A multilevel spatial model to investigate voting behaviour in the 2019 UK general election

Kevin Horan^1*, Chris Brunsdon² and Katarina Domijan³

¹ Hamilton Institute, Maynooth University, Maynooth, Ireland.
² National Centre for Geocomputation, Maynooth University, Ireland.
³ Department of Mathematics and Statistics, Maynooth University, Maynooth, Ireland.
^* contact author: kevin.horan.2021@mumail.ie

Full paper published in Applied Spatial Analysis and Policy (11 January 2024) with open access available at this link.

This repository contains the code used to produce the findings discussed in this paper.

Abstract

This paper presents a modelling framework which can detect the simultaneous presence of two different types of spatial process. The first is the variation from a global mean resulting from a geographical unit’s ‘vertical’ position within a nested hierarchical structure such as the county and region where it is situated. The second is the variation at the smaller scale of individual units due to the ‘horizontal’ influence of nearby locations. The former is captured using a multi-level modelling structure while the latter is accounted for by an autoregressive component at the lowest level of the hierarchy. Such a model not only estimates spatially-varying parameters according to geographical scale, but also the relative contribution of each process to the overall spatial variation. As a demonstration, the study considers the association of a selection of socio-economic attributes with voting behaviour in the 2019 UK general election. It finds evidence of the presence of both types of spatial effects, and describes how they suggest different associations between census profile and voting behaviour in different parts of England and Wales.

Data visualisation

Dependent variable and guide map

Left: Values of dependent variable, Butler swing to the Conservatives, mapped across constituencies of England and Wales. The vast majority of constituencies recorded a positive swing. Figures projected as Dougenik cartograms such that equal populations occupy equal area while maintaining constituency contiguities.

Right: Guide map of the regions of England and Wales under a similar projection.

Swingometers

Graphical representation of swing by region and county. Swing is scaled relative to circle such that the maximum swing is 90 degrees from the vertical. In this regional map, the North East swing of 8.36% is the maximum.

Contiguities

Create contiguity matrix of constituencies, with some manual alterations to account for islands, bridges, ferries and tunnels.

First order queen contiguity structure of constituencies in England and Wales, shown as edges radiating from nodes at the centroid of constituencies.

Explanatory variables

Explanatory variable	Calculation from census	Justification/theory
degree educated	percentage of population with level 4 qualification or higher	post-industrial / knowledge-economy / peripherality
health not good	percentage of the population self-reporting ‘poor’, ‘bad’, or ‘very bad’ health	life outcomes / young people
white	percentage of population of white ethnicity	ethnic / cultural diversity / values

Description of explanatory variables

Values of independent variables mapped across England and Wales. Figures projected as Dougenik cartograms such that equal populations occupy equal area while maintaining constituency contiguities.

Modelling

Simplest model

Map of residuals from a simple linear model which does not take geography into account. Regions such as the South West and Merseyside appear to be almost completely red (overprediction of swing), the North East show a block of red alongside a block of blue, while a blue pattern of underprediction spreads across the boundary between the East Midlands and Yorkshire and the Humber.

Model variations

Alternative autoregressive components

This framework, however, also allows for more complex structures than this. For example, in addition to the random intercepts and slopes provided for in the hierarchical component, we have the option of using either

1. spatially autocorrelated random slopes for each covariate in each constituency,
2. a spatially autocorrelated random intercept at constituency level, or
3. both together.

To decide which of these three options was most suitable for this particular dataset, their performances can be compared. The fitting of such spatial models using the mgcv package requires the tuning of a parameter k which is the number of basis functions used to generate the autoregressive smoothing. Lower values of k lead to a smoother result. This is because k represents the number of components from the eigen decomposition of the variance-covariance structure which are to be used. Not all can be used because there are not enough data points for this to be computable. The k value has been optimised for each model such that the Akaike information criterion (AIC) is minimised, striking a balance between goodness of fit and model complexity.

Model comparison

Shown below are performance metrics for each of these model combinations, named models 1-3. Of these three potential structures, model 2, which we have been discussing, has the best performance metrics and was deemed the most suitable structure for modelling this particular dataset. Such a process can be used to find the most suitable structure for any potential dataset.

Model	Autoregressive spatial process(es)	AIC	RMSE	adjR2	Loglik
1	constituency component	2336	1.43	0.76	-1015
2	varying coefficients	2373	1.62	0.73	-1086
3	constituency component + varying coefficients	2381	1.64	0.73	-1094

Optimal model

The structure of the model is outlined below:

$$ \begin{aligned} y_{ijk} &= \beta_0 + \beta_1 degree_{ijk} + \beta_2 health_{ijk} + \beta_3white_{ijk}\\ &+ b_{0i} + b_{1i} degree_{ijk} + b_{2i} health_{ijk} + b_{3i}white_{ijk}\\ &+ b_{0ij} + b_{1ij} degree_{ijk} + b_{2ij} health_{ijk} + b_{3ij}white_{ijk}\\ &+ \gamma_{l}|\gamma_{m}, l\neq{m}\\ &+ \epsilon_{ijk} \end{aligned} $$

where y_ijk is the swing in constituency k in county j in region i for

• i = 1, ..., 11 regions,

• j = 1, ..., J_i counties within region i,

• k = 1, ..., K_i**j constituencies within county j within region i, and

• l = 1, ..., 571 individual constituencies.

• β₀, β₁, β₂, β₃ are fixed effects.

• b_0i, b_1i, b_2i, b_3i are the random effects (intercept and three slopes) associated with region i,

• b_0i**j, b_1i**j, b_2i**j, b_3i**j are the random effects (intercept and three slopes) associated with county j in region i.

• ϵ_ijk are independent normally distributed error terms.

Rather than estimate each of the random effect coefficients directly, the variance of each random effect is instead estimated. For the region and county level random effects, each is assumed to be independent of the others within its level, and to be normally distributed with mean of zero. This independence is a key restriction in multi-level modelling with mgcv as opposed to other packages.

The γ_l’s are constituency level random effects which model the spatial interactions at the lowest level of the model, based on an ICAR distribution. Let there be m = 1, ..., M potential neighbouring constituencies, where M = L = 571. Each γ_l is conditional on the sum of the weighted values of its neighbouring γ_m’s (w_l**mγ_m) and has unknown variance. As a constituency is not a neighbour to itself, the full conditional distribution can be written as follows: $$\gamma_l | \gamma_m,l\neq{m} \sim\text{N}\bigg(\frac{\sum_{l\neq{m}}{\gamma_l}}{d_l},\frac{\sigma_{l}^2}{d_l}\bigg)$$ where the term d_l represents the number of neighbours. Thus the mean of each γ_l is equal to the average of its neighbours, while its variance decreases as the number of neighbours increases.

The joint specification of the ICAR random vector γ when centred at 0 with common variance 1 rewrites to the pairwise difference formulation: $$\gamma \propto\text{exp}\bigg(-\frac{1}{2}\Sigma_{l\neq{m}}(\gamma_l-\gamma_m)^2\bigg)$$ To overcome the problem of unidentifiability, the constraint Σ_Lγ_l = 0 is added to centre the model.

Results

The aim of this modelling structure was to enable us

1. to test for the presence of spatial effects resulting in different associations between covariates and the dependent variable according to geography, taking into account both hierarchical and autoregressive spatial processes (spatial heterogeneity),

2. and also to estimate the relative variance associated with each type of process at different spatial scales (‘analysis of variance’ of spatial processes). Plot of fixed or global intercept and coefficients from combined model, coloured according to direction of association with swing. A higher proportion of people of white ethnicity in an average constituency is associated with a swing to the Conservatives while the opposite is true for increases in the proportion of degree-educated.

Fixed effects

Region randoms

Regions of England and Wales, coloured according to direction and magnitude of region-level random effects of each covariate with swing to the Conservatives. The global fixed effect from which these divergences occur is shown above each map. Unlike ‘health not good’ and ‘white’, ‘degree educated’ does not show significant divergence at this level from its global coefficient.

County randoms

Random effects of ‘degree educated’ variable at the county level of England and Wales. Particularly strong within-county variation can be observed in the East.

Random effects of ‘white’ variable at the county level of England and Wales. Particularly strong within-county variation can be observed in the East Midlands.

Table of variances

Net hierarchical effects

Net hierarchical effects of each variable. These are the sum, for each coefficient, of its fixed effect and two random effects (at region and county level), showing the spatial heterogeneity accounted for by the hierarchical component of the model. ‘Degree educated’ is negatively associated with swing to the Conservatives across England and Wales, albeit to different extents. ‘Health not good’ and ‘white’ show not only different magnitudes but also different directions of association with swing in different regions and counties across the study area.

Constituency level

The spatially autoregressive (in this case, ICAR) component at the lowest level (constituency) of the model. It shows a blue area of increased tendency to swing to the Conservative party, surrounded by a paler band, and red areas to the south, west and parts of the north where the tendency is to swing to Labour, controlling for the covariates and hierarchical effects. Areas of clear cross-regional spillover of effects are highlighted with red circles.

Spatial diagnostics of model

Left: Residuals from model, mapped by location, which appear by visual inspection to be randomly distributed.

Right: dot-plot of residuals of constituencies against their spatially lagged neighbours which shows neither a positive nor negative association between a constituency’s residuals and those of its neighbours.