---
title: "Binary Outcomes"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Binary Outcomes}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

This vignette shows how to use CMR when the pilot outcome is binary. Binary
outcomes support the exact folded-binomial variance bounds in addition to the
general bounded-outcome bounds. The outcome should be coded as `0` and `1` for
the exact Bernoulli method.

```{r setup}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
library(cmrdesign)
```

## Simulate a binary pilot

```{r simulate-binary-pilot}
set.seed(202)

n_pilot <- 220
d <- rbinom(n_pilot, size = 1, prob = 0.5)
response_prob <- ifelse(d == 1, 0.38, 0.55)
y <- rbinom(n_pilot, size = 1, prob = response_prob)

table(treatment = d, outcome = y)
```

## Exact Bernoulli CMR

For 0/1 outcomes, `method = "auto"` resolves to the exact Bernoulli variance
confidence set.

```{r bernoulli-cmr}
fit_auto <- cmr_binary(y, d, method = "auto", alpha = 0.05)

fit_auto$method
fit_auto$pi
round(fit_auto$rectangle, 4)
fit_auto$pilot$n
```

The exact Bernoulli rectangle is also available explicitly with
`method = "bernoulli"` or `method = "bernoulli_exact"`.

```{r binary-methods}
methods <- c("auto", "bernoulli", "bernoulli_exact", "bounded", "mtr")

comparison <- do.call(rbind, lapply(methods, function(method) {
  fit <- cmr_binary(y, d, method = method, alpha = 0.05)
  c(
    pi = fit$pi,
    U_CMR = fit$U_CMR,
    v_l1 = fit$rectangle[["v_l1"]],
    v_u1 = fit$rectangle[["v_u1"]],
    v_l0 = fit$rectangle[["v_l0"]],
    v_u0 = fit$rectangle[["v_u0"]]
  )
}))
rownames(comparison) <- methods

round(comparison, 4)
```

Use the exact Bernoulli option when the observed outcome is genuinely binary.
Use the bounded-outcome options when the outcome is continuous or an index
already scaled to `[0, 1]`.

As in the other workflows, `$pi` is the main-wave assignment share and
`$U_CMR` is a regret certificate, not a treatment-effect confidence interval.

## Inspect the folded-binomial sufficient statistic

The exact Bernoulli bounds are functions of the folded count in each arm.

```{r folded-counts}
fit_auto$confidence_set$treatment$statistic
fit_auto$confidence_set$control$statistic
```