Getting started

1. Get into groups of 3-4.
   
2. You need one (and only one) laptop, organise yourselves so the laptop can be seen be everyone, and designate a driver.
   

3. Start by reviewing your answers to the questions:

   • What feature of Figure 1 do the authors argue is evidence for election fraud?

   • For a particular year, say 2018, how do the authors measure the statistical significance of the feature in Figure 1? What statistic do they calculate? What do they compare that statistic to?

Class Discussion Review answers and narrow focus to voter turnout in 2018.

Get the project

1. Access the class workspace on RStudio cloud: http://bit.ly/st541-f18.
2. Open russia-elections project and save a copy.
3. Knit 01-data-import.Rmd.

• What is reported by the polling stations?
• How is the turnout calculated from the available data?
• Compare the figures, does this data seem to match the paper?

• What is reported by the polling stations?

• How is the turnout calculated from the available data?

• Compare the figures, does this data seem to match the paper?

Find the number of integer turnouts

Open 02-explore-analysis.Rmd

Task 1: How many polling stations reported an integer turnout in 2018?

Try with a small example first. Imagine x contains these 4 turnouts:

(x <- c(85, 85.2, 100*(((0.8 - 0.2) * 2000))/2000, 100*1562/1838))
## [1] 85.00000 85.20000 60.00000 84.98368

What should the result be? How can you calculate it with code?

Why is this hard?

Monte Carlo Approach

Find the null distribution of the test statistic by:

• simulating many datasets under the null hypothesis, and
• calculating the test statistic on each.

If observed test statistic is far from those simulated, we have evidence against the null hypothesis.

Simulate a turnout

Task 2: How would you simulate the turnout percentage for a polling station in the absence of fraud?

What distribution would you use? What would you use for its parameter values?

Simulate a turnout for the first polling station:

# A tibble: 1 x 14
  turnout_percent leader_percent region       tik      uik voters leader
            <dbl>          <dbl> <chr>        <chr>  <int>  <int>  <int>
1            96.1           91.1 Республика … 1 Ады…     1   2256   1977
# ... with 7 more variables: ballots_inside <int>,
#   ballots_outside <int>, ballots_early <int>, invalid <int>,
#   valid <int>, given <int>, received <int>

The authors approach

Let:

• \(V_i\) be the number or people on the voter list
• \(G_i\) be the number of ballots given

To simulate the turnout percentage, \(T^\text{MC}_i\), first simulate the number of ballots given (i.e. the number of people who voted): \[ G_i^{\text{MC}} \sim \text{Binomial}(n = V_i, p = G_i/V_i) \]

Then convert to a turnout percentage \[ T^\text{MC}_i = G_i^\text{MC}/V_i \cdot 100\% \]

In words: simulate the number of voters as Binomial based on the number of people on the voter list and the observed turnout proportion.

Turnout % for first station:

100*(rbinom(n = 1, size = 2256, prob = 2169/2256)/2256)

## [1] 96.76418

Turnout % for all stations

with(ok_stations, 
  100*(rbinom(n = region, size = voters, prob = given/voters)/voters)
)

Simulate turnouts for all polling stations

Find a count of integer stations based the simulated turnout at all stations.

A larger simulation

In 03_simulation.Rmd if you are curious.

What would an alternative approach be?

Find sampling distribution of test statistic analytically

Very large sum of random variables (integer or not indicator).

Some version of Central Limit Theorem (but not the usual one, why?).

Need to work out expectations and variances for random variables.

Main themes of course:

• Computational alternatives to analytical techniques: specific algorithms commonly encountered in statistics.
• Programming for statistical computation: code that is correct, clear, and if necessary, fast.
• Best practices for computational projects: organization and workflows for projects so they are easy to reproduce, share and collaborate on.

Lab

First lab tomorrow 3pm-3:50pm in Bexell Hall 324

Important to attend: You will learn workflow for getting and completing homeworks on github.

Before you get there:

• Get a github account
• Get an RStudio cloud account

Both are free. More details as first step in Lab 01.

Next week

Pre-lecture reading (links on class website):

• Section 3.1 Pseudorandom Number Generation in Simulation (1 page)
• Section 4.1 The Inverse Transform Method in Simulation up to and including Example 4a (~ 2 pages)
• Section 5.1 The Inverse Transform Algorithm in Simulation up to and including Example 5b (~ 2 pages)

Be prepared to answer:

• Why are random numbers generated by a computer called pseudo-random numbers?
• How would you use a Uniform(0, 1) random variable to simulate a Bernoulli( \(\pi\) ) random variable?
• How would you use a Uniform(0, 1) random variable to simulate a Exponential( \(\lambda\) ) random variable?