# Distance versus direction: the illusory defeat of the proximity theory of electoral choice.

In an extensive series of articles, two of which were published in the Review, George Rabinowitz, Stuart Elaine Macdonald, and Ola Listhaug have launched a new spatial theory of electoral choice.(1) The new theory is presented as a direct alternative to the classical spatial model introduced by Downs (1957), on the basis of earlier work by Hotelling (1929).(2) Both models try to account for the same phenomenon (electoral choice) on the basis of the same information (the issue stands of voters and parties) but conceptualize the independent variable and, by implication, its effect in different ways.Judging from the results hitherto presented, it would seem that the new theory has already superseded the old. In a large number of tests based on data from multiple countries and periods, Rabinowitz, Macdonald, and Listhaug consistently find their directional model superior to the classical proximity approach. This massive evidence notwithstanding, I shall argue that the superiority of the directional model is illusory rather than real. When properly evaluated, the proximity model remains the better theory.

My conclusion is based on a comparative evaluation of the two theories. Each theory is examined in the light of two basic criteria. The first is its degree of falsifiability, that is, the proportion of logically conceivable empirical outcomes that it rules out as incompatible with its propositions. A highly falsifiable theory has greater empirical content (i.e., it tells us more about reality) and is therefore more valuable than a less falsifiable one (Popper 1980, chapter 6). Yet, a high degree of falsifiability is obviously not the only hallmark of a good theory. Of great importance is the second criterion, namely, the theory's degree of compatibility with empirical evidence. In short, a good theory should maximize the risk of being proved wrong but nevertheless should prove right. The results of my examination indicate that the directional model is inferior to the proximity model in both respects.

My analysis proceeds in five steps. I begin by describing the main differences and similarities between the two theories. In so doing, I also spell out the implications for the first of the two criteria. The three subsequent sections focus on the second criterion. I first consider how the two theories have been, and should be, tested. based on the conclusions from this section, I then reanalyze a significant portion of the data on which Rabinowitz, Macdonald, and Listhaug base their verdict. Finally, I specify the reasons why my results differ from those previously obtained. In the concluding section, I summarize my results and discuss their implications.

THE TWO THEORIES

The key distinction between the proximity and the directional model rests with the conceptualization of issue dimensions. In the proximity theory, issues are conceived as ordered and continuous sets of policy alternatives (e.g., a continuous set of alternative tax rates). An issue stand thus represents the preferred position on a policy continuum. In the directional theory, by contrast, policy alternatives are assumed to be dichotomous (e.g., low taxes versus high taxes), and issue stands are assumed to represent the intensity with which either of the two sides or directions is supported. The main argument advanced by Rabinowitz and Macdonald (1989, 93-6) for this issue conception is that it provides a more realistic account of voter policy preferences. In their view, voters do not discriminate between policy gradations as assumed by the proximity model but only possess a diffuse sense of direction coupled with a smaller or greater sense of urgency.

Research based on the proximity model has long given prominent attention to the hypothesis that the salience of an issue conditions its effect on the vote.(3) Hence, the point made by the directional conception is not simply that intensity matters. Rather, the new idea is that policy alternatives are always understood in binary terms, which leaves intensity as the only continuum. Thus, when voters are asked to indicate their issue stands on a continuous or semicontinous scale, the differences between responses falling on either half of the scale will express variations in intensity rather than variations in policy position. It follows that separate questions designed to measure issue salience are redundant, since the intensity component is already in hand.

An example will help clarify the implications of these ideas for the relationship between issue stands and party preference formation. Figure 1 shows the stands taken by two voters (1 and 2) and four parties (A to D) on an issue dimension. If we take the dimension to represent a policy continuum, party preferences will depend on how close the various parties are to the points representing voter ideals. If, by contrast, the dimension is assumed to represent two directions or sides (indicated by the sign of the score, with zero as a neutral point) combined with the intensity with which either side is supported (indicated by the absolute value of the score), party preference formation will follow somewhat different rules (see Rabinowitz and Macdonald 1989, 97-8). First, voters always have reason to prefer a party favoring their own side to one favoring the opposite side. I will refer to this as the side rule. Second, voters also have reason to prefer a more intense party on their own side to a less intense one since, under the directional model, the intensity of a party's stand is assumed to represent the probability that it will pursue the preferred policy alternative (Macdonald and Rabinowitz 1993a, 63). I will call this the party-intensity rule.

As shown in Figure 1, the extent to which the two rules discriminate depends on voter location. For voters far to the extremes, the preference orders predicted under the two issue conceptions are very similar. Beyond the last interparty midpoint, there is no distinction at all, as illustrated by voter 1. For voters in the middle ground, the predictions are more likely to diverge. Thus, if voter 2 uses the directional conception, A and B will be preferred to C and D according to the side rule, although C is closer than B and D closer than A. In addition,A will be preferred to B according to the party-intensity rule, although B is more proximate.

Mathematically, both rules are embodied in a single principle stating that the utility any one voter associates with each of a set of parties increases monotonically with the product of voter and party issue stands. In a multidimensional space, the simple product is replaced by the scalar product, that is, the sum of the dimensionwise products. Thus, if we let rank-order([x.sub.1], [x.sub.2], . . ., [x.sub.n]) denote a vector containing the rank order of a series of quantities, [x.sub.1], [x.sub.2], . . ., [x.sub.n], the directional model proposes that for any one voter, i,

rank-order([U.sub.iA], [U.sub.iB], . . ., [U.sub.iJ])

= rank-order(i [multiplied by] A, i [multiplied by] B, . . . . i [multiplied by] J), (1)

where [U.sub.iA] is the utility that voter i associates with party A and i [multiplied by] A the scalar product of the issue stands taken by the same voter and party. The proximity model, by contrast, posits that

rank-order([U.sub.iA], [U.sub.iB], . . ., [U.sub.iJ])

= rank-order(-[d.sub.iA], -[d.sub.iB], . . ., -[d.sub.iJ]), (2)

where -[d.sub.iA] is the proximity (i.e., distance inverted) of voter i's issue stands to those of party A.

Implicit in the transition to the multidimensional case made in these formalizations is a third rule distinguishing the directional from the proximity model, namely, that the weight of a dimension in the decision calculus is directly proportional to the voter's deviation from its midpoint. This third distinctive feature of the directional model can be named the voter-intensity rule.(4)

The Region of Acceptability

Thus far, it would seem that the two models are structurally equivalent in that they both rest on a single, unifying principle: scalar product in the one case, proximity in the other. Such, however, is not the case. The directional model incorporates a second principle without counterpart in the proximity theory, namely, that of the region of acceptability.

This region is a portion of the issue space falling within a directionally invariant Euclidean distance, r, from the origin. In a unidimensional space it is thus the interval -r to +r. In a two-dimensional space it is the area of a circle centered on the neutral point with radius r. In a space with three or more dimensions it is the interior of a sphere or hypersphere with the same attributes. All members of an electorate are assumed to have a common understanding of the size and location of the region, independently of their own locations.

For parties inside the region, utility is assumed to be a function of the scalar product alone. Parties outside its boundary, however, may incur a penalty that detracts from the utility derived on the basis of the scalar product. Equation 1 thus holds only in the special case in which the parties are all inside the region. More generally, the directional model posits that

rank-order([U.sub.iA], [U.sub.iB], . . ., [U.sub.iJ]) = rank-order(i [multiplied by] A

- [P.sub.iA], i [multiplied by] B - [P.sub.iB], . . ., i [multiplied by] J - [P.sub.iJ]), (3)

where [P.sub.iA] is the penalty voter i imposes on party A, the value of which is zero ira is inside the region.

The rationale offered for the region of acceptability and the penalty varies somewhat. Rabinowitz and Macdonald (1989, 108) argue - in the context of U.S. politics - that a candidate "must convince voters of his or her reasonableness. Voters are wary of candidates who seem radical and project harshness or stridency. The label 'extremist' can attach to such candidates and severely hamper the enthusiasm of potential supporters." Macdonald, Listhaug, and Rabinowitz (1991, 1110) emphasize - in the context of Norwegian politics - the extent to which voters feel "the party can function effectively in government. A party that is viewed as overly intense or excessively strident on issues will be evaluated less favorably than a party regarded as responsible and more able to govern effectively."

Implications for Falsifiability

The size of the region of acceptability as well as the magnitude of the penalty for parties beyond its boundary are left largely unspecified by the theory.(5) No size constraints are imposed on the region itself. The only restrictions on the penalty are that parties inside the region will not be penalized and that a party located outside the region will always be electorally inferior to one placed on the boundary in the same direction (Rabinowitz and Macdonald 1989, 108, 117; Rabinowitz, Macdonald, and Listhaug 1991, 152).(6) Hence, for parties outside the region, the penalty can vary freely across parties as well as voters as long as it stays above the threshold set by the second of the two restrictions.

The unspecified nature of the region of acceptability and the penalty, absolutely as well as in relation to the scalar product, renders the directional theory considerably less falsifiable than the proximity theory. As shown by Popper (1980, chapter 7, Appendix *viii), the number of freely adjustable parameters in a theory is directly and negatively related to its degree of falsifiability.

The basic point can easily be illustrated. Consider again the case of voter 2 in Figure 1. The proximity model predicts that the preference order of this voter will be CBDA, whereas the directional model predicts ABCD. Suppose the preference order actually observed is BCDA. As far as the proximity model is concerned, our only option is to concede that the observed order is in partial disagreement with the predicted one. With respect to the directional model, by contrast, we still have some cards to play. The only modifications required in order to make the prediction fit the observation is to suggest that (1) the boundary of the region of acceptability falls at a distance of somewhere between two and four units from the origin and (2) the penalty imposed on party A by voter 2 exceeds three units in the currency of the scalar product. In fact, by suitably modifying the size of the region and the penalties, the theory can be made to comply with any preference order. In applications involving an entire electorate rather than a single voter, the possibilities in this regard are restricted by the requirement that the region be the same for all members of the electorate. Nevertheless, the degree of plasticity remains considerable (see Appendix c).

A Theoretical Summary

The two principles on which the directional model rests allow us to distinguish between two versions of the theory: a reduced version based on the scalar product alone and a complete version including the region of acceptability and the penalty. The reduced version resembles the proximity model in several important respects. First, both are rational-choice models in the sense that party preferences are assumed to be a rational response to the issue stands taken by voters and parties. Second, by implication, both are also proximity models in the sense that voters are assumed to act so as to maximize the agreement between the desired policies and those enacted. Finally, both possess a fairly high and equal degree of falsifiability.

The key difference lies in the conceptualization of issue dimensions. This difference, in turn, gives rise to a distinction between the spatial functions linking voter and party issue stands to voter utility. If issue stands are taken to represent policy positions, then party preferences will be a function of proximity. If they are taken to represent direction combined with intensity, then party preferences will instead be a function of the scalar product. The way in which the decision calculus embodied by the scalar product differs from that of proximity can be expressed by means of three rules: the side rule, the party-intensity rule, and the voter-intensity rule.

In the complete version of the directional theory, utility is no longer a function of the scalar product alone but also of the region of acceptability and the penalty. The weak specification of these additional constructs makes the complete version inferior to the proximity model in terms of falsifiability.

PREPARING THE TEST BED

The basic premise for the empirical analyses of the two models reported by Rabinowitz, Macdonald, and Listhaug is the measure of issue stands introduced in the U.S. national election studies in 1968. This question format presents the respondent with a scale whose endpoints are labeled with two policy alternatives. The remaining points are unlabeled. The respondent is thus free to interpret the scale in either a positional or a direction-intensity sense. By comparing the observed candidate or party preferences with those predicted under the two issue conceptions, one can infer the extent to which the conceptions are actually used by the voters and thereby determine the degree of support enjoyed by the two models.

While the analyses I will present match those of Rabinowitz, Macdonald, and Listhaug in this fundamental respect, they differ in some other vital regards. In the following subsections, I consider six choices facing us on the way toward a proper test of the two theories. In three of the six cases, I argue that the choice should be another than that made by Rabinowitz, Macdonald, and Listhaug. In the remaining three, I find it important to present my arguments for other reasons.

In later empirical sections, I first test the two theories in the fashion I consider most appropriate. As will become evident, this leads to an outcome diametrically opposed to that previously obtained. I then specify the extent to which this outcome is dependent on each of the three ways in which my methods differ from those of Rabinowitz, Macdonald, and Listhaug. Finally, I try to explain the effect of the most significant choice, that between intrapersonal and interpersonal comparisons.

Intrapersonal versus Interpersonal Comparisons

Suppose we ask two voters, 1 and 2, to express their likes and dislikes with respect to two parties, A and B. Suppose in addition that our instrument is the thermometer scale commonly used in survey research, with the maximum score of 100 indicating a very positive evaluation and the minimum score of 0 a very negative evaluation. Let us finally assume that our results are those displayed in Table 1.

TABLE 1. Illustration of Intrapersonal versus Interpersonal Comparisons of Evaluation Scores Party A Party B Voter 1 70 60 Voter 2 40 50

The table basically allows two kinds of comparison. By comparing across columns for a given row we can investigate how the same voter relates to different parties. By comparing across rows for a given column we can investigate how different voters relate to the same party. The first is an intrapersonal comparison, the second an interpersonal one.

It is not difficult to see which of the two comparisons we should make in order to test our two theories. Both are theories of individual choice. They thus predict the utility any one person will associate with each alternative relative to other alternatives and thereby the preference order with respect to the whole set of alternatives (see equations 1-3). They say nothing whatsoever about how the utility of a certain alternative to one person compares with its utility to another person. The reason is that interpersonal comparisons of utility are completely irrelevant to individual choice.(7) Voter 2 gives party B a lower rating than does voter 1. Nevertheless, B is the preferred party of voter 2 but not of voter 1. What voter 1 thinks about the parties is of no consequence to the decision of voter 2 and vice versa. Only the intrapersonal comparison matters.

In view of this, it is highly remarkable that none of the seven contributions by Rabinowitz, Macdonald, and Listhaug that deliver the primary individual-level evidence in support of the directional model over the proximity model contains a single empirical test of the proper kind. All analyses are based on interpersonal rather than intrapersonal comparisons.(8)

In the prototypical case, the evaluation scores for each party or candidate are analyzed separately, so that all the variation is contained between rather than within individuals. In some cases, pooled analyses combining evaluation scores for multiple parties or candidates are also reported. Yet, since no attempt is made to isolate the intrapersonal element in these analyses, they become for all practical purposes just another interpersonal comparison.

In terms of the example presented in Table 1, the type of analysis performed by Rabinowitz, Macdonald, and Listhaug thus amounts to an explanation of why voter 1 provides a higher evaluation of party A as well as party B than does voter 2. What the two theories purport to explain, however, is why voter 1 prefers A to B and why the opposite holds for voter 2. The bulk of empirical work on the relative performance of the two theories thus tests them on predictions that neither makes. It follows that the results can neither refute nor corroborate either theory.

How, then, can we perform a test of the proper kind? Equations 1-3 above suggest an immediate solution. The theories predict that the rank order of the dependent variable within individuals should match the corresponding rank order of the independent one. Hence, all we have to do in order to assess the fit is to transform the data to intrapersonal ranks and then compute the correlation between them (Spearman's rho applied intrapersonally).

While this solution has the advantage of following directly from theoretical specifications without any additional assumptions, it does not readily extend to the multivariate case. Hence, it is not well suited to tests of the complete version of the directional model, where the lack of theoretical precision makes it necessary to treat the penalty factor associated with the region of acceptability as a separate independent variable. Furthermore, it cannot provide a precise estimate of the relative empirical strength of the two theories, particularly since the scalar-product and proximity measures by definition are strongly intercorrelated. The obvious solution to this problem is to switch to interval-level methods (as do Rabinowitz, Macdonald, and Listhaug in their analyses). This, in turn, calls for a somewhat more specific assumption of functional form (e.g., linear) than that required by the theories (monotonically increasing).

Just as in the ordinal case, however, we must make sure that the variation analyzed is theoretically relevant, that is, intrapersonal rather than interpersonal. To get a better grasp of this problem as well as its solution, consider the two diagrams displayed in Figure 2, which illustrate the relationship between a dependent variable, such as party evaluation, and a predictor, such as proximity or scalar product. Figure 2A shows the intrapersonal relationship for two different voters. In both cases, the relationship is perfect. Once the independent variable is held constant, however, voter 1 provides more positive evaluations than does voter 2. Due to this interpersonal difference, which is irrelevant with respect to voter choice, the regression line for the two voters combined shows a less than ideal fit. Hence, it does not correctly reflect the perfect correspondence between theory and data.

The problem can be resolved by subtracting the individual mean value from the scores of each respondent. This purges the data of the interpersonal variance and leaves only the intrapersonal component. As shown in Figure 2B, the joint regression line now overlaps completely with those of the two voters, thereby mirroring the strength of the intrapersonal relationships we want to assess.

Formally, the operation described corresponds to the following regression equation, which models the dependent variable as an intrapersonal function of the predictor:

[Mathematical Expression Omitted] (4)

The symbol [Y.sub.ij] denotes voter i's evaluation of party j, [Mathematical Expression Omitted] voter i's mean evaluation across all parties, [X.sub.ij] the score of voter i for party j on the independent variable whose effect we want to assess (e.g., proximity or scalar product), and [Mathematical Expression Omitted] the mean for voter i across all parties on the same variable. Of course, multiple predictors can be inserted in the same fashion. Note that no intercept term is required: If added, its estimated value is always nil since both [Mathematical Expression Omitted] and [Mathematical Expression Omitted] have a mean of zero.

Evaluation Scores versus the Vote

As already mentioned, evaluation scores have served as the dependent variable in most prior tests of the relative performance of the two theories. In view of the fact that both theories claim to be theories of voting, one is prompted to ask: Why not the actual vote?

My answer, which differs from that of Rabinowitz, Macdonald, and Listhaug although we both prefer evaluation scores,(9) is based on theoretical as well as methodological considerations. Both theories predict the entire preference order of each voter, not just the most preferred alternative (see equations 1-3). Evaluation scores allow a complete test of these predictions, whereas the vote limits us to the peak preference. Evaluation scores also provide information about the size of the utility differences, which is useful if, on methodological grounds, we would like to proceed on the basis of interval-level assumptions. The vote, by contrast, contains no information of this kind. Consequently, while both measures can and will be used to test the two theories, evaluation scores permit a more comprehensive, precise, and robust assessment.(10)

My considerations on this score accord with those motivating the use of evaluations as a mediator between the vote and the factors of substantive interest (e.g., issue stands) in general models of voting (e.g., Markus and Converse 1979, Page and Jones 1979). In view of the superior measurement characteristics of evaluation scores and the efficiency with which they are long since known to predict the vote (Brody and Page 1973), there seems to be much to gain and little to lose from such a strategy.

Individual versus Aggregate Images of Party Issue Stands

To construct the independent variables for the models, two types of information are required. Not only do we need to know each voter's own views on the issues but also his or her beliefs as to where the parties are located. Although voters may at times be mistaken about these locations, it is their personal beliefs, whether right or wrong, that will guide preference formation. The best source for both types of information is therefore the individual voter.

This conclusion is not uncontroversial. Instead of using the information on party locations held by each voter, as I propose, Rabinowitz, Macdonald, and Listhaug base their analyses on the mean party placements for the entire aggregate of voters. Their main arguments in support of this approach are twofold (Rabinowitz and Macdonald 1989, 119, n7; Macdonald, Listhaug, and Rabinowitz 1991, 1130, n12; Macdonald, Rabinowitz, and Listhaug 1995b, 460). First, both theories require each party to have a single location on each issue. If not, it becomes impossible to link party strategy with popular support. Second, the images of party issue stands held by individual voters are subject to projection effects (wishful thinking) based on the voters' own issue stands and their party evaluations. Using such images as part of the independent variable therefore entails a risk of reverse causality.

While both statements are correct, they are irrelevant as arguments in the case at issue. As for the first argument, both theories do require that we, as scientific observers, are willing to accept the notion of objectively determinable party locations for the purpose of assessing party strategy. Neither theory demands, however, that all voters have identical and perfectly accurate information about those locations. If this were indeed a requirement, we could immediately write off both theories as empirically unreasonable. To maintain the link between electoral choice and party strategy, it is sufficient to assume that the issue stands actually taken by parties have some effect on the images of those stands held by voters. This, of course, is a much more realistic assumption.(11)

The position taken by Rabinowitz, Macdonald, and Listhaug on this matter is particularly difficult to understand in view of the cognitive shortcomings they ascribe to ordinary voters as well as their claims to having developed a model more attuned to these shortcomings (see, e.g., Rabinowitz and Macdonald 1989, 94-5). It is not entirely convincing to argue that a good model of electoral choice should be based on the presumption that the "vast preponderance of citizens operate with low levels of information" and that their policy preferences do not "go beyond a diffuse sense of direction" but also on the assumption that they are nevertheless perfectly aware of where parties or candidates are located.

As for the second argument, the presence of projection effects certainly constitutes a problem if the principal task is to estimate the absolute degree to which either theory is able to explain variations in the dependent variable. This, however, is not the main objective in the case at issue. The foremost concern is the performance of the two theories relative to each other. Given this purpose, the potential presence of projection processes constitutes no problem. The reason is simple. Projection takes place because people sometimes find it easier to alter their cognition to fit their evaluation rather than the other way around. Regardless of the causal direction, however, the notion of fit should be the same. There is no psychological reason to expect a directional voter to project on the basis of proximity inasmuch as this would not make his or her mental state more balanced. On the contrary, projection for such a voter should be based on scalar product and penalty. For a proximity voter, we would of course expect the opposite, that is, projection based on proximity but not on scalar product and penalty. It follows that the unwarranted increase in statistical association due to projection should affect the results for the two theories in direct proportion to their true explanatory power. Hence, their estimated strength relative to each other should be the same regardless of whether projection is present or not.(12)

Consequently, individual party placements do not, in this context, possess the drawbacks with which they have been charged. Mean party placements, by contrast, give rise to several problems, the exact nature of which depends on our starting point.

Suppose, on the one hand, that we would like to proceed on the assumption that voters have perfect information about party issue stands. The target of our operationalization would then be the true party locations. The mean placements, however, are not a particularly good measure of these locations. Although they are likely to mirror some properties of the true account, such as the order of the parties with respect to each other, they are apt to be more or less severely distorted in other respects. For example, Powell (1989) demonstrates a bias toward the middle of the scale due to the fact that the party placement data partly consist of random guesses. This bias, furthermore, can be shown to have more serious consequences for the proximity than for the directional model.(13)

Suppose, on the other hand, that we wish to pursue what I consider to be the most fruitful objective, namely, to assess how voters actually think rather than how they might think if they all had perfect information. Our target would then be the actual cognitive map of each voter. The contorted pictures of those maps that result if we use mean rather than individual party placements cause two different types of problems.

First, we have no guarantees that the distortions will affect a test of the two theories in a fair and unbiased manner. On the contrary, there are several reasons to think that will not be the case.(14) Second, and equally serious, we risk losing sight of a perceptual process that may well be of vital importance for our understanding of the interaction between party systems and electorates. If we are seriously interested in that interaction, then we need to study the perceptual process, not hide it through aggregation or assume it away.

In his now classical comment on the Downsian theory, Stokes (1963, 376) concluded that insight into the applicability of that model (and others of its kind) would best be furthered by treating "as explicit variables the cognitive phenomena that the prevailing model removes from the discussion by assumption." In my view, there are many good reasons for, and few against, following his advice.(15)

Specifying the Concept of Proximity

I have hitherto spoken rather indifferently about distance and its opposite, proximity, as though the meaning of these concepts were self-evident. As long as the space under consideration is unidimensional, such is indeed the case. In a multidimensional space, by contrast, many notions of distance are conceivable. To my knowledge, however, most prior work on spatial choice models restricts itself to two varieties, Euclidean distance and city-block distance, and I find no compelling reasons to proceed beyond these alternatives here. Formally, the city-block distance between two points representing a voter, i, and a party, j, in an n-dimensional space can be written as

[summation of] [absolute value of] [i.sub.k] - [j.sub.k] where k = 1 to n

that is, the simple sum of the dimensionwise distances, whereas the formula for Euclidean distance is

[-square root of] [[summation of]([i.sub.k] - [j.sub.k]).sup.2 where k = 1 to n,

that is, the square root of the sum of the squared dimensionwise distances.

Neither the proximity theory itself nor the general concept of rationality requires or favors either definition. In the absence of such theoretical guidelines, I shall argue that there are three reasons to prefer the city-block conception. First, as Ordeshook (1986, 22-3) puts it, the city-block metric "nicely summarize[s] the intuition that, when comparing candidates, voters look separately at how different each candidate is from their most preferred position on each election issue. They then vote for the candidate whose total distance across all issues is the least." According to the Euclidean conception, voter information processing would follow a more complex logic.

Second, the city-block metric makes the transition from the unidimensional to the multidimensional case mathematically parallel for the two theories. In the unidimensional case, the directional model is based on the product of the voter's and the party's score, the proximity model on the unsigned difference between them. Given a city-block conception of proximity, the multidimensional versions are both based on the simple sum of the respective unidimensional quantities.

Third, the city-block conception offers methodological advantages. City-block distances present no difficulties in moving from a constrained proximity model, that is, one in which the relative weights of the dimensions are predetermined, to an unconstrained model, that is, one in which the weights are left free for estimation. Under the Euclidean conception, by contrast, the unconstrained case yields a regression equation which is nonlinear in the parameters. While such equations can still be estimated by resorting to special techniques, it is prudent to avoid them unless there are strong reasons to the contrary. Of course, the problem may also be resolved by modeling utility as a quadratic rather than linear function of distance, but this is appropriate only if the quadratic form can be shown to fit the data at least as well as the linear.

In their tests of the proximity model, Rabinowitz, Macdonald, and Listhaug switch back and forth between different distance conceptions and functional forms. In some cases they model utility as a function of Euclidean distance, in others as a function of squared Euclidean distance (e.g., to accommodate the demands of their so-called mixed model), and in still others as a function of city-block distance (when estimating unconstrained models). For obvious reasons, it is desirable to avoid such ad hoc shifts between specifications. In the present study, the city-block specification is used for all analyses of the proximity model except those whose purpose is to assess the effect of differences in method.

Specifying the Region of Acceptability and the Penalty

To test the complete version of the directional model, we must operationalize the region of acceptability and the associated penalty. One difficulty in accomplishing this task is that the theory provides no criterion whereby we can decide the size of the region. Another problem is that the size of the penalty for a party outside the region is left largely unspecified by the theory.

Uncertainty is further increased by the absence of justifications for the decisions taken when initially designing these constructs. One of the most remarkable features of the region of acceptability is that it is defined as a property of the entire electorate rather than as an attribute of the individual voter. This immediately prompts several questions. Why, for example, should an extremist voter see the same region as a voter located in the center or at the opposite extreme? And why should we expect such a voter to penalize a party sharing his or her own direction and degree of extremism? The region of acceptability is, as it were, a region in search of its acceptor.(16)

While the aggregate nature of the region of acceptability remains a puzzle, a partial response is offered by Rabinowitz, Macdonald, and Listhaug (1991, 152), who introduce the idea that the size of the penalty may vary with voter location although the region itself stays constant. More specifically, they suggest that "voters who are themselves intense and sympathetic to [a party outside the region] might apply no penalty or a very small one, while other voters might apply a larger penalty." They also present a class of mathematical functions corresponding to this idea but do not attempt to apply it empirically. Regrettably, the formalization they provide is inconsistent with their verbal suggestion as well as with other requirements of the directional theory. As shown in Appendix A, the inconsistency is rooted in a contradiction between the requirements themselves and therefore not so easily remedied. Hence, the theoretical line of development proposed by Rabinowitz, Macdonald, and Listhaug (1991) cannot serve as a guideline for the operationalization of the penalty.

In view of this situation, I have chosen to take my point of departure in operationalizations actually used by Rabinowitz, Macdonald, and Listhaug in empirical application. This, of course, does not resolve the problem. of justifying these particular operationalizations in comparison with other alternatives equally compatible with the theory. Nor does it imply that I will refrain from modifying them when necessary in order to meet theoretical or methodological requirements. Nevertheless, they make it possible empirically to apply the region of acceptability and the penalty, and to do so in a manner which Rabinowitz, Macdonald, and Listhaug themselves have previously found acceptable.

Choosing the Testing Ground

The two theories have already been tested on a considerable number of data sets. It is not possible, of course, to offer a reanalysis of all these data within the confines of a single article. Nor is such an extensive task required in order to refute the claims made by Rabinowitz, Macdonald, and Listhaug, since for reasons already discussed their prior analyses have failed to test the theories on the predictions they actually make.

It remains, however, to find a good starting point. In my view, the best alternative is that through which Macdonald, Listhaug, and Rabinowitz (1991) introduce the directional model as a general theory of electoral choice in multiparty systems: the Norwegian 1989 election study.(17) First, this study meets two basic requirements: a rich set of items assessing respondent as well as party issue locations and a prior operationalization of the region of acceptability and the penalty based on these items. Second, it constitutes a more representative case than other conceivable alternatives, such as the United States and Britain. As Rabinowitz, Macdonald, and Listhaug (1991, 148) put it: "Most of the world's democracies are multiparty proportional systems. If basic models of voting are to have broad relevance for the understanding of democratic politics, they must be applicable to multiparty systems."

In some instances, the quest for a representative case runs counter to the quest for a crucial case. Not so with respect to Norway, where the relatively large number of parties distinctly improves our ability to discriminate between the two theories. Recall the side rule and the party-intensity rule, both of which help us tell the two theories apart. Certain conditions must be fulfilled in order for them to do so, however. The side rule discriminates only when a party on the voter's side is more distant than a party on the opposite side. The party-intensity rule yields different predictions only when there are at least two parties on the voter's side, one of which is more extreme than that to which the voter is closest. Both of these conditions are likely to occur with much greater frequency in a system with many parties than in one with only two or three contenders.

A final reason for focusing on a multiparty system such as the Norwegian is that it offers a more robust test. The relatively long preference orders that such a system puts at our disposal make the test less sensitive to arbitrary decisions regarding the operationalization of the region of acceptability and the penalty. In addition, they reduce the risk that factors outside the investigated issue space, which might favor some contenders over others, will seriously distort the outcome of the test. Note, however, that these advantages are contingent on the use of evaluation scores (full preference order) rather than actual vote (partial preference order) as the dependent variable. Only then is the greater robustness of the test fully realized (see Appendix C).

PROXIMITY REHABILITATED

Seven major parties competed for seats in the 1989 Storting elections: the Socialist Left Party, the Labor Party, the Liberals, the Center Party, the Christian Liberal Party, the Conservatives, and the Progressive Party. The proximity theory predicts that voter preferences across these parties will be based on party-voter distances. The directional theory predicts that they will instead be based on the respective scalar products minus any penalty for parties outside the region of acceptability.

In the subsection on intrapersonal versus interpersonal comparisons, I described two basic ways to test these predictions: an ordinal-level method, which follows directly from theoretical specifications but only allows a partial analysis, and an interval-level method, which requires auxiliary assumptions of functional form but permits a more complete analysis. Both methods will be applied in turn.

According to the ordinal-level method, we assess the predictions through the correlations between the intrapersonal rank orders of the variables involved (Spearman's rho applied intrapersonally). In the present case, the dependent variable, party evaluation, is based on thermometer ratings. The independent variables, Proximity and Scalar Product, are based on six issue items plus placement on a left-right scale.(18) Following the practice of Rabinowitz, Macdonald, and Listhaug, the seven items are weighted equally.

The prior results obtained by Macdonald, Listhaug, and Rabinowitz (1991) from the same data set would lead us to expect greater predictive power for the scalar product than for proximity.(19) The present test, however, shows the opposite. The intrapersonal rank-order correlation between party evaluation and proximity is .55, whereas that between party evaluation and scalar product amounts to .49, the difference between the two coefficients being statistically significant at the .001 level.(20)

While the absolute magnitude of the difference is relatively small, this does not imply that it is of little theoretical significance. As previously mentioned, the proximity and scalar product measures are by definition strongly intercorrelated (.76 for the intrapersonal rank orders in the present case). Thus, even if one of the models had no explanatory power of its own, its spatial function would be nearly as strongly related to the dependent variable as that of the model actually accounting for the variation.

The high correlation between the proximity and the scalar-product measure is a reflection of the fact that the two models make similar, or even identical, predictions in many cases. Yet, the degree of similarity depends on voter location. Recall that two of the three prediction rules distinguishing the reduced version of the directional model from the proximity model - the side rule and the party-intensity rule - are more likely to result in different predictions for voters in the middle ground than for those at the extremes. To the extent that these two rules account for the observed difference in performance, we would therefore expect the gap between the two theories to vary with the voter's distance from the center.

That such is actually the case is shown in Figure 3, which relates the predictive advantage of the proximity model (i.e., the intrapersonal rho for proximity minus the intrapersonal rho for the scalar product) to the degree of voter eccentricity (i.e., the extent to which the voter is off center).(21) For voters taking very eccentric stands, the difference between the models is nearly nil. From there, the advantage of the proximity model increases monotonically to a value just below .20 for voters located close to the center.

The results hitherto presented permit the conclusion that the proximity model outperforms the reduced version of the directional model, but we have yet to consider the region of acceptability and the penalty. By means of these constructs, the directional model may still be able to show its strength, although a potential improvement in its compatibility with the evidence is bought at the expense of a reduced degree of falsifiability.

The multivariate framework required by the penalty factor mandates a switch to interval-level methods. This simultaneously allows us to determine the net explanatory power of one theory after controlling for that of the other. Table 2 presents the results of modeling party evaluation as a linear intrapersonal function (see equation 4) of proximity, scalar product, and penalty.(22) The operationalization of the penalty factor follows that of Macdonald, Listhaug, and Rabinowitz (1991) save for modifications due to the use of individual rather than mean party placements. This means that the variable is scored zero for parties placed inside the region and by a value corresponding to the Euclidean distance from the boundary for parties placed outside the region.(23)

[TABULAR DATA FOR TABLE 2 OMITTED]

Table 2 contains estimates for six statistical models. The first two are bivariate regressions with proximity and scalar product, respectively, as the only predictor. They thus constitute the interval-level equivalents of the ordinal-level analyses already performed. As expected under the assumption that a linear function is a reasonable approximation, the standardized regression coefficients (which in the bivariate case are identical to product-moment correlations) are both slightly higher than their ordinal-level counterparts, and the difference between them is slightly larger.

Model 3 includes both the proximity and the scalar-product measure. As a result, the effect attributable to the directional model drops to near zero, whereas that accounted for by the proximity model remains almost as high as in the bivariate case. Despite the high correlation between the independent variables (.87), the standard errors indicate that the coefficient estimates are quite precise. The relative strength of the two predictors is therefore by no means coincidental. Note also that adding the scalar product to a pure proximity model yields no perceptible growth in the proportion of variance explained, whereas adding proximity to a pure scalar-product model yields a substantial increase in predictive power (up by .086).

The three remaining regressions test the impact of the penalty. When combined with proximity alone, the penalty effect is close to zero, exactly as we would expect. When combined with the scalar product alone, it shows a substantively significant negative effect, which is also in line with prior expectations. The proportion of variance explained grows perceptibly in comparison with the pure scalar-product model (up by .052) but still trails that of the pure proximity model. Note that this is a generous test of the directional model in that it does not enforce the requirement that the predicted utility for a party outside the region of acceptability always be lower than for a party located on the boundary in the same direction. With this constraint enforced, the predictive power is lower.(24)

The sixth and final regression includes all three predictors. The results are essentially the same as for model 3. The coefficients for the scalar product and the penalty both drop to near zero, the coefficient for proximity remains on the same level as in the bivariate case, and there is no visible gain in predictive power compared to a pure proximity model. Thus, the proximity theory receives much stronger empirical support even when confronted with the complete version of the directional model.(25)

Although, for reasons previously explained, actual vote choice offers a less comprehensive, less precise, and less robust test of the two theories than the full range of evaluation scores, the results in this case generalize perfectly. The proximity model correctly predicts the vote in 52.5% of the cases as against 14.3% for a random model. The results for the directional model are 36.9% based on the scalar product alone and 49.7% with the penalty factor included.(26)

EXPLAINING THE OUTCOME

One question still remains. Why, exactly, are the results obtained here diametrically opposed to those previously reached by Macdonald, Listhaug, and Rabinowitz (1991) on the basis of the same data? Obviously, the answer must be sought in differences of method. Three [TABULAR DATA FOR TABLE 3 OMITTED] such differences separate my analysis from theirs: (1) its focus on choice and, consequently, on intrapersonal rather than interpersonal comparisons, (2) its use of individual rather than mean party placements, and (3) its consistent use of city-block rather than Euclidean or squared Euclidean distance in tests of the proximity model.

The analyses presented in Table 3 allow us to specify the effect of these differences. The table shows the results of testing the proximity model against the reduced and complete version of the directional model, respectively, under each of the 12 (2 x 2 x 3) method combinations derivable by means of the three factors. All 24 tests are multivariate regressions, with party evaluation as the dependent variable.

The first distinction - that between analyses based on an intrapersonal function and those based on raw scores - turns out to be the most important. The proximity model stands up well to extremely well against the directional model in the upper half of the table but loses most contests in the lower half. Note also that every regression in the upper half shows a better fit (higher [R.sup.2]) than the corresponding regression in the lower haiti This result is upheld in 11 out of 12 cases, even when the regressions are compared in terms of the absolute amount of variance explained rather than in terms of [R.sup.2].(27)

The two remaining distinctions are of some importance as well, although their effect does not rival that of the first. As expected, the use of mean party placements reduces the proportion of variance explained and tends to favor the directional model at the expense of the proximity model in most instances. Regressions taking squared Euclidean distance as a predictor - as in Rabinowitz, Macdonald, and Listhaug's multivariate confrontations of the two theories - consistently yield a worse fit than those specifying a linear distance effect. As a rule, they also put the proximity model at a disadvantage compared to the directional model (cf. Merrill 1995, 283). The contest between the two linear distance functions is more even, with the city-block specification showing a slightly better fit in tests based on individual party placements, and the Euclidean specification having a small edge when mean party placements are used.

In earlier sections, I showed that the intrapersonal comparisons required by choice theory may lead to a different outcome than do the interpersonal comparisons performed by Rabinowitz, Macdonald, and Listhaug. The above analysis demonstrates that they actually do so. Only one question remains: Why, specifically, does the one type of comparison not produce the same results as the other?

A first clue to the answer is given by Figure 4, which shows the relationship between party evaluation and party location on a left-right scale for each of ten different voter locations.(28) According to the proximity theory, the curves should peak at the point representing the voter's stand. The data show that they actually do so in nine out of ten cases, the tenth case falling but a hair's breadth off target. This result is far from self-evident. Rabinowitz, Macdonald, and Listhaug have repeatedly reported their failure to unravel curves of the kind displayed in Figure 4.(29) On the contrary, they have always found evaluation to be a monotonic and more or less linear function of location, so as to conform to the pattern predicted by the scalar product.

The key to the paradox is largely to be sought in the choice of perspective. The diagrams in Figure 4 keep voter location constant while letting party location vary, thereby mirroring the idea of a voter choosing between rivaling parties. Those presented by Rabinowitz, Macdonald, and Listhaug, by contrast, keep party location constant while letting voter location vary, which is less informative from a choice point of view. Figure 4 also reveals why this shift of perspective makes a difference. Although voters generally prefer a party sharing their own stand, the absolute level of their enthusiasm varies considerably by voter location. For voters occupying one of the two most extreme slots on either side of the midpoint, the peak value is in the range of 82-90. For voters taking one of the two innermost stands on either side, the peak reaches no higher than 61-66. A closer look reveals that the pattern is not restricted to the peak values. In fact, voters far from the center tend to provide generally more positive evaluations once proximity is held constant.

The likely cause of this phenomenon, which we may call the eccentricity effect will be discussed shortly. However, let us first, by means of Figure 5, consider its implications for the results derived on the basis of the two perspectives. The leftmost diagram keeps voter location constant (at -1.5) while letting party location vary, just as in Figure 4. The rightmost diagram keeps party location constant (at -1.5) while letting voter location vary, as in the analyses by Rabinowitz, Macdonald, and Listhaug. Note how the eccentricity effect conceals the proximity pattern in the latter case, making evaluation increase almost monotonically from right to left rather than climb to a peak and then decline.(30) As expected, a linear regression based on proximity fits the points of the leftmost diagram better than one based on the scalar product, whereas the opposite holds for the diagram to the right.

We may thus conclude that the eccentricity effect constitutes a plausible reason for the difference between my results and those of Rabinowitz, Macdonald, and Listhaug. Why, however, does this effect occur? The most credible explanation rests with the nature of the evaluation scores. In all likelihood, most respondents understand the thermometer items in agreement with the main design objective, that is, as a way of expressing their appreciation of each party relative to the appreciation of other parties. Consequently, each respondent adjusts the ratings so that a party located at a middling distance (compared to other parties) obtains a score near the neutral point of the scale (50). Parties closer than this benchmark are given higher ratings and those more distant lower. What is easily overlooked, however, is that the distance to the benchmark is not the same across respondents. Those far from the center will be farther away from their respective benchmarks than those in the middle ground. When proximity is held constant, the ratings produced by eccentric respondents will therefore be higher than those of centric ones.

An illustration is provided by the two voters presented in Figure 2A. Assume that the y axis represents party evaluation and the x axis proximity. In keeping with the above rating scheme, the means of the evaluations supplied by the two voters are the same. However, voter 1 is located farther from the center than voter 2, which implies a lower mean as well as a higher variance in voter-party proximity. Due to the difference in mean proximity, the ratings provided by the first voter fall to the left of, and above, those provided by the second. Note also that the greater variation in the proximity scores of voter 1 yields a higher top rating and a lower bottom rating than for voter 2.

It remains but to subject my argument to a systematic test based on the full set of issue dimensions. If my reasoning up to this point is valid, then one simple modification of the analytical procedure normally used by Rabinowitz, Macdonald, and Listhaug should do much to redress the balance between the two theories. That such is actually the case is shown by the results presented in Table 4, which contains estimates for two [TABULAR DATA FOR TABLE 4 OMITTED] series of statistical models. The first series emulates the type of pooled regression analysis commonly employed by Rabinowitz, Macdonald, and Listhaug. Consequently, the analysis is based on the original scores, without any subtraction of individual averages. Furthermore, it uses mean rather than individual party placements and a measure of proximity based on Euclidean distance. The second series is identical but for one modification: Proximity is measured in terms of its deviation from the individual mean.

As expected, the first series of estimates consistently favors the directional model. In the second series, however, the proximity model fares much better since most of the problems caused by the relative nature of the evaluation scores are now appropriately corrected. In fact, the simple adjustment of the proximity measure alone is in this case sufficient to shift the weight of evidence from one model to the other.

CONCLUSION

Few would deny that the Downsian theory has its shortcomings. As many have observed before me, however, a model is not likely to be replaced by anything but another model. According to Rabinowitz, Macdonald, and Listhaug, the directional theory is the solution for which we have been waiting. My analysis indicates that such is not the case. A worthy successor to the classical spatial model of electoral choice has yet to be found.

This main conclusion is based on four central observations: one theoretical, one methodological, and two empirical. The theoretical observation is that substantive theories cannot be evaluated merely by the extent to which they are compatible with the evidence. Their degree of falsifiability, that is, the extent to which they can, logically speaking, be refuted by empirical observations must also be taken into account. While this idea is hardly new, and while I cannot lay any claims to its origin however much I would want to, experience shows that it bears repeating. The largely unspecified nature of the region of acceptability and the penalty renders the directional theory less falsifiable than the proximity theory. The implication is that even if the directional model were found to be more compatible with empirical evidence, this would not be sufficient to demonstrate its general superiority.

The methodological observation is that we must learn to take choice theories seriously. A crucial feature distinguishing these from other types of theories is that the predictions they make concern the way an individual relates to a set of alternatives rather than the way an alternative relates to a set of individuals. The predictions should be tested accordingly, that is, by means of comparisons within rather than across individuals. Most of the tests performed by Rabinowitz, Macdonald, and Listhaug do not meet this requirement and therefore can neither refute nor corroborate either theory. The one exception to that rule does not qualify as serious evidence for other reasons (see Appendix C).

While the methodological point is worth making in its own right, irrespective of its consequences for the outcome, my empirical analyses demonstrate that it does matter. When the two theories are retested based on more appropriate methods, the results are different although the data are the same. This time, the evidence clearly supports the proximity model over the reduced and complete version of the directional model alike. By implication, there are reasons to believe that voters do not generally see issues as a dichotomous choice between a black and a white policy alternative, as assumed by the directional model, but are able to distinguish as well as express at least some shades of gray.

The final empirical section specifies the major reason why the intrapersonal perspective required by choice theory yields a different outcome than does the interpersonal perspective used by Rabinowitz, Macdonald, and Listhaug. A closer look at the data reveals an eccentricity effect, the consequence of which is a bias against the proximity model in tests based on interpersonal comparisons. Once this effect is taken into account, the proximity model fares much better even in analyses otherwise similar to those performed by Rabinowitz, Macdonald, and Listhaug.

Whereas Rabinowitz, Macdonald, and Listhaug have looked at evidence on the individual as well as the system level in their comparisons of the two theories, my own analyses have been confined to the individual level. My reasons for staying on that level are simple. Both theories are predicated on the idea that voter choice determines party strategy. Therefore, the extent to which the two theories are consonant with system-level evidence is of interest only insofar as they succeed on the individual level. Even if the directional model were found to be more compliant with system-level properties than the proximity model - a matter which, in my view, is still open for debate - it would lack interest as an explanation on that level unless it can assert itself as a model of voter choice.

Its failure to do so raises the question of how the shortcomings of the proximity model should instead be addressed. I shall resist the temptation to outline some possible answers to that question because my undertaking is not well suited to that end. As Popper (1968) points out, both conjectures and refutations are needed in the pursuit of scientific progress. Rabinowitz, Macdonald, and Listhaug contributed a conjecture. I have contributed a refutation. If Popper is right, the two efforts combined have brought us one small step nearer to the truth.

APPENDIX A: INCONSISTENCY OF THE PENALTY FUNCTION

Rabinowitz, Macdonald, and Listhaug (1991, 152) suggest that the penalty a voter applies to a party located outside the region of acceptability may vary with the voter's own stand. Voters who are more directionally sympathetic to a party, that is, those whose scalar product with the party is greater, might apply a weaker penalty than those who are less directionally sympathetic. They also present a class of utility functions that, in their view, satisfy this idea (Rabinowitz, Macdonald, and Listhaug 1991, 182, n5). For a party located outside the region of acceptability, this class of functions is defined as

[U.sub.ij] = (r/absolute value of j])i [multiplied by] j - k([absolute value of j] - r)/(1 + [e.sup.i [multiplied by] j]),

where [U.sub.ij] is the utility that voter i associates with party j, i [multiplied by] j the scalar product between the vectors extending from the origin to the points representing the issue stands of the voter and the party, [absolute value of j] the length of the party vector, r the radius of the region of acceptability, and k an arbitrary positive constant. The penalty component contained in this function can be extracted by means of the general utility function of the directional model, that is, [U.sub.ij] = i [multiplied by] j - [P.sub.ij], where [P.sub.ij] is the penalty applied by voter i to party j (Macdonald, Listhaug, and Rabinowitz 1991, 1110). By substitution we obtain

i [multiplied by] j - [P.sub.ij] = (r/[absolute value of j]) i [multiplied by] j - k([absolute value of j] - r)/(1 + [e.sup.i [multiplied by] j])

which after rearrangement yields the penalty function:

[P.sub.ij] = (1 - r/[absolute value of j]) i [multiplied by] j] + k([absolute value of j] - r)/(1 + [e.sup.i [multiplied by j]).

According to Rabinowitz, Macdonald, and Listhaug (1991, 151-2, 182, n5), the penalty function should meet three conditions for any voter and any party outside the region:

1. i [multiplied by] j - [P.sub.ij] [less than] (r/[absolute value of j])i [multiplied by] j)i (which represents the condition that the utility associated with a party located outside the region of acceptability must always be lower than that associated with a party located on the boundary of the region in the same direction).

2. [P.sub.ij] [greater than or equal to] 0 (which represents the condition that the penalty must be a penalty, not a reward).

3. [Delta][P.sub.ij]/[Delta](i [multiplied by] j) [less than or equal to] 0 for all i given any particular j (which represents the condition that the penalty a voter applies to a certain party decreases as the voter becomes increasingly directionally sympathetic to the party).

The penalty function does satisfy the first condition, but it can easily be shown that it does not satisfy the latter two. To that end, it is assumed in the following that the location of party j remains constant at any given point outside the region of acceptability (in agreement with what is stipulated in condition 3) and that only voter location varies.

Consider first condition 2. The factor (1 - r/[absolute value of j]) in the first term of the penalty function is always positive since r [less than] [absolute value of j] for any party outside the region of acceptability. Hence, (1 - r/[absolute value of j]l)i [multiplied by] j takes on increasingly large negative values with increasingly large negative values of i [multiplied by] j. The second term, by contrast, approaches the positive constant k([absolute value of j] - r) under the same conditions. Thus, at some point [P.sub.ij] always falls below 0.

Condition 3 stipulates that the derivative of [P.sub.ij] with respect to i [multiplied by] j must not be positive for any i [multiplied by] j. The derivative can be written as:

[Mathematical Expression Omitted].

The first term is a positive constant. The first bracketed factor in the second term approaches 0 as i [multiplied by] j grows increasingly large, whereas the second bracketed factor approaches unity under the same condition. Hence, the entire second term approaches 0 as i [multiplied by] j grows increasingly large. The entire expression will, therefore, at some point always become positive.

That the function does not satisfy conditions 2 and 3 can also be shown through a simple example. Let us for convenience assume that the space is unidimensional. Assume in addition that the score of party j is 5, with r = 4, and k = 1. The penalty applied by a voter with a score of -5 would be

[P.sub.ij] = (4/5 - 1)25 + 1(5 - 4)/(1 + [e.sup.-25]) [approximately equal to] -4,

and by a voter with a score of 5:

[P.sub.ij] = (1 - 4/5)25 + 1(5 - 4)/(1 + [e.sup.25]) [approximately equal to] 5.

In other words, the penalty increases rather than decreases as the voter becomes more directionally sympathetic, and the first of the two examples does in fact result in a reward, not a penalty.

In all likelihood, it is impossible to devise a function that simultaneously satisfies all three conditions except by introducing thresholds, discontinuities, or boundary conditions that seem hard to justify on theoretical grounds. The root of the problem is that conditions 1 and 3 are in direct contradiction with each other. To satisfy condition 1, the penalty should increase with increasing positive values of i [multiplied by] j, but to satisfy condition 3 it should decrease under the same circumstance.

APPENDIX B: REDUNDANCY OF THE MIXED MODEL

In order to perform a multivariate test of the two theories, Rabinowitz, Macdonald, and Listhaug suggest that we use a special statistical model, which they call a mixed model. The mixed model takes its point of departure in the formula for the squared Euclidean distance between a voter and a party, that is, [[absolute value of i].sup.2] + [[[absolute value of j].sup.2] 2i [multiplied by j], where the first two terms represent the squared lengths of the voter and party vectors and the third twice the

scalar product. The squared Euclidean distance can thus be separated into two components: a length component, [[absolute value of i].sup.2] + [[absolute value of j].sup.2], and a scalar-product component, -2i [multiplied by] j. The mixed model takes these components as its independent variables, albeit with their signs reversed so that the theoretically expected value of the corresponding regression coefficients becomes positive rather than negative. The model can thus be written as:

[Y.sub.ij] = [b.sub.1] - [b.sub.2] ([[absolute value of i].sup.2] + [[absolute value of j].sup.2]) + [b.sub.3]2i [multiplied by] j + [u.sub.ij].

According to Rabinowitz, Macdonald, and Listhaug, the empirical strength of the two theories can be evaluated by comparing and combining the coefficients of the model in various ways. If [b.sub.2] = [b.sub.3] [greater than] 0, we have a pure proximity model. If [b.sub.2] = 0 and [b.sub.3] [greater than] 0, we have a pure directional model. If [b.sub.2] [greater than] 0 and [b.sub.3] [greater than] 0 and [b.sub.3] [greater than] [b.sub.2], both theories enjoy at least a modicum of support. In order to assess their relative strength, Rabinowitz, Macdonald, and Listhaug suggest that we form the model ratio [b.sub.3]/[b.sub.2]. Although this ratio can in principle take on values less than 1, the interesting range is from 1, in the case of a pure proximity model, to infinity, in the case of a pure directional model.

When the mixed model was first introduced (Rabinowitz and Macdonald 1989), no rationale was provided for preferring it to a more conventional approach, that is, one taking (-squared or un-squared) distance and scalar product as the predictors. In later works (Rabinowitz, Macdonald, and Listhaug 1993; Macdonald, Rabinowitz, and Listhaug 1995a, b), it is argued that the mixed model avoids the colinearity problems resulting from the fact that the scalar product is mathematically a component of Euclidean distance.

As I will demonstrate below, however, there are no reasons for preferring the mixed model to the conventional solution. On the contrary, the latter is superior on several counts. In order to keep my demonstration as short and simple as possible, I will ignore the special problems associated with the penalty factor (which is always left aside when Rabinowitz, Macdonald, and Listhaug use the model) as well as those related to the distinction between intrapersonal and interpersonal comparisons. Given this starting point, the alternative to the mixed model, which I will call the standard model, can be written as

[Y.sub.ij] = [b.sub.4] - [b.sub.5][d.sub.ij] + [b.sub.6]i [multiplied by] j + [u.sub.ij],

where [d.sub.ij] is the measure of (-squared or un-squared) distance and i [multiplied by] j the scalar product. Compared to this alternative, the mixed model has three disadvantages.

First, in contrast to the standard model, the mixed model requires that we estimate the dependent variable as a quadratic function of Euclidean distance. Other (and better fitting) functional forms (e.g., linear) or distance conceptions (e.g., city block) cannot be accommodated or meet with difficulties. If, for example; we would like to estimate the dependent variable as a linear rather than quadratic function of Euclidean distance, the mixed model returns an equation which is nonlinear in the parameters. While such equations can still be estimated by resorting to special techniques, one is well advised to avoid them unless there are solid reasons to the contrary. The mixed model hardly constitutes such a reason.

Second, Rabinowitz, Macdonald, and Listhaug fail to specify a criterion whereby the mixed model and its associated model ratio lets us decide which of the two theories enjoys the strongest support. The standard model, by contrast, provides an unambiguous breakpoint. If the standardized version of [b.sub.5] equals that of [b.sub.6], support for the two theories is evenly balanced. If not, the theory corresponding to the higher coefficient has the upper hand.(31)

Third, in the standard model, the coefficients b5 and b6 represent the two theories in a direct and uncomplicated fashion. In the mixed model, by contrast, [b.sub.3] always has to be compared to [b.sub.2] before we can interpret its theoretical significance. This indirect relationship between the theories and the coefficients in the mixed model makes for needless difficulties of interpretation.

Let us finally consider the one argument put forth in support of the mixed model over the standard model, namely, that it is preferable as an estimation strategy because of the reduction of colinearity. If we let [d.sub.ij] represent squared Euclidean distance, the standard model can be rendered in a form similar to that of the mixed model:

[Y.sub.ij] = [b.sub.4] - [b.sub.5]([[absolute value of i].sup.2] + [[absolute value of j].sup.2]) + [b.sub.5] + 0.5[b.sub.6]) 2i [multiplied by] j + [u.sub.ij].

Comparing this version of the standard model with the mixed model, we can immediately conclude that [b.sub.1] = b4, [b.sub.2] = [b.sub.5], and [b.sub.3] = [b.sub.5] + 0.5b6. It follows that all six coefficients can be estimated on the basis of either model. Irrespective of which model we use for estimation purposes, the estimates will be exactly the same, as will the associated standard errors. Furthermore, if we compare the standard errors of the coefficients of the two models, we find that they differ only in one case: The standard error of [b.sub.3] will be lower than that of [b.sub.6]. This improvement is bought at the expense of the interpretability of [b.sub.3], however, and therefore yields no true benefit. Once we perform any of the operations required to interpret [b.sub.3], such as calculating 2([b.sub.3] - [b.sub.2]) to isolate the effect of the scalar product, the standard error of the resultant value becomes exactly the same as for the corresponding value derived on the basis of the standard model (in this case, the value of Ds).

APPENDIX C: PRIOR EVIDENCE REFUTED

The individual-level tests of the two theories reported by Rabinowitz, Macdonald, and Listhaug (1991) differ in several important respects from those the three authors have presented elsewhere. First, these tests are based on the proper form of comparison: intrapersonal rather than interpersonal. This is why they merit special consideration here. Second, the dependent variable is the actual vote rather than party evaluations. Third, and largely as a consequence of the first two differences, the region of acceptability and the penalty factor play a much more crucial role than they do in all other tests performed by Rabinowitz, Macdonald, and Listhaug. No special motivation for these departures from their normal testing strategy is provided.

The data come from the 1985 Norwegian and 1979 Swedish election studies.(32) The independent variables are based on placement on a left-right scale only. The region of acceptability is drawn so as to exclude the Socialist Left Party and the Progressive Party in Norway and the Left Party Communists in Sweden. The exact location of the border of the region is left unspecified. The penalty is operationalized so as to imply that no voter prefers a party outside the region. None of these choices are discussed or motivated, aside from the observation that the parties excluded are wing parties; see Figure C-1.(33)

In Rabinowitz, Macdonald, and Listhaug's view, the directional model proves superior in both countries. According to their analyses, it correctly predicts the vote in 64% of the Norwegian and 62% of the Swedish cases, versus 31% and 48%, respectively, for the proximity model (Rabinowitz, Macdonald, and Listhaug 1991, 171, Table 1, 177, Table 2). The following additional observations can be made.

1. The outcome is entirely dependent on how the region of acceptability and the penalty factor are operationalized. If, for example, all parties are considered to be inside the region, the proportion of votes correctly predicted by the directional model drops from 64% to 9% for Norway and from 62% to 20% for Sweden.

2. The operationalization decisions made ensure that all predictions of the directional model will benefit the two largest parties in either system: the Labor Party and the Conservatives in Norway, the Social Democrats and the Conservatives in Sweden. Of course, placing all bets on the known winners substantially increases the chance of obtaining a large proportion of correct predictions on the individual level. (On the party-system level, of which no account is taken in this part of Rabinowitz, Macdonald, and Listhaug's analysis, one simultaneously obtains the prediction that both Norway and Sweden have two-party systems, which obviously does not square very well with the evidence.)

3. Although the directional model provides a lot of leeway for the operationalization of the penalty factor, the one chosen here is inconsistent with its requirements. As repeatedly stressed by Rabinowitz, Macdonald, and Listhaug, the theory does not imply that a party outside the region will necessarily lose a contest, let alone every single vote, to any competitor inside the region.(34) The operationalization used in this case, however, forces such an outcome.

4. While the theory also provides a lot of latitude for the operationalization of the region of acceptability, the one used here is nevertheless inconsistent with it. The theory imposes but a single requirement, namely, that the region be located at a directionally invariant distance from the neutral point. Still, in the Norwegian case, the Socialist Left Party is placed outside the region but the Conservatives inside it, despite the former being closer to the neutral point than the latter according to Rabinowitz, Macdonald, and Listhaug's own choice of measurement [ILLUSTRATION FOR FIGURE C-1 OMITTED].

5. The Swedish Conservatives are in fact more distant from the neutral point than the Norwegian Socialist Left and Progressive parties [ILLUSTRATION FOR FIGURE C-1 OMITTED]. Still, the former is placed inside the region, while the latter two are placed outside. Cross-national differences in the size of the region are not ruled out by the theory, but one would have expected Rabinowitz, Macdonald, and Listhaug to motivate explicitly and convincingly the presence of such a difference in the case at issue in order to comply with the demand of fair testing.

6. Drawing the boundary of the region of acceptability so as to separate parties that are as similar in terms of distance from the neutral point as are the Socialist Left, the Progressive, and the Conservative parties in Norway or the Left Party Communists and the Conservatives in Sweden must be considered a questionable testing strategy, at least when the consequences of placing a party inside or outside the region are as drastic as in the present case.

7. Although the directional theory predicts that respondents placing themselves at the neutral point should be indifferent to all parties, Rabinowitz, Macdonald, and Listhaug contrive a prediction that such respondents should, in the Swedish case, all prefer the Social Democrats. (In Norway; the scale used did not allow respondents to place themselves at the neutral point.) The prediction is based on the presumption that all the neutral respondents are in fact slightly left-wing, since there are more respondents in the category just to the left of the neutral point than in the one just to the right. Rabinowitz, Macdonald, and Listhaug thus ensure that 42% of the predictions for the neutral respondents will be correct as opposed to 9% if these respondents had instead been counted as slightly right-wing. As pointed out by Granberg and Gilljam (n.d., n4), the same reasoning would have led to but 9%, 7%, and 5% correct predictions, respectively, if applied to the 1985, 1988, or 1991 Swedish election study. In those years, the respondents just to the right of the neutral point outnumbered those just to the left, whereas the proportion of neutral respondents voting for the Social Democrats and Conservatives was approximately the same as in 1979.

8. Rabinowitz, Macdonald, and Listhaug's reliance on mean rather than individual party placements in constructing the main independent variables - proximity and scalar product - leads, in combination with other factors, to absurd consequences in the test of the proximity model. For example, it becomes theoretically impossible for a Norwegian voter to occupy such a position that Labor becomes the most proximate party. At -2.5, the Socialist Left Party is the closest, and at -1.5 the most proximate location is already taken by the Liberals [ILLUSTRATION FOR FIGURE C-1 OMITTED].(35)

Due to the above weaknesses, I find it difficult to accept the results reported by Rabinowitz, Macdonald, and Listhaug (1991) as serious evidence in the contest between the directional and the proximity model. The methods described in the main text of this article (those underlying Table 2) allow a more reliable verdict. Among other things, these methods limit (although they cannot completely eliminate) the effect of more or less arbitrary decisions regarding the boundary of the region of acceptability by making use of the entire preference order rather than the vote alone. Furthermore, the region and the associated penalty are operationalized in agreement with theoretical requirements.(36) No ad hoc predictions of the type described under point seven above are added. Finally, artifacts of the type exemplified for Norwegian Labor voters are avoided.

The results are shown in Table C-1.(37) The substantive conclusions are the same as those reached in the main text. Irrespective of whether we look at Norway 1985 or Sweden 1979 and whether we include the penalty factor or not, the evidence clearly favors the proximity model. Again, this result also generalizes to the actual vote, The proximity model correctly predicts the vote in 54% of the Norwegian cases and 61% of the Swedish. The corresponding results for the directional model are 20% and 34% based on the scalar product alone. With the penalty factor taken into account, the figures are 49% and 56%.(38)

[TABULAR DATA FOR TABLE C-1 OMITTED]

1 The list of references includes eleven contributions with Rabinowitz, Macdonald, or Listhaug as the first author. All but one (Listhaug, Macdonald, and Rabinowitz 1994b) focus on the new theory.

2 See, for example, Rabinowitz and Macdonald 1989, 93, the abstract; Macdonald, Listhaug, and Rabinowitz 1991, 1107, the abstract. Virtually all Rabinowitz, Macdonald, and Listhaug's writings on the new theory pit it against the classical model.

3 Some studies have found support for this hypothesis, others have not. Krosnick (1988) provides a good review of past research in the area along with new tests and some methodological explanations for the failure to distinguish a salience effect in some of the prior attempts.

4 It can be argued that the proximity model contains an implicit weighting scheme of a somewhat similar kind in that the variance in voter-party proximities on any one dimension tends to decline as the voter approaches the center of the party spectrum. Yet, the effect of this tendency is neither as strong nor as straightforward as that of the voter-intensity rule in the directional case.

5 Examples and operationalizations are sometimes provided, but these are not presented as being part of, or fully implied by, the theory.

6 This, in turn, implies that all voters, irrespective of their locations, will reject a party A outside the region, in favor of a party B on the boundary in the same direction. If A would be preferred by voters in some locations, there would be no guarantee that B would perform better thanA in the electorate as a whole, since the theory makes no assumptions about the spatial distribution of voters. It should be observed that this partial specification of the penalty function in Rabinowitz, Macdonald, and Listhaug's first two articles on the topic is not repeated in later contributions, where they leave the shape of the function more open (see, e.g., Macdonald, Listhaug, and Rabinowitz 1991, 1113, 1127). To the extent that this can be understood as a rejection of the prior specification or a denial of its universal applicability, it implies a further reduction in the falsifiability of the directional model.

7 As a rule, interpersonal comparisons of utility are completely irrelevant to positive choice theory. As pointed out by Elster and Roemer (1991), there are cases in which the observers must take into account that the observed individuals sometimes make such comparisons, as in the study of envy and of bargaining. Yet, even in these special cases - none of which is relevant here - there is no need for the observers themselves to compare utilities across individuals. In normative choice theory, by contrast, it is often desirable to make such comparisons but difficult to find a justifiable way of doing so. For a modern discussion of this problem, see the contributions to the volume of which Elster and Roemer 1991 is part.

8 The seven contributions I have in mind are: Rabinowitz and Macdonald 1989; Macdonald, Listhaug, and Rabinowitz 1991; Macdonald and Rabinowitz 1993b; Rabinowitz, Macdonald, and Listhaug 1993; Listhaug, Macdonald, and Rabinowitz 1994a; Macdonald, Rabinowitz, and Listhaug 1995a and b. In an eighth article - Rabinowitz, Macdonald, and Listhaug 1991 - which in general is less ambitious than any of the aforementioned when it comes to individual-level testing one does find a few analyses based on intrapersonal comparisons. These analyses are discussed in Appendix C.

9 In Rabinowitz, Macdonald, and Listhaug's early contributions, the question is never raised. In some of their later pieces, however, it is explicitly addressed (Macdonald and Rabinowitz 1993b, 63; Macdonald, Rabinowitz, and Listhaug 1995b, 475-6). The answer provided is that (1) vote choice is based on the utility difference an individual sees between two parties and (2) the two theories are, under some provisions, indistinguishable through interpersonal comparisons of such differences. For reasons already discussed, however, a test of the two theories should be based on intrapersonal comparisons, in which case the theories remain distinguishable irrespective of which measure we use. The choice between the two measures must therefore be made on other grounds.

10 Some further specifications of the reasons favoring evaluation scores over the actual vote are given in the subsection "Choosing the Testing Ground" below. See also Appendix C for an illustration. Note, finally, that the loss of information incurred by taking the actual vote as the dependent variable is likely to prompt simplification of the independent variables as well. For reasons spelled out in the last two sentences of note 14, the ensuing loss of precision in the predictors increases the risk of faulty inferences in simultaneous tests of the two theories.

11 The accuracy of voter images of party or candidate issue stands has been studied by, for example, Granberg and Holmberg (1988, chapter 3), Powell (1989), and Listhaug, Macdonald, and Rabinowitz (1994b). A general conclusion from these studies is that voter images are far from perfectly accurate or unanimous but nevertheless are clearly related to what can realistically be considered the objectively true locations.

12 The same kind of reasoning applies to persuasion effects, that is, the effect of party locations on the voters' own issue stands. Note, in addition, that the dynamic estimates of Markus and Converse (1979, 1061) show both effects to exist but neither to be of overwhelming magnitude. This result is echoed in an analysis by Merrill (1995, 284-5), whose findings additionally indicate that proximity-driven projection does not threaten the validity of comparative tests of the two theories based on individual party placements.

13 With the midpoint scored 0, the bias due to guessing can be described by the function [x.sub.j] = (1 - p)[X.sub.j], where [x.sub.j] is the estimated location of party j, [X.sub.h] the true location of the same party, and p the proportion of guesses. It follows that the estimated scalar product for all party-voter combinations will equal the true scalar product multiplied by (1 - p). This in turn implies that the predictions made on the basis of the estimated scalar products will be the same as those based on the true scalar products. The estimated proximities, however, will not be a simple linear transformation of the true proximities, and the predictions made on the basis of the former will as a rule be different than those based on the latter.

14 Prior research (e.g., Aldrich and McKelvey 1977) indicates that all respondents do not interpret the end-points and units of response scales in the same fashion. Mean party placements do not take proper account of the ensuing variations in respondent scale use. For reasons similar to those discussed in the previous note, this is likely to have more serious consequences for the proximity than for the directional model. A second problem is that the mean-placement method allows parties to be located anywhere on the scale, whereas voters are forced into discrete slots. This may lead to severe consequences in tests of the proximity model. For instance, a party may obtain a mean that makes it theoretically impossible for any voter to take a stand so as to make the party in question the closest one (see Appendix C for an illustration). Finally, since proximity is by definition strongly correlated with the scalar product, it is generally of great importance to keep unreliability in the predictors at a minimum in simultaneous tests of the two models. There is otherwise a considerable risk that one theory will be credited with effects actually attributable to the other (cf. Blalock 1979, 477; Westholm 1987).

15 Others have arrived at similar conclusions, although their arguments are not necessarily the same as mine. See, for example, Gilljam 1997; Merrill 1993, 1994, 1995; Pierce 1997.

16 An obvious solution to this problem would be to accord each individual a private region of acceptability. Once this step is taken, however, the distinction between the two theories becomes very tenuous. A directional model thus modified is more or less tantamount to a hybrid model in which utility is partly determined by party - voter distance and partly by the scalar product (cf. Iversen 1994a and b; Merrill and Grofman 1997).

17 The study, conducted by Henry Valen and Bernt Aardal, is based on a nationally representative sample of all members of the electorate aged 18-80. Data were successfully collected, predominantly by means of personal interviews, from 2,195 respondents, which corresponds to a response rate of 73.2%. The data set is available from the Norwegian Social Science Data Services (NSD).

18 The six issue items are: agricultural policy (maintain versus reduce government support for agriculture), environmental policy (give versus do not give priority to environmental protection over standard of living), immigration policy (more liberal versus more restrictive), health care policy (private versus public health care), alcohol policy (more liberal versus more restrictive), and crime policy (milder versus tougher punishment of violent crime). Responses to all issue items as well as left-right placement were obtained on a ten-point scale, originally scored 1-10 but rescored to range from -4.5 to 4.5 in all analyses presented here. Proximity scores were computed as - [Sigma] [absolute value of] [i.sub.k] - [j.sub.k]/[n.sub.ij] and scalar product scores as [Sigma][i.sub.k][j.sub.k]/[n.sub.ij], where [i.sub.k] is the score of voter i on item k, [j.sub.k] the score of party j on the same issue, and [n.sub.ij] the number of items with valid data for voter i and party j. Thermometer ratings were obtained on a 0-100 scale. An English translation of the question wording for all items used is provided by Macdonald, Listhaug, and Rabinowitz (1991, 1127-8, 1130, n10).

19 According to Macdonald, Listhaug, and Rabinowitz (1991, 1121, Table 7, note b), the directional model outperforms the proximity model even before the penalty measure is taken into account.

20 The number of cases is 13,503 and the number of respondents 1,929. The test of significance used is described in Blalock 1979, 424-5, and its applicability justified by the observation that Spearman's rho is defined as the product-moment correlation applied to ranks. The significance level reported obtains even under the conservative assumption that N equals 1,929 rather than 13,503 minus 1,929.

21 Eccentricity is operationalized, here as well as in subsequent analyses, as the respondent's average deviation from item midpoints, that is, as [Sigma][absolute value of [i.sub.k]/[n.sub.i], where [i.sub.k] is the score of voter i on item k, and [n.sub.i] is the number of items with valid data for voter i. Alternative definitions of the central point on each dimension are, of course, conceivable; for example, the population mean or the location of the median party. In the case at issue, however, the practical difference between these alternative specifications is very small. The eccentricity values used in the figure have been rounded to the closest half unit.

22 Rabinowitz, Macdonald, and Listhaug argue that a special statistical model, which they call a mixed model, should be used when estimating the relative effect of proximity and scalar product within the framework of a single, multivariate, statistical model. A description of this model is given in Appendix B. As shown in the same appendix, however, it offers no advantages but several disadvantages compared to a more conventional approach. There is thus no reason to depart from the latter. Furthermore, Macdonald, Listhaug, and Rabinowitz (1991) report results for statistical models including as well as excluding controls for factors outside the scope of the two choice theories, for example, social class and sex. The controls leave the outcome of their theory comparison substantively unaffected. In view of space constraints and in the absence of compelling empirical or theoretical reasons to the contrary, I focus solely on the variables singled out by the two theories.

23 The region of acceptability itself is operationalized as being the same for all voters since this is a requirement of the directional theory. Party placements are allowed to vary across respondents, however, just as in the construction of the proximity and scalar-product measures (see the subsection "Individual versus Aggregate Images of Party Issue Stands"). Macdonald, Listhaug, and Rabinowitz (1991, 1130, n15) draw the border of the region at a Euclidean distance from the origin of 4.305 based on six items. Expressed in the currency of the normalized distance measure (distance relative to the number of dimensions) used in the present analysis, the value becomes [-square root of][4.305.sup.2]/6 [approximately equal to] 1.758. The score of the penalty factor is [Mathematical Expression Omitted] if the value is positive (i.e., if the party is placed outside the region) and zero if not. The symbol [j.sub.k] denotes the score of party j on item k and [n.sub.j] the number of items with valid data for party j. According to Macdonald, Listhaug, and Rabinowitz (1991, 1130, n14), one of the seven items used in constructing the proximity and scalar-product measure (crime policy) should not be included when defining the boundary of the region and the penalty factor since it is shown in their tests to have little effect on party evaluation. I have chosen to follow their practice in order to avoid a redefinition of the radius of the region.

24 See footnote 6. In formal terms, the condition can be written as - [b.sub.2] [greater than or equal to] [b.sub.1] max([absolute value of i]), where [b.sub.1] and [b.sub.2] are the unstandardized regression coefficients for the scalar-product and the penalty variable, respectively, and max([absolute value of i]) the theoretical maximum length of any voter (or party) vector in the issue space under consideration. In this case, -[b.sub.2] = 7.58 and [b.sub.1]max([absolute value of i]) = 2.99 [multiplied by] 4.50 = 13.46, which violates the condition. Constraining the parameter estimates so that the condition is met lowers the adjusted [R.sup.2] to .325.

25 Models in which the weight of each issue was set free for estimation were also analyzed (cf. Macdonald, Listhaug, and Rabinowitz 1991, 1120-1). The results continue to support the proximity model over both versions of the directional model.

26 For the proximity model and the directional model without penalty, the predicted party is the one having the highest proximity and scalar-product score, respectively. For the directional model with penalty, it is the party with the highest predicted score according to the estimates for the fifth model in Table 2. If several parties shared the top score, a share of the case was attributed to each of the tied parties. The analyses include only those respondents (numbering 1,674) who (1) reported voting for one of the seven parties for which there is party placement data, (2) actually took part in the election (according to public records), and (3) had valid scores for proximity, scalar product, and penalty across all seven parties.

27 Multiplying the values of [R.sup.2] in the upper half of Table 3 by .86 makes them comparable to those in the lower half in terms of the absolute amount of variance explained.

28 For another analysis of the two theories based on the idea of keeping voter location constant while letting party location vary, see Granberg and Gilljam n.d.

29 Rabinowitz and Macdonald 1989, 98-9, Figure 2, and 105-7, Figure 4; Rabinowitz, Macdonald, and Listhaug 1993, 5-6, Figure 2, and 11-2, Figure 3 to Figure 7; Listhaug, Macdonald, and Rabinowitz 1994a, 114-6, Figure 2, and 123-41, Figure 4 to Figure 9; Macdonald, Rabinowitz, and Listhaug 1995a, 3-4, Figure 2, and 10-3, Figure 3 to Figure 7.

30 In many cases, including the one illustrated here, the pattern is somewhat sigmoid (S-shaped) or inverted sigmoid. See Granberg and Gilljam (n.d.), who, on the basis of Swedish data, report curves resembling a logistic function. See also the graphs presented by Merrill (1995).

31 Standardizing the coefficients of the mixed model, as in Rabinowitz and Macdonald 1989, is of no help here since the model's rules of interpretability, e.g., the equality between [b.sub.2] and [b.sub.3] in the case of a pure proximity model, hold for the unstandardized coefficients only. Standardizing [b.sub.5] and [b.sub.6] in the standard model is a meaningful operation, however, since these two coefficients represent the two theories and their associated measures in a direct fashion. A solution to the lack of a parity criterion for the mixed model is proposed by Merrill and Grofman (1997) in the course of developing a unified model for testing spatial choice theories. Regrettably, their unified model, like the mixed model, operates on the basis of squared Euclidean distance only.

32 The two studies - conducted by Henry Valen and Bernt Aardal in Norway and by Soren Holmberg in Sweden - are both based on nationally representative samples of all members of the electorate aged 18-80. Data were successfully collected, predominantly by means of personal interviews, from 2,180 respondents in Norway and 2,816 in Sweden, which corresponds to response rates of 72.7% and 80.5%, respectively. The data sets are available from the Norwegian Social Science Data Services (NSD) and the Swedish Social Science Data Service (SSD).

33 The Norwegian scale is originally scored from 1 to 10 and the Swedish from 0 to 10. Both were rescored to range from -4,5 to 4.5 in all analyses presented here.

34 According to Rabinowitz, Macdonald, and Listhaug (1991, 152), a party outside the region is sure to lose only against a competitor located on the boundary in the same direction. Since they do not specify the precise location of the boundary, it is unclear whether such a party is assumed to exist in the present case. What is clear, however, is that boundary parties cannot exist on both sides simultaneously, since the theory requires that the boundary fall at a directionally invariant distance from the origin. This suffices to make the operationalization untenable.

35 This problem passes unnoticed in the analysis of Rabinowitz, Macdonald. and Listhaug (1991). The prediction for a voter score of -2.5 is, without comment, accorded the Labor Party rather than the Socialist Left.

36 In agreement with the practice followed by Rabinowitz, Macdonald, and Listhaug in their analysis of the Norwegian 1989 data set, the border of the region of acceptability is drawn so as to just enclose the central cluster of parties based on the mean party placements, that is, those located at a distance of between 0.60 and 1.91 from the neutral point according to Figure C-1. The border is thus drawn at 1.92.

37 Party evaluations in Sweden were obtained on a dislike-like scale ranging from -5 to 5 rather than by means of the 0-100 feeling thermometer used in Norway.

38 The methods underlying these percentages are analogous to those described in footnote 26.

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Anders Westholm is Associate Professor, Department of Government, Uppsala University, Box 514, S-751 20 Uppsala, Sweden.

The author thanks Torbjorn Berglund, Stefan Bjorklund, Hans-Joachim Fuchs, Mikael Gilljam, Jorgen Hermansson, Leif Lewin, Lena Lundstrom, Mats Lundstrom, Franco Mattei, Samuel Merrill III, Richard G. Niemi, Lennart Nordfors, Aleksandra Opalek-Westholm, Henrik Oscarsson, Olof Petersson, Lynda Powell, Jan Teorell, and three anonymous reviewers for valuable help in the course of preparing the manuscript. The data sets were obtained from the Swedish Social Science Data Service (SSD) and the Norwegian Social Science Data Services (NSD). Of course, only the author can be held responsible for the analyses and interpretations presented here.

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