2009 Iran Election

On June 12 2009, the Republic of Iran held an election where President Mahmoud Ahmadinejad sought re-election against three challengers. When it was announced that Ahmadinejad had won with 62% of the vote, there were widespread allegations of election fraud. There are many methods, both quantitative and qualitative, to detect election fraud. In this lab we will explore just one proposed method.

Benford’s Law

Benford’s law observes that in many naturally-occuring phenomena with a quantitative measurement, some patterns can be observed in the digits of that measurement. Specifically, the law states that the first digit of such observations, rather than being distributed uniformly between the numbers 1-9, takes a decreasing log distribution. The code and plotted distribution for this are below. Investments with interest, populations of cities, and election results have all been observed by different groups to follow Benford’s law.

benfords_p <- data.frame(1:9, log10(1 + 1/1:9))

One purported method of detecting election fraud, then, is seeing if the first digit of the results for each reporting region fits the distribution specified by Benford’s law. Try this out with the Iran data.

Getting Started

Load packages

For this lab we will read in the data using the readr package, explore it using the dplyr package and visualize it using the ggplot2 package for data visualization. The data can be found in the oilabs package.

library(readr)
library(dplyr)
library(ggplot2)
library(oilabs)

Data

The election data from Iran contains 366 observations of 9 variables, where every row is a city. The first two variables identify the province and city the vote totals come from, the next four give vote totals for the four candidates (Mahmoud Ahmadinejad, Mohsen Rezaee, Mehdi Karroubi, and Mir-Hossein Mousavi), and the final three give total and procedural vote totals.

data(iran)

Elections

Iran, 2009

  1. Add a column to the iran dataset called first_digit that contains the first digit of the votes cast for the winner, Mahmoud Ahmadinejad.

Hint: The get_first() function in the oilabs package will be useful here. If you’re wondering how it works under the hood, try running the following code line at the command line and inspecting the output.

substr("A long time ago, in a galaxy far far away...", 21, 28)
substr(387, 1, 2)
as.numeric(substr(387, 1, 2))
get_first(387)
  1. Create a bar plot like the one above, showing the frequencies of first digits in the total vote count. Does this plot appear to match the ideal Benford’s distribution? Where does it deviate?

This result might be due to election fraud, but it might just be a result of a small sample size or other innocent factors. Try a similar method on data collected from the 2016 US presidential election.

United States, 2016

The OpenElections project is obtaining and standardizing precinct-level results from the 2016 US presidential election, among other US elections. To access the data, search OpenElections’ GitHub page and pick an election that is marked “CSV” or “baked raw”.

  1. Pick a state and use this data to create a plot of that state’s first digit distribution by precinct. Use the number of votes cast for Hillary Clinton in each precinct. Make sure you remove rows for “total” counts if your state has them.

  2. Does the election you chose appear to fit the distribution better or worse than the Iran election?

Election Assessment

We’ve used plots to assess the difference between the observed distribution and the distribution expected by Benford’s Law, but it can be useful to quantify that difference with a statistic. We can start by building a dataframe that directly compares the proportions for each of the digits.

  1. Using the dplyr techniques from the last lab, reformulate the iran data into the following dataframe and call it iran_props.
## # A tibble: 9 x 3
##   first_digit   obs_prop   ben_prop
##         <dbl>      <dbl>      <dbl>
## 1           1 0.25956284 0.30103000
## 2           2 0.23224044 0.17609126
## 3           3 0.14754098 0.12493874
## 4           4 0.08196721 0.09691001
## 5           5 0.08743169 0.07918125
## 6           6 0.06284153 0.06694679
## 7           7 0.04644809 0.05799195
## 8           8 0.05464481 0.05115252
## 9           9 0.02732240 0.04575749

  1. Formulate your own statistic to measure the distance between the observed proportions (obs_prop) and those expected by Benford’s Law (ben_prop). This exercise is similar to the first quiz, where you were asked to propose your own statistic to measure variability; there are many many possible choices, but some are more useful than others. Describe this statistic in words (or write out the formula for it if you are comfortable using LaTeX), then calculate it for this data.

  2. Repeat exerises 5 and 6 for the US state data. Which distribution is judged to be farther from the ideal Benford’s distribution using your distance statistic? What do you conclude about the presence of fraud in these elections?