The Polls

23 June 2019

Outline

Jackman (2005; 2009); Jackman & Mansillo (2018).
Let \(t\) index the \(T\) = 1,051 days between 2016 election and 2019 election.
Poll \(p\) fielded on day \(t\) by polling company \(j\) yields a estimated voting intention, a proportion \(\color{cyan}{y_p} \in [0,1]\), with sample size \(\color{cyan}{n_p}\). Variance of this estimate is approximately \(\color{cyan}{V_p = y_p (1-y_p)/n_p}\).
True, latent voting intentions on day \(t\) are \(\color{orange}{\xi_t}\). These are observed exactly on Election Days 2016 and 2019, \(\color{orange}{\xi_1}\) and \(\color{orange}{\xi_T}\), respectively.
Polling company \(j\) has a time-invariant “house effect” \(\color{orange}{\delta_j}\), such that \(E(\color{cyan}{y_p}) = \color{orange}{\xi_{t(p)}} + \color{orange}{\delta_{j(p)}}\).

Measurement model: \(\color{cyan}{y_{p}} \sim N(\color{orange}{\xi_{t(p)}} + \color{orange}{\delta_{j(p)}} \, , \, \color{cyan}{V_p})\)
Dynamic model: \(\color{orange}{\xi_t} \sim N(\color{orange}{\xi_{t-1}}, \color{orange}{\omega^2})\), with the endpoint constraints from election results observed on \(\color{orange}{\xi_1}\) and \(\color{orange}{\xi_T}\).
Given published polls, \(\color{cyan}{\boldsymbol{y}}\), sample sizes, field dates and identity of polling companies — and the model — recover

trajectory of latent voting intentions \(\color{orange}{\boldsymbol{\xi}} = (\color{orange}{\xi_1}, \ldots, \color{orange}{\xi_T})'\)
house effects: \(\color{orange}{\boldsymbol{\delta}} = (\color{orange}{\delta_1}, \ldots, \color{orange}{\delta_J})'\)
“pace of change” parameter (innovation variance), \(\color{orange}{\omega^2}\).

Augment model with unknown step/discontinuity \(\color{orange}{\gamma}\) in \(\color{orange}{\boldsymbol{\xi}}\) trajectory, for Turnbull/Morrison turnover on 28 August 2018 (day 788).

for election polling, the truth will out; commerical implications
survey houses have choices about weighting etc, by survey houses, typically after data collection, before publishing results.
better to be wrong with others than wrong on your own, but apparently not for Ipsos.

suppose true level of voting intentions is \(\color{orange}{\pi} \in (0,1)\)
special case of the Central Limit Theorem (de Moivre 1738, Laplace 1820): under unbiased, simple random sampling (SRS), survey based estimates of \(\color{orange}{\pi}\) will be distributed normally around \(\color{orange}{\pi}\) with variance \(V(\color{orange}{\pi}) = \color{orange}{\pi}(1-\color{orange}{\pi})/\color{cyan}{n}\).
put the question of bias to one side (dealt with previously with house effects estimates)
we compare observed dispersion of the polls with theoretically expected dispersion, given (1) stated sample sizes \(\color{cyan}{n}\); (2) assumption about \(\color{orange}{\pi}\)

assume true voting intentions are \(\color{orange}{\pi}\) (e.g., the level observed on Election Day)
assume no change in voting intentions for \(d\) days prior to the election
\(\color{cyan}{\mathcal{D}}\) are polls fielded within \(d\) days of the election, with standard deviation \(\color{cyan}{s_\mathcal{D}}\). Repeat the following over range of values of \(d\).
simulate poll results for each \(\color{cyan}{p} \in \color{cyan}{\mathcal{D}}\), \(\color{red}{y_p} \sim N(\color{orange}{\pi}, \color{cyan}{V_p})\), \(\color{cyan}{V_p} = \color{orange}{\pi} (1 - \color{orange}{\pi})/\color{cyan}{n_p}\). Round \(\color{red}{y_p}\) to the same degree of precision as in reported polls. Let \(\color{red}{s^*}\) be the standard deviation of the \(\color{red}{y_p}\).
Over many simulations how often do we observe \(\color{red}{s^*} > \color{cyan}{s_\mathcal{D}}\)? That is, results of actual polls \(\color{cyan}{\mathcal{D}}\) are underdispersed relative to what we should see under SRS.