Modelling voter preferences: a multilevel, longitudinal approach Dr. Edward Fieldhouse, Jerry Johnson, Prof. Andrew Pickles, Dr. Kingsley Purdam, Nick Shryane Cathie Marsh Centre for Census and Survey Research University of Manchester UK

Some limitations in modelling voter preferences Dichotomous response models, ‘minor parties’ and non-voting Handling complexity of voter preferences and party positions in ideological space Assumption of Independence of Irrelevant Alternatives Contextual Influences on voting

A Simplified conceptual model

Data and methods British Election Panel Study, Eight waves Information on preferences, voting, rankings of parties, left-right placement, tactical voting Multilevel design (occasion/person/location) Random Utility Models Generalised Linear Latent and Mixed Models

Some Assumptions of Party Identification Stable – even when vote switching takes place Enduring – across several consecutive years Resilient – to ephemeral political events Only relevant to only a small proportion of the electorate

BEPS 2001 Party ID

Party ID and vote

Party ID and SoF

U, the subjective value of a choice, i.e. utility, is modelled as being comprised of two parts: V, measured characteristics of the chooser or choice alternative, e.g. age, cost , a random component representing unmeasured idiosyncrasies Random Utility Models

There will be a utility associated with each choice-alternative. For example, with two alternatives: Binary choice Utility maximisation Alternative 1 will be chosen if U 1 > U 0 or equivalently if

If  1 and  0 have type-1 extreme value (Gumbel) distributions then  1 -  0 has a logistic distribution, and therefore the probability that U 1 is greater than U 0 is Utility Logit

Choice Logit When V is parameterised as a linear combination of subject-specific covariates X, the coefficients for the reference category are set to zero (for identification), yielding the familiar logit model: i.e. the probability that alternative 1 is chosen in preference to the reference (alternative 0)

When choosing among more than two alternatives, utility can be decomposed as before, e.g. for three alternatives: Polytomous choice

Assuming (  1 -  0 ) and (  2 -  0 ) are independent logistic distributions yields the familiar multinomial logit model: Multinomial logit

Assuming (  1 -  0 ) and (  2 -  0 ) are independent logistic distributions allowed specification of the multinomial logit model Independence from irrelevant alternatives This assumption of independence is known as “independence from irrelevant alternatives” (IIA) However, it is usually implausible to assume that (  1 -  0 ) and (  2 -  0 ) are independent.

Latent random variables The correlation between random components due to violation of IIA can be modelled by introducing shared random effects, u :

Latent variable distribution We assume that (  1 -  0 ) and (  2 -  0 ) have logistic distributions The latent variables are specified as  1 = ( u 1 - u 0 )  2 = ( u 2 - u 0 ) and are distributed bivariate normal The latent variables reflect the propensity to favour one choice over another when the effect of the explanatory variables ( X) has been accounted for.

Multinomial model with latent variables Allowing for correlation among utilities with latent variables gives the following model

Multinomial model with latent variables In general, the latent variables that give rise to the correlation among choices can be poorly identified This can be overcome using ranked preferences instead of first-choices

A model of ranked preferences The Luce model for ranked preferences is a direct extension of the random utility derivation of the multinomial choice model With three alternatives; first choice probabilities are as for the original model Second choice probabilities, conditional on the first choice, are given by the same multinomial form, but with the first-choice excluded from the choice set

For example, with three alternatives, the probability that choice 1 will be ranked first, followed by choice 2 second (with the final choice redundant) is: Multinomial logit for rankings

Multinomial logit for rankings with latent variables Allowing for correlations among utilities with latent variables gives:

GLLAMM Such models can be estimated using GLLAMM (Generalized Linear, Latent and Mixed Models; Rabe-Hesketh, Pickles & Skrondal, 2001) GLLAMM is a STATA programme freely available from

Latent variables structure and political theory A fundamental way by which political parties are characterised is where they fall along a uni-dimensional, “left-right” continuum (cf. spatial models of political preference by Downs [1957] and Black [1958])

Latent variables structure and political theory Conventionally, in the UK the Conservative party is seen as the most right-wing of the major parties, with Labour as the most left- wing. The Liberal Democrats are seen as occupying the middle ground, but closer to Labour than the conservatives.

Latent variables structure and political theory If this is so, ranked preferences for Labour and the Liberal Democrats should be clustered together to a greater extent than preferences for Conservative and Liberal Democrats (or indeed, for Conservative and Labour)

Latent variables structure and political theory In terms of the latent variables, those who prefer Labour to the Conservatives will have positive u lab – u con. The same people are also likely to prefer Liberal Democrat over the Conservatives, and thus also have positive u libdem - u con Therefore, the latent variables should be positively correlated

Political preference in the UK Data: British Election Panel Survey, 2001 wave (N = 1560 voting age respondents living in England [excludes Scottish- and Welsh-based respondents]) Party approval ratings were used to construct ranked preferences for the three major parties; Conservative, Labour, Liberal Democrat. (First place ties were split where possible by the respondents’ stated party ID. 80 first-placed ties remained after this)

Political preference in the UK Example party ranking data IDNoConrankLabrankLibDem_rank

Political preference in the UK Party preference ranks were modelled in GLLAMM using multinomial logistic regression with two latent variables. Covariates included were age and sex First, though, a multinomial logit model with no latent variables was fit, for comparison

Baseline category: Conservative Log Likelihood: Model 0: Multinomial logit of ranked party preference ParameterEst.SESig. LabourIntercept <.001 Age <.001 Sex <.05 LibDemIntercept.68.17<.001 Age ns Sex ns

Baseline category: Conservative Log Likelihood: Model 1: ranked preference with two latent variables ParameterEst.SESig. LabourIntercept <.001 Age <.01 Sex ns LibDemIntercept <.001 Age <.05 Sex ns LatentVar(1) VariablesVar(2) Corr(1,2)1.00

Model 1: ranked preference with two latent variables Model 1 is a massive improvement in fit over model 0 The latent variables are both significant, indicating a tendency to rank both Labour and Liberal Democrats differently from the Conservatives. The variance for Lab. vs. Con is greater than that of LD. vs. Con. – Lab. is more ‘distant’ from Con. than is LD. The two latent variables are highly correlated. The tendency to choose Labour over conservatives is related to the tendency to choose LibDems over Conservatives This violates IIA, invalidating Model 0

Uni-dimensional preference structure The strong correlation between latent variables implies that only one latent dimension is required to model ranked party preferences (the “left-right” dimension?)

A single-factor model was fitted to the data, whereby the second latent variable,  2 (the propensity to choose LibDem over Conservative) was defined as a function of  1 (Labour vs. Conservative) where is a ‘scale’ factor, to account for the different ‘distances’ between Lab-Con and LD-Con Model II: On factor model of ranked party preference

Baseline category: Conservative Log Likelihood: Model II: one-factor model of ranked party preference ParameterEst.SESig. LabourIntercept <.001 Age <.01 Sex ns LibDemIntercept <.001 Age <.05 Sex ns LatentVar(1) Variables.65.03

Model II fits at least as well as model I (difference in log-likelihoods is not significant) Coefficients are virtually identical to model I The scale factor ( ) is less than one, indicating that the Liberal Democrats are closer to the Conservatives than is Labour Model II: one-factor model of ranked party preference

A traditional multinomial logit model, fitted to political party preference in the UK, provided a poor fit of the data by failing to account for violation of IIA – the correlation between choices Latent variables were included to account for this A model with one latent variable fitted the data as well as the model with two, indicating that UK party preferences seem to fit a one- dimensional spatial model Summary