The Kim-Fording Measure

The KF measure of the voter median assumes that each voter and party in a given political system can be allocated a position along a single dimension or scale of political belief. It need not be a left-right dimension, but it must be the basis for vote choice – that is, voters are assumed to vote for the party closest to them on it. This proximity voting rule defines ‘party segments’ of the scale: each party's segment ranges from the midpoint between its position and that of its closest neighbour on one side to the midpoint between its position and that of its closest neighbour on the other side. With these segments defined, it is possible to stipulate the final assumption, which is that voter opinion is uniformly distributed along the segment containing the median voter.

The segments are crucial because they structure the calculation of the voter median. The essence of the procedure is to start with the lower bound of the segment that contains the median voter and then calculate how far along that segment one would have to go to get to that voter's position (assuming a uniform distribution of voters in the segment). The general formula is:

where L is the lower end of the interval containing the median, C is the percentage of voters below the interval containing the median, F is the percentage of voters in the interval containing the median and W is the width of that interval.Footnote 16

What factors might affect the ability of the KF median to find the middle of voter opinion accurately? Consider first a case in which the measure's assumptions hold: there is only one salient policy dimension, voters know where the parties are located on this dimension, voters invariably vote for the closest party and their distribution within the median party's interval is uniform.Footnote 17 In these circumstances, the KF median will be accurate regardless of the number or distribution of parties in the system. This is the case even though an irregular distribution of party positions along the scale would cause their segments to be both unequal in length and ‘lopsided’, that is, the segments would not be centred on party positions, which means that party positions would not be centred on the positions of their voters. These asymmetries do not affect the accuracy of the KF median because the party positions themselves do not enter into the KF formula; the only positional data are the endpoints of the segments, which in this scenario are accurate.

If we move away from the uniform distribution stipulation and adopt the more realistic assumption that voter opinion is normally distributed, error will begin to appear. The KF formula can be expected to produce reasonable results if the median party's segment is located in the centre of the voter distribution, since this region of the distribution is relatively flat and hence not too far from uniformity (the same is true at the extremes, although this situation is much less likely). The normal distribution is steepest in the regions that are moderately left of centre and moderately right of centre; here, the uniform distribution assumption will be especially vulnerable. These tendencies will be conditioned by the degree of spread in voter positions: a concentration in the middle may heighten the contrast, while a relatively broad spread may ameliorate it. Much may also depend on where the true median falls in the median party's segment. If it lies close to the lower or upper bound of the segment, the proportion of voters whose positions are inferred erroneously from a uniform distribution will be relatively small. Violations of the uniform distribution assumption would then be relatively innocuous, even though the segment itself is not well centred on the true median.

Error will also appear to the extent that the assumption of one-dimensional proximity voting does not hold. There are several ways in which this might occur. The most obvious is if vote choices involve some degree of random error. This situation may occur because voters make errors in determining which party is closest to them on the scale or because they introduce idiosyncratic criteria of judgment; that is, they apply their own evaluative criteria. It stands to reason that, other things being equal, the KF median will tend to become unreliable as proximity voting diminishes, although one would not expect it to become biased. Error that is independent of positioning should not shift the KF scores more often in one direction than the other.

Another kind of non-proximity influence on vote choice is valence. The term ‘valence’ has been given many meanings,Footnote 18 but as used here, it refers to any influence that is shared among voters and hence unrelated to voter-party distances. Candidate appeal is one commonly cited type of valence, as are judgments that one party is more competent to govern than another. The potential impact of valence is different from the other factors considered to this point because it may systematically bias the KF median. For instance, if the parties on one side of the true median receive more votes than their position alone would indicate because they are considered more competent to govern, the KF median will shift in that direction.

Such a shift may be more than just an occasional or minor occurrence. In his recent investigation spanning nine countries across twenty-two years, Clark found that party valence, as indicated by the occurrence of events that reflect party competence, integrity or unity, has a pronounced effect on subsequent electoral performance – perhaps as much as policy proximity itself.Footnote 19 Since the KF median depends on vote choices, this tendency for votes to reflect valence will cause the KF median to do the same. This effect would be fine if voters change their own left-right positions in line with their changing evaluations of party competence, but if they maintain some distinction between where they stand ideologically and how they evaluate parties, the KF median will be biased accordingly.

Voters who use more than one policy dimension to make their vote choices may also unwittingly interfere with the KF median's capacity to do its intended job. Let us assume, for simplicity, that there are two policy dimensions that play a significant role in vote choices. If the dimensions are independent of each other, the impact of the second dimension on the KF median should resemble that of random error, since it will be unconnected with the party configuration that structures the KF measure. Thus the measure will tend to become inaccurate as second-dimension considerations loom larger in vote choices. To the extent that the dimensions are correlated (that is, party positioning on the second dimension resembles that on the first), this inaccuracy should diminish (other factors being equal).

Although other possible influences on vote choice that would distort the KF measure may be suggested,Footnote 20 the present analysis will focus on these influences, which are either inherently true or have a great deal of evidence and plausibility to support them. As noted earlier, our approach is two-pronged: we investigate the impact of these factors using both simulated and real data. The first prong comes in the next section, which details the results of a series of simulation experiments that explore the susceptibility of the KF median to variations in the factors mentioned above.

Simulating Violations of the KF Assumptions

The main advantage of using simulations is that we can compare the KF median with the true median and thereby assess its accuracy in a wide variety of situations. We will examine a number of simulated environments, each of which captures variation within a particular trait or cluster of traits. While different environments incorporate different features of political systems, they all have certain features in common. Specifically, each simulated environment is explored by means of 10,000 elections, each held in a distinct (unrelated) political system that contains 10,000 voters. These voters have positions x_i on a left-right dimension that are drawn randomly from a truncated normal distribution with range [0, 1] and mean = 0.5. The standard deviation of this distribution varies randomly across systems. So do the numbers and positions of parties. Each system has a randomly selected number of parties ranging from three to seven, and their positions are drawn randomly from the same distribution that provided the voters’ positions for that system.Footnote 21 Further details on the specifications for each set of simulations are provided in the online appendix.

The output from these simulations includes a variety of measures that will be discussed in due course. One point of commonality is that in all of the analyses, the principal variable of interest is the KF deviation, that is, the deviation of the KF median from the true median. The absolute value of this variable is used when the focus is solely on inaccuracy in the KF measure; where it is possible to anticipate the direction of this accuracy, the variable is used in its original or signed form.

Assumption Violations in Practice

Violations of the assumptions that underlie the KF measure clearly have the potential to introduce inaccuracy in its estimates of the median, but to what extent is this potential realized with real-world data? Using real data means that we cannot isolate and manipulate the circumstances that produce error as we did in the simulation experiments, but it does allow us to examine the aggregate impact of assumption violations. We shall be particularly concerned with the extent to which: (a) the one-dimensional proximity voting principle is violated and (b) these violations can be linked to inaccuracy in the KF measure.

One means of assessing the extent to which the proximity principle is violated is simply to examine the proportion of voters that does not vote for the party closest to them on the first or left-right dimension. In itself, this proportion can be misleading, however, since it does not take off-setting deviations from the proximity assumption into account. A better approach would be to consider whether these deviations favour one direction over the other. For each voter, the amount of directional bias would be indicated by the extent to which the party voted for is to the left or right of the closest party to the voter's position. For instance, a vote for a party 0.2 units to the left of the voter's position when the closest party is 0.1 unit to the right of that position would indicate a directional bias of −0.3 units. Averaged across all voters in a given election, the resulting variable would register the degree to which the electorate as a whole has voted more to the left or right than proximity voting would predict.

This measure's potential relevance can be gauged from the simulations data. A good place to look for directional bias is in the simulated data that involve valence. Consider, for example, the data set that incorporates skew-normal valence, one-dimensional proximity voting and random error. The mean level of directional bias across the 10,000 simulated elections is very close to zero (0.0001), as one would expect. But the standard deviation across elections is 0.045, suggesting a fair degree of variation in directional bias. This bias, in turn, correlates very highly with the KF deviation (r = 0.870). In fact, regressing the KF deviation on mean directional bias produces an intercept near zero (α = −0.001, SE = 0.0002) and a slope of one (β = 1.000, SE = 0.0052), indicating a near-perfect one-to-one relationship: when voters on average vote one unit to the left (right) of the party closest to them, one can expect the KF median to shift approximately one unit to the left (right) of the true voter median.

Another way to approach the issues of violations in one-dimensional proximity voting and their impact on the KF measure is at the level of parties. The advantage of this approach is that we can focus exclusively on the parties that are involved in the calculation of the KF measure (the median party and parties to its left). The KF formula indicates that both the total percentage of the vote that goes to parties to the left of the median and the vote percentage that goes to the median party have negative effects on the KF median: the larger each of these is, the lower (more to the left) the KF measure will be (ceteris paribus). These properties imply that if these parties receive more votes than pure proximity voting would allocate to them, then the KF measure will be biased to the left of the true voter median; if they receive fewer votes, the KF median will be biased to the right.

We can test this intuition using the same simulated data set. The test requires that two variables are constructed: (1) the difference between the proportion of votes that parties below the median party received and the proportion that pure proximity voting would have given them and (2) the same difference for the median party itself. These variables will be labelled vote differential – parties below median party and vote differential – median party, respectively. The expectation is that both variables will negatively affect the KF deviation (that is, higher than expected values will shift the KF measure to the left). The first model of Table 2 shows that this is indeed the case: both variables show negative impacts that jointly account for about 80 per cent of the variance in KF deviations.

Table 2

The Impact of Deviations from Perfect Proximity Voting in Simulated and Real (CSES) Data

Note: The dependent variable is the KF deviation. The simulated data set is that based on a skew-normal valence term.

†p < 0.001 in a one-tailed test.

Let us turn now to real-world data. As noted earlier, the survey data that will be used for this investigation consist of the first two modules of the CSES collection of national election surveys. This collection is particularly useful because most of the surveys it contains asked respondents to place both themselves and the parties of their systems on an eleven-point left-right scale. Generally speaking, the vote percentages that each party received in elections to the lower house of the national legislature are also included. This information will be used to construct both the KF median and the ‘true’ voter median, as discussed below. With one exception, the analysis will include all national elections for which these variables were provided.Footnote 24

Calculating the KF median requires information on party positions and vote percentages. In the present analysis, party positions will be estimated using the mean positions given to them by survey respondents. Despite the demanding nature of this task, it appears that respondents, on average, did a very good job of locating parties in left-right terms.Footnote 25 In the present data, for instance, party positions estimated in this manner correlate very highly (r=0.90) with the positions estimated by the CSES investigators; there is reason, moreover, to suspect that some of the discrepancy can be traced to shortcomings in the latter.Footnote 26 Another consideration favouring the use of voter-based estimates is that error in these estimates tends to be less damaging to the KF measure's accuracy.Footnote 27 The vote percentages will be those provided in the CSES data for elections to the lower house of the national legislature.

The true voter median in an election will be measured by respondents’ median left-right self-placement in the national survey (omitting suspect or clearly meaningless responses as indicated in note 25). Although the accuracy of these responses may also be questioned, the fact that respondents were able, in the aggregate, to provide very accurate estimates of party positions suggests that they are likely to have been at least as adept, again in the aggregate, at locating their own median or central position.Footnote 28 One obstacle to using these responses, however, is that they are all in the form of integer values; thus the position of the median respondent is likely to be a relatively uninformative 5 or 6 in all systems. This obstacle can be circumvented by adopting the interpolation method that Kim and Fording employed.Footnote 29 In this application, the segments are defined by the eleven evenly spaced integers of the left-right scale presented to respondents, rather than by the positions of the parties. The same assumption of a uniform distribution of positions in the median segment is required, although here the median segment is likely to be narrower (since there are more segments) and more centred in the middle of the distribution. Both properties should reduce the distorting impact of the assumption relative to its impact on the KF median.Footnote 30

Let us begin with the KF deviations themselves. Across the fifty-nine elections for which we have the necessary data, the average KF median is 5.30. Applying the procedure outlined above for estimating the true voter median produces a very similar average value of 5.16. This similarity in average values does not mean that the two methods produce similar values in individual elections, however. In fact, the average extent to which the KF measure deviates from the true value is 0.50 units.

What generates this difference? Indeed, what justifies considering the KF measure ‘deviant’ and the opinion-based measure ‘true’? The simulation experiments described in the previous section established that the KF measure is particularly prone to produce deviant values when valence or other factors cause overall voting behaviour to deviate systematically from what the proximity rule would predict. Let us therefore examine how well the assumption of left-right proximity voting holds up in these data.

In the fifty-nine elections under examination here, more than two-thirds (68.7 per cent) of voters, on average, did not vote for the party that is closest to their position on the left-right scale. This would appear to be a high rate of violation of the key assumption underlying the KF measure but, as noted earlier, higher rates of non-proximity voting do not automatically lead to higher KF deviations: violations of the assumption may offset one another or be irrelevant to the calculation of the KF median. This is borne out by the total absence of a relationship between KF deviations and the proximity voting rate (r = −0.007, p = 0.956). What matters instead is the degree of directional bias. Across the fifty-nine elections, the mean directional bias averages 0.03, with a standard deviation of 0.41, which (taking into account the difference in scales) is very similar to that reported above for the skew-normal valence simulations. More importantly, the KF deviation and the mean directional bias are correlated in these data as well (r = −0.547, p < 0.001). In fact, the regression of the former on the latter also produces a one-to-one relationship (α = 0.011, SE = 0.008; β = 0.991, SE = 0.201), suggesting that electorates that vote, on average, one unit to the left (right) of their proximity choices can be expected to produce KF medians that deviate one unit to the left (right) of the true median.

Let us now focus the analysis a little more by moving to the level of parties, particularly those that matter for the calculation of the KF median. Our concern is the extent to which these parties receive larger or smaller vote shares than would be expected under pure proximity voting. With the simulated data, it was straightforward to compare the vote shares that parties actually received with the vote shares they would have received under left-right proximity voting; here, a more indirect procedure is required. We have noted that, with the elimination of some highly suspect responses, the distributions of left-right positions in all fifty-nine elections appear to be roughly normal. We assume that all distributions are in fact normal and can therefore be characterized by their means and standard deviations. On this basis (and given the party positions in each system), it is possible to calculate the vote shares that each party should have received under strict proximity voting. With this information, we can calculate the two key variables: the vote differential for parties below the median party and the vote differential for the median party itself. As before, both variables should have a negative impact on the KF median: the more the votes of these (sets of) parties exceed proximity-based expectations, the more the KF median will be shifted to the left.

The second model of Table 2 shows the results. In this case, the results are very similar to those produced using the skew-normal set of simulations (Model 1): both variables have the expected negative relationships with the KF deviation, with the vote differential for the parties below the median exerting the dominant influence. Even the proportion of explained variance, at 77.4 per cent, is approximately the same.

In sum, the analysis of the CSES data reveals that the KF median deviates from the opinion-based median in ways that highly resemble its deviations in some of the simulations. It is not simply that voters frequently violate the proximity voting assumption; rather, voters in these elections tend to shift either to the left or the right of their most proximate choices, which shifts the KF median to the left or right, accordingly. The KF median captures these shifts because it is based not on opinion per se but on voting behaviour, which clearly reflects more than left-right proximity. The implications of these findings are explored more fully in the concluding section.

Implications for Research

Kim and Fording's measure was originally intended to locate the electorate's median ideological position. The fact that the measure ignores non-voters may not be much of a problem; in the fifty-nine CSES elections covered here, the opinion-based citizen median is virtually identical to the opinion-based voter median (r = 0.99). But the fact that it is based on vote choice rather than opinion itself does matter. As the simulation experiments reported here have shown, when factors other than left-right policy distance enter into the vote calculus – for instance, the perceived competence of a party or the appeal of its leader, the influence of other policy dimensions, errors in identifying the best choice given one's preferences or idiosyncratic judgments – the KF median can become noticeably inaccurate. More specifically, when these factors cause overall vote choices to deviate to the left or the right of what left-right proximity would predict, then the KF median will follow.

For some applications, this may not matter very much. If the objective is to measure policy demands as expressed in votingFootnote 31 or simply to control for the ‘political centre of gravity’,Footnote 32 the KF median may be an appropriate, or at least an acceptable, choice. However, if the goal is to assess the degree of ideological congruence or policy responsiveness, using the KF median can be far from inconsequential. It is easy to see why. Suppose that a rising level of popularity for parties on the right causes votes to drift rightward in an election. This in turn will (a) cause the KF median to shift to the right and (b) raise the odds that the next government formed will be right wing, or at least more right wing than the present one (since the more right-wing parties will have received increased vote shares). If this happens, the voter median and government policy will appear to have co-varied, consistent with the ideological congruence thesis – even though the fundamental left-right positioning of voters may not have changed at all.

Denmark provides an example of this scenario. The 2001 election in that country produced a considerable movement to the right from the 1998 election results, with the Venstre party picking up fourteen seats (an increase of one-third) and the Social Democrats (SD) losing eleven. As a result, the KF median moves from 5.29 to 6.71 on the CSES scale. The rightward shift in votes also resulted in the SD-led government being replaced by a Venstre-led one, generating a shift in government left-right position from 4.15 to 7.30.Footnote 33 Thus, government position appears to have moved rightward with a shift in median left-right opinion, suggesting responsiveness, if not congruence. In fact, however, the median citizen position based on respondent self-placements scarcely changed at all: it was 5.36 in the 1998 survey and 5.39 in the 2001 survey.

While this example may be atypical, in that it shows a total disconnect between KF and true medians, its exceptionalism may be one of degree only. In Warwick's analysis of the CSES data, KF medians were found to relate much more closely to government positions than those calculated directly from respondent self-placements.Footnote 34 This difference suggests that the use of the KF median exaggerates the strength of the opinion/policy connection in a systematic fashion.

The adverse consequences do not stop here; the KF measure may also cause the nature of the opinion/policy connection to be misunderstood. When measured using survey data, relatively small changes in median position appear to generate much larger changes in government policy stances.Footnote 35 Given that left-right position is usually seen as a relatively fixed property of individuals – evolving over time, certainly, but not bouncing back and forth with every change in government direction – changes in the left-right median that are small relative to changes in intended government position are precisely what one would expect. But the presence of such a ‘multiplier effect’, if confirmed by further research, would point to a very different understanding of the opinion-policy relationship: one in which congruence no longer figures either as the central reality or as the standard for democratic evaluation.

Awareness of these dangers has been increasing recently. A good case in point is Kang and Powell's investigation of the connection between left-right preferences and welfare spending.Footnote ³⁶ They observe that the KF measure, which they utilize, is ‘not quite the same thing as the unmediated preference of the median citizen’;Footnote ³⁷ apart from the fact that not all citizens vote, ‘median voter preferences [so measured] may be affected by party offerings. One might consider these ‘induced’ preferences rather than authentically pre-political ones’.Footnote ³⁸

But is this enough? Kang and Powell's relegation of this discussion to a footnote and their use of ‘expressed’ as a synonym for ‘induced’ suggest that they believe these induced preferences will not be very much different from unmediated ones. We have seen that this could be so: if left-right proximity is the only factor that drives vote choices, the KF measure can estimate the left-right median opinion with reasonable accuracy, even through the filter of party choice. It is much more likely, however, that vote choices involve other considerations as well – valence, other party dimensions, personal criteria, misjudgments of party positions and so on. If any of these are operative, the KF measure will pick them up. The KF median therefore runs a considerable risk of incorporating considerations that have nothing to do with policy positions or that involve policy areas independent (or largely independent) of the left-right dimension. If the concern is with ideological congruence or responsiveness on a dominant left-right dimension, as it so often is, these properties make the KF measure a problematic tool to rely on.