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doi:10.1016/j.electstud.2004.10.009    How to Cite or Link Using DOI (Opens New Window)  
Copyright © 2005 Elsevier Ltd All rights reserved.

An entropy measure of uncertainty in vote choicestar, open

Jeff GillCorresponding Author Contact Information, E-mail The Corresponding Author

University of California, Davis, Political Science, One Shield Avenue, Davis, CA 95616, USA

Available online 20 January 2005.


Abstract

We examine voters’ uncertainty as they assess candidates’ policy positions in the 1994 congressional election and test the hypothesis that the Contract with America reduced voter uncertainty about the issue positions of Republican House candidates. This is done with an aggregate evaluation of issue uncertainty and corresponding vote choice where the uncertainty parameterization is derived from an entropy calculation on a set of salient election issues. The primary advantage is that it requires very few assumptions about the nature of the data. The entropic model suggests that voters used the written and explicit Republican agenda as a means of reducing issue uncertainty without substantially increasing time spent evaluating candidate positions.

Keywords: Entropy measure; Uncertainty; Vote choice; Heteroscedastic probit; Spatial preferences


Article Outline

1. Introduction: information, uncertainty, and voting
1.1. Objectives
2. A spatial model of vote choice and uncertainty
2.1. Entropy as a measure of political uncertainty
3. Entropy in information theory
4. An entropy model of vote choice
4.1. Heteroscedasticity and the likelihood function
4.2. Testing the entropy model of uncertainty
4.3. Findings
5. Conclusion
Appendix A. Data Appendix
References


1. Introduction: information, uncertainty, and voting

Complexity, obfuscation, vagueness, and uncertainty are permanent features of American electoral politics (Aldrich et al., 1982). As modern social and economic issues become more complicated and legislatures increasingly dodge controversial and divisive votes, a greater number of poorly explained and widely misunderstood issues are thrust onto the ballot (Gafke and Leuthold, 1979). Complex or lengthy ballot measures can also confuse voters (Gerber, 1996). These are commonly worded in stern legal language and many voters lack the time or analytical background necessary to determine policy consequences. For example, the 1996 ballot initiative in California to eliminate state supported affirmative action programs, Proposition 209, confused many voters on its actual intent (Los Angeles Times, October 31, 1996, p. A22). The words “affirmative action” do not appear anywhere in the bill's language, and the bill is disingenuously entitled the “California Civil Rights Act.”

Candidates often have motivations to make their issue positions deliberately vague (Downs, 1957, Franklin, 1991, Glazer, 1990 and Shepsle, 1972). This can be out of a desire not to alienate certain constituency groups or simply that the candidate lacks a strongly defined position. There is evidence of this phenomenon even in very high salience elections. Bob Dole made deliberately vague statements on his abortion position in the 1996 presidential election after conservatives in the Republican party successfully placed strong anti-abortion rights language into the party platform at the San Diego convention (San Francisco Chronicle, August 5, 1996, p. A1). In campaigns it is a well-worn strategy to recast the language of an opponent's issue position into something less popular with voters (Nimmo, 1970), and with frequent iterations of this practice has a generally confusing effect on the electorate.

So why do many voters endure high levels of uncertainty and cast their ballot? The answer is elusive, though well studied in the literature (Downs, 1957, Ferejohn and Fiorina, 1974, Olson, 1965, Wolfinger and Rosenstone, 1980, Abramson and Aldrich, 1982, Burnham, 1987, Palfrey and Rosenthal, 1985 and Teixera, 1992). Feddersen and Pesendorfer, 1999, Feddersen and Pesendorfer, 1997 and Feddersen and Pesendorfer, 1996 find that “asymmetric information” critically changes the calculus of voting by sufficiently blurring the distinctions between candidates for some citizens such that they are better off not voting at all. This work gives two findings of interest here. First, lower levels of (“private”) information possessed by voters affect their ability or willingness to use perceived issue distance to sort-out preferred vote choices. Second, the observation that many vote, but some do not, implies that levels uncertainty must vary across individuals.

Rather than address the effect of uncertainty on the decision to turnout, this work evaluates the effect of uncertainty on the vote choice of those that do turnout. This focus stems from the observation that vote choice models developed without incorporating a voter uncertainty component miss an important criterion of the vote choice: “Models that rely on the assumption of complete information may offer a misleading view of the democratic process.” (Ordeshook, 1986, p. 187).

While considerable attention has been paid to voting under uncertainty in two-candidate elections where voters abstain due to indifference or ignorance, we still do not have a complete theory leading to equilibrium conditions (Coughlin, 1990) across a more complete range of circumstances. Furthermore, Hinich (1977) found that even a small amount of uncertainty is enough to destroy Black's elegant medium voter equilibrium result. Bartels (1988) used the “Don't Know” response as a measure of uncertainty, but this does not indicate the level of uncertainty for respondent provided responses. Also, Coombs and Coombs (1976) identify a difference in “Don't Know” answers: researcher created scale dependencies, and actual issue ambiguity. Furthermore, this issue ambiguity can be from ignorance, uncertainty, or just indecision (Sanchez and Morchio, 1992).

Alvarez and Franklin (1994) note the additional problem that there is not a consensus in the literature on how uncertainty is measured, although they were able to design a survey that provided very specific information on issue-level uncertainty to reveal subtle patterns in the form of the responses. More recently, Alvarez (1998) takes a comprehensive look at this question focusing on presidential elections. He finds that voters do, in fact, seek and acquire sufficient information to make informed vote decisions, at least at the presidential level.

Bartels (1988) studies the 1980 presidential election and finds that voters in general dislike uncertainty and this uncertainty, measured as variance around candidate positions, can exceed perceived spatial distances between the candidates. Like the Bartels uncertainty model, the approach developed here rests on the work of Enelow and Hinich, 1981 and Shannon, 1948 in that issue positions are not singular and fixed. Instead they are represented by probability distribution functions with manifest observations (samples). It is precisely the quality of these samples, which provide information about the underlying true distribution of candidate positions, that affects levels of voter certainty. Such a theoretical model leads directly to a statistical framework where the mean is used as a point estimates of issue position and the variance represents uncertainty.

1.1. Objectives

The approach developed here leverages a unique aspect of the 1994 midterm election to evaluate how voters interpret differing levels of information about House candidate positions. In this election the Republican leadership in the House developed a well-defined and explicit issue-agenda. Republican incumbents and challengers signed the Contract with America promising that if elected to a majority they would initiate eight congressional reforms and bring ten specified issues (eventually requiring many more actual bills) to the House floor for a vote within the first 100 days of the 104th Congress. The voting behavior model here focuses on the uncertainty difference between Republican and Democratic candidates in this election, and asks whether there exist differing levels of information uncertainty between the party that controlled the House of Representatives for forty years, and a minority party making explicit electoral promises.

Unlike previous measures of uncertainty, which come from individual survey based indicators or respondent self-assessments, the entropy form of uncertainty uses measures of respondent uncertainty that are produced by the format and content of the survey as assessed by the full sample. For purely methodological reasons researchers have been forced into the model assumption that all people have the exact same indifference to uncertainty and that this applies to all candidates evaluated. By replacing the variance term with an entropy term in the respondent's utility function, uncertainty is tied to the structure of the question choice and the full sample evaluation of the particular issue complexity. Therefore, since it is unrealistic to assume homogeneity of uncertainty threshold in respondents, the gain in moving away from the statistical variance term as a measure of issue uncertainty to an entropy measure of issue uncertainty can be substantial.

2. A spatial model of vote choice and uncertainty

In this section a model of candidate vote choice is developed using the entropy concept as a relative measure of voter uncertainty between Democratic and Republican congressional candidates. This model is based on the well-known proximity spatial model developed by Downs, 1957, Davis et al, 1970, Hinich and Pollard, 1981, Enelow and Hinich, 1981, Enelow and Hinich, 1984a, Enelow and Hinich, 1984b and Enelow and Hinich, 1990, and others. Like many works, the model relies on the assumption that voters seek a candidate choice that minimizes the squared Euclidean distance between their own issue preference and the perceived position of the candidate. However, issue distance is not a complete explanation of voter preferences and empirical models generally need to incorporate non-policy and attitudinal covariates as well.

Define a utility function, Uij, that specifies the value of candidate j to voter i such that utility reduces for increasing distance between the perceived candidate position on issues and the respondents own position on these issues, and for increasing uncertainty about placing the candidates positions. So voters dislike candidates who are both further from their preferred policy positions and less clear on where they stand on these issues. Obviously we would also want to include terms that estimate the effects from non-issue contributions such as party identification, political interest, media exposure, and demographics. If we assume that the assessment of issue difference is squared, then this utility function for the ith respondent and the jth takes the form:


Click to view the MathML source(1)

where Cijk is respondent i's placement of candidate j on policy issue k, Rik is respondent i's self-placement on issue k, and Oij is the effect of non-issue factors that contribute to respondent i's utility for candidate j. Voters are assumed to see candidates’ issue positions at any moment in time as imprecise approximations due to candidate personality, campaign, and psychological effects, so we can therefore model this perception as realizations from a probability distribution (Bartels, 1988, Enelow and Hinich, 1981 and Shepsle, 1972). Now Uij is also a realization from some underlying probability distribution, and voters will make decisions based on their expected utility (Grafstein, 1991). This produces, with minor algebra (Bartels, 1988, p. 711), an expression for the expected utility of candidate j to voter i:

Click to view the MathML source(2)

where E[Cijk] is the expected position of the candidate and Var[Cijk] is the variance of the voters perception, both for issue k. So voters will prefer a candidate with a higher expected utility subject to the possibility of making an error in placing the candidate's issue positions.

The methodological problem with (2) is that the Var[Cijk] term cannot be directly observed except with specialized panel data. Bartels finesses this problem by assuming that uncertainty is a function of observable candidate and respondent characteristics provided in the survey data. He then posits a variance threshold (constant across candidates, issues, and respondents) which defines when respondents will make placements, and estimates from a linear model a quantity that is proportional to the probability that the respondent refuses to make a placement. Unfortunately this approach requires the assumption of homogeneity of the uncertainty (variance) threshold, and leads to a more complex, two-stage estimation procedure.

2.1. Entropy as a measure of political uncertainty

Suppose instead of estimating respondent uncertainty by respondent and candidate characteristics we use the empirical distribution of placements across the survey. For example, if an issue was particularly complicated or it was one in which candidates generally avoided taking specific positions, then we would expect the distribution of candidate placements by respondents to be somewhat uniform across the range of answers reflecting aggregate voter uncertainty. Conversely, if the issue was one that was either very clear to the electorate or one where candidates tend to take unambiguous positions, then we would expect a highly concentrated distribution of respondent placements. What is required is a single value that can be substituted into (2) to replace Var[Cijk] based on this empirical measurement of uncertainty from the survey.

As an example of the effects of uncertainty, Fig. 1 shows how dramatically different respondents view candidate ideology depending on perceived uncertainty and party in the 1994 congressional election (data from the American National Election Study). The different empirical distributions demonstrate the value of measuring uncertainty. Not surprisingly, respondents indicating that they were not certain about ideology placement tended to place the candidates in the middle of the scale, clustered around the moderate category. Conversely respondents that were very sure about ideology placement were generally not reluctant to label candidates as more extreme. Fig. 1 also indicates that there is less variance in the distribution of ideology placement for Republicans than Democrats.


Enlarge Image
(63K)

Fig. 1. Ideology placement by certainty, respondent counts by 7-point scale.

The method proposed herein is to empirically assign the categorical probabilities (call these pi for the probability of being in the ith category) for a given issue from the observed response structure in the dataset, and then calculate the entropy for that issue. So the measure is based on the issue, characteristics of the candidates, the wording of the question, and the aggregated responses. Thus the decision uncertainty is constant across respondents but not across candidates or issues.

The entropy approach makes absolutely no assumptions about the distribution of the variance of uncertainty (Shannon, 1948, Jaynes, 1957 and Jaynes, 1968). Furthermore, empirical distributions that satisfy given linear constraints (side conditions), pi=1 in our case, concentrate around the maximum entropy distribution which satisfies the same constraints (Van Campenhout and Cover, 1981 and Robert, 1990). This finding, formalized as Jaynes’ Entropy Concentration Theorem (1968), provides a motivation for treating the unknown distribution of candidate uncertainty as an entropy term because it builds directly on Shepsle's (1972) idea that observed candidate positions are realizations from an underlying probability distribution.

The entropy concept has been applied to a wide variety problems concerning uncertainty, yet there is no known application of entropy to measuring the lack of knowledge political actors have about each other's positions and intentions. Uncertainty about candidates and ballot issues is a fundamental problem for voters. Information is necessarily limited by transmission means (Shannon, 1948), candidates and issues can be confusing at the source (Franklin, 1991 and Gerber, 1996), and voters have limited time and attention and might search for shortcuts for even the most salient election issues (Carmines and Stimson, 1980). The search for a measure of uncertainty is also disadvantaged in that uncertainty in individuals is a subjective and changing phenomenon (Alvarez and Franklin, 1994).

The entropy approach to measuring voter uncertainty not only has few assumptions, but also applies to survey data where there are no specific uncertainty questions (Alvarez and Franklin, 1994). Furthermore, there is no need to estimate a uniform variance threshold (Bartels, 1988), and also no need to exploit some unique attribute of the data (Franklin, 1991) if one exists.

3. Entropy in information theory

There are many definitions of information in various literatures but all of them have the same property of distinction from a message. If a message is the physical manifestation of information transmission (language, character set, salutations, headers/trailers, body, etc.), then the information is the substantive content independent of transmission processes. For example, in baseball the third base coach transmits a message to the runner on second base through the use of hand gestures. The information contained in the hand gestures is independent of the actual gesture to the base runner. If you could condense a set of messages down to a minimal finite length, then the information content would be the number of digits required by these alternate sets of messages. In this case information and message would be identical. Information and message are never exactly the same. For example, the DNA strand required to uniquely determine the biological characteristics of a human being contains far more codes than is minimally sufficient to transmit the required genetic information. In information theory entropy is a measure of the average amount of information required to describe the distribution of some random variable of interest (Cover and Thomas, 1991). More generally, information can be thought of as a measure of reduction in uncertainty given a specific message (Shannon, 1948 and Ayres, 1994, p. 44).

Suppose we wanted to identify a particular voter by serial information on this person's characteristics. We are allowed to ask a consecutive set of yes/no questions (i.e. like the common guessing game). As we get answers to our series of questions we gradually converge (hopefully, depending on our skill) on the desired voter. Our first question is: does the voter reside in California. Since about 9.7% of voters in the United States reside in California, a yes answer gives us different information than a no answer. Restated, a yes answer reduces our uncertainty more than a no answer because a yes answer eliminates 90.3% of the choices whereas a no answer eliminates 9.7%. If Pi is the probability of the ith event (residing in California), then the improvement in information is defined as:


Click to view the MathML source(3)

The probability is placed in the denominator of improvement.formula because the smaller the probability, the greater the investigative information supplied by a yes answer. The log function is required to obtain some desired properties (discussed below), and is justified by limit theorems (Bevensee, 1993, Jaynes, 1982 and Van Campenhout and Cover, 1981). The log is base-2 since there are only two possible answers to our question (yes and no) making the units of information bits. In this example Hi=−log2(0.097)=3.321928 bits, whereas if we had asked: does the voter live in Arkansas then an affirmative reply would have increased our information by: Hi=−log2(0.01)=6.643856 bits, or about twice as much. However, there is a much smaller probability that we would have gotten an affirmative reply had the question been asked about Arkansas. What Shannon (1948) refined was the idea that the “value” of the question was the information returned by a positive response times the probability of a positive response. So the value of the ith binary-response question is just:

Click to view the MathML source(4)

And the value of a series of n of these questions is:

Click to view the MathML source(5)

where fi is the frequency distribution of the ith yes answer and k is an arbitrary scaling factor that determines choice of units. The arbitrary scaling factor makes the choice of base in the logarithm unimportant since we can change this base by manipulating the constant.1 The function (5) was actually first introduced in the early Twentieth Century, although similar forms were referenced by Boltzmann (1877) where fi is the probability that a particle system is in microstate i.

We can see that the total improvement in information is the additive value of the series of individual information improvements. So in our simple example we might ask a series of questions narrowing down on the individual of interest. Is the voter in California? Is the voter registered as a Democrat? Does the voter reside in an urban area? Is the voter female? The total information supplied by this vector of yes/no responses is the total information improvement in units of bits since the response-space is binary. Its important to remember that the information obtained is defined only with regard to a well-defined question having finite, enumerated responses.

The link between information and entropy is often confusing due to differing terminology and usage in various literatures. In the thermodynamic sense, information and entropy are complementary: “gain in entropy means loss in information – nothing more” (Lewis, 1930). So as uncertainty about the system increases, information about the configuration of the microstate of a system decreases. In terms of information theory, the distinction was less clear until Shannon (1948) presented the first unambiguous picture. To Shannon the function (5) represented the expected uncertainty. In a given communication, possible messages are indexed m1,m2,,mk, and each have an associated probability p1,p2,,pk. These probabilities sum to one, so the scaling factor is simply 1. Thus, the Shannon entropy function is the negative expected value of the natural log of the probability mass function, with a scaling factor c:


H=−cpiln(pi)(6)

Shannon defined the information in a message as the difference between the entropy before the message and the entropy after the message. If there is no information before the message, then a uniform prior distribution2 is assigned to the pi and entropy is at its maximum.3 In this case any result increases our information. Yet, if there is certainty about the result, then a degenerate distribution describes the mi, and the message does not change our information level.4 So according to Shannon, the message produces a new assessment about the state of the world and this in turn leads to the assignment of new probabilities (pi) and a subsequent updating of the entropy value.

The simplicity of the Shannon entropy function belies its theoretical and practical importance. Shannon (1948, Appendix 2) showed that (6) is the only function that satisfies the following three desirable properties:

1. H is continuous in (p1,p2,,pn).

2. If the pi are uniformly distributed, then H is at its maximum and is monotonically increasing with n.

3. If a set of alternatives can be reformulated as multiple, consecutive sets of alternatives, then the first H should equal the weighted sum of the consecutive H values: H(p1,p2,p3)=H(p1,1−p1)+(1−p1)H(p2,p3).

These properties mean that the Shannon entropy function is well behaved with regard to relative information comparisons.5

There has been considerable debate about the nature of information with regard to entropy (Ayres, 1994, Jaynes, 1957, Ruelle, 1991, Tribus, 1961 and Tribus, 1979). A major question arises as to whose entropy is it? The sender? The receiver? The transmission equipment? These are often philosophical debates rather than practical considerations as Shannon's focus was clearly on the receiver as the unit of entropy measure. Jaynes (1982), however, considers the communication channel and its limitations as the determinants of entropy. Some, like Aczél and Daróczy (1975), view entropy as a descriptor of a stochastic event, and the traditional thermodynamic definition of entropy is as a measure of uncertainty in microstates. For the purposes of this work, the Shannon formulation (6) is used.

4. An entropy model of vote choice

For each issue in which there exists in the survey a candidate assessment and a self-placement, it is possible to provide an entropy measure. The entropy term for a given issue is created from the empirically observed counts in each response category and then normalized into probabilities. Returning to the 1994 American National Election Study data, the Democratic candidates’ respondent placement distribution on support for Clinton is the normalized vector:


[0.077,0.014,0.000,0.899,0.000,0.007,0.003]

These are just the normalized probabilities that a respondent selected the corresponding categorization for the Democratic candidates perceived support of Clinton (from almost always supports Clinton to almost never supports Clinton). This produces an entropy value of HDemocrat, Support.Clinton=0.404 by applying (6). In contrast, a uniform distribution produces the maximum entropy value for a seven category discrete random variable: H=1.946. Note that like the variance, the entropy measure of uncertainty is “direction-less” since [0.5,0,0.5,0,0,0] and [0,0,0,0.5,0,0.5] return the same value. Unlike the variance, the entropy measure retains uncertainty in the placement of the scale, [0.5,0,0,0,0,0.5], [0,0,0.5,0,0.5,0,0] return the same value. In both of these latter cases, respondents should be equally uncertain as to where the candidates stand; just because the second candidate has closer modal values does not imply less uncertainty in the context of a 7-point scale.

For each of the K issue terms and candidates, an entropy function, Hjk is created to indicate the corresponding decrease in utility from uncertainty. We can directly substitute the entropy term for the variance term in (2) to give the expected utility of candidate j to voter i as:


Click to view the MathML source(7)

where the Hjk is not indexed by i since it comes from the entire dataset. Entropy and issue distance are additive here, so they must be standardized on the same scale for each issue. Furthermore, since every variable was standardized on the 7-point scale, relative comparisons across these covariates in the model results also make sense.

The highlighted difference between (7) and Bartels’ formulation in (2) is that the entropy measure reflects the full sample attitude about uncertainty varying by issue and candidate. The uncertainty component is provided by a measure of all respondents’ summed assessments through the candidate/issue entropy term in (7). Unless one is willing to assert that uncertainty is systematically related to observed respondent characteristics, then there is literally nothing in the data to obtain information about respondents’ level of candidate position uncertainty.

The entropy measure takes advantage of questions inserted into the ANES 1995 Pilot Study directed at assessing how certain a respondent is about the candidates positions on a subset of issues. Respondents could indicate whether they were “very certain,” “pretty certain,” or “not very certain” after answering a particular substantive question. This allows us to build an aggregate measure of uncertainty by candidate and by issue that captures the effects in the survey sample without assuming direct relationships to covariates. While an individual's level of uncertainty guides their vote choices, the entropy approach gives an aggregated picture of the vagueness or specificity from issues and candidates that feeds into that. Thus, it is a valuable component for the statistical model.

The form of (7) implies that a particular issue can reduce respondent utility either through squared issue distance from the candidate or from increasing the level of uncertainty expressed through the entropy term. The j and k subscript on the entropy term specify the corresponding candidate and issue distribution from which the entropy term is calculated. So this term is constant across respondents but not candidates and issues.

After calculating the complete utility function for two competing candidates, a measure of quality of the model and assumptions is the degree to which prediction agrees with observed vote choice. Specifically, we would expect for a particular voter i, if E[Ui1]>E[Ui2], then voter i selects candidate 1. The form of (7) incorporates the threshold existence property: issue based utility preference for one candidate over another is subject to mitigation by high levels of uncertainty (Brady and Ansolabehere, 1989), where the uncertainty component is measured here by entropy rather than the unobtainable statistical variance threshold.

A key feature of this structural specification is that the entropy and issue distance are normalized to be on the same scale and thus allowed to compete. The question we are asking, therefore, is whether issue distance is sufficient to overcome uncertainty, or whether uncertainty can be substantial enough to overwhelm issue distance. The empirical test of these questions involves building a statistical model in which each candidate is assessed across issues where the respondent places this candidate and themselves, and where we can obtain the entropy distribution from aggregate uncertainty controlling for demographics and other personal factors. Distance matters, therefore, because of both issue placements and candidate entropy. So for the issues assessed in this way (five actually, given the data at hand), there will be a Democratic term and a Republican term, which are actually directly comparable. The advantage to this approach is that it not only allows direct issue-by-candidate evaluation where the estimated coefficient provides an importance weight in the decision making process, but also that it avoids the problems described above in obtaining variance thresholds.

In both (2) and (7) the issues are equally weighted with regard to their contribution to total utility. This is not actually a requirement. If there was some reason for assuming that certain issues were more important or more topical than others, then deliberately specified salience weights could easily be assigned to each of the K issue terms reflecting their a priori relative importance. This is not done for the five issue distance variables considered below, because there is no substantive justification for claiming that any one of the policy areas supersedes the others in terms of relevance to the election and we will estimate their relative importance empirically (the coefficients can be thought of as posterior weights in our context). Also, these weights could be estimated in the more general context of a Bayesian hierarchical model where their values are stipulated as conditional on additional criteria such as individual voter characteristics. This approach is not particularly difficult if one is willing to adapt the tenets of Bayesian inference which have considerable power and elegance (Gill, 2002).

4.1. Heteroscedasticity and the likelihood function

Attitudes and assessments about political figures are likely to vary across respondents such that the residuals from a simple generalized linear model are heteroscedastic. A graphical diagnostic test of the model described above indicates that this is the case here. It is not surprising to find that respondents perceive choices and candidate issue differences in this particular election with non-uniform variation.

Because we are interested in an expected utility based choice between two candidates, a dichotomous model specification is appropriate. Heteroscedasticity is particularly troublesome in qualitative choice models as the resulting maximum likelihood estimates of the coefficients are often inconsistent (Amemiya, 1981 and Davidson and MacKinnon, 1984) and the asymptotic covariance matrix (generally computed from the inverse of the negative Hessian at the maximum likelihood coefficient values) is biased. Yatchew and Griliches (1984) also point out that the biases are particularly problematic when an explanatory variable is correlated with the heteroscedasticity (p. 138).

An effective way of compensating for heteroscedasticity in probit models is to identify explanatory variables that introduce widely varying response patterns and specifically associate them in the model with the dispersion term. This method, based on Harvey (1976), is illustrated in Greene (2003, p. 680–681), and was employed by Alvarez and Brehm, 1995 and Alvarez and Brehm, 1997 to address response heteroscedasticity from issue ambivalence. The functional form is changed from:


Click to view the MathML source(8)

where Φ is the standard notation for the cumulative standard normal distribution, and the exponent function in the denominator is just a convenience that prevents division by zero. This approach distinguishes between the standard treatment of the estimated coefficients β with corresponding matrix of explanatory observations X, and the set of dispersion determining estimated coefficients α with a corresponding matrix of explanatory observations Z. By this means the dispersion term is reparameterized to be a function of a set of coefficients and observed values that are suspected of causing the differences in error standard deviations: Click to view the MathML source The log likelihood of the probit model is now:

Click to view the MathML source(9)

where the inclusion of a separate parameterized dispersion component in (9) addresses the heteroscedasticity problem. A formal test for heteroscedasticity is a likelihood ratio test using the value of the log likelihood in the restricted model (the restriction is that α=0) and the unrestricted (9) evaluated at the maximum likelihood estimates of the parameters: β and α. The test is simply: LR=−2[restrictedunrestricted]=2[(Xβ,Zα)−(Xβ)], where LR is (as usual) distributed χ2 but with degrees of freedom equal to the number of columns of the Z matrix (the number of dispersion estimation parameters in the Harvey specification).

4.2. Testing the entropy model of uncertainty

The 1994 congressional election represents an interesting and perhaps unique opportunity to test the effect of uncertainty on vote choice. Republican House candidates signed the Contract with America on September 9, 1994 with the goal of capturing the majority in Congress by specifically identifying their policy positions through a set of ten promised House floor votes. It is reasonable to predict that a clearly defined, written agenda that is widely reported in the media increases voter certainty about the Republican candidates’ positions. We test this claim by applying the entropy model described above using survey data from the 1994 American National Election Study (data collected between November 9, 1994 and January 9, 1995).

The data include placements for 1795 respondents on several currently salient issues as well as standard demographic information. Fortunately, for five of these issue questions the respondent is asked not only to place themselves, but also to place the Republican and Democratic House candidates running in their district (both only if this is a contested race). The five issues are: support for Clinton, concern about crime, the government's role in providing economic opportunities to citizens, appropriate level of government spending, and the appropriate level of government involvement in the healthcare system. These variables are particularly useful since they are policy areas addressed by the 103rd Congress and the respondent places themself and the two candidates for the House. Each of these fifteen variables (five times each for the respondent, the Republican, and the Democrat) are rescaled to the seven-point format, and those responding “Don't Know” for the issue variables are placed at the midpoint of the scale in order to minimize their squared error distance and subsequent effect on the model (details provided in the Data Appendix). The scales are standardized such that higher values are those traditionally associated with Republican positions. For instance, the question that asks about increasing or decreasing government spending for the provision of domestic services was recoded to make “7” the response indicating government should provide far fewer services.

Four general demographic variables are included in the model: Education (in years), Gender, Race (white or non-white), and Age (in years). The model also includes two variables which are intended to capture the respondents’ levels of interest and information. The data include a question which asks the respondents to place themselves on a three-point scale in terms of their attention to recent campaigns. Respondent information is measured by mean-averaging respondents’ number of days reading the newspaper per week, number of days watching television per week, and number of days listening to news on the radio per week.

Three of the explanatory variables are specified as belonging to the Z matrix in (Xβ/e use for the exponential base of the natural logarithmsZα): Party Identification Scale, Political Interest Scale, and Education. It hypothesized that self-reported partisanship tends to vary the effects of issue-distance since respondents will incorporate party loyalties differently in assessing distances. For different reasons, levels of interest in politics and levels of education should alter the quantity and quality of political information that an individual obtains. All of these variables are subsequently centered in the analysis so that exp(Zα)=1 at the mean. This reduces the sensitivity of the β parameters to scale standardization.

Multiple imputation is used to replace the explanatory variable missing values resulting from refusals and survey errors (Little and Rubin, 1983 and Rubin, 1987). Cases where the House race is for an open seat were dropped since the survey therefore does not provide data on support for Clinton and vote on the Crime bill for non-incumbent candidates. These considerations and the treatment of the outcome variable (described below) reduce the utilized sample size to 1335.6

The outcome variable is candidate choice in the 1994 House election by party, where Democrats are coded 0, and Republicans are coded 1. Respondents who did not claim to have voted were excluded from the analysis, although very similar results were obtained by replacing their missing votes with their stated vote preferences and including these respondents in the sample. So there are exactly two utility functions from (7) for each respondent: UiD for the ith respondent's utility from the Democratic candidate, and UiR for the ith respondent's utility from the Republican candidate. The inferential value of this qualitative choice model is assessed by the frequency with which the respondent selects the candidate with the greater utility function. Table 1 summarizes the results from the heteroscedastic probit regression model, (9), on these data.

Table 1.

Parameter estimates for entropy model of vote choice 1994 ANES
Parameter estimateStandard error95% CI lower bound95% CI upper bound
Choice parameters (Xβ)
Intercept−0.47770.2096−0.8885−0.0669
Democratic support for Clinton0.01150.0063−0.00080.0238
Republican support for Clinton0.01120.0074−0.00340.0257
Democratic crime concern0.01020.0065−0.00250.0230
Republican crime concern−0.02060.0064−0.0331−0.0081
Democratic government help for disadvantaged0.02850.00800.01280.0442
Republican government help for disadvantaged−0.01550.0085−0.03220.0012
Democratic government spending0.09960.01850.06320.1359
Republican government spending−0.09040.0141−0.1180−0.0628
Democratic federal healthcare0.01500.00540.00450.0256
Republican federal healthcare−0.01450.0060−0.0262−0.0028
Media exposure−0.02020.0068−0.0336−0.0068
Gender−0.33320.0769−0.4838−0.1826
Race−0.26150.1010−0.4595−0.0635
Age0.00190.0021−0.00230.0061

Dispersion parameters (eZα)
Party identification scale0.19510.01680.16230.2280
Political interest scale−0.23070.0332−0.2957−0.1657
Education0.04950.02400.00250.0966
Likelihood ratio test: LR=82.35, p<0.001 for Click to view the MathML sourceOutcome variable: Democrats=0, Republicans=1
Correctly classified: 946/1335 (using the mean-criteria)
Residual deviance=1282.6

4.3. Findings

The heteroscedasticity likelihood ratio test indicates that we are justified in developing the Harvey heteroscedastic probit model for these data over a standard specification: a test statistic of 101 Click to view the MathML source is clearly sufficient evidence to reject the hypothesis that α=0. Furthermore, the residual deviance, 1282.6, is not in the tail of a χ2 distribution for 1317 degrees of freedom, justifying the model fit.

The five pairs of issue terms (the first ten terms after the constant in Table 1 are interpreted as follows. Take, for instance, Support for Clinton as the k=1 issue and j=1 for the Democratic House candidate, to be considered by the ith respondent. The first column of the X matrix (save for the column of ones) is constructed by: −(Ci11Ri1)2H11. So this column of data is constructed from the negative square of the distance between self-placement and candidate placement, minus the entropy uncertainty for this issue and partisan candidate. Thus, the explanatory variable's value is the degree to which it explains variation of the outcome variable (on the probit scale), and the degree to which it can overcome corresponding entropy uncertainty.

The individual coefficient results in Table 1 suggest that the model does reflect vote choice under uncertainty in a reasonable manner. Nine of the thirteen choice coefficients in the numerator of (8), have 95% confidence intervals bounded away from zero. Of the partisan issue distance pairs, all are signed in the anticipated direction, negative for Republicans and positive for Democrats, except for Support for Clinton (both of which are not reliable anyway). Recall that a positive coefficient pushes respondents away from Democrats and a negative coefficient pushes respondents away from Republicans since these are based on squared issue distance plus entropy uncertainty.

Government spending policy was an important issue in the 1994 campaign, and the model produces results exactly as we would predict. Spending continues to be a potent campaign topic because it allows House members to easily run against Congress as an institution (Fenno, 1977). The finding that the Democratic and Republican coefficients are of equal magnitude is interesting as it suggests that spending distance, in the presence of uncertainty, affects both parties in similar fashion.

Part of the Republican agenda enshrined in the Contract with America was the idea of reigning-in the Clintons’ healthcare initiative. We find nearly identical effects of issue distance for Democrats and Republicans, indicating again that issue increasing distance drives voters away in the same fashion.

The three choice covariates in the numerator of the Harvey specification produce reliable coefficients with expected effects: females and non-whites are more likely to vote Democratic, and increased media consumption leads to more Democratic voting. While these are control variables in our context, such reliability and predictability is certainly reassuring about the overall quality of the model.

Unfortunately Age is not statistically reliable here. This may be because we are controlling for enough of the other covariates related to the respondents’ age, such as education, political interest, and healthcare, that age lacks differentiated explanatory power.

5. Conclusion

It appears that uncertainty matters. This model demonstrates that in the aggregate, including a measure of uncertainty affects the importance of individual issue-distances. Uncertainty matters because it does or does not mitigate issue distance on a per-issue basis in the classical sense from models such as those given by Enelow and Hinich, 1981, Shannon, 1948 and Bartels, 1988. The implication is that politicians gain by clarifying policy positions provided that those positions are not too far out of line with those widely supported by the electorate. Or, at a minimum, there is evidence that politicians need to trade off the cost of uncertainty with the cost of alienating specific groups of voters.

The three dispersion covariates in the denominator of the Harvey specification also produce reliable coefficients. As expected, the Party Identification coefficient is large, positive, and has a 95% confidence interval which does not cover zero. There is a vast literature to support this finding (Nie et al., 1976, Schulman and Pomper, 1975, Page and Jones, 1979 and Wattenberg, 1990). It appears that increased levels of political interest decrease model variability and subsequently increase the reliability of the choice parameters since the coefficient for Political Interest Scale is negative and reliable. This makes intuitive sense since those with greater interest are more likely to seek candidate and issue information than less-interested individuals. The finding for Education is curious since the corresponding coefficient is positive and reliable, and therefore gives the opposite effect.

Two other issues give only partially satisfactory results. In the partisan coefficient pairs for Government Help for Disadvantage and Crime Concern, one coefficient is reliable at the 95% level and the other is not possibly because the contribution from the entropy uncertainty term exceeds that of the spatial distance term, or for purely statistical reasons (lack of sample information, given this model).

Crime has been a bread-and-butter Republican campaign issue for decades, and the model indicates that increased distance, factoring in uncertainty, reduces the probability that the respondent will vote for the Republican candidate. Notice also that while the standard error for the Democratic candidate is identical (subject to rounding), the Republican coefficient is about twice that of the other, which for this reason is not statistically reliable. Taken together, this suggests that Republicans are almost certainly punished for failing to take a position close to the electorate while there is no evidence that Democrats must do so (although the coefficient is signed in the direction we would expect).

Interestingly, the bread-and-butter Democratic campaign issue of help for societies shows exactly the pattern as seen with crime, except that the roles are reversed. The coefficient for the Democratic candidate is reliable while that for the Republican is not (and once again the standard errors are comparable). In addition, just like crime, the reliable coefficient is twice that of the other suggesting that Democrats are punished for deviating sharply from positions close to the electorate but there is no evidence that Republicans suffer the same fate.

The model provides interesting and somewhat perplexing results with regard to the Support for Clinton variable. Both the Democratic and the Republican variables have 95% confidence intervals which bound zero so we cannot place much faith in these findings, but the Republican variable is signed oppositely than expected: greater distance between the Republican candidate and respondent provides greater probability of voting for that candidate. We would normally expect that as the distance increases, a respondent is increasingly dissatisfied with the Republican candidate and more likely to vote for the Democratic candidate. It is possible, but not able to be shown here, that there is something special happening with this issue, and the lack of clarity supports Clinton's status as somewhat of an enigma. Clinton often embraced traditional Republican positions as a “New Democrat” (reinventing government, a focus on the economy, protection of programs for the elderly, reevaluation of welfare policies), but he also alienated many Republicans with his stance on gays in the military and other issues.

Entropy as a measure of uncertainty in political communication indicates a lack of substance in transmissions. Entropy captures the cost of uncertainty in this calculus by measuring this uncertainty through analysis of discrete issue placements. The entropy measure can be superior to other tools because it calculates uncertainty with reasonable assumptions: diffusion is equivalent to uncertainty, information is the substantive component independent of the transmission process, and the receiver (i.e. voter) is the judge of information quality and quantity. Conversely, measures of information uncertainty based on variances require indirect estimation of unobservable parameters and therefore further specification.

Rather than find evidence that the Contract with America made a substantive difference in the electoral fortunes of the Republican candidates, the conclusion from this research is that combined issue distance and uncertainty disproportionately affect Democrats, thus diminishing the importance of the Republican campaign pledge. The asymmetry of the signed coefficients slightly favors Republicans over Democrats. It is possible that forty years of House control created a high level of perceived certainty about the policy positions of Democratic candidates and for them issue distance was more important than vagueness or clarity.

The Contract with America was a Republican effort to stake out and explicate a set of specific policy positions. The political sophistication of the Republican leaders in the House during this time remains open to question. If the Contract with America was developed to increase the clarity of Republican candidates’ positions nationwide, then it would imply a high level of strategic foresight with regard to analyzing voter preferences and uncertainty. However, there are other reasons that the leadership developed an explicit agenda and required members to publicly sign the document. These include control of the agenda by the leadership, control of the freshmen members, and a strategy for opposing policies advocated by the president. It is entirely possible that the Republican leadership was not aware of the possible advantage available by reducing uncertainty, but they simply benefited by luck.


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Appendix A. Data Appendix

This section summarizes the data format and coding decisions applied to data from the 1994 American National Election Study (study No.6507 conducted between November 9, 1994 and January 9, 1995). Categorical sums given for each outcome exclusive of missing values. For the five pairs of issue variables below (k=1:K), three scales are created: one for the respondent's self-placement (Rik), and one each for the respondent's placement of their corresponding Democratic candidate and Republican candidate (Cijk,j=1,2). The aggregated candidate placements are used to create the corresponding entropy value (Hjk). Demographic covariates (Oij,j=3:9) are left in their original measurement, with the adjustments described below.

1. Support for Clinton. The respondent measure is specified as a seven-point scale by combining the approve/disapprove question (VAR 201) with the strength of approval/disapproval question (VAR 202). The seven-point Democratic Candidate Support for Clinton variable and Republican Candidate Support for Clinton variables were created by combining House incumbent supports Clinton Half the time (VAR 647) with House incumbent supports Clinton almost always (VAR 645) and House incumbent supports Clinton almost never (VAR 646) then cross-referencing by incumbent's party (VAR 17).

2. Concern About Crime. These measures for the respondent and the two candidates are created by rescaling the voted-for or supported the Crime Bill question (VAR 1040 for each respondent, VAR 648 and VAR 649 for the incumbent, cross referenced again by incumbent's party), where the respondents’ interior points are derived from the question asking about appropriate level of spending on programs dealing with crime (VAR 825). This procedure gives the desired seven-point scale.

3. Government Help for Disadvantaged. The three measures come directly from self-assessment (VAR 930) as well as placement of the Democratic (VAR 932) and Republican (VAR 933) candidates on a seven-point scale where 1 indicates “the government should see to a job and good standard of living,” and 7 indicates “government should let each person get ahead on their own.” No rescaling is necessary.

4. Government Spending. The three measures of government spending on social programs come directly from self-assessment (VAR 940) as well as placement of the Democratic (VAR 942) and Republican (VAR 943) candidates on a seven-point scale where 1 indicates “the government should many fewer services,” and 7 indicates “the government should many more services.” No rescaling is necessary, but the scale direction is reversed to put the more typically Republican position on the high end of the scale.

5. Federal Healthcare. The three healthcare measures also come from self-assessment (VAR 950) as well as placement of the Democratic (VAR 952) and Republican (VAR 953) candidates on a seven-point scale where 1 indicates that the provision of healthcare nationally should be completely through a “government insurance plan” and 7 indicates it should be completely through a “private insurance plan” No rescaling is necessary.

6. Party Identification Scale. This is given as a seven-point summary scale (VAR 655) with the categories: (0) strong Democrat, (1) weak Democrat, (2) independent, leaning Democrat, (3) independent, (4) independent, leaning Republican, (5) weak Republican, (6) strong Republican.

7. Political Interest Scale. Political interest is measured directly on the points: (1) very interested, (2) somewhat interested, (3) not much interested (VAR 124).

8. Media Exposure. This variable is specified by averaging daily consumption across three media sources. The respondent reports the typical number of days per week that they: read the paper (VAR 205), watch TV news (VAR 206), and listen to radio news (VAR 207).

9. Education. The ANES summary measure (VAR 1209) is used in its provided form with seven categories: 8th grade or less, 9th to 11th grade, high school or GED, more than 12th, associate degree, bachelors degree, and advanced degree.

10. Gender. Gender is switched in order from VAR 1434 such that females are coded 0 and males are coded 1.

11. Race. Race is dichotomized according to white (0) and not-white (1) from VAR 1435.

12. Age. Actual reported age is used (VAR 1203), ranging from 18 to 91, with mean 46.5, and standard deviation 18.

13. Vote Choice. The outcome variable is dichotomized by collapsing the information from VAR 613 such that voting for the Democrat is coded 0, and voting for the Republican is coded 1.



star, openLarry Dodd, Bob Huckfeldt, Bob Jackman, Renee Johnson, Gary King, Michael Martinez, Lee Walker, and three anonymous reviewers made useful comments on earlier drafts. The data are available from The Inter-university Consortium for Political and Social Research (http://www.icpsr.umich.edu/), formatting instructions and computer code are available at the author's webpage: http://psblade.ucdavis.edu.
Corresponding Author Contact InformationTel.: +1 530 752 3077; fax: +1 603 804 7773.



1 For instance if the entropy form were expressed in terms of the natural log, but log2 was more appropriate for the application (such as above), then setting k=1/ln2 converts the entropy form to base 2.
2 The uniform prior distribution as applied provides the greatest entropy since no single event is more likely to occur. Thus, the uniform distribution of events provides the minimum information possible with which to decode the message. This application of the uniform distribution does not imply that this is a “no information” assumption since equally likely outcomes is certainly a type of information. A great deal of controversy and discussion has focused around the erroneous treatment of the uniform distribution as a zero-based information source (Dale, 1991 and Stigler, 1986).
3 H=−∑(1/n)ln(1/n)=ln(n), so entropy increases logarithmically with the number of equally likely alternatives.
4 Click to view the MathML source.
5 Only the discrete form of the entropy formulation has been discussed. Since entropy is also a measure of uncertainty in probability distributions, the following entropy formula also satisfies all of the properties enumerated above: Click to view the MathML source This form is often referred to as the differential entropy of a random variable X with a known probability density function f(x). There is one important distinction: the discrete case measures the absolute uncertainty of a random variable given a set of probabilities, whereas the continuous case this uncertainty is measured relative to the chosen coordinate system (Ryu, 1993). So transformations of the coordinate system require transformation in the probability distribution function with the associated Jacobian. In addition, the continuous case can have infinite entropy unless additional, “side” conditions are imposed (see Jaynes, 1968).
6 All data, code, and imputations for replication purposes are archived at the website: http://psblade.ucdavis.edu.



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