Ratios in causal inference

deans_example.R

library("ggplot2")
library("dplyr")

# Population size
n <- 2500

# Sessions per user if assigned to test
sessions_test <- as.integer(exp(rnorm(n, 0.5, 1)))

# Sessions per user if assigned to control
sessions_control <- as.integer(exp(rnorm(n, 0.0, 1)))

# Has sessions variables are transformations of these.
has_hessions_test <- as.integer(sessions_test > 0)
has_hessions_control <- as.integer(sessions_control > 0)

# Given assignment, estimate the estimand in a naive way
analyze <- function(
    sessions_test,
    sessions_control,
    assignments
) {
    ratio_test <- (
        (
            sum(sessions_test * assignments)
        )
        /
        (
            sum(sessions_test * assignments > 0)
        )
    )
    ratio_control <- (
        (
            sum(sessions_control * (1 - assignments))
        )
        /
        (
            sum(sessions_control * (1 - assignments) > 0)
        )
    )
    estimator <- ratio_test / ratio_control
    return(estimator)
}

monte_carlo <- function(
    sessions_test,
    sessions_control,
    n_sims=10000L
) {
    estimators <- rep(NA, n_sims)
    for (s in 1:n_sims) {
        assignments <- rbinom(n, 1, 0.5)
        estimators[s] <- analyze(
            sessions_test,
            sessions_control,
            assignments
        )
    }
    return(estimators)
}

estimand <- (
    (
        sum(sessions_test) / sum(has_hessions_test)
    )
    /
    (
        sum(sessions_control) / sum(has_hessions_control)
    )
)

estimators <- monte_carlo(sessions_test, sessions_control)

hist(estimators - estimand)

mean(estimators)

estimand

> mean(estimators)
[1] 1.350879
>
> estimand
[1] 1.350425