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Causal Inference Course

Discussion 4. Interference and Stat Review

STSCI/INFO/ILRST 3900: Causal Inference

September 17, 2025

Interference

You can download the slides for this week’s discussion.

Stat review reminder

  • \(\pi^a= P(Y^a=1)\)) under treatment condition \(a\)
  • Let \(\hat\pi^a\) be the estimate of that unknown probability
  • \(\hat\pi^a = \frac{1}{n_a}\sum_{i:A_i=a} Y_i^a\), proportion of people in that treatment condition to have outcome 1
  • To make a confidence interval on \(\hat\pi^a\) we need the standard error of \(\hat\pi^a\)

Standard error

  • Let \(Y^a\) be a Bernoulli random variable
  • The variance of \(Y^a\) is \(V(Y^a) = \pi^a (1-\pi^a)\)
  • We have estimated by an average: \(\hat\pi^a\)
  • If we did this many times in many hypothetical samples, we would get different estimate
  • The estimates sampling variance is \(V(\hat\pi^a) = \frac{\pi^a(1-\pi^a)}{n_a}\)
  • The standard error is the square root of the sampling variance: \[SE(\hat\pi^a) = \sqrt\frac{\pi^a(1-\pi^a)}{n_a}\]
  • We can estimate this SE by plugging in our estimate \(\hat\pi^a\)
  • In R: we translated the standard error formula into code
se_binary <- function(p, n) {
  se <- sqrt( p * (1 - p) / n )
  return(se)
}

Sampling distribution

  • \(\hat\pi^a\) is a sample mean
  • By the Central Limit Theorem: as \(n \to \infty\), across hypothetical repeated samples the distribution of \(\hat\pi^a\) estimates becomes Normal
  • Across repeated samples, the middle 95% of estimates will fall within a known range: \[\pi^a \pm \Phi^{-1}(.975) \times SE(\hat\pi^a)\]
  • where \(\Phi^{-1}()\) is the quantile of the standard Normal distribution
  • \(\Phi^{-1}(.975) \approx 1.96\), so the number 1.96 might be familiar to you
  • Plug in the estimates \(\hat\pi^a\) and \(\widehat{SE}(\hat\pi^a)\) to get a 95% confidence interval
  • The CI is centered on the estimate \(\hat\pi^a\)
  • If we repeatedly made a CI using hypothetical samples from the population, the CI would contain the unknown true parameter \(\pi^a\) 95% of the time
  • In R: we translated the confidence interval formula into code
ci_lower <- function(estimate, standard_error) {
  estimate - qnorm(.975) * standard_error
}
ci_upper <- function(estimate, standard_error) {
  estimate + qnorm(.975) * standard_error
}