estimating the choice model in (8a) –
(8b). The estimation for the conjunctive screening process, equation (10) is described first.
Extension to the disjunctive process, equation (11) is straightforward. Changes necessary to
estimate the compensatory process, equations (12) and (9a') – (9c') are then described. The
standard hierarchical Bayes probit model (Rossi, et. al. 1996) and structural heterogeneity
models (Yang and Allenby , 2000) have been described elsewhere. The augmented data problem
for the conjunctive model is:
yhij=1 if zhij > zhjk for all k such that I(xhik,γh)=1
(A1)
zhij = x 'hij βh+ εhij
(A2)
ε ~ N(0,σε=1)
I(xhij,γh)=1 if ∏m[I(xhijm > γhm)] = 1
(A3)
Where 'h' indexes the respondent, 'i' the choice set, 'j' the alternative in the choice set, and 'm' the
attributes in each alternative. Heterogeneity on the individual household parameters is introduced
hierarchically with the following distributions:
βh ~ MVN( β , Σβ)
γhm ~ Multinomial(H, θm1, …..,θmn) for discrete attributes γ1, … γ10
γh11 ~ N( γ , σ γ2 ) for γ11 a continuously scaled attribute
The hierarchy is completed by specifying the prior distributions on the hyper-parameters:
β ~ MVN(0,100I)
Σβ ~ IW(ν,∆)
θm ~ Dirichlet(α)
for m = 1,…,10 corresponding to discrete attributes
γ ~ N(0,100)
σ 2γ ~ IG(a,b)
for γ11, continuous attribute
where IW is the inverted Wishart distribution, ν = k+8, and ∆ = νI. θm is a vector of dimension
'n' where 'n' indexes the possible values for γm, and α is a conforming vector with each element
set equal to 6. IG is the inverted gamma distribution and a=10 and b=.1.
A Gibbs type sampler is constructed to draw from the conditional posteriors described in (9a) –
(9c) as well as the distributions for heterogeneity.
27
1. Generate zhi|yhi, Xhi, I(Xhi,γh), βh, σε=1 for i=1, …. I and h=1,…,H
Start with j=1,
if yhij = 1 then zhij ~ TN(xhij'βh, σε=1, zhij > zhjk for all k such that I(xhik,γh)=1)
if yhij=0 and I(xhij,γh)=1 then zhij ~ TN(xhij'βh, σε=1, zhij < zhim, where yhim=1)
else zhij ~ N(xhij'βh, σε=1)
increment j and return to top.
2. Generate γhm|zh, yh, Xh for m = 1, ….10 (discrete attributes) and h=1,…,H
a
Conditioned on the current values for zh and the observed data, the set of allowable γ hmn
p
must be identified from the set of possible γ hmn
. Here, 'n' indexes the number of possible
a
values of γ for each attribute as listed in table 5 (e.g. n=4 for Body Style). Let I( γ hmn
)=1
indicate that γhmn is a member of the set of allowable γhm.
Start with n = 1, test each j=1,…J, i=1,….I
a
If yhij = 1 and ∏m[I(xhijm > γhm*)] = 0, then I( γ hmn
)=0
If yhij = 0 and ∏m[I(xhijm > γhm*)] = 1, then
a
)=0
If zhij > zhik where yhik=1, then I( γ hmn
a
)=1
Else, I( γ hmn
increment n and return to the top.
p
a
Here γhm* includes γ hmn
for the appropriate attribute. Now choose γhm from γ hmn
using a
Metropolis-Hastings type step:
a
γhm = γ hmn
with probability =
a
I(γ hmn
)θ mn
a
∑ I(γ hmn )θ mn
n
This is an example of the "Griddy Gibbs" as described by Tanner (1993).
28
3.
Generate γh11|zh, yh, Xh for (continuous attribute) h=1,…,H
This is a more traditional random walk Metropolis-Hastings step, but the candidate γ (n)
h11 ,
must be checked to make sure it is a member of the allowable set.
(o)
Let γ (n)
h11 = γ h11 + e where e ~ N(0, σe=.5). The value σe is chosen to ensure the acceptance
rate is approximately 50%. γ (o)
h11 is the value of γh11 from the previous iteration of the
Gibbs sampler.
(n)a
(n)
(o)
Determine I( γ (n)a
h11 ) following the steps outlined in 2; if I( γ h11 ) = 0, then γ h11 = γ h11 . If
(n)
I( γ (n)a
h11 )=1, then accept γ h11 with probability:
min:
2
exp[-(2σ 2γ ) −1 ( γ − γ (n)
h11 ) ]
,1
2
exp[-(2σ 2γ ) −1 ( γ − γ (o)
h11 ) ]
4. Generate βh | Xh, zh for h=1, …, H
βh ~ MVN(b, (Xh ‘Xh + Σβ-1)-1)
b = (Xh ‘Xh + Σβ-1)-1 (Xh‘zh + Σβ-1 β )
5. Generate β | {βh}, Σβ
β ~ MVN( b , ((Σβ/H)-1 + (100I)-1)-1)
H
b = ((Σβ/H)-1 + (100I)-1)-1)( Σβ-1 ∑ β h + (100I)-1(0))
h =1
6. Generate Σβ | {βh}, β
Σβ ~ IW(ν + H ,∆ +
H
∑ (β
h =1
h
− β )′( β h − β ) )
29
7. Generate θm| {γhm} for m=1,…,10 (discrete attributes)
Define shmn = 1 if γhm = γhmn or shmn = 0, otherwise. Then,
θm ~ Dirichlet(
H
∑s
h =1
H
hm1 +α1, …., ∑ s hmn +αn)
h =1
See Allenby, Arora, and Ginter (1998) for additional details on drawing θm.
8. Generate γ | {γh11}, σ 2γ
γ ~ N(c, d)


 H

γ h11  
2
∑

σγ
100

 h =1
(0) + 2
c =  2
σ + 100
σ γ + 100  H  
 γ
 



d=
100σ 2γ
σ 2γ + 100
9. Generate σ 2γ |{γh11}, γ
H
σ ~ IG(H/2 + a, {b + ½ ∑ (γ h11 − γ ) 2 }-1 )
2
γ
h =1
We now describe the changes necessary to estimate the compensatory consideration set model.
In our application of the model, the deterministic component of utility Vhij = xhij'βh and (A3)
from above equals:
I(xhij,γh)=1 if I(xhij'βh > γh) = 1
(A3')
where γh is a continuous scalar for each household with heterogeneity and prior distributions:
γh ~ N( γ , σ γ2 )
γ ~ N(0,100)
σ 2γ ~ IG(a,b)
Steps 3 and 7 from the previous procedure are no longer needed. Only steps 2 and 4 need to be
altered:
30
2a.
Generate γh|zh, yh, Xh, βh for h=1,…,H
Again, conditioned on the current values of zh and βh and the observed data, the key is to
draw γh from the set of allowable values. The allowable values of γh define consideration
sets such that the chosen alternative always has the maximum value of z.
If yhij = 1 then xhij'βh > γh; let B* = min(xhij'βh for all i and j where yhij = 1)
If yhij = 0 and zhij > zhik where yhik = 1, then xhij'βh < γh;
let A* = max(xhij'βh for all i and j with yhij = 0 and zhij > zhik where yhik = 1)
Then A* < γh < B* defines the allowable range for γh. A* requires that all chosen
alternatives be part of the consideration set. B* requires that any alternative that is not
chosen but has a value of z greater than the chosen alternative, be excluded from the
consideration set.
Following Chib and Greenberg (1995) we perform a Metropolis-Hastings step by
drawing γh directly from the allowable range of the prior; in our case the prior is the
heterogeneity distribution and we draw from the truncated normal distribution:
γh ~ TN( γ , σ γ2 , A*<γh<B*)
4a. Generate βh | yh, Xh, zh for h=1, …, H
Generate a candidate βh:
β hc ~ MVN(b, (Xh ‘Xh + Σβ-1)-1)
b = (Xh ‘Xh + Σβ-1)-1 (Xh‘zh + Σβ-1 β )
Define A* and B* as in step 2a with β hc . If A* > B* reject β hc , draw a new candidate. If
A*<B* then βh = β hc .
This is an example of acceptance/rejection sampling.
31
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34
Table 1
Camera Attributes and Levels
Attribute/Level
A Basic body style and standard
features
A1 Low
A2 Medium
A3 High
B Mid-roll Change1
B1 None
B2 Manual
B3 Automatic
C Annotation1
C1 None
C2 Pre-Set List
C3 Customized List
C4 Custom Input method 1
C5 Custom Input method 2
C6 Custom Input method 3
D Camera Operation Feedback1
D1 No
D2 Yes
E Zoom
E1 None
E2 2X
E3 4X
F Viewfinder
F1 Regular
F2 Large
G Camera Settings Feedback
G1 None
G2 LCD
G3 Viewfinder
G4 Both LCD & Viewfinder
Available in
"Low" Body
Style
Available in
"Medium"
Body Style
Available in
"High" Body
Style
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
1
×
×
×
×
×
×
Unique to APS camera at time of the study; all other features available on both APS and 35mm
cameras at the time of the study.
35
Figure 1
Example for Specifying Cut-offs for Conjunctive & Disjunctive Model
Discrete Attributes
Attribute
Zoom Lens
Levels
Recoded
Values (x)
If γ = -0.5:
Probability
of Each
Cut-off
None
0
γ(1)= -0.5
θ(1)
2x
1
γ(2)= 0.5
θ(2)
4x
2
γ(3)= 1.5
θ(3)
γ(4)= 2.5
θ(4)
Scenario #1:
Screening
rule:
Possible
Cut-offs
Scenario #2:
Screening
rule:
I(x > γ) = 1
I(0 > -0.5) = 1
I(1 > -0.5) = 1
I(2 > -0.5) = 1
If γ = 0.5:
All attribute levels
are acceptable
I(x > γ) = 1
I(0 > 0.5) = 0
I(1 > 0.5) = 1
I(2 > 0.5) = 1
Camera must have
2x zoom lens to be considered
For the conjunctive model, if γ = -0.5 then Zoom Lens is not
being used to screen alternatives.
Scenario #3:
Screening
rule:
If γ = 1.5:
Scenario #4:
Screening
rule:
I(x > γ) = 1
I(0 > 1.5) = 0
I(1 > 1.5) = 0
I(2 > 1.5) = 1
If γ = 2.5:
Camera must have 4x
zoom lens to be considered
I(x > γ) = 1
I(0 > 2.5) = 0
I(1 > 2.5) = 0
I(2 > 2.5) = 0
None of the attribute levels
are acceptable
For the disjunctive model, if γ = 2.5 then Zoom Lens is not
being used to screen alternatives.
37
Table 2
Model Results
In Sample
Model
HB Probit
Compensatory
Conjunctive
Disjunctive
Structural Heterogeneity
1
LMD
-3468.8
-3449.6
-2990.8
-3586.2
-3012.9
Hit 2
Probability
0.525
0.527
0.583
0.528
0.583
Predictive
Hit
Probability
0.391
0.393
0.418
0.39
0.417
Hit3
Frequency
266
267
276
267
277
1
Log marginal density of the data calculated using the importance sampling method of
Newton and Raftery (1991, p. 21).
2
Posterior mean of predicted probability for selected alternative. Calculated from a
sample of 300 posterior point estimates.
3
Posterior mean of correct predictions for two observations for each respondent; a total of
604 hold-out choice sets. Calculated from a sample of 300 posterior point estimates.
38
Table 3
Part-worth Estimates for the Probit Model
Attribute
Level
Body Style
Low Body
Medium Body
High Body
Mid-Roll Change Manual Change
Automatic Change
Pre-set List
Annotation
Customized List
Input Method 1
Input Method 2
Input Method 3
Beta
Posterior
Mean
1
2
3
4
5
6
7
8
9
10
0.325
2.843
2.224
0.059
0.178
0.362
0.678
-1.036
0.767
-0.985
Attribute
Level
Camera Operation
Feedback
Operation Feedback
2x Zoom
Zoom
4x Zoom
Large Viewfinder
Viewfinder
Settings Feedback LCD
Viewfinder
LCD & Viewfinder
ln(price)
Price
Beta
Posterior
Mean
11
12
13
14
15
16
17
18
0.357
1.122
1.477
-0.158
0.759
0.757
0.902
-0.71
Covariance Matrix - Σβ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
19.605
17.952
15.303
-0.548
-2.077
-0.575
-0.584
-8.252
-1.921
-5.553
-0.808
-0.207
-0.811
-1.394
-0.434
-0.289
-0.472
-2.789
2
20.258
17.674
-0.906
-2.232
-0.309
-0.366
-7.975
-1.52
-5.628
-0.67
0.651
0.54
-1.054
0.06
0.035
-0.042
-3.459
3
17.845
-1.218
-2.153
-0.333
-0.356
-6.735
-1.285
-4.753
-0.665
0.895
1.08
-0.859
0.324
0.161
0.12
-3.278
4
1.803
0.507
0.097
0.147
0.371
0.139
0.432
0.061
-0.009
-0.225
-0.011
-0.075
0.011
0.019
0.144
5
1.4
0.171
0.231
1.134
0.395
0.854
0.173
0.022
-0.08
0.149
0.04
0.044
0.057
0.287
6
0.629
0.255
0.3
0.237
0.238
0.019
0.106
0.225
0.051
0.106
0.083
0.109
-0.045
7
0.685
0.387
0.26
0.224
0.033
0.121
0.173
-0.021
0.12
0.067
0.118
-0.054
8
9
10
11
12
13
14
15
16
17
18
5.392
0.974
2.734
0.494
0.134
0.418
0.592
0.167
0.129
0.255
1.22
1.096
0.726
0.225
0.167
0.251
0.231
0.181
0.179
0.173
0.12
3.315
0.347
-0.056
-0.021
0.487
0.04
0.083
0.084
0.923
1.055
0.115
-0.008
0.064
0.092
0.123
0.106
0.046
1.42
1.07
0.122
0.343
0.298
0.321
-0.412
2.21
0.13
0.454
0.336
0.427
-0.52
0.947
0.141
0.119
0.074
0.142
0.966
0.525
0.526
-0.216
0.976
0.538
-0.206
0.937
-0.192
1.079
Estimates in bold have more than 95% of the posterior mass away from 0.
39
Table 4
Part-worth Estimates for the Conjunctive Model
Attribute
Level
Body Style
Low Body
Medium Body
High Body
Mid-Roll Change Manual Change
Automatic Change
Pre-set List
Annotation
Customized List
Input Method 1
Input Method 2
Input Method 3
Beta
Posterior
Mean
1
2
3
4
5
6
7
8
9
10
-0.34
0.577
0.024
0.257
0.169
0.289
0.604
-0.369
0.655
-0.619
Attribute
Level
Camera Operation
Feedback
Operation Feedback
2x Zoom
Zoom
4x Zoom
Large Viewfinder
Viewfinder
Settings Feedback LCD
Viewfinder
LCD & Viewfinder
ln(price)
Price
Beta
Posterior
Mean
11
12
13
14
15
16
17
18
0.413
0.841
1.231
-0.139
0.374
0.373
0.52
0.056
Covariance Matrix - Σβ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
16.183
15.035
14.839
-1.35
-1.903
-0.637
-0.667
-6.744
-2.246
-5.42
-0.641
0.443
0.106
-1.292
-0.648
-0.65
-0.662
-2.338
2
15.918
15.176
-1.218
-1.821
-0.527
-0.531
-6.458
-2.069
-5.306
-0.479
0.499
0.427
-1.256
-0.611
-0.745
-0.717
-2.463
3
16.419
-1.398
-1.854
-0.591
-0.604
-6.301
-2.022
-5.148
-0.589
0.491
0.387
-1.289
-0.494
-0.736
-0.704
-2.464
4
1.88
0.625
0.117
0.198
0.653
0.319
0.662
0.1
0.035
-0.101
0.057
0.005
0.125
0.15
0.137
5
1.523
0.148
0.194
0.956
0.481
0.876
0.169
-0.051
-0.179
0.154
0.07
0.104
0.112
0.233
6
0.632
0.241
0.349
0.229
0.318
0.007
-0.005
0.112
0.065
0.045
0.043
0.088
0.048
7
0.742
0.399
0.26
0.305
0.009
0.006
0.08
-0.047
0.079
0.059
0.113
0.042
8
4.828
1.191
2.752
0.451
-0.119
0.084
0.616
0.28
0.341
0.381
0.957
9
1.269
1.073
0.205
-0.048
0.014
0.281
0.125
0.143
0.141
0.258
Estimates in bold have more than 95% of the posterior mass away from 0.
40
10
11
12
13
14
15
16
17
18
3.892
0.287
-0.161
-0.234
0.695
0.186
0.302
0.285
0.813
1.122
0.126
-0.015
0.022
0.087
0.065
0.077
0.057
1.056
0.557
-0.049
0.097
0.105
0.122
-0.184
1.653
-0.112
0.139
0.056
0.161
-0.25
1.213
0.102
0.065
0.014
0.209
0.864
0.365
0.38
0.007
0.903
0.422
0.011
0.878
0.026
0.774
Table 5
Threshold Estimates for the Conjunctive Model
Posterior Means
Attri bute
Body Style
Levels
Recoded
Values
Low
0
Medium
1
High
2
Possible
Cut-offs
Probability of
Each Cut-off
γ 1(1)= -0.5
θ1 =.598
γ 1(2)= 0.5
θ2 =.324
γ 1(3)= 1.5
θ3 =.059
γ 1(4)= 2.5
θ4 =.015
γ 2(1)= -0.5
Mid Roll -
None
0
Change
Manual
1
Auto
2
Annotation1
Annotation2
Annotation3
None
0
Pre-set
1
Custom
2
None
0
Input #2
1
None
0
Input #3
1
Attri bute
Annotation4
θ5 =.926
γ 2(2)= 0.5
θ6 =.027
γ 2(3)= 1.5
θ7 =.028
γ 2(4)= 2.5
θ8 =.020
γ 3(1)= -0.5
θ9 =.940
γ 3(2)= 0.5
θ10 =.022
γ 3(3)= 1.5
θ11 =.018
γ 3(4)= 2.5
θ12 =.020
γ 4(1)= -0.5
θ13 =.960
γ 4(2)= 0.5
θ14 =.020
γ 4(3)= 1.5
θ15 =.020
γ 5(1)= -0.5
θ16 =.961
γ 5(2)= 0.5
θ17 =.019
γ 5(3)= 1.5
θ18 =.020
Levels
Recoded
Values
None
0
Input #4
1
Operation
None
0
Feedback
Feedback
1
Zoom Lens
Viewfinder
None
0
2x
1
4x
2
Regular
0
Large
1
41
Probability of
Each Cut-off
γ 6(1)= -0.5
θ19 =.961
γ 6(2)= 0.5
θ20 =.019
γ 6(3)= 1.5
θ21 =.020
γ 7(1)= -0.5
θ22 =.947
γ 7(2)= 0.5
θ23 =.034
γ 7(3)= 1.5
θ24 =.020
γ 8(1)= -0.5
θ25 =.808
γ 8(2)= 0.5
θ26 =.140
γ 8(3)= 1.5
θ27 =.032
γ 8(4)= 2.5
θ28 =.019
γ 9(1)= -0.5
θ29 =.961
γ 9(2)= 0.5
θ30 =.019
γ 9(3)= 1.5
θ31 =.019
γ 10(1)= -0.5
θ32 =.783
γ 10(2)= 0.5
θ33 =.157
Settings
None
0
Feedback
LCD
1
Viewfinder
2
γ 10(3)= 1.5
θ34 =.021
Both
3
γ 10(4)= 2.5
θ35 =.020
γ 10(4)= 3.5
θ36 =.019
Continuous Attribute:
-ln(price)
All estimates have more than 95% of the posterior mass away from 0.
Possible
Cut-offs
γ11 = -6.269
Figure 2
Distribution of Price Threshold
Conjunctive Model
70%
59%
60%
50%
41%
40%
30%
30%
20%
20%
10%
10%
0%
<$276
<$356
<$425
42
<$499
>$499
Figure 3
Proportion of Respondents
Screening on Each Attribute
Conjunctive Model
45%
41%
40%
40%
35%
30%
25%
22%
19%
20%
15%
10%
8%
6%
4%
5%
4%
5%
4%
4%
*Indicates attributes only available on APS cameras at the time of the study.
43
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