Calculating the total vote where the district magnitude is greater than one: A test of some algorithms using British local election data

Electoral analysis using aggregate data relies on the availability of accurate voting statistics. One vital piece of information, often missing from official electoral returns, particularly British local government elections, is the total number of valid ballot papers. This figure is essential for the calculation of electoral turnout. When voters have a single vote and official information about the number of ballot papers issued is missing, a figure for the total vote can still be derived. However, local elections in Britain frequently use a system of multiple-member wards, where voters have as many votes as there are seats to be filled. In such cases, calculating the total vote and, hence, the turnout does present a real problem. It cannot be assumed that all voters will use their full quota of votes or that voters will cast a ballot in favour of a single party. This paper develops and tests diff erent algorithms for calculating the total vote in such circumstances. We conclude that the accuracy of an algorithm is closely related to the structure of party competition. The findings of this paper have a number of important implications. First, the difficulties in calculating the turnout in multiple-member wards are identified. This will inform the debate about public participation in the local electoral process. Second, the method for deriving a figure for the total vote has an important bearing on a number of other statistics widely employed in electoral analysis.     

Forecasting on British election night 1997

An account is given of methods used to predict the outcome of the 1997 general election from early declared results, for use by the British Broadcasting Corporation (BBC) in its election night television and radio coverage. Particular features of the 1997 election include extensive changes to constituency boundaries, simultaneous local elections in many districts and strong tactical voting. A new technique is developed, designed to eliminate systematic sources of bias such as differential refusal, for incorporating prior information from the BBC's exit poll. The sequence of forecasts generated on election night is displayed, with commentary.     

Estimating Parliamentary composition through electoral polls

Summary.  All electoral systems have an electoral formula that converts proportions of votes into Parliamentary seats. Pre-electoral polls usually focus on estimating proportions of votes and then apply the electoral formula to give a forecast of Parliamentary composition. We describe the problems that arise from this approach: there will typically be a bias in the forecast. We study the origin of the bias and some methods for evaluating and reducing it. We propose a bootstrap algorithm for computing confidence intervals for the allocation of seats. We show, by Monte Carlo simulation, the performance of the proposed methods using data from Spanish elections in previous years. We also propose graphical methods for visualizing how electoral formulae and Parliamentary forecasts work (or fail).     

The dependence of voter shares under a two-vote system

In an electoral system where each voter is allowed to select up to two candidates in a double-seat constituency, the combination of votes forms a special kind of contingency table. The usual hypothesis of complete independence of the voter shares for the candidates is of interest, but it is shown to be almost unattainable. The correlation coefficient of the voter shares is constructed. It is shown to be equivalent to Pearson's phi coefficient. The limitations of the coefficient are discussed. It is found that the use of conditional probabilities gives a better understanding of the dependence of the voter shares under the system considered.     

Latent class models for ecological inference on voters transitions

This paper introduces some new models of ecological inference within the context of estimation of voter transitions across elections. In particular, we assume that voters of a given party in a given occasion may be split into two latent types: faithful voters, who will certainly vote again for the same party and movers, who will reconsider their choice. Our models allow for unobserved heterogeneity across polling stations both in the weights of the two latent classes within each party and also when modelling the choice of unfaithful voters. Different ways of modelling the unobserved heterogeneity are considered by exploiting properties of the Dirichlet-multinomial distribution and the Brown Payne model of voting transitions can be seen as a special case within the class of models presented here. We discuss pseudo-maximum likelihood estimation and present an application to recent elections in Italy.     

Evidence of bias in the Eurovision song contest: modelling the votes using Bayesian hierarchical models

The Eurovision Song Contest is an annual musical competition held among active members of the European Broadcasting Union since 1956. The event is televised live across Europe. Each participating country presents a song and receive a vote based on a combination of tele-voting and jury. Over the years, this has led to speculations of tactical voting, discriminating against some participants and thus inducing bias in the final results. In this paper we investigate the presence of positive or negative bias (which may roughly indicate favouritisms or discrimination) in the votes based on geographical proximity, migration and cultural characteristics of the participating countries through a Bayesian hierarchical model. Our analysis found no evidence of negative bias, although mild positive bias does seem to emerge systematically, linking voters to performers.     

Modelling Political Popularity: an Analysis of Long-range Dependence in Opinion Poll Series

A simple model of political popularity, as recorded by opinion polls of voting intentions, is proposed. We show that, as a consequence of aggregating heterogeneous poll responses under certain assumptions about the evolution of individual opinions, the time series of poll data should exhibit long memory characteristics. In an analysis of the monthly Gallup data on party support in the UK, we confirm that the series have long memory and further show them to be virtually pure 'fractional noise' processes. An explanation of the latter result is offered. We study the role of economic indicators in predicting swings in support, perform event analyses and use our estimates to generate post-sample forecasts to April 1997.     

Asymptotic Equivalence of Seat Bias Models

Apportionment methods round vote proportions to integer numbers of seats in a parliament. An important issue is whether a given apportionment method treats larger and smaller parties equally or gives rise to seat biases. In this paper two models of quantifying seat biases of popular apportionment methods are compared to each other. The models are found to result in the same asymptotic behaviour when the number of seats available for apportionment grows large.     

Extended voting measures

This paper extends a number of voting measures used to quantify voting power. The extension is based on the recognition that individuals sometimes vote in coalitions. This observation gives rise to a statistical model which considers past voting patterns of subsets of eligible voters. The model is then used to obtain estimates of the probabilities of all voting combinations from which empirical measures are calculated. The calculation of the estimated probabilities may involve high-dimensional integrations. An example is given based on past decisions arising from the Supreme Court of Canada.     

The Scottish spoiled ballot: where was statistics?

In May 2007, Scotland went to the polls to elect both local constituency and regional members to the Scottish Parliament. Astonishingly, 100000 votes were rejected as "spoiled". Voters had misunderstood the new form of ballot paper—3% had marked a single cross despite having two votes. Parties contesting the election described it as "a debacle", "a shambles", "totally unsatisfactory". One American called it more flawed than Florida's notorious "hanging chad" ballot that gave the 2000 Presidential election to George Bush. What went wrong? Sheila Bird looks into the Scottish ballot paper.     

Quick counts from non-selected polling stations

Countless examples of misleading forecasts on behalf of both campaign and exit polls affecting, among others, British, French, and Spanish elections could be found. This has seriously damaged their image. Therefore, procedures should be used that minimize errors, especially on election night when errors are more noticeable, in order to maintain people's trust in surveys. This paper proposes a method to obtain quick and early outcome forecasts on the election night. The idea is to partly sample some (whatever) polling stations and use the consistency that polling stations show between elections to predict the final results. Model accuracy is analysed through simulation using seven different types of samples in four elections. The efficacy of the technique is also tested predicting the 2005 Eusko Legebiltzarra elections from real data. Results confirm that the procedure generates highly reliable and accurate forecasts. Furthermore, compared with the classical quick count strategy, the method is revealed as much more robust and precise.     

Poll of Polls: A Compositional Loess Model

A large number of polls on party preferences are published today. In order to obtain an estimate of the changes in political opinion, the polls may be combined into a poll of polls. We discuss a method for combining polls using the fact that they are compositions and respecting the properties of the compositional sample space (the simplex). The method is easily implemented and the estimate may be computed in linear time. We provide an example with Swedish data from year 2007 to 2010. The method also allows us to present the deviations between the estimated compositions and the observed. In the data set, we note, for example, differences between different polling institutes.     

How many ways can you look at a proportion?: cross-community vote transfers in Northern Ireland after the Belfast Agreement

Summary.  The single transferable vote is a method of election that allows voters to mark candidates in order of preference. Votes that are not required to elect a candidate are passed to the next candidate in the voter's order of preference. Results of this kind of election give us data about the degree to which voters of a given persuasion are willing to pass their vote to a candidate of a different persuasion. Measures of voters' willingness to pass a vote to a candidate of a different persuasion are of particular interest in places such as Northern Ireland, where communities differ by religion and national aspiration, and agreed new political institutions are based on cross-community power-sharing. How we quantify this voting data may depend on the questions that we want to answer, of course. But, to understand changes in how the voter orders her or his preference, one may need to ask several questions, and to quantify the results of the election in more than one way.     

Analysis Of Variance In A Descriptive Context: A Geographic Example

This paper is designed to introduce students in the 16–19 group to Analysis of Variance. It takes as its subject a political-geographic issue, namely electoral districting, or gerrymandering. The topic of gerrymandering is outlined, and the distribution of boundaries for the 1979 EEC elections in London examined. Different gerrymanders are then explored, using Analysis of Variance to describe various types of solution, according to the distribution of the Labour vote both within and between constituencies.     

A first-digit anomaly in the 2009 Iranian presidential election

A local bootstrap method is proposed for the analysis of electoral vote-count first-digit frequencies, complementing the Benford's Law limit. The method is calibrated on five presidential-election first rounds (2002–2006) and applied to the 2009 Iranian presidential-election first round. Candidate K has a highly significant (p<0.15% ) excess of vote counts starting with the digit 7. This leads to other anomalies, two of which are individually significant at p～ 0.1% and one at p～ 1%. Independently, Iranian pre-election opinion polls significantly reject the official results unless the five polls favouring candidate A are considered alone. If the latter represent normalised data and a linear, least-squares, equal-weighted fit is used, then either candidates R and K suffered a sudden, dramatic (70%±15% ) loss of electoral support just prior to the election, or the official results are rejected (p～ 0.01% ).     

Deprivation, ill-health and the ecological fallacy

The use of ecological studies in explaining the relationship between deprivation and ill-health is widespread in many health applications. However, inferences drawn from these studies about individuals are susceptible to serious bias known as the ecological fallacy. Our paper demonstrates the ecological fallacy effect in this context but also shows how it can be considerably reduced by taking into account different population structures at the aggregate level. Two regression analyses of limiting long-term illness are performed, one at the individual level and one at the electoral ward level, using the 1991 UK census sample of anonymized records and the small area statistics. The analyses compare several measures of deprivation including the standard Carstairs index, with the separate variables which make up the indices, to determine their effectiveness in explaining rates of illness. Two of the deprivation scores are constructed using latent variable modelling techniques which enable a score to be generated at the individual level as well as at the ward level. It is shown that, given the right choice of socioeconomic variables and taking into account the age structure of the population, it should be possible to construct a single aggregate deprivation index that will explain most of the variation in rates of illness across the study region.     

Ecological inference techniques: an empirical evaluation using data describing gender and voter turnout at New Zealand elections, 1893–1919

Summary.  The difference, if any, between men's and women's voting patterns is of particular interest to historians of gender and politics. For elections that were held before the introduction of opinion surveying in the 1940s, little data are available with which to estimate such differences. We apply six methods for ecological inference to estimate men's and women's voting rates in New Zealand (NZ), 1893–1919. NZ is an interesting case-study, since it was the first self-governing country where women could vote. Furthermore, NZ officials recorded the voting rates of men and women at elections, making it possible to compare estimates produced by methods for ecological inference with known true values, thus testing the efficacy of different methods for ecological inference for this data set. We find that the most popular methods for ecological inference, namely Goodman's ecological regression and King's parametric method, give poor estimates, as does the much debated neighbourhood method. However, King's non-parametric method, Chambers and Steel's semiparametric method and the Steel, Beh and Chambers homogeneous approach all gave good estimates that were close to the known values, with the homogeneous approach performing best overall. The success of these methods in this example suggests that ecological inference may be a viable option when investigating gender and voting. Moreover, researchers using ecological inference in other fields may do well to consider a range of statistical methods. This work is a significant NZ contribution to historical politics and the first quantitative contribution, in the area of NZ gender and politics.     

Hierarchical related regression for combining aggregate and individual data in studies of socio-economic disease risk factors

Summary.  To obtain information about the contribution of individual and area level factors to population health, it is desirable to use both data collected on areas, such as censuses, and on individuals, e.g. survey and cohort data. Recently developed models allow us to carry out simultaneous regressions on related data at the individual and aggregate levels. These can reduce 'ecological bias' that is caused by confounding, model misspecification or lack of information and increase power compared with analysing the data sets singly. We use these methods in an application investigating individual and area level sociodemographic predictors of the risk of hospital admissions for heart and circulatory disease in London. We discuss the practical issues that are encountered in this kind of data synthesis and demonstrate that this modelling framework is sufficiently flexible to incorporate a wide range of sources of data and to answer substantive questions. Our analysis shows that the variations that are observed are mainly attributable to individual level factors rather than the contextual effect of deprivation.     

Small area variation in hospital admission rates: Bayesian adjustment for primary care and hospital factors

Hospital admission rates are often used as a proxy to reflect patterns of morbidity or health need in population subgroups or across geographic areas. This paper considers the estimation of small area variations in relative health need, as measured by routinely collected hospital admissions data, after allowing for variation in general practice (primary care) and hospital (supply) effects. A fully Bayesian hierarchical modelling framework is adopted, using combinations of electoral ward populations and general practice patients' lists to define catchment groups for analysis. Hospitals create a further stratum, with flows of patients between catchment groups and hospitals being represented by a gravity model. Variations in health outcomes are modelled by using a range of random-effects structures for each cross-classification of strata, together with a consideration of ward, practice, hospital and crossed level covariates. The approach is applied to case-studies of child respiratory and total emergency hospital admissions for residents in a London health authority.     

Finite mixture of regression models for a stratified sample

Despite the popularity and importance, there is limited work on modelling data which come from complex survey design using finite mixture models. In this work, we explored the use of finite mixture regression models when the samples were drawn using a complex survey design. In particular, we considered modelling data collected based on stratified sampling design. We developed a new design-based inference where we integrated sampling weights in the complete-data log-likelihood function. The expectation–maximisation algorithm was developed accordingly. A simulation study was conducted to compare the new methodology with the usual finite mixture of a regression model. The comparison was done using bias-variance components of mean square error. Additionally, a simulation study was conducted to assess the ability of the Bayesian information criterion to select the optimal number of components under the proposed modelling approach. The methodology was implemented on real data with good results.