MaxG87 / clear-or-close

Decide if an election was close in a statistical sense

clear-or-close

This project allows you to check whether an referendum was close in a statistical sense. Currently it only supports referendums with exactly two choices.

The project was born when media coverage claimed that the Brexit referendum was "close". Since I was preparing a course about statistical testing I took the chance to present a Monte Carlo test with recent relevance.

Statistical Model

The performed statistical test is based on MonteCarlo testing. Here, $N$ samples of a suitable model are drawn and checked wheter they are at least as extreme than the referendum in question.

Let $N$ be the number of entitled voters, $0 < M <= N$ be the number of actual voters and blue and red the options to vote for.

The statistical model is as follows. It assumes that every voter is decided on exactly one of the options. Who actually attends the referendum is up to chance. However, there is no correlation between the choice of a voter and whether she will attend the referendum.

Thus, the outcome of the referendum, according to the statistical model sketched above, can be modelled by drawing $M$ balls from an urn with $\frac 12 N$ red and $\frac 12 N$ blue balls. This is a special case of the Hypergeometric Distribution.

A sampled result is considered to be at least as extreme than the result that is tested for significance if one option got at least as many votes than the option with a majority share of votes. Thus, when the actual poll resulted in $(13, 37)$ and the sampling resulted in $(37, 13)$, the sampled result is considered to be at least as extreme.

Sampling from the Statistical Model

The implementation to sample the Hypergeometric Distribution closely mimicks drawing balls from an urn. It maintains a vector counts which tells how many examples for each option remain and samples options with probability proportional to these counts. The sampling repeats this process until $M$ elements were drawn.

The runtime is prohibitive for large values of $M$.

Todo:

• Find clever approximations so sampling gets faster. An $O(k)$ way to sample from a k-dimensional Hypergeometric dimension would be awesome. HINT: https://wikis.hu-berlin.de/mmstat/Approximation_von_Verteilungen#Approximation_durch_die_Normalverteilung_2

• Write tests!

• Add command line parser so no recompilation is required for different values.

• Add support for general elections. Needs a better $H_0$ and a better measure of extremness, though.

• Generate confidence intervals to indicate how reliable the results are.

Result

Usually election results are significant according to the statistical model described above. Even for 2500 eligible voters the Brexit result would be significant.