Missing Voters 17May10

Missing Voters, Missing Data: Using Multiple Imputation to Estimate the

Effects of Low Turnout

17 May 2010

Patrick Bernhagen

Department of Politics and International Relations

University of Aberdeen

Michael Marsh

Department of Political Science

Trinity College Dublin

Address for correspondence:

Patrick Bernhagen, University of Aberdeen, Department of Politics and International

Relations, Edward Wright Building, Dunbar Street, Aberdeen, AB24 3QY, United Kingdom.

E-mail: [email protected].


Effects of Low Turnout

ABSTRACT: In recent years, different methods have been proposed to estimate the political

effects of low voter turnout. This article contributes to the discussion by assessing the

performance of multiple imputation in estimating the partisan effects of low turnout. Using

the 2002 Irish General Election as a case study, we demonstrate how multiple imputation can

be used to fill in the vote choices of non-voters. We verify simulations and reported turnout

against official data and compare the results to those from alternative, maximum likelihood

methods. While the methods differ in their ability to simulate vote choice correctly, these

differences are generally not large enough to affect the counterfactual estimation of election

results under universal turnout. To asses the generality of this finding, we also compare the

different methods across 30 elections in the Comparative Study of Electoral Systems dataset.

Multiple imputation produces on average higher turnout effects than multinomial logit

methods and the differences increase as turnout goes down. System variables such as the

number of parties do not affect the differences in results between methods.

Acknowledgements

Previous versions of this article were presented at the annual conference of the Political

Studies Association Specialist Group on Elections, Public Opinion and Parties (EPOP), 8-10

September 2006, Nottingham, and at the 65th Annual National Conference of the Midwest

Political Science Association, April 12-15, 2007, Chicago, IL. We would like to thank Brad

Gomez and three anonymous reviewers for the Journal of Elections, Public Opinion and

Parties for very helpful comments. Patrick Bernhagen would like to acknowledge grant

support from the British Academy.

2


Effects of Low TurnoutIn his 1996 presidential address to the American Political Science Association, Arend

Lijphart (1997) drew attention to the problem that less than full turnout might affect election

results and lead to the under-representation of the political interests of some groups of

citizens, such as ethnic minorities or low-income groups. Recent elections in Germany and

the USA revived the notion in public discourse that decreasing turnout is bad for social

democratic parties or that higher turnout would benefit the Democratic candidate. The

political consequences might extend well beyond the realm of electoral representation. For

example, class differences in voter mobilization have been shown to affect welfare spending

(Hill, Leighley and Hinton-Andersson, 1995). Numerous attempts have been made to assess

the veracity and relative importance of this claim: If turnout went up, would it affect the

election result? Which parties would gain and which would lose? Different methods have

been proposed to estimate the political effects of low turnout, often yielding different answers

to these questions.1 This article contributes to the methodological discussion by assessing the

performance of multiple imputation in the estimation of the partisan effects of low voter

turnout.

We proceed as follows: After a brief overview of the main strategies for detecting

turnout effects, we introduce the idea of treating vote abstainers as missing data. Using the

2002 Irish General Election as a case study, we first demonstrate how a statistical model of

multiple imputation can be used to fill in the vote choices of non-voters. We compare the

results to those generated by more traditional vote-propensity methods and demonstrate the

validity of the multiple imputation method by verifying estimates and reported turnout

1 A recent special issue of Electoral Studies contains studies of turnout effects in the context of large numbers of

national and supranational elections as well as referendums. Lutz and Marsh (2007) provide a thorough review

of the previous literature on turnout effects.

3

against official data. Both the Irish polity and the Irish National Election Study have a

number of useful properties that enable us to verify our estimates against professed vote

choices and official records and evaluate the performance of different methods for

ascertaining turnout effects. Furthermore, we asses the generality of our results by comparing

turnout effects at 30 elections in 25 countries in the Comparative Study of Electoral Systems

dataset and identifying the institutional and structural circumstances in which the results from

the different simulation methods diverge. We conclude by evaluating the utility of multiple

imputation vis-à-vis traditional methods of estimating turnout effects and discuss the

implications for political scientist’s efforts to assess the political consequences of low

turnout.

Does Turnout Matter? An Overview of Methods and Findings

Political preference revelation at elections and other polls will be biased whenever turnout is

short of 100 percent and abstainers’ preferences are non-randomly distributed among the

electorate. When everyone votes there can be no bias in the representation of party

preferences. And if non-voters are a representative cross-section of the electorate, their

abstention will not increase the influence of any one group of voters at the expense of

another. But vote abstention seems far from random and for some time now political

scientists have been able to identify those segments of the electorate that are least likely to

vote (Brady, Verba and Schlozman, 1995; Norris, 2002, 83-100; Rosenstone and Hansen,

1993). At the same time, the ability to estimate the size and direction of the political bias that

results from non-random voter abstention is severely hampered. We frequently know how

voters differ from non-voters with respect to various sociodemographic characteristics or

political attitudes. We can also determine the preferences of those voters that are in many

important social and political respects like non-voters and in that way estimate the

4

hypothetical vote choices of nonvoters. Finally, we can ask non-voters how they would vote

if they voted. But we cannot know with certainty what the vote choice of non-voters would

be if they actually turned into voters. Together, these hypothetical vote choices make up what

we refer to as turnout effects – the effects that higher or lower turnout would have on an

election outcome, where election outcome is defined as the vote shares of the parties (cf. Lutz

and Marsh, 2007: 541). Any attempt to estimate such turnout effects is subject to the

fundamental methodological problem that the demonstration of political inequalities resulting

from unequal turnout requires knowledge of how non-voters might behave were they not

non-voters. The question is, how can we know the parameters that govern this counterfactual

state of the world?

In the presence of these limitations, three strategies can be distinguished among

efforts to estimate counterfactual voter behaviour and its effects on election outcomes. One

approach uses opinion polls to consider whether voters and nonvoters differ in any significant

way on the dimension of partisan identification or with respect to various policy-related

issues. Some studies use survey data to compare the attitudes of voters and non-voters with

respect to social and economic policy issues or general attitudinal dispositions (e.g., Bennett

and Resnick, 1990), while others ask more specifically about partisan identifications and

preferences (e.g., Highton and Wolfinger, 2001). But querying nonvoters as to whether their

attitudes differ from those of voters is not the same as to estimate how these people would

behave in actual elections: the propensity to vote is part of an individual’s attitudinal make-up

and it is possible that, as this aspect changes, other elements such as party or policy

preferences change too.

A second approach examines election studies and official election results for evidence

of turnout effects. This involves regressing the vote share of certain types of parties and

candidates (usually left-of-centre ones) on aggregate turnout and a variety of control variables

5

(DeNardo, 1980; Radcliff, 1994; Pacek and Radcliff, 1995). But the use of aggregate data by

this approach poses an ecological inference problem: the fact that turnout rates in a group of

elections are similar does not imply that in each of the elections the types of individuals

abstaining are the same (Herron, 1998: 6-7).

A third approach to estimating turnout effects, finally, tackles these problems by

analyzing individual-level behaviour using the information contained in reported electoral

behaviour to estimate the propensity to vote for a particular party and then compare (self-

reported) voters and non-voters (e.g., Brunell and DiNardo, 2004; Citrin et al., 2003;

Martinez and Gill, 2005). In this approach, used for various analyses of US elections, the vote

choices of nonvoting individuals are predicted using coefficients from maximum likelihood

regression analysis of the behaviour of voters who share similar sociodemographic and,

sometimes, political characteristics.2 These votes are then added to the observed vote choices

after a weighting has been applied to account for differential propensity to vote.

This approach has its own drawbacks. Firstly, in common with the previous approach

it works on the assumption that nonvoters would behave like those voters with whom they

share a set of sociodemographic and attitudinal characteristics. This assumption is

problematic. Sociodemographic and attitudinal correlates of vote choice alike may provide a

weak basis for estimating the preferences of abstainers, not least because abstainers might be

those with preferences that run counter to the norm in their respective social group. Secondly,

simulations of the vote choices of nonvoters are often based on a restricted set of variables,

consisting mainly of demographics (e.g., Citrin et al., 2003; Brunell and DiNardo 2004).

Martinez and Gill’s study (2005) improves on this by employing an expanded set of predictor

variables of vote choice and turnout in their model. But attitudinal variables tend to have

more missing values than demographics and, with listwise deletion as the default method of

2 This strategy is also followed by Tóka’s (2002) analysis of elections in the Comparative Study of Electoral

Systems dataset, although he uses discriminant analyses to estimate his vote choice models.

6

dealing with missing data, more variables usually mean fewer cases. Thus, the multinomial

logit methods employed by these authors involve a trade-off between the richness of the

model on one hand and the number of cases to be used as a basis for prediction on the other.

This is a choice between a loss of information and potential bias incurred through an

underspecified model and a loss of information and potential bias stemming from deletion of

a considerable number of cases available for fitting the model.3 By opting for the former,

Citrin et al. (2003) avoided data attrition at the expense of a richly specified model for

predicting vote choice.4 By contrast, Martinez and Gill (2005) use a more fully specified vote

model but at the cost of losing a considerable amount of data. At best, this leads to a loss of

information and an increase in variance around the predictions. At worst, missingness is

correlated with either vote choice or vote propensity, or both. Hence, Martinez and Gill’s

(2005) model for predicting vote choice may also be mis-specified as it omits the selection

variable, and their estimates of turnout effects may be incorrect. In the following section we

describe an alternative method that addresses these limitations.

Multiple Imputation

Voting is an individual choice, which makes the sample of voters a self-selected one (Dubin

and Rivers, 1989). The reason why we do not have data on how some citizens voted is simply

because they chose not to vote. This means that the problem of less-than-full turnout is

analogous to the problem of missing observations in any statistical analysis. If missing data

3 In fact, because the problem of sample selection is one of omitted variable bias (Heckman, 1979; Dubin and

Rivers, 1989), the loss of observations is strictly speaking also a problem of model specification.

4 This is not to insinuate that Citrin et al. made a deliberate decision in favor of larger N and against a fully

specified model. Rather, as Citrin et al. analyze state-level elections in the USA, their data restrictions result

from the census data they used, which are the best data available for their purpose but do not contain attitudinal

variables.

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points are uncorrelated with the error term or with the variables of interest, they pose few

problems beyond reducing the number of observations and thereby increasing the variance of

the estimates of parameters or quantities of interest. But if the sampling fractions are

correlated with the errors or the dependent variable, estimates may be biased. In the case of

voter turnout, we have reason to suspect such correlation: while studies of turnout effects

may have failed to establish a robust and directional link between turnout and vote choice,

their findings on the whole suggest that abstention is not random in relation to vote choice

(e.g., Norris, 2002, 83-100).

Treating the unobserved vote choices of non-voters as missing data points means that

we assume the vote choices of non-voters constitute data that actually exist but have not been

observed or recorded because the potential voters’ proclivity to turn out the vote has not

exceeded a certain threshold. Those who find this assumption difficult to accept might find it

more agreeable to think in terms of preferences instead of vote choices: if we assume that

individual preferences over parties and candidates are measured by votes, data on these

preferences are missing for non-voters (Dubin and Rivers, 1989, p. 383). The question of

turnout effects is whether or not the decision to vote and the decision which party to vote for

are related. This question can be tackled using available techniques for the imputation of

missing data. “Ad-hoc” methods of addressing missing data such as filling in means or

imputing predicted values from regression analysis based on the observed data points

(“conditional means imputation”) at best understate variability and at worst induce bias

(Allison, 2002; Horton and Kleinman, 2007). Imputations arrived at by these methods will be

unbiased only if the probability of missing data on any variable is unrelated to the values of

the variable itself or the values of any other variables in the data. In the context of turnout

effects, predictions from regression analysis therefore run the risk of producing wrong

8

estimates of nonvoters’ party choices while at the same time suggesting a false sense of

accuracy of these estimates.

The risk of biased estimates can be greatly reduced by the use of maximum likelihood

estimation. But maximum likelihood predictions of vote choice such as those reported by

Citrin et al. (2003) or Martinez and Gill (2005) may still be biased if the probability of any

data point missing depends on both the dependent and independent variables – which is of

course precisely the conjunction that is suspected to render turnout problematic for election

outcomes. Given the robust findings in the literature on the sociodemographic and attitudinal

determinants of vote choice (Nie, Verba and Petrocik, 1980) and equally robust evidence

concerning the individual-level correlates of turnout (Wolfinger and Rosenstone, 1980), the

possibility of missingness being jointly contingent on both dependent and independent

variables cannot be dismissed.

In this situation, multiple imputation (MI) is an efficient method of arriving at

estimates of non-recorded vote choices. Bernhagen and Marsh (2007) use this method to

estimate turnout effects at 30 elections in 25 countries from the Comparative Study of

Electoral Systems (Module I). While their model specifications closely resemble Tóka’s

(2002) and Martinez and Gill’s (2005) models, their reported changes in the parties’ fractions

of the vote are generally higher than those generated by the multinomial logit or discriminant

analyses of these studies. They find the change in parties’ vote share before and after

imputation is mainly a function of turnout, that is, of the scope for change, which increases

with declining turnout. At the level of parties, Bernhagen and Marsh found no evidence for

either left, right, or centre parties gaining from full turnout scenarios. However, they did find

that non-governing parties typically benefit from full turnout, as was originally suggested by

DeNardo (1980), and that smaller parties would gain from full turnout. More generally,

Bernhagen and Marsh found that full turnout would on average reduce the gap between the

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strongest and second-strongest party. In one case, the US Congressional election of 1996, this

would have led to a different party coming in first in the election.

Originally proposed by Rubin (1976), MI involves three steps. First, plausible values

for missing observations are created that reflect uncertainty about the nonresponse model.

These values are used to “fill-in” the missing data points. This process is repeated, resulting

in the creation of a number (usually 5-10) of “completed” datasets. In a second step, each of

these datasets is analyzed using standard methods. In the case of estimating turnout effects,

this simplifies to estimating the vote proportions of the different parties. Thirdly, the results

are combined, which allows the uncertainty regarding the imputation in step one to enter the

final result. To apply this method to the problem of low turnout, we have to assume that the

probability of an individual’s vote choice remaining unrecorded may depend on the observed

values of other variables, but, after controlling for these variables, is independent of any other

missing information. In the terminology established by Rubin’s (1976) classification of data

missingness, that is to assume the data are missing at random (MAR), i.e., missingness is

random after controlling for missingness due to observed quantities. Formally, if there are

two variables X and Y, where X is always observed and Y is sometimes missing, MAR

means,

.

Of course, most multivariate datasets will contain missing values on several, perhaps

even on most, variables. Moreover, it is impossible to test whether the MAR condition is

actually satisfied (Allison, 2002: 4). However, if at least one element in a vector of

independent variables X is fully observed, we can assume that the data are MAR, conditional

on the imputation model (King et al., 2001: 53). Furthermore, the MAR assumption can be

made more realistic by including more informative variables in the imputation process

(Collins, Schaffer and Kam, 2001).

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To estimate turnout effects by way of multiple imputation we use the Amelia II

program written by Honaker, King and Blackwell.5 Amelia II uses an expectation-

maximization (EM) algorithm to generate values in place of missing observations. While the

likelihood conditional on the observed (but incomplete) data cannot be easily constructed, the

likelihood of a rectangularized data set (i.e., one for which all cells are treated as observed) is

easy to construct and maximize, especially under the assumption of multivariate normality.

The EM algorithm rectangularizes the data set by filling in estimates of the missing elements,

generated from the observed data. In the E-step, missing data points are filled-in using linear

regression, with their expected values conditional on the current estimate of the sufficient

statistics and the observed data. In the M-step, a new estimate of the sufficient statistics is

computed from the current version of the completed data (see Honaker, James and King,

2006 for a detailed exposition).

The multiple imputation procedure is not intended to create causal explanation or

parameter interpretation (King et al. 2001). The algorithm imputes values to all empty cells in

a dataset loaded by it and does not discriminate between variables according to their status as

dependent or independent variables in a regression model. Thus, MI imputes values on the

independent variables as much as on the dependent variable, and the imputed values of the

dependent variable are then used to improve the imputation of independent variables, and

vice versa.

EM always starts with the full covariance matrix, which means that it uses all the

available variables as predictors for imputing the missing data (Allison, 2002: 20). Thus, the

MI model uses more information, both in terms of more variables and more observations than

the approaches of Citrin et al. (2003) and Martinez and Gill (2005), each of which delete

considerable amounts of existing data. This may well alter the simulation of non-voters’ vote

5 The software, Amelia II: A Program for Missing Data (version 1.1-6 beta, July 18, 2006) is freely available at

<http://gking.harvard.edu/amelia/ >.

11

choices and lead to different simulations of election outcomes. It also means that the standard

errors of the resulting estimates will generally be too low, as the estimator assumes that there

are complete data for all cases. The solution to this problem is to repeat the EM-based

imputation process m times to produce m “complete” datasets. If random draws from the

residual distribution of each imputed variable are made and added to the imputed values,

estimates of the parameters of interest will be slightly different depending on which imputed

dataset is used. This variability can be used to adjust the standard errors upward by averaging

the parameters of interest and combining their standard errors according to a formula devised

by Rubin (1987). For the analysis of turnout effects, this involves obtaining the probability of

voting for a particular candidate or party for each imputed dataset (j = 1, ..., m) and averaging

the m values. The standard error for the estimate is obtained in three steps: (1) the standard

errors for the m point estimates are squared and then averaged; (2) the sample variance in the

point estimates across the data sets is calculated; (3) the results from (1) and (2) are added

together, weighted by a factor that corrects for the bias resulting from m < ∞, and the square

root is taken (Allison, 2002: 29-30). As vote choice is a categorical variable measuring

choices among k candidates, as is the case in most elections, we generate multiple

imputations for k dummy variables created from the categories of the original vote choice

variable.

Data: The 2002 Irish National Election Study and the Comparative Study of Electoral

Systems

The first data we use to demonstrate the MI method of simulating higher voter turnout are

from the 2002 Irish National Election Study.6 Several aspects make Ireland an important case

study for the assessment of the political effects of low voter turnout. Firstly, among European

6 The 2002 Irish National Election Study was funded under the PRTLI/National Development Plan: see

www.tcd.ie/ines.

12

countries, the Republic of Ireland has fared notably badly in terms of election turnout.

Average turnout at general elections has been only slightly above 70 percent since the 1970s,

hitting a low at the 2002 election with 63 percent. Secondly, Ireland’s low turnout facilitates

the estimation of tangible turnout effects, as any biases will be larger the bigger the share of

vote abstainers among the electorate. Together with the previous point this means there is

both more chance of observing a bias in the first place and more chance that such a bias can

have a significant impact on the result (Kohler and Rose, 2008). Thirdly, the country’s STV

electoral system leads to a fairly proportional translation of seats into votes, which reduces

the amount of strategic voting among the observed votes that provide much of the

information for simulating the behaviour of nonvoters.

There are also a number of desirable properties of this particular election study that

facilitate our efforts of estimating turnout effects and gaining a sense of how valid our

method is. The Irish election study contains official data on individual turnout, allowing the

validation of voters and weight corrections for over reporting. Furthermore, it contains the

stated preferences of those who reported a vote, even if the official record indicated that the

vote was not cast, as well as the hypothetical preferences of many who said they had not

voted but told the interviewers how they would have voted. This will be useful in evaluating

the MI method for assessing the impact of low turnout on the outcome. The dataset contains a

range of demographic and political variables for 2,663 individuals. 2,391 respondents have

reported their vote choices at the 2002 election by either agreeing to fill out a ballot paper

13

during the interview or simply reporting their first preference vote.7 Of these, 1,835 have

been officially validated as having voted.

The MI procedure can make use of an extensive set of variables that can plausibly be

suspected to be related to vote choice. We include gender and age, as these variables are often

found to be influential in determining candidate or party preference. Additional

sociodemographic variables, such as union membership, education, income, urban versus

rural residence, religious denomination and language are also included, because it has been

conjectured that these sociodemographic variables influence either turnout or vote choice or

both. Beyond these sociodemographic characteristics, we include respondents’ evaluations of

the economy and of other policy areas (health and housing), as well as a measure of political

knowledge. Above all, however, we are able to impute missing data points based on the

reported party and party leader preferences of voters and non-voters as recorded by reported

probability to vote (PTV) and thermometer scales. Thus, the extensive set of data employed

here enables us to go well beyond the sociodemographic correlates of turnout. The inclusion

of party and leader preferences and evaluations at least allows for the possibility that

abstainers might be those with preferences that run counter to the norm in their social groups,

thus providing a significant improvement over simulations based solely on demographics.

Furthermore, the political variables add to the overall richness of the set of predictors in the

model, thus making the MAR assumption more realistic. A list of the variables included in

the multiple imputation model is provided in Appendix A.

7 Respondents were asked to fill out a ballot paper for the relevant constituency. Those who sad they voted in

May 2002 were asked, “Please fill it in as you did on polling day – as best you can remember.” Respondents

who said they did not vote were asked, “Suppose you had voted in the May 2002 election, how would you have

filled in the ballot paper on election day?” Respondents who declined to fill out the ballot paper were asked,

“Could I ask which party you gave your first preference vote to?”.

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The particular properties of the Irish election study may mean that findings based on

this case are not necessarily generalizable to other contexts with different characteristics,

such as political systems with fewer parties or larger incentives for strategic voting. To better

judge the generality of our findings from the Irish case, we therefore replicate Bernhagen and

Marsh’s (2007) analysis of turnout effects at 30 elections in 25 countries from the

Comparative Study of Electoral Systems (Module I). These surveys were conducted between

1996 and 2002 and at a functionally equivalent point in time: when a national election was

taking place. This maximizes unit homogeneity across elections that differ in respect of

important variables such as the number of parties or extent of voter participation.

Analysis and Results

Existing theories and evidence about the reasons why people do not vote suggest a number of

patterns that can be expected. A well-established argument claims that non-voters are non-

voters because they have little contact with agencies of mobilization (Rosenstone and

Hansen, 1993; Brady, Verba and Schlozman, 1995). Following this logic, we might expect

the larger Irish parties, Fianna Fáil (FF) and Fine Gael (FG), to be more attractive to potential

supporters than smaller parties, as well as being more effective at mobilizing any latent

support. Consequently, those who stay at home might be more likely to have a preference for

smaller parties, and smaller parties such as Greens, Labour, and Progressive Democrats (PD)

should benefit most from “complete” turnout.8 A second argument sees non-voting as a sign

of disaffection (Crozier, Huntington and Watanuki, 1975; Gurr, 1970). Non-voters are more

detached from the established political system, and if they did vote it would tend to be for

8 We do not expect Sinn Féin to fall within the domain of this argument because this party is renowned for its

resourced and resourceful local campaigns that match those of the larger parties.

15

more radical parties, both right and left. Applying this logic to the 2002 Irish case we might

expect more radical parties, such as the Greens or Sinn Féin (SF) to benefit from 100 percent

turnout. Furthermore, we expect parties with a strong working class support base to benefit

most from complete turnout. In many European countries these will be left-wing, socialist or

Labour parties, but given what we know abut party affiliations in Ireland, we expect Sinn

Féin to benefit most from higher working class turnout, followed by Fianna Fáil.

While we have referred so far to voters and non-voters as separate groups of people,

people may move both into and out of the electorate over time. For example, habitual voters

may abstain because they are acutely unhappy with the incumbent performance of their

traditional party. This may be due to the state of the economy or the likely failure by any

government to deliver on some of their election pledges. At any rate we know that governing

parties tend to lose votes (Nannestad and Paldam, 2002), and some of that loss will be due to

abstention by erstwhile supporters. This would imply that incumbent parties (Fianna Fáil,

Progressive Democrats) benefit most from “complete” turnout.9

[Figure 1 about here]

Figure 1 displays the vote shares of each party and the residual group of

independents/others as quadruplets of bars. For each party, the first bar represents the party’s

vote share as recorded from the 1,835 validated voters in the dataset, weighted by

demographic corrections for survey bias. This is contrasted with the second bar based on

counterfactual full (100 percent) turnout. The full turnout figures are composed of the 1,835

vote choices of validated voters, the 148 imputed vote choices of people that did vote, but did

not divulge their choice, and the imputed vote choices of 680 non-voters. The imputations

9 We should also allow for the opposite effect of turnout increases identified by DeNardo (1980) for a series of

US Congressional elections in the 1960s and 1970s. According to DeNardo, turnout boosts tend to harm the

incumbent party as they involve the mobilization of “peripheral” voters, who respond in a rather fickle fashion

to short term campaign effects.

16

have been obtained using information on the full range of variables listed in Appendix A.10

The error bars represent 95 percent confidence intervals. To assess the robustness of the

imputations to different imputation models, we also obtained imputations based only on,

firstly, socio-demographic variables and, secondly, utility and thermometer scales of parties

and party leaders. Imputations on the vote variable are influenced by the observed votes when

either of these subsets of variables is used in the imputation process, while imputed votes

differ more from the observed votes when the maximum set of information is used (see

Appendix B).

In order to compare the MI results to maximum likelihood simulations of full-turnout,

we replicate the procedures employed by Citrin et al. (2003) and Martinez and Gill (2005).

This involves modelling vote choice at the 2002 election and using the estimated coefficients

from multinomial logit estimation to calculate for each non-voter the predicted probabilities

of vote choice for the various parties. To obtain the parties’ vote shares under a full turnout

scenario, we then add the estimated vote choice of non-voters to the observed vote

proportions, weighting the components by the actual turnout rate of 63 percent.11 Following 10 The models employed here replicate those used in Bernhagen and Marsh (2007) and are fully justified there.

This is because in a later stage of this analysis we use the same CSES data sets as used by that study and want to

maintain comparability. We could have used a different model for the Irish analysis here but chose not to do so

on grounds of consistency. In doing so, we ensure that our imputation model corresponds to with previous

research on Irish election turnout. There are very few models of turnout. Lyons and Sinnott (2002) and Marsh et

al. (2008) detail mobilization and resource effects as well as socio-demographics – all key elements of the

model used here. While these studies make the point that non-voting is heterogeneous, with some abstaining for

circumstantial reasons and some more deliberately, this argument, if correct, is not only appropriate to Ireland

(Blondel et al., 1998). In any case, the CSES data sets do not include the variable operationalizing this

categorization.

11 As we use survey data, the “observed” vote choices have not been observed in the strict sense. For clarity and

ease of presentation, we will use the following nomenclature: “observed” vote choice is the vote choice reported

by respondents in a national election study who have been verified as having turned out to vote according to

17

Citrin et al. (2003), the first vote choice model includes gender, education, income, age, and

urban-versus-rural residence.12 In a second step, we add a comprehensive set of non-

demographic variables to specify a model analogous to Martinez and Gill’s (2005). This list

of variables is identical with the one used for the MI simulation procedure above. Thus, the

third bar represents the simulated vote share based on predictions from multinomial logit on

demographics only (Citrin et al.’s approach). The fourth bar represents the simulated vote

share based on predictions from multinomial logit using the full set of information available

in the election study for all cases with no missing data on variables other than vote choice

(Martinez and Gill’s approach).

The most notable result is that the simulated increase in turnout from 69 (in the

survey) to 100 percent does not lead to any radical changes in the vote shares of the parties –

regardless of the simulation method used. The single biggest change through full turnout is a

loss of about two percentage points for Fianna Fáil, matched by an increase of roughly one

percentage point each for Fine Gael and Sinn Féin. Even the two percent drop in the Fianna

Fáil vote share, however, is well within the five-percent margin of uncertainty indicated by

the error bars. Thus, while the literature offers many reasons to expect the costs and benefits

of full turnout to be unevenly distributed across parties, this analysis of the 2002 Irish

General Election suggests the impact would be marginal, and that, indeed, we cannot be sure

that it would have any differential impact at all.

official records. “Professed” vote choice is the vote choice reported by respondents who said they cast a vote in

the actual election but did not actually do so. This latter category also included the choice of non-voters

indicating how they believe they would have voted had they gone to the polls.

12 Citrin et al. also include race as a key variable. However, race is not a noteworthy issue in Irish electoral

behavior and the Irish election study contains no information on ethnic or race variables. Instead, we include

urban-versus-rural residence, as this variable is a key correlate of Irish voter behavior (cf. Marsh et al., 2008:

164-79).

18

Comparing the MI simulations with simulated election results based on predictions

using multinomial logit coefficients from a demographic turnout model, we can see that the

latter are quite similar to those generated by MI. Full turnout simulations using this method

are slightly less “full” (N=2,570) than those arrived at by MI. This is due to small amounts of

missing observations on several predictor variables, which led to the loss of almost 100 cases

by way of listwise deletion. For some parties, the demographics-based multinomial logit

simulations are between the MI simulations and the actual result; sometimes they are a little

above or below the one or the other. While the turnout effects suggested by MI are overall

slightly more pronounced than those picked up by the different multinomial logit models, the

deviations from either the actual or the MI results do not follow a systematic pattern. If

anything, the demographics-based multinomial logit simulations suggest slight gains for

smaller parties from full turnout, at the expense of the larger parties. These observations also

hold for the richer multinomial logit simulations as suggested by Martinez and Gill. Because

of the expanded number of independent variables in this vote model, many of which have

missing observations, listwise deletion leads to the loss of over 400 cases.

Just how much faith can we have in the MI method and how good is it at estimating

the vote choices of non-voters? We cannot answer that question definitively, for the same

reasons that make this or any other simulation strategy necessary in the first place. But, to

begin with, the MI results are confirmed by multinomial logit estimates following the

strategies of Citrin et al. (2003) and Martinez and Gill (2005). However, as the multinomial

logit estimator is no more efficient than MI, different standards have to be applied. Therefore,

we conduct two further tests to explore the validity of the imputations. First, the 2002 Irish

Election Study records how non-voters said they had voted or said they would have voted.

Thus, we know their survey responses to the question of how they voted (if they falsely claim

they had voted) or how they think they would have voted had they actually voted. After

19

verifying that these respondents really are non-voters, we can compare their professed vote

choices with the simulated votes of the same individuals obtained through MI. The sub

sample for this analysis comprises of those 556 respondents that are known not to have voted,

but who reported a vote choice during the interview.

The results are displayed in Figure 2. The general picture that emerges is consistent

with the differences between the observed and simulated full turnout. As before, the

simulations from multinomial logit generally trail the prevailing pattern. For the larger parties

the demographics-based predictions are “between” the professed and the imputed vote shares,

while they tend to give larger gains to Sinn Féin and independent candidates than either what

respondents would claim or what the imputation algorithm would fill in. The simulations

based on the full multinomial logit model follow that random pattern, being at times closer to

respondents’ assertions than either MI or predictions from the demographic multinomial logit

model, at times further away, and sometimes in between. Even when comparing purely

simulated voting behaviour we find that the differences between MI and either of the

multinomial logit alternatives are not on the whole significant. The only exception is the

simulated vote share of the Progressive Democrats, which according to the simulations based

on Martinez and Gill’s model is 5.8 percent and lies outside the 95 percent margin of error of

the MI estimate (3.4 percent +/- 2 percentage points).


It might be argued that the reported vote of actual non-voters is not necessarily a reliable

indication of how they would have voted had they done so. There may be a bias in favour of

the winners, for instance, and against the most obvious losers. Indeed, this appears to be the

case with respect to Fianna Fail and Fine Gael respectively. It might also be that these were

people who paid less attention to the campaign and gave less thought to their choice than

actual voters. In this respect the test is not an ideal one, as it cannot tell us whether it is the

20

MI procedure that was unreliable or the imagination of the respondents (Karp and

Brockington 2005). We can gain a better idea of the performance of the imputation procedure

by artificially creating a further set of missing values on the vote variable and using MI to re-

estimate the votes of these fake “nonvoters”. To do so, we first truncate the dataset to one that

contains only confirmed voters (N=1,835). We then impose the survey turnout rate of 69

percent anew by removing observations at random, before re-imputing the deleted

observations using Amelia. By cross-tabulating the distributions of observed and re-imputed

preferences we can gauge how close the individual imputations are to the recorded votes.

Lastly, we can obtain similar cross-tabulations for imputed and professed votes reported in

Figure 2 above and compare the closeness of the two sets of cross-tabulations. This will

indicate how MI performs as an approximation of actual vote choices vis-à-vis professed

votes.

The results are displayed in Table 1. The general picture that emerges from panel (a)

is that the distributions of observed and re-imputed preferences are quite similar. This is all

the more significant as the re-imputations of artificial non-voters have to make do with much

smaller numbers of cases. However, we are primarily interested in the accuracy of imputation

within each party group: to what extent does the imputation procedure correctly identify

those who vote Fianna Fáil, or Fine Gael and so on?

[Table 1 about here]

The table can be read in two directions: down and across. Reading across, the

accuracy seems poor. Except in the case of Fianna Fáil and Fine Gael, less than half of the

voters for any party are correctly identified in either panel of the table. However, this may

partly be an artefact of the EM algorithm, which will be biased towards the larger parties.

More appropriate is to read downwards: it can then be seen that even in the case of the

smaller parties the imputations are much more likely to predict the true choice with accuracy

21

than it is to make any other prediction. Lastly, we might expect the imputations to be closer

to the “real” ones in the case of the artificially created non-voters than in the case of the

actual non-voters (panel b), on the basis that the choice made by the former was a realized

intention and that of the latter at best an intention and at worst something respondents make

up. This difference in closeness is shown in panel (c): The bigger the positive difference, the

more our re-imputation of real vote choices outperforms our imputation of the professed

choices of non-voters. Negative figures indicate that our implications are closer to the

professed vote choices than they are to the actual ones. The expectation that MI simulations

resemble actual votes more than professed ones bears out for most parties, but not for Fianna

Fáil or the Progressive Democrats.13

As the “artificial” abstainers were chosen at random from among actual voters, the

MAR assumption is arguably more likely satisfied than in the context of the other MI

simulations reported in this article.14 The fact that our re-imputation of the vote choices of

artificial non-voters outperforms our simulation of the professed votes of actual non-voters

(who have been presented with the problematic task of answering counterfactual questions)

further validates the imputation model. While potentially extraneous information about

socioeconomic determinants of turnout decision and vote choice may not aid the imputation

algorithm in making the right imputations, it appears at worst irrelevant.


How do the multinomial logit strategies of estimating turnout effects perform in this

respect? Table 2 repeats the cross-tabulations for predictions based on Citrin et al.’s model.

Here never more than half of the voters for any party are correctly identified in either panel of

13 These were the two government parties both before and after the 2002 election. It is unclear to what extent

this pattern reflects respondents’ impulse to side with the winners.

14 However, as these are random draws from a sub sample that can itself not be considered random with respect

to vote choice, the “missing” vote choices of the newly created non-voters’ are not missing completely at

random.

22

the table. And only in the case of Sinn Féin are the simulations notably more likely to predict

the true choice with accuracy than to make any other prediction (panel a). Furthermore, for

most parties the simulations are closer to the professed votes of non-voters than they are to

the real votes of the artificially created non-voters, although the differences are very small

(panel c). In other words, while MI obtains its best results when estimating what people really

did, multinomial logit estimation using demographics is hardly better at replicating what

people did rather than what they say they did, and it matches either data rather poorly.


This does not necessarily mean that MI performs per se better than multinomial logit.

While the model replicating Citrin et al.’s method had to make do with a significantly smaller

number of predictor variables, relying on demographics only, a fairer comparison might be

one between MI and multinomial logit based on the full set of demographic and political

variables as used by Martinez and Gill (2005). The predicted vote choices of real and

artificially created non-voters from multinomial logit estimation of a comprehensive model

are presented in Table 3. Despite the inevitably considerable attrition, these multinomial

logit-based simulations of vote choice perform well, attributing the correct vote choice more

than two thirds of the time for all parties (but only about half the time for the independent

candidate). And while the simulations are also closely correlated to non-voters’ professed and

hypothetical votes, the match is again best for re-estimation of the deleted votes of actual

voters.15 Indeed, the match for re-estimation of the deleted votes of actual voters is overall

better for multinomial logit than it is for MI, although the latter has been able to make use of

a significantly larger number of observations.

Despite these differences, the main finding that emerges from this analysis of the

2002 Irish election is that the three methods for simulating the vote choices of non-voters

15 Again, we find that the edge of re-estimation of the deleted votes of actual voters over simulating professed

votes is somewhat reduced in the cases of the two government parties.

23

lead to substantially similar results. To gauge to what extent this finding is particular to this

case, we also use the three methods to simulate full turnout at 30 elections in 25 countries

from the Comparative Study of Electoral Systems (Module I).16 These elections capture the

full variation of systems of government, electoral systems, party systems and voter turnout. In

order to express the net differences between the observed and the hypothetical vote at the

national level, we use the Gallagher Index of Disproportionality. Originally designed to

measure the difference between the distributions of votes and seats in an election, this index

uses squared differences of the proportions, thereby avoiding the problem that changes to

party vote share cancel each other out while giving larger weight to the big vote share

changes for individual parties (Gallagher, 1991).17 The index provides an ideal tool for

comparing the observed-versus-imputed vote bias across diverse elections.

Figure 3 shows the distribution of the resulting disproportionality scores across the 30

elections in the CSES data as well as the 2002 Irish election. Looking at this range of

elections from different countries, many of the simulated election results under full turnout

are less similar across the three methods than in the Irish case. Firstly, MI leads to bigger

turnout effects on average. The mean disproportionality score using MI is 4.25 (S.D. = 2.25),

compared to 1.95 (S.D. = 1.33) when the Martinez and Gill method is used and .9 (S.D.

= .74) in the case of the method used by Citrin et al. Bivariate correlations are moderately

high and statically significant between disproportionality scores based on MI and Martinez

16 The study contains data on two successive elections in Mexico (1997 and 2000) and Spain (1996 and 2000).

To control for distinct electoral cleavages, data on Wallonia, East Germany, and Scotland are treated as separate

elections.

17 The index is calculated as the square root of

,

where vc denotes the vote share based on 100 percent (“complete”) turnout and vo denotes the observed vote

share of the n competing parties.

24

and Gill’s method (r = .41, p = .02) and between the disproportionality scores using Martinez

and Gill’s and Citrin et al.’s methods, respectively (r = .37, p = .04), but not between the MI

and Citrin et al. based disproportionality scores (r = .27, p = .14). This suggests that the

model used by Martinez and Gill assumes a middle position between simulation on logit

predictions and MI.


Looking at the differences in correspondence of the three disproportionality scores

across elections, few systematic sources of variation can be discerned. Instances in which

election results amended by MI simulations differ starkly from simulations using multinomial

logit predictions include proportional representation systems such as the Spanish one as well

as first-past-the-post systems like in Canada. However, cases like the Spanish elections or

New Zealand in 1996 suggest that discrepancies between the different simulation methods

might vary with the number of parties. To investigate this possibility, we regress differences

between MI and each of the two multinomial logit-based simulations on the number of

parties fielded in each election (Figure 4). The measure for the difference between any two

disproportionality scores is again Gallagher’s Index of Disproportionality. The mild positive

association between the number of parties and discrepancy between disproportionality scores

is not statistically significant, suggesting that the differences between MI and multinomial

logit predictions pertain as much to elections in two-party systems as to those in multi-party

systems.


The only systematic pattern that seems to appear is that the three disproportionality

scores differ less in the context of elections with high turnout. We test this more directly by

regressing the disproportionality scores expressing the difference between MI and each of the

two multinomial logit disproportionality scores, respectively, on turnout. Figure 5 shows that

25

this expectation bears out: A ten percent increase in turnout reduces the difference between

MI and multinomial logit based simulations by almost a full unit on both disproportionality

scores (which range from .66 to 9.28 in the demographics only case and from .5 to 10.23 in

the case of the full information model). Thus, the differences between MI simulations of

counterfactual voting behaviour and multinomial logit predictions of the same matter most in

the context of low turnout elections.


Discussion and Conclusions

The question of whether turnout matters for election results can only be answered if we can

say with some degree of certainty how the abstainers would have voted had they voted. In

this article we have assessed and compared multiple imputation as a method of ascertaining

the impact of turnout on election results. Our findings suggest that we can have reasonable

confidence in the MI method of estimating turnout effects: two validity tests produced good

results. The MI results are also partly matched by simulations using multinomial logit models

as proposed by Citrin et al. (2003) and Martinez and Gill (2005).

Beside a relative similarity to the results from multinomial logit estimation of full

turnout election results, MI simulation has a number of advantages. Firstly, it provides us

with a measure of confidence reflecting the uncertainty of the imputation method as well as

uncertainty fundamentally inherent in the world. Secondly, simulation by MI rests on

assumptions about the relationship between turnout and vote choice that are less demanding

and therefore more realistic than those underlying simulations based on multinomial logit

predictions. Thirdly, multinomial logit strategies face a trade-off between the richness of the

model and the number of cases to be used as a basis for prediction on the other. In the cases

of the Citrin et al. and Martinez and Gill studies, this amounts to a trade-off between a model

26

based on demographics only, which tend to have fewer missing values than attitudinal

variables, and a fully specified model. This trade-off necessitates a choice between the loss of

information and potential bias incurred through an underspecified model or the loss of

information and potential bias stemming from deletion of a considerable number or cases. By

imputing missing data among all variables including the vote choice variable, MI simulation

avoids this trade-off. The MI approach is therefore likely to be more efficient than

multinomial logit to the extend that it can utilize a wider range of variables, including, where

available, party measures of voter utilities as given by party and candidate thermometers and

PTV scores, without incurring a loss of observations. Comparing the predictions from models

following Citrin et al., and Martinez and Gill’s specifications, respectively, the lesson is to

prioritize a full model over thee need to maximize observations if that is the only choice

available. MI ensures that a third option is available.18

In line with recent studies on turnout effects at elections around the world (cf. Lutz

and Marsh 2007), the main finding from the Irish case study is that the fortunes of the Irish

political parties at the 2002 election would have remained virtually unaffected by universal

turnout. One might speculate about the difference that a few more votes might have made to

the distribution of seats: Fine Gael might not have suffered quite such the meltdown (the

18 The latest version of Stata (release 11) provides MI routines that offer a range of multiple imputation methods.

For imputation of missing values on only a single variable, multinomial logit regression is offered. This method

enables researchers to simulate higher turnout using the method proposed by Martinez and Gill (2005) but with

the added benefit of estimating the uncertainty of the imputations based on averaging across a number (e.g. five)

sets of imputed values. To the extent that this remedies the biggest weakness of the Martinez and Gill method –

the absence of a measure of uncertainty – Stata 11 can implement an “enriched” version of their method.

Alternatively, an iterative Markov chain Monte Carlo method is available to impute missing values on all

variables in the dataset, resembling the procedure implemented by Amelia and evaluated in this article. Through

iteratively improving imputations on all (dependent and independent) variables, this option continues to have the

added benefit of using all the information available in the data.

27

party incurred by far the largest losses of all parties from the previous election), and Fianna

Fáil might have needed support from independent TDs to form a government with the

Progressive Democrats. But while the empirical differences between MI and the two sets of

multinomial logit simulations are negligible in the case of the Irish election, an extension of

the comparison to the first CSES module suggests that discrepancies are the norm rather than

the exception. Differences become more marked the more nonvoters there are whose

behaviour has to be simulated in order to detect the political effects of high and low election

turnout.

Finally, a caveat. What we have done here is simply to assess the impact of full

turnout, which in practice can only ever be approximated even trough the use of compulsory

voting rules. Our analysis does not say what would happen if turnout rose by 5, 10 or 15

percentage points, and there are good reasons to expect that it would depend on what the

agency of mobilization is. The simulation of election results under full turnout is nonetheless

important – this is after all at the heart of the argument in Lijphart’s (1997) seminal article on

the topic. Future analyses should examine how MI performs relative to other approaches in

simulating incremental increases or decreases in turnout.

28

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Appendix A. Variables in Imputation Model for the 2002 Irish General Election

N Mean S.D. Min. Max.Number of FF candidates in constituency 2663 2.63 0.62 2 4Number of FG candidates in constituency 2663 2.14 0.7 1 4Number of Green candidates in constituency 2663 0.78 0.41 0 1Number of Labour candidates in constituency 2663 1.11 0.58 0 3Number of PD candidates in constituency 2663 0.49 0.65 0 3Number of SF candidates in constituency 2663 0.86 0.48 0 2Number of independent candidates in constituency 2663 3.47 1.8 0 7How likely ever to vote for Fianna Fáil 2625 6.74 3.2 1 10How likely ever to vote for Fine Gael 2603 5.11 3.05 1 10How likely ever to vote for Green Party 2586 4.69 2.81 1 10How likely ever to vote for Labour 2602 4.81 2.78 1 10How likely ever to vote for Progressive Democrats 2592 4.76 2.75 1 10How likely ever to vote for Sinn Féin 2595 3.37 2.83 0 10How likely ever to vote for an independ. candidate 2599 5.68 2.97 0 10Thermometer degree, Bertie Ahern 2612 65.55 24.24 0 100Thermometer degree, Mary Harney 2595 51.07 23.54 0 100Thermometer degree, Ruairi Quinn 2558 42.87 20.71 0 100Thermometer degree, Trevor Sargent 2419 42.18 21.87 0 100Thermometer degree, Michael Noonan 2562 36.74 23.07 0 100Thermometer degree, Gerry Adams 2579 38.85 26.6 0 100Thermometer degree, Fianna Fáil 2591 63.92 25.6 0 100Thermometer degree, Green Party 2543 47.71 21.92 0 100Thermometer degree, Fine Gael 2567 47.03 23.34 0 100Thermometer degree, Labour 2557 45.29 20.97 0 100Thermometer degree, Progressive Democrats 2553 47.22 22.34 0 100Thermometer degree, Sinn Féin 2538 33.36 25.8 0 100Evaluation of economy over last 5 years 2657 1.9 1.08 1 6Evaluation of health services over last 5 years 2655 3.5 1.26 1 6Evaluation of housing situation over last 5 years 2651 3.13 1.48 1 6Age 2640 46.9 17.12 18 100Female 2663 0.52 - 0 1Urban 2592 0.29 - 0 1Class 2498 2.53 1.7 1 5Education 2654 3.84 1.37 1 6Union member 2326 0.35 0.48 0 1Left/right self placement 2347 2.1 0.61 1 4Satisfaction with democracy 2341 6.91 2.82 0 11Efficacy 2660 2.54 1.66 1 7Frequency of attending religious service 2393 3.09 1.84 1 8Political knowledge 2663 3.39 1.28 0 5Party identification 2663 0.28 - 0 1Did Fianna Fáil contact? 2663 0.33 - 0 1Did Fine Gael contact? 2663 0.24 - 0 1Did Greens contact? 2663 0.02 - 0 1Did Labour contact? 2663 0.12 - 0 1Did Progressive Democrats contact? 2663 0.04 - 0 1Did Sinn Féin contact? 2663 0.06 - 0 1

32

Appendix B. Party vote shares under validated and full turnout using alternative imputation models and multinomial logit predictions

Observed Vote (N=1,835) Simulation of Full Turnout

MI (full set of variables, N=2,663)

MI (demographics only, N=2,663)

MI (preferences only, N=2,663)

M-logit (demographics only, N=2,570)

M-logit (all variables, N=2,240)

Full sampleN Prop. Prop. S.E. Prop. S.E. Prop. S.E. Prop. Prop.

FF 820 45% 43% 0.012 44% 0.012 44% 0.012 44% 43%FG 406 20% 21% 0.011 20% 0.009 21% 0.010 20% 20%Greens 80 4% 4% 0.006 5% 0.006 4% 0.006 5% 4%Labour 173 10% 10% 0.007 10% 0.007 10% 0.007 10% 10%PD 58 3% 3% 0.004 3% 0.004 3% 0.005 3% 4%SF 90 6% 7% 0.006 6% 0.007 6% 0.006 7% 8%Independent 208 11% 11% 0.008 12% 0.008 11% 0.007 11% 11%

Nonvoters only Professed vote (N=556) (N=556) (N=556) (N=556) (N=496) (N=305)

N Prop. Prop. S.E. Prop. S.E. Prop. S.E. Prop. Prop.FF 261 45% 42% 0.036 43% 0.032 43% 0.033 43% 40%FG 107 19% 23% 0.033 20% 0.025 24% 0.025 19% 21%Greens 30 7% 4% 0.016 5% 0.011 4% 0.017 5% 5%Labour 63 13% 10% 0.020 10% 0.016 10% 0.019 10% 11%PD 22 3% 3% 0.010 3% 0.011 3% 0.013 3% 6%SF 22 4% 7% 0.012 6% 0.019 6% 0.012 8% 10%Independent 51 10% 11% 0.025 13% 0.018 10% 0.017 12% 8%

Figure 1. Observed versus full (100%) turnout vote

Figure 2. Professed versus simulated vote

35

Figure 3. Differences between observed and simulated vote shares based on MI and multinomial logit predictions (Gallagher Index)

Figure 4. Differences between MI and multinomial logit simulations by number of parties

USA 1996

CAN 1997

TWN 1996

Scotland 1997

Britain 1997

PRT 2002KOR 2000

MEX 2000

AUS 1996

MEX 1997

IRL 2002

SWE 1998ICE 1999

BELF1999

SLV 1996

EastG 1998

DEU 1998

CZH 1996

NZL 1996DNK 1998

HUN 1998

BELW1999

PRU 2001

ROM 1996

ESP 2000

ESP 1996

ISR 1996

NTH 1998

UKR 1998

SWZ 1999

RUS 1999

02

46

810

5 10 15 20 25

b = .081; se = .085; R-sq = .03; N = 31

(a) Difference between MI and Citrin et al.

USA 1996

Britain 1997

Scotland 1997

PRT 2002

TWN 1996

CAN 1997

KOR 2000

AUS 1996

MEX 2000

IRL 2002

MEX 1997

SLV 1996

SWE 1998ICE 1999BELF1999

EastG 1998

DEU 1998

CZH 1996

NZL 1996

HUN 1998

DNK 1998

ROM 1996PRU 2001ISR 1996

ESP 1996

ESP 2000

BELW1999

NTH 1998

UKR 1998

SWZ 1999

RUS 1999

02

46

810

5 10 15 20 25

b = .092; se = .082; R-sq = .04; N = 31

(b) Difference between MI and Martinez and Gill

Dis

prop

ortio

nalit

y

Number of Parties

Figure 5. Differences between MI and multinomial logit simulations by turnout

SWZ 1999

USA 1996

MEX 1997

CAN 1997

KOR 2000

Scotland 1997

Britain 1997

HUN 1998

PRT 2002

RUS 1999

IRL 2002

PRU 2001MEX 2000

UKR 1998

NTH 1998TWN 1996

SLV 1996

CZH 1996

ESP 2000

ROM 1996

ESP 1996

SWE 1998

DEU 1998

EastG 1998

AUS 1996

NZL 1996DNK 1998

ICE 1999

ISR 1996

BELW1999BELF1999

02

46

810

40 50 60 70 80 90

b = -.084; se = .030; R-sq = .21; N = 31

(a) Difference between MI and Citrin et al.

SWZ 1999

USA 1996

MEX 1997

CAN 1997

KOR 2000

Scotland 1997

Britain 1997

HUN 1998

PRT 2002

RUS 1999

IRL 2002

PRU 2001

MEX 2000

UKR 1998

NTH 1998

TWN 1996SLV 1996

CZH 1996

ESP 2000

ROM 1996

ESP 1996

SWE 1998

EastG 1998

DEU 1998

AUS 1996

NZL 1996DNK 1998

ICE 1999

ISR 1996

BELF1999BELW1999

02

46

810

40 50 60 70 80 90

b = -.087; se = .029; R-sq = .24; N = 31

(b) Difference between MI and Martinez and Gill

Dis

prop

ortio

nalit

y

Actual Turnout

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Table 1. MI Vote by Observed and Professed Vote

Re-imputed, N=569 (of 1,835)(a) Observed Vote FF FG Greens Labour PD SF Independ.

FF 0.64 0.17 0.02 0.03 0.03 0.00 0.11FG 0.17 0.58 0.03 0.06 0.04 0.00 0.12Greens 0.39 0.18 0.31 0.10 -0.10 0.01 0.11Labour 0.17 0.20 0.05 0.36 0.02 0.02 0.18PD 0.37 0.35 0.01 0.04 0.10 0.09 0.05SF 0.21 0.09 0.09 0.12 -0.04 0.32 0.21Independent 0.25 0.31 0.09 0.06 0.03 0.02 0.23

Total 0.40 0.30 0.05 0.08 0.03 0.02 0.14

Imputed, N=556 (of 2,663)(b) Professed Vote FF FG Greens Labour PD SF Independ.

FF 0.70 0.12 0.02 0.04 0.03 0.02 0.07FG 0.28 0.47 0.03 0.06 0.03 0.01 0.11Greens 0.27 0.21 0.19 0.15 0.05 0.03 0.11Labour 0.30 0.16 0.05 0.27 0.01 0.07 0.13PD 0.33 0.23 0.00 0.14 0.17 0.02 0.11SF 0.24 0.23 0.04 0.06 0.02 0.30 0.12Independent 0.33 0.18 0.04 0.12 0.01 0.07 0.23

Total 0.49 0.21 0.04 0.09 0.03 0.04 0.10

(c) Difference in Party Match -6 11 12 9 -7 2 0

Note: Cell entries are the average probabilities of a vote for the column party. Imputations are based on MI model including all available variables related to vote choice.

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Table 2. Multinomial logit-simulated Vote by Observed and Professed Vote (Citrin et al. model)

Re-simulated, N=513 (of 1,835)(a) Observed Vote FF FG Greens Labour PD SF Independ.


Total 0.45 0.25 0.03 0.08 0.02 0.04 0.12

Simulated, N=496 (of 2,663)(b) Professed Vote FF FG Greens Labour PD SF Independ.


Total 0.43 0.21 0.05 0.10 0.03 0.06 0.12

(c) Difference in Party Match 2 2 -4 -2 -2 -3 0

Note: Cell entries are the average probabilities of a vote for the column party. Simulations are based on multinomial logit model including only demographics.

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Table 3. Multinomial logit-simulated Vote by Observed and Professed Vote (Martinez and Gill model)

Re-simulated, N=313 (of 1,835)(a) Observed Vote FF FG Greens Labour PD SF Independ.


Total 0.43 0.28 0.02 0.08 0.02 0.03 0.14

Simulated, N=305 (of 2,663)(b) Professed Vote FF FG Greens Labour PD SF Independ.


Total 0.42 0.20 0.05 0.11 0.06 0.08 0.09

(c) Difference in Party Match 7 18 31 10 15 -3 19

Note: Cell entries are the average probabilities of a vote for the column party. Simulations are based on multinomial logit model including all available variables related to vote choice.

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Missing Voters 17May10