Choice Set Logistic Regression Analysis With
Screening Rules
Marco R. Steenbergen
Dominik Hangartner
Paper prepared for the Annual Meeting of the
Swiss Political Science Association
St. Gallen, Switzerland, January 9-11, 2009
Abstract
Manski (1977) proposed that choice procedes in two stages: First, decision makers narrow down the universe of alternatives into a choice
set; next, they choose an alternative from wthin this set. We adapt
the basic model to allow for both compensarory and non-compensatory
screening rules in the rst or consideration stage. We apply the model
to vote choice in the 2002 Dutch parliamentary elections using issue
proximity on four issues as screening rules. We estimate two types of
non-compensatory screening: (1) a conjunctive screening rule whereby a
party must perform satisfactory on all four issues and (2) a disjunctive
rule that requires a party to perform well on at least one issue. The
results show that a compensatory rule performs best, followed by the
conjunctive rule. The disjunctive rule performs worst.
Institut fur Politikwissenschaft, Universitat Bern, Lerchenweg 36, CH-3009 Bern,
Switzerland. Email: [email protected] or [email protected].
1
Introduction
Choice is an essential aspect of politics. It is no wonder, then, that political
scientists have devoted considerable eort in modeling choice behavior (Alvarez and Nagler 1998; Born 1991; Burden 1999, Dow and Endersby 2004,
Glasgow 2001; Lacy and Burden 1999;. Quinn, Martin, and Whitford 1999;
Whitten and Palmer 1996). What has been missing from the literature so
far, however, is an emphasis on choice set models. Choice set models view
choice as a two-stage process, whereby decision makers (e.g. voters) rst
narrow down the universe of alternatives into a choice set (the consideration
stage) and then select one alternative from within this choice set (the choice
stage). The idea of choice set modeling goes back to a seminal paper by
Manski (1977), which inspired a great deal of interest in consumer and marketing research (e.g. Ben-Akiva and Boccara 1995; Gilbride and Allenby 2004)
and transportation economics (e.g. Basar and Bhat 2005) but not political
science. Nevertheless, we believe that choice set models oer a number of
distinct advantages that could benet political science. These include psychological realism through an explicit consideration of bounded rationality, a
exible specication of the size and composition of choice sets, the ability to
specify dierent choice mechanisms in dierent stages, and the relaxation (to
some extent) of the independence-of-irrelevant-alternatives (IIA) assumption
(see Steenbergen, Hangartner, and Wilson 2007).
In this paper, we extend the Manksi-type choice set model by allowing for
both compensatory and non-compensatory screening rules in the consideration
stage. Here screening rule refers to a criterion that should be satised before
an alternative is added to the choice set. Two types of non-compensatory
screening rule are being considered|a conjunctive rule, whereby several criteria
need to be satised simultaneously, and a disjunctive rule, whereby at least one
criterion should be satised before an alternative is considered. We dene these
rules, as well as the compensatory rule, in probabilistic terms. By allowing for
screening rules to be non-compensatory, we add still greater psychological
realism to choice set modeling.
We apply our models to the 2002 Dutch parliamentary elections, focusing
on ve alternatives and on four screening rules in the form of issue distances.
The application is meant as an illustration of what, in principle, is a very
general model. Still the Dutch case is interesting because the complexity of
the 2002 elections|a new party entrant in the form of List Pim Fortuyn, a
political assassination (of Fortuyn), collapse of the previous government, and
a tumultuous campaign (as a result of the former events)|makes it likely that
2
Dutch voters would rely on, potentially non-compensatory, screening rules to
make their vote decision more manageable.
The paper is organized as follows. In the next section, we lay out our
Manski-type choice set model with screening rules, which is based on work
by Ben-Akiva and Boccara (1991) and Gilbride and Allenby (2004). We then
discuss our application and the results. We conclude by discussing possible
extensions of our model that allow for a better incorporation of voter heterogeneity (but that so far have eluded successful estimation).
Choice Sets with Screening Rules
Imagine a voter trying to decide between m > 2 political parties. Most extant
vote models, including multinomial logit (e.g. Witten and Palmer (1996),
conditional logit (e.g. Alvarez and Nagler 1998), multinomial probit (Alvarez
and Nagler 1998; Lacy and Burden 1999; Quinn, Martin, and Whitford 1999),
and mixed multinomial logit (e.g. Glasgow 2001), assume that voters consider
all of these alternatives simultaneously.1 This assumption seems reasonable
when m is small. But all that we know about bounded rationality (e.g. Simon
1955, 1985) suggests that the assumption is unrealistic when m is large. How
can one realistically model vote choice in that case?
In an important paper on random utility theory, Charles Manski (1977)
proposed that choice can be conceived of in terms of a selection process from
within choice sets that are probabilistic in nature. Alternatives are selected
from the universe of available options and grouped together in choice or consideration sets that vary in size and composition. Each choice set has a particular probability of being selected by the decision maker. Then an alternative is,
again probabilistically, selected from within each choice subset. The probability
that decision maker q chooses alternative i 2 M is thus
qi =
X
C 2G
q (i jC ) q (C )
(1)
Here M is the universal choice set, i.e. the choice set consisting of all m
available alternatives. Further, G is the set of all non-empty subsets (C )
of M , i.e. G is the powerset of M , which consists of 2m 1 alternatives.2
An exception is McFadden's (1978) nested logit model (see Born [1991] for an application to political science).
2
The model assumes that the choice set contains at least one element, so that the empty
choice set is ruled out. In electoral research, this turns out to be a mild assumption because
1
3
The probability that individual q has choice set C is given by q (C ) and the
probability that alternative i is chosen conditional on choice set C is q (i jC ).
We call the model in (1) the probablistic choice set model or PCSM.
Consideration Stage
The consideration stage is where alternatives get screened and either accepted
into or rejected from the consideration set. One can think of this process as
one in which decision makers apply thresholds that the alternatives have to
exceed before they are considered further (see Hauser and Wernvelt 1990).
The very idea of screening is that it simplies decision making. Rather than
having to evaluate all alternatives on all attributes, the decision maker utilizes
a small number of criteria by which some alternatives can be eliminated. No
further eort is spent on those alternatives in the choice stage. The nature
of the screening mechanism can vary. The screening rules can be applied
in a compensatory manner, allowing for trade-os between dierent criteria.
They may also be applied in a non-compensatory manner, whereby trade-os
do not occur. Here an alternative is considered only if it meets at least one
criterion. Note that the usage of compensatory or non-compensatory screening
rules does not distract from the fact that, across the consideration and choice
stages, the Manski (1977) model clearly species a non-compensatory decision
making logic.
Compensatory Screening Rules
Imagine a decision maker who uses P attributes, xqip , to screen alternatives.
One way in which this can be done is to weight these attributes, create a linear
prediction, and determine if this prediction exceeds some cuto, . Letting
Iqi be an indicator that takes on the value 1 if alternative i meets the cuto
for decision maker q and 0 otherwise, the screening rule can be formalized as
Iqi = 1 if Vqi =
X
p
p xqip > (2)
where Vqi is the overall value of the alternative. This screening rule is compensatory in nature in that, in principle, strong attributes of an alternative
can oset weak ones. Allowing for heterogeneity among decision makers, we
abstention|the decision not to choose|can easily be included among the alternatives (e.g.
Paap et al. 2005).
4
include
a stochastic component to P
the screening rule, which now becomes
P
x
+
>
=
>
qi
qi
p p qip
p p xqip . Assuming that the stochastic component follows a standard logistic distribution, the probability that q
includes i in his or her choice set is given by
!qi = P r Iqi = 1
=
1 + exp 1
p p xqip
P
(3)
The idea of compensatory screening rules is consistent with several models
of choice behavior. For example, Hauser and Wernvelt (1990) have argued
that decision makers will consider an alternative only if its expected utility exceeds the cognitive cost (see also Roberts and Lattin 1991). Thus decision
makers try the alternative out by considering a small number of attributes
and determining if the alternative is worth incurring further information processing costs that would result from considering other attributes. Swait and
Ben-Akiva (1987) proposed a theory of \random constraints" that limit the
available alternatives (see also Ben-Akiva and Boccara 1995). These can operate in a compensatory as well as non-compensatory manner. An example of
a compensatory constraint is that of a consumer who wishes to simultaneously
consider the price and quality of a product and explicitly allow for trade-os
between these attributes. Basar and Bhat's (2005) study of airport usage in
the San Francisco Bay Area is an example of a compensatory screening rule.
Non-Compensatory Screening Rules
Much of human decision making appears to be of a non-compensatory nature
(see Baron 2007) and this holds also for electoral behavior (Herstein 1981;
Lau and Redlawsk 2006; Taber and Steenbergen 1995). One can build a noncompensatory screening rule by considering a sequence of indicator functions,
one for each attribute that is used in the screening, such that
Iqip = 1 if xqip > p
(4)
(assuming, without loss of generality, that xqip is coded such that higher value
are better). Under a conjunctive non-compensatory rule, all of the indicator
functions have to equal 1, i.e. all screening criteria have to be met. Thus,
5
Y
p
Iqip = 1
(5)
Such a principle is akin to the elimination-by-aspects model of Tversky (1974),
although that model would stipulate continuation of the screening process until
only one alternative remains. Under a disjunctive rule, at least one criterion
should be met so that
X
p
Iqip 1
(6)
We can again specify these rules in probabilistic terms by allowing a stochastic component to enter in each rule so as to capture heterogeneity. Thus,
xqip + qip > p or qip > p xqip . Assuming that the stochastic components are independent across alternatives and attributes, and follow a standard
logistic distribution, we can model the probability that Iqip = 1 in the following
manner:
P r Iqip = 1
=
1
1 + exp p
xqip
(7)
Under the conjunctive rule, the inclusion probability of i for q is then
!qi = Pr Iqi 1 = 1 \ Iqi 2 = 1 \ \ IqiP = 1
=
Y
p 1 + exp
1
p xqip
(8)
Under the disjunctive rule, the inclusion probability is given by
!qi = 1 Pr Iqi 1 = 0 \ Iqi 2 = 0 \ \ IqiP = 0
= 1
Y
p 1 + exp
1
xqip p
6
(9)
Choice Set Probabilities
Assuming that the inclusion probabilities are independent from one another,
the probability of a particular choice set, C , is given by
q (C ) =
Q
i 2C !qiQ j 2= C (1 !qj )
1
i (1 !qi )
Q
(10)
where the denominator is a normalizing constant to accommodate the exclusion of the empty choice set. The assumption of independence among the
!s amounts to saying that the decision maker decides on the acceptability of
an alternative independent from all other alternatives. While one could surely
relax this assumption, it is consistent with a large number of psychological
decision models that make similar assumptions (see Baron 2007).
Choice Stage
In the choice stage, we use a conditional logit specication (McFadden 1973)
to capture the conditional probability of selecting i from C . Let Uqi bet the
utility of alternative i 2 C . We assume that q is a utility maximizer, so that i
is selected if and only if Uqi > Uqj 8j 6= i 2 C . We assume that U contains an
observed component, V , and an unobserved stochastic component, . Thus,
the utility maximization assumption may also be stated as qj qi < Vqi Vqj .
Assuming that the errors are independent and follow a Gumbel distribution, it
is easily demonstrated that
q (i jC ) =
exp(Vqi )
j 2C exp(Vqj )
P
(11)
for i 2 C and q (i jC ) = 0 for i 2= C .
The choice stage model is completed by specifying the nature of V . For
the sake of simplicity, we assume a simple linear function such that Vqi = 0 zqi .
In principle, the predictors included in zqi may overlap with those used in the
screening stage.
Estimation
In principle, model estimation can proceed via maximum likelihood. The loglikelihood function is given by
7
` =
XX
q
i
wqi ln qi
(12)
where wqi takes on the value 1 if q chooses i and 0 otherwise. This can be
optimized with respect to the , , and parameters. However, a practical
problem with this log-likelihood function is that it contains near-at regions,
causing diculties with numerical optimization. In this paper, we solve this
problem via a Bayesian approach.3 We stabilize the posterior by including
weakly informative priors in the form of independent Cauchy (0, 2.5) distributions (see Gelman et al. 2008). If anything, these priors make the estimates
more conservative, pulling them closer toward zero. Optimization then occurs
via the BFGS algorithm.
Application
We apply the choice-set model with screening rules to the 2002 Dutch parliamentary elections. These elections stand as some of the most tumultuous
in Dutch history and as a stark departure from electoral politics as usual (e.g.
Van Holsteyn and Irwin 2003). The sitting government, the so-called \purple coalition" consisting of PvdA, VVD, and D66, collapsed just a short time
before scheduled elections. The main opposition consisted of the ChristianDemocrats (CDA) and a new party, the List Pim Fortuyn (LPF). Historically,
the CDA had been a central player in coalition government, but it had been
standing at the side lines since it suered heavily at the polls in 1994. The
LPF was an o-shoot of Leefbaar Nederland, a party that had done remarkably
well in local elections and was marked by erce opposition to immigration and
a tough stance on crime. Leefbaar Nederland, however, lacked the charisma
of Pim Fortuyn who, as leader of the LPF, became the most successful populist leader of post-WWII Dutch politics and arguably of Europe. LPF became
the second largest party in the 2002 elections, surpassed only by the CDA.
Fortuyn himself did not get to enjoy his party's success, however, as he was
assasinated just a week before the elections. The ocial campaign was halted
3
See Steenbergen, Hangartner, and Wilson (2007) for an alternative approach that is
based on a mixture of derivative-free genetic algorithms and standard numerical optimization methods. Note however, that our approach, although Bayesian, is closely related to
Firth's (1993) penalized likelihood approach, where the maximum a posteriori estimates are
proportional to the product of the likelihood times negative the square root of the expected
Hessian (Jereys prior).
8
after this event, but unocially it continued as opinion leaders increasingly
singled out the left as the culprit behind the assassination. A common claim
was that the \bullet came from the left," implying that the criticisms of Fortuyn by PvdA leader Melkert and leader of the Greens, Paul Rosemoller, had
created an atmosphere in which a political assassination had become possible.
Wether because of these allegations or because of the decreasing popularity
of the government, the PvdA suered badly in the 2002 elections. It was put
in opposition, as CDA, LPF, and VVD forged a new coalition, which was itself
to be short-lived.
The unusually tumultuous nature of the 2002 elections|the fact that
this was denitely not politics as usual|is likely to have increased decision
diculty for Dutch voters. In such an environment, the use of screening
rules is particularly likely, wether they would be of a compensatory or noncompensatory nature. Political issues may have played a particularly large role
in the screening process. Van Holsteyn and Irwin (2003) have argued that
political issues such as asylum seekers played an important role in the 2002
elections. We go one step beyond this claim by arguing that Dutch voters used
these issues to screen the alternatives, doing this in either a compensatory
or non-compensatory manner. More specically, we expect that proximity
on the issues of euthanasia, income dierences, asylum seekers, and crime
played a critical role in screening the alternatives, where we focus on ve such
alternatives: PvdA, CDA, VVD, D66, and LPF.
Data
Our data come from the 2002-03 Dutch Parliamentary Election Study (Irwin,
Van Holsteyn, and Den Ridder 2003). This study employs a cluster-sample
design of municipalities with face-to-face interviews in pre- and post-election
waves. A total of 90 municipalities was selected. The number of respondents
that completed the pre- and post-election surveys was 1574.4
Ideally, our model would consider the complete set of available alternatives. Unfortunately, we can focus only on ve alternatives because the Dutch
Parliamentary Election Survey did not ask respondents to place a number of
parties on all of the issues that we consider in our screening rules. The ve
parties for which we have data include the four largest parties and garnered
80.5 percent of the vote and 82 percent of the seats.
4
The post-election wave is critical for our model, since estimation is based on the actual
vote choices that respondents made.
9
Specication and Measures
Respondents where asked to place themselves and the PvdA, CDA, VVD, D66,
and LPF on four issues using 7-point scales. For euthanasia the scale ranged
from (1) forbid euthanasia to (7) allow euthanasia; for income dierences it
ranged from (1) dierences should be increased to (7) dierences should be
decreased; for asylum seekers it ranged from (1) admit more asylum seekers
to (7) send back as many asylum seekers as possible; and for crime it ranged
from (1) the government is too tough on crime to (7) the government should
get tougher on crime. We computed proximity measures by computing jR
P j, where R reects the respondent's position on an issue and P the party's
position as perceived by the respondent.
We stipulate that voters used their proximity to the parties on the four
issues as a screening rule. In the compensatory case, a linear composite of the
four proximity measures has to exceed a certain threshold. In the conjunctive
non-compensatory case, a party has to pass muster on all four issues before
it is included in the choice set. Finally, in the disjunctive compensatory case,
the party has to be acceptable on at least one issue before it is added to the
choice set.
In the choice stage, we include two attributes of the voters, to wit satisfaction with the government and religion. Government satisfaction should help, in
particular, to decide between parties that were in and out of the government.
Religion should help to decide between the CDA and the other alternatives.
In addition to these substantive predictors, we add party-specic constants to
the choice stage model. For identication purposes, the coecients for CDA
are set to zero; this alternative thus serves as the baseline.
Results
Conjunctive Rule
Table 1 shows the results of a choice-set model with a conjunctive screening
rule. All four issue proximity measures play a signicant role in the consideration stage with the cuto on income dierences and crime being more stringent
than those on euthanasia and asylum seekeres. Considering that issue proximity ranges from -6 (most distant) to 0 (most proximate), our estimates suggest
that parties as much as four units removed from voters on euthanasia would
still be acceptable, but that a distance of only two units would be tolerated on
income dierences and crime.
It might be relatively dicult for multiple parties to pass a conjunctive
10
Table 1: PCSM With Conjunctive Screening Rule
Variables
Coef SE
jz j Stage
Dist: Euthanasia
4:73 0:40 11:94
1
Dist: Income
2:55 0:23 11:02
1
Dist: Asylum
3:26 0:32 10:33
1
Dist: Crime
2:59 0:26 10:10
1
GovSat x PvDA
0:95 0:36 2:66
2
GovSat x VVD
0:97 0:35 2:81
2
GovSat x D66
0:79 0:43 1:85
2
GovSat x LPF
0:48 0:39 1:25
2
Rel x PvDA
4:77 0:97 4:93
2
Rel x VVD
3:68 0:88 4:17
2
Rel x D66
4:56 1:16 3:92
2
Rel x LPF
4:62 1:08 4:26
2
PvDA
0:51 0:73 0:71
2
VVD
0:61 0:66 0:93
2
D66
2:13 0:95 2:24
2
LPF
0:66 0:65 1:01
2
N
633
max(posterior)
704
Correct Pred
80:0%
AP CP
43:4%
Table entries are Bayesian BFGS estimates with standard errors based on
quadratic approximation of the posterior. \Correct Pred" equals one if the party with
the highest estimated probability is actually the party voter q voted for, and zero otherwise. AP CP is the average probability of correct prediction, i.e. the average predicted
probability that voter q votes for party i given she actually voted for this party.
Note:
11
screening rule. This is born out by the choice set probabilities shown in Figure
1. This gure shows a high degree of screening with the probability of considering all ve parties being quite low and the probability of only one party passing
muster being comparatively high. Under a conjunctive rule, the LPF and CDA
clearly stand out as parties with a particularly high inclusion probability, either
on their own or as part of a choice set including other alternatives.
The choice stage of the choice-set model with a conjunctive rule shows
signicant positive eects of government satisfaction for the parties in government. Thus, individuals who were satised with the \purple" coalition tended
to be more likely to select PvdA, VVD, or D66 than CDA from a choice set.
There is no evidence that the LPF beneted more from government dissatisfaction than the CDA. The choice stage model also reveals that religious
individuals where less likely to vote for PvdA. VVD, D66, or LPF compared to
CDA.
As an example, we illustrate the eect of the conjunctive screening rule on
party selection and vote shares with a small simulation study where we compare
the predicted probabilities as a function of minimal versus maximal distance
scores, while holding everything else constant. In the minimal distance condition, we replace each decision maker's distance measure score with zero for the
LPF|the new kid on the block (again, holding the distance measures for all
other four parties constant). Using the parameters estimated from the original sample, we predict vote shares and consideration probabilities. We perform
the same predictions under the maximal condition, where distance is set to its
maximal (absolute) value (-6) for the LPF. The dierences in predicted vote
shares and consideration probabilities under these two conditions are shown
in Table 2. Consistent with the parameter estimates, moving from one pole
to the other has a huge impact on both the consideration probability and the
(marginal) vote choice probability of the LPF. All consideration probabilities
and vote shares get more than tripled when the issue measure is moved from
the minimum to the maximum (of the absolute value). The eect is biggest
for income dierences, but only slightly smaller for the other issues.
Disjunctive Rule
Results for the disjunctive model are shown in Table 3. The consideration
stage of this model shows acceptability thresholds that are statistically indistinguishable from zero or well above it. Given that the issue distances range
from -6 to 0, these estimates suggest that no party passes any of the screening
rules except by chance. Because of this, the disjunctive choice set probabilities
12
0.08
0.06
0.04
0.02
0.00
PvdA, CDA, VVD, D66, LPF
CDA, VVD, D66, LPF
PvdA, VVD, D66, LPF
PvdA, CDA, D66, LPF
PvdA, CDA, VVD, LPF
PvdA, CDA, VVD, D66
VVD, D66, LPF
CDA, D66, LPF
PvdA, VVD, LPF
CDA, VVD, D66
PvdA, D66, LPF
PvdA, VVD, LPF
PvdA, VVD, D66
PvdA, CDA, LPF
PvdA, CDA, D66
PvdA, CDA, VVD
D66, LPF
VVD, LPF
VVD, D66
CDA, LPF
CDA, D66
CDA, VVD
PvdA, LPF
PvdA, D66
PvdA, VVD
PvdA, CDA
LPF
D66
VVD
CDA
PvdA
Figure 1: Distribution of Choice Sets for PCSM with Conjunctive Screening Rule
13
Table 2: Predicted Probabilities Based On Conjunctive Screening Rule
PrC (0) PrC ( 6) Pri (0) Pri ( 6)
Dist: Euthanasia
0:44
0:12
0:18
0:05
Dist: Income
0:55
0:03
0:23
0:01
Dist: Asylum
0:50
0:04
0:20
0:02
Dist: Crime
0:47
0:02
0:19
0:01
PrC () is short for Pr(i 2 C j Dist = ), and Pri () is short for Pr(i
All predicted probabilities correspond to the LPF.
Note:
j
Dist = ).
are more nearly uniformly distributed as can be seen in Figure 2.
The results for the choice stage are quite similar to those observed in the
conjunctive model. This is a pattern we observe across the dierent models,
suggesting that the choice stage results are relatively robust to changes in
screening logic.
Again, Table 4 illustrates the impact of the issue distances. Consistent
with the rather small absolute value of the estimated screening parameters,
moving from one pole to the other on the distance measures has|compared
to the estimates from the conjunctive model|much smaller eects on both
the consideration probability and the vote choice probability of the LPF.
Compensatory Rule
Table 5 shows the results for the compensatory rule. Here the totality of
the four issues inuences whether a party is considered or not. It is clear
that income dierences and crime carry the greatest weight in the screening
of alternatives. The threshold of inclusion is -2.75, which means that the
weighted um of the issue distances cannot be too large.
The resulting choice set probabilities are displayed in Figure 3. This gure
is similar to that for the conjunctive rule, except that the probability of more
comprehensive choice sets tends to be larger due to the possibility of trade-os
between the issues.
The simulated consideration probabilities and voting probabilities are displayed in Table 6. Here, the magnitude of the dierences in the predicted
probabilities are somewhere between the ones from the conjunctive and the
disjunctive models. Again, the dierence is largest for the income dierences
issue.
14
Table 3: PCSM With Disjunctive Screening Rule
Variables
Coef SE jz j Stage
Dist: Euthanasia
1:72 0:58 2:96
1
Dist: Income
0:02 0:31 0:06
1
Dist: Asylum
0:26 0:29 0:91
1
Dist: Crime
0:90 0:43 2:11
1
GovSat x PvDA
0:93 0:26 3:52
2
GovSat x VVD
0:88 0:26 3:44
2
GovSat x D66
0:67 0:30 2:20
2
GovSat x LPF
0:36 0:28 1:31
2
Rel x PvDA
4:34 0:72 6:06
2
Rel x VVD
3:37 0:64 5:29
2
Rel x D66
4:28 0:85 5:03
2
Rel x LPF
4:08 0:73 5:62
2
PvDA
0:33 0:52 0:63
2
VVD
0:40 0:50 0:79
2
D66
1:45 0:64 2:25
2
LPF
1:08 0:54 1:99
2
N
633
max(posterior)
736
Correct Pred
78:8%
AP CP
39:8%
Table entries are Bayesian BFGS estimates with standard errors based on
quadratic approximation of the posterior. \Correct Pred " equals one if the party with
the highest estimated probability is actually the party voter q voted for, and zero otherwise. AP CP is the average probability of correct prediction, i.e. the average predicted
probability that voter q votes for party i given she actually voted for this party.
Note:
Table 4: Predicted Probabilities Based On Disjunctive Screening Rule
PrC (0) PrC ( 6) Pri (0) Pri ( 6)
Dist: Euthanasia
0:54
0:46
0:17
0:15
Dist: Income
0:69
0:37
0:22
0:13
Dist: Asylum
0:67
0:41
0:21
0:13
Dist: Crime
0:58
0:40
0:18
0:13
PrC () is short for Pr(i 2 C j Dist = ), and Pri () is short for Pr(i
All predicted probabilities correspond to the LPF.
Note:
15
j
Dist = ).
0.04
0.03
0.02
0.01
0.00
PvdA, CDA, VVD, D66, LPF
CDA, VVD, D66, LPF
PvdA, VVD, D66, LPF
PvdA, CDA, D66, LPF
PvdA, CDA, VVD, LPF
PvdA, CDA, VVD, D66
VVD, D66, LPF
CDA, D66, LPF
PvdA, VVD, LPF
CDA, VVD, D66
PvdA, D66, LPF
PvdA, VVD, LPF
PvdA, VVD, D66
PvdA, CDA, LPF
PvdA, CDA, D66
PvdA, CDA, VVD
D66, LPF
VVD, LPF
VVD, D66
CDA, LPF
CDA, D66
CDA, VVD
PvdA, LPF
PvdA, D66
PvdA, VVD
PvdA, CDA
LPF
D66
VVD
CDA
PvdA
Figure 2: Distribution of Choice Sets for PCSM with Disjunctive Screening Rule
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Table 5: PCSM With Compensatory Screening Rule
Variables
Coef SE jz j Stage
Dist: Euthanasia
0:36 0:06 6:39
1
Dist: Income
0:74 0:08 9:50
1
Dist: Asylum
0:47 0:07 6:54
1
Dist: Crime
0:64 0:08 7:68
1
Constant
2:75 0:30 9:22
1
GovSat x PvDA
0:87 0:29 2:99
2
GovSat x VVD
0:87 0:28 3:15
2
GovSat x D66
0:73 0:34 2:14
2
GovSat x LPF
0:25 0:30 0:85
2
Rel x PvDA
3:78 0:71 5:36
2
Rel x VVD
2:90 0:66 4:40
2
Rel x D66
3:65 0:83 4:41
2
Rel x LPF
3:74 0:79 4:72
2
PvDA
0:62 0:61 1:02
2
VVD
0:70 0:56 1:24
2
D66
1:96 0:77 2:55
2
LPF
0:27 0:55 0:50
2
N
633
max(posterior)
674
Correct Pred
80:8%
AP CP
46:8%
Table entries are Bayesian BFGS estimates with standard errors based on
quadratic approximation of the posterior. \Correct Pred" equals one if the party with
the highest estimated probability is actually the party voter q voted for, and zero otherwise. AP CP is the average probability of correct prediction, i.e. the average predicted
probability that voter q votes for party i given she actually voted for this party.
Note:
17
Figure 3:
0.08
0.06
0.04
0.02
0.00
PvdA, CDA, VVD, D66, LPF
CDA, VVD, D66, LPF
PvdA, VVD, D66, LPF
PvdA, CDA, D66, LPF
PvdA, CDA, VVD, LPF
PvdA, CDA, VVD, D66
VVD, D66, LPF
CDA, D66, LPF
PvdA, VVD, LPF
CDA, VVD, D66
PvdA, D66, LPF
PvdA, VVD, LPF
PvdA, VVD, D66
PvdA, CDA, LPF
PvdA, CDA, D66
PvdA, CDA, VVD
D66, LPF
VVD, LPF
VVD, D66
CDA, LPF
CDA, D66
CDA, VVD
PvdA, LPF
PvdA, D66
PvdA, VVD
PvdA, CDA
LPF
D66
VVD
CDA
PvdA
Distribution of Choice Sets for PCSM with Compensatory
Screening Rule
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Table 6: Predicted Probabilities Based On Compensatory Screening Rule
PrC (0) PrC ( 6) Pri (0) Pri ( 6)
Dist: Euthanasia
0:52
0:22
0:19
0:09
Dist: Income
0:67
0:06
0:24
0:03
Dist: Asylum
0:59
0:17
0:21
0:07
Dist: Crime
0:55
0:07
0:20
0:03
PrC () is short for Pr(2 C j Dist = ), and Pri () is short for Pr(i
All predicted probabilities correspond to the LPF.
Note:
j
Dist = ).
Model Comparison
Which of the three models performs best? We can answer this question in a
number of ways. First, we can compare the maximum of the posteriors. This
statistic not only captures t but also penalizes for lack of parsimony.5 This is
important because the compensatory model contains one more parameter than
the two non-compensatory models. Judged by the maximum of the posteriors,
it is clear that the compensatory model out-performs the conjunctive model,
which in turn out-performs the disjunctive model.
Another criterion by which we can judge model performance is to consider
the incidence of correct predictions. One way to do this is to predict a choice
for that alternative that receives the highest predicted probability by equation
(1) and to contrast this with the actual vote choice. By this standard, all
screening rules perform relatively well. However, the best performance is again
for the compensatory rule, followed by the conjunctive and disjunctive rules,
respectively. Another criterion is the average probability of correct prediction
(APCP), which is the average predicted probability that voter q voted for party
i given that she actually voted for this party. The APCP is also highest for
the compensatory rule, again followed by the conjunctive and disjunctive rules,
respectively.
This is so because the posterior combines the log-likelihood and the log of the priors, one
for each estimated parameter. The way we have specied the Cauchy priors, their logarithm
is always negative. Thus adding an extra parameter means adding an additional negative
term that works against improvements in the log-likelihood.
5
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Conclusions
Summary
In this paper, we have presented a Manski-type choice-set model with dierent
kinds of screening rules that allow decision makers, in our case voters, to
narrow down the eld of alternatives, in our case political parties, they consider.
Using the 2002 Dutch parliamentary elections as an illustration, we showed how
a set of issues can be used in compensatory or non-compensatory fashion to
screen alternatives. The evidence presented in this paper suggests that, for
the 2002 Dutch elections, a compensatory model performs best. Here the
issues are weighted and then summed into an overall evaluation, which allows
poor party performance on one issue (i.e. a large distance from the voter on
that issue) to be compensated by good performance on another. However, a
conjunctive non-compensatory rule performed also reasonably well. Under this
rule, a party would be considered only if it performed satisfactory on all issues,
thus ruling out trade-os between issues. The worst performance came from
a disjunctive rule that stipulates that a party should perform well on at least
one issue. This is a relatively easy criterion but, in order to accommodate the
data, the cut-os on the issues had to be made so strict that parties would
manage to be included in the choice set only by chance. This alone is sucient
reason to call into question the disjunctive model and to prefer the other two
models.
Model Extensions
The model described in this paper can be extended to accommodate voter heterogeneity in a more comprehensive manner. First, one could let the threshold
of acceptability vary across voters. For example the compensatory screening
rule could be written as
Iqi = 1 if
X
p
p xqip > q
with q N (; 2 ). This model suggests that voters have dierent thresholds
of acceptability, which can be considered realizations from a normal distribution
with a common threshold and a variance of 2 . Similar heterogeneity can
be specied in the cutos for the non-compensatory screening rules.
Second, one could allow for heterogeneity in the choice stage, either by
moving toward a multinomial probit specication (e.g. Alvarez and Nagler
20
1998; Lacy and Burden 1999; Paap et al. 2005; Quinn, Martin, and Whitford
1999) or a mixed logit specication (cf. Glasgow 2001). Such additional
heterogeneity is likely to make choice-set models more realistic but also more
dicult to estimate.
Another area in which choice set models can be extended is to incorporate
auxiliary data. In this paper, we have only used actual choice data to draw
inferences about screening rules. One could, however, easily use information
search data to model the screening process in the consideration stage. Indeed,
Meyer (1979) and Richardson (1982) provide denitions of consideration sets
that are closely tied to information gathering behavior. Since information
search data have somewhat of a legacy in political science (e.g. Herstein
1981; Lau and Redlawsk 2006), this may be a particularly fruitful direction to
go.
Regardless of how one wishes to extend the basic model, we hope this
paper has demonstrated that something can be learnt from conceptualizing
electoral choice behavior in terms of choice sets with screening rules. Their
greater psychological realism and the additional insights they provide seem
worth the price in estimation costs and render choice set models a useful tool
for political science.
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