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 ResearchUniversity of ManchesterUK

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

Party preference

Vote

Contest

Abstention

Policy preferences

Perception of parties

Covariates e.g Social class Sex Age etc

Data and methods

British Election Panel Study, 1997-2001 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

General Election Vote (1997 & 2001) and Voting intention (1998-2000) in the years 1997-2001

0

5

10

15

20

25

30

35

40

45

1997 1998 1999 2000 2001

Year

Per

cent

of B

EPS

res

pond

ents

Didn’t / wouldn't vote

Conservative

Labour

Liberal democrat

Don't Know

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

Identification Frequency Percentage

None 168 7.2

Conservative 622 26.7

Labour 1055 45.2

Liberal Democrat 318 13.6

Scottish National Party 96 4.1

Plaid Cymru 12 .5

Other 39 1.7

Refused 8 .3

Don't know 15 .6

Total 2333 100

BEPS 2001 Party ID

Party voted for in 2001

general election

Party identification (2001)

Total

None C L LD SNP PC Other

Didn’t vote 95 109 223 49 23 4 7 510

Conservative (C) 22 451 11 10 2 . 4 500

Labour (L) 12 17 723 28 3 . 4 787

Liberal Democrat (LD) 24 31 69 226 1 . 6 357

Scottish National Party

(SNP) 1 2 10 . 65 . . 78

Plaid Cymru (PC) 1 3 . . . 7 . 11

Other 10 9 15 3 2 . 11 50

Total 165 622 1051 316 96 11 32 2293

Party ID and vote

Party identification Party of first preference

based on ‘strength of feelings’

ratings

None

C

L

LD

SNP

PC

Other

Total

Conservative (C) 13 436 5 2 . . 1 457

Labour (L) 12 15 655 1 . 1 2 686

Liberal Democrat (LD) 29 20 28 234 1 . 15 327

Scottish National Party (SNP) 3 . 6 1 70 . 2 82

Plaid Cymru (PC) . 1 . . . 9 . 10

Party ID and SoF

Party ID in 1997 Party ID Changes till 2001a Total % 0 1 2 3 4

None 42 109 49 27 13 240 11 Conservative 464 70 77 38 6 655 29 Labour 721 87 144 25 11 988 43 Liberal Democrat 148 49 58 25 14 294 13 Scottish National Party 55 15 9 6 3 88 4 Plaid Cymru 3 1 1 1 0 6 0 Green Party 2 3 4 0 0 9 0

Total 1435 334 342 122 47 2280b 100 % 63 15 15 5 2 100

Stability of Party ID

0% 20% 40% 60% 80% 100%

None

Conservative

Labour

Liberal Democrat

Scottish National Party

Plaid Cymru

Green Party

All

0

1

2

3

4

No. of changes in rank of party…

C L LD

Conservative (C) -

Labour (L) .05 -

Liberal Democrat (LD) .20 .16 -

Stability in ranked preferences

0% 20% 40% 60% 80% 100%

Conservative rank

Labour rank

LD rank 0

1

2

3

4

Stability of Party ID

0% 20% 40% 60% 80% 100%

None

Conservative

Labour

Liberal Democrat 0

1

2

3

4

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

VU

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 U1 > U0 or equivalently if

000 VU

111 VU

0)( 0101 VV

If 1 and 0 have type-1 extreme value (Gumbel) distributions then 1 - 0 has a logistic distribution, and therefore the probability that U1 is greater than U0 is

Utility Logit

)exp()exp(

)exp()Pr(

01

101 VV

VUU

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:

)exp(1

)exp()1ePr(

X

Xchoic

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

000 VU

111 VU

222 VU

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

Multinomial logit

)exp()exp(1

)exp()1ePr(

21

1

XX

Xchoic

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:

0000 uVU

1111 uVU

2222 uVU

Latent variable distribution

We assume that (1 - 0) and (2 - 0 ) have logistic distributions

The latent variables are specified as

1 = (u1 - u0)

2 = (u2 - u0)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

)1choicePr(

212211

11

21)exp()exp(1

)exp(

dXX

X

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

) 2choice 2nd 1,choice1st Pr(

)exp(1

)exp(

)exp()exp(1

)exp(

2

2

21

1

X

X

XX

X

Multinomial logit for rankings with latent variables

Allowing for correlations among utilities with latent variables gives:

) 2choice 2nd 1,choice1st Pr(

2122

22

2211

11

21)exp(1

)exp(

)exp()exp(1

)exp(

dX

X

XX

X

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

www.gllamm.org

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 ulab – ucon. The same people are also likely to prefer Liberal Democrat over the Conservatives, and thus also have positive ulibdem - ucon

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 dataIDNo Conrank Labrank LibDem_rank

1 1 2 3

2 1 2 3

3 2 2 1

4 3 1 2

5 3 1 2

6 3 1 2

7 2 1 1

8 3 2 1

9 1 1 3

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: 2671.00

Model 0: Multinomial logit of ranked party preference

Parameter Est. SE Sig.

Labour Intercept 1.32 .18 <.001

Age -.18 .05 <.001

Sex -.25 .10 <.05

LibDem Intercept .68 .17 <.001

Age -.08 .05 ns

Sex -.02 .10 ns

Baseline category: Conservative

Log Likelihood: 2281.64

Model 1: ranked preference with two latent variables

Parameter Est. SE Sig.

Labour Intercept 2.56 .41 <.001

Age -.31 .11 <.01

Sex -.34 .23 ns

LibDem Intercept 1.42 .33 <.001

Age -.19 .09 <.05

Sex -.05 .18 ns

Latent Var(1) 11.35 1.51

Variables Var(2) 4.93 .92

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

12

Baseline category: Conservative

Log Likelihood: 2279.07

Model II: one-factor model of ranked party preference

Parameter Est. SE Sig.

Labour Intercept 2.57 .41 <.001

Age -.31 .11 <.01

Sex -.34 .23 ns

LibDem Intercept 1.42 .33 <.001

Age -.19 .09 <.05

Sex -.05 .18 ns

Latent Var(1) 11.45 1.51

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