Stephen Fisher, Jane Holmes, Nicky Best, Sylvia RichardsonDepartment of Sociology, University of OxfordDepartment of Epidemiology and Biostatistics

Imperial College, London

http://www.bias-project.org.uk

Combining individual and aggregate data to improve estimates of ethnic voting in Britain in 2001 and 2005

Outline Question of interest Model we will use Analysis

Target analysisIndividual exposure

Aggregate exposure

Individual outcomeyijxij

Zi, Xi

Aggregate exposure YiAggregate outcome

Ecological regressionZi, Xi

Aggregate outcome

Individual exposure

Aggregate exposure

Individual outcomeyijxij

Yi

Hierarchical Related Regression (HRR)

Zi, Xi

A decline in ethnic minority support for Labour? From 1974 to 2001 around 80% of ethnic minorities vote Labour Between 2001 and 2005 there were

Islamic terrorist attacks US and UK led invasions of Afghanistan and Iraq Heightened security and suspicion of non-whites Unlawful detention of foreign terror suspects Convictions of British soldiers for Iraqi prisoner abuse

These and other events are thought to have undermined support for Labour among ethnic minorities.

On the other hand, harsh stance on immigration in Conservative 2005 election campaign may have alienated ethnic voters

A decline in Muslim support for Labour? Initially

We found that the gap in Labour vote between whites and non-whites narrowed between 2001 and 2005.

Results presented at PSA 2009 Audience opinion was interesting, but really wanted to know

whether the same was true of Muslims So

We tested whether the gap in Labour vote between Muslims and non-Muslims narrowed between 2001 and 2005.

Individual-level model British Election Study post-election survey (BES)

Cross-sectional survey carried out after every general election For subject j in constituency i,

yij = voted Labour (1) / didn’t vote Labour (0) xij = Muslim (1) / non-Muslim (0)

But 1,898 subjects with validated data, only 20 Muslims

Area-level random effect

Probability subject j votes Labour

Log odds ratio of Muslim voting Labour compared with non-Muslim

Aggregate data However, we have data at the aggregate level for entire population

2001 Census data on % who are Muslim Number of people who vote Labour in each constituency from

General election results Data viewed as a 2x2 table. For constituency i:

yi = number who vote Labour ni = number who are eligible to vote xi = number who are Muslim

Vote Labour Don’t vote Labour

Non-Muslim ? ? 1- xi

Muslim ? ? xi

yi yi - ni ni

Ecological bias Standard analysis of this data will probably lead to biased results Bias in ecological studies can be caused by:

Confounding Confounders can be area-level (between-area) or

individual-level (within-area) include control variables and/or random effects in model

Non-linear covariate-outcome relationship, combined with within-area variability of covariate

No bias if covariate is constant in area (contextual effect) Bias increases as within-area variability increases … unless models are refined to account for this hidden

variability

Improving ecological inference Alleviate bias associated with within-area covariate variability Data at area-level, for constituency i:

Area-level outcome yi = number of people who vote Labour Area-level predictor = proportion who are Muslim

Then yi ~ Binomial(ni , pi ) where the area-level probability pi is calculated by integrating

individual-level probabilities given by individual-level model with respect to the within-area joint distribution fi(x) of all individual-level predictors

pi = pij(x) fi(x) dx pi is average group-level probability (of voting Labour) pij(x) is individual-level probability given covariates x fi(x) is distribution of covariate x within area i

The model for a single binary covariate Consider a single binary covariate x, e.g. Muslim/non-Muslim fi(x) is the proportion of individuals with x = 1 in each area, i.e. the

proportion Muslim in each constituency Individual-level model

pij = g(i + xij), where g() = e/(1+e) pij = g(i) if person j is non-Muslim pij = g(i + ) if person j is Muslim

Integrated group-level model = proportion Muslim in constituency i (mean of xij) pi = average probability (proportion) of voting Labour in area i

Prob. Muslimvotes Labour

Prob. of beingMuslim

Prob. non-Muslimvotes Labour

Prob. of beingnon-Muslim

Hierarchical Related Regression The parameters of the aggregate model have been derived from

an underlying individual-level model So the exposure-outcome relationship is assumed to be the same

in both the aggregate data and the individual-level data This means that the individual and aggregate data can be used

simultaneously to make inference on the underlying individual-level model.

The likelihood for the combined data is simply the product of the likelihoods of each set of data

This combined model is termed a hierarchical related regression (HRR). (Jackson, Best and Richardson, 2006)

Recap Question of interest

How do Muslims vote? And did they change their voting behaviour between the 2001 and 2005 general elections?

i denotes constituency, j denotes subject within a constituency

Individual-level data Aggregate dataOutcome yij = 1 if subject j votes Labour

0 if don’t vote Labouryi = number who vote Labourni = electorate

Explanatory variable

xij = 1 if subject j is Muslim 0 if subject j is not Muslim

= proportion who are Muslim

Proportion of electorate who voted Labour in 2001 and 2005, by constituency

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

Proportion who are Muslim

Prop

ortio

n of

ele

ctor

ate

who

vote

Lab

our

20012005

Analyses To start, various models are fit to the 2001 general election only

Simple model with only an individual Muslim effect

Analyses To start, various models are fit to the 2001 general election only

Simple model with only an individual Muslim effect Add a contextual effect of Muslim as well as an individual

effect

Analyses To start, various models are fit to the 2001 general election only

Simple model with only an individual Muslim effect Add a contextual effect of Muslim as well as an individual

effect Add an interaction term

Analyses To start, various models are fit to the 2001 general election only

Simple model with only an individual Muslim effect Add a contextual effect of Muslim as well as an individual

effect Add an interaction term Include socio-economic status as a confounder

Partly motivated by the apparent interaction

Socio-economic status coded as manual/non-manual

More than one individual-level binary covariate For the integrated group-level model, when we have more than

one binary covariate we need to know the cross-classification of individuals between covariate categories within each area, e.g. number of Muslims who have a manual job

Then average probability of voting Labour in area i,

Estimate p(xij, zij) by proportion in area i with covariates xij, zij

Census does not contain these cross-classifications Estimate by product of the 2 marginals, Lasserre et al

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Comparison of predictions for all models for 2001

Proportion who are Muslim

Prop

ortio

n of

ele

ctora

te w

ho vo

te L

abou

r

Ind. MuslimInd. + cont. MuslimInd. + cont. Muslim + interactionInd. cont. Muslim + ses

Odds ratio of voting Labour for Muslims = 9.45 (3.20, 19.81)

Comparison of voting behaviour in 2001 and 2005 What we are really interested in is whether Muslims changed their

voting behaviour between the 2001 and 2005 general elections

Individual model for 2001 election

Individual model for 2005 election

Results – odds ratios

Individual Muslim effect, 2001 8.32 (3.99, 16.47)

Individual Muslim effect, 2005 3.55 (1.48, 6.73)

Difference in individual Muslim effect 2.51 (1.18, 4.61)

Socio-economic status 0.52 (0.45, 0.59)

Conclusions Muslims are more likely to vote Labour than non-Muslims Muslims did significantly change their voting behaviour between

2001 and 2005 In 2005 they were less likely to support Labour than in 2001

We need to find and include more individual Muslim data in our analysis

Jackson, C. H, Best, N. G. and Richardson, S. (2006). Improving ecological inference using individual-level data. Statist. Med., 25, 2136-2159

Lasserre, V., Guihenneuc-Jouyaux, C. and Richardson, S. (2000). Biases in ecological studies: utility of including with-area distribution of confounders. Statist Med., 19, 45-59