Simulation check to show halving p-values for chi-squared test _could_ be a valid statistical procedure

sim_chisq_halving.R

library(ggplot2)

N_treatment <- 709
N_control <- 699
N_sims <- 1e4
prob_event <- 0.1

# Assuming prob(event|treatment) = prob(event|control) 
# The case prob(event|treatment) > prob(event|control) produces fewer false positives (type-I errors)
pos_control <- rbinom(N_sims, size = N_control, prob = prob_event)
pos_treatment <- rbinom(N_sims, size = N_treatment, prob = prob_event)

p_values <- array(NA_real_, N_sims)
false_positives <- data.frame(alpha = seq(0.01, 0.1, by = 0.01), 
                              fp_direct = 0, fp_halved = 0)
for(i in 1:N_sims) {
  cases_table <- matrix(c(pos_control[i], N_control - pos_control[i], pos_treatment[i], N_treatment - pos_treatment[i]), ncol = 2)
  if(pos_treatment[i] >= pos_control[i]) {
    # Effect in the null direction, don't reject, no false positives
    next
  } else {
    p_values[i] <- chisq.test(cases_table, correct = FALSE)$p.value
    reject_direct <- p_values[i] < false_positives$alpha
    reject_halved <- p_values[i] / 2 < false_positives$alpha
    false_positives$fp_direct <- false_positives$fp_direct + reject_direct
    false_positives$fp_halved <- false_positives$fp_halved + reject_halved
  }
}

ggplot(false_positives, aes(x = alpha, y = fp_direct/N_sims)) + geom_point() + geom_abline(color = "blue")
ggplot(false_positives, aes(x = alpha, y = fp_halved/N_sims)) + geom_point() + geom_abline(color = "blue")