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

This vignette shows the basic two-arm CMR workflow with a simulated bounded
pilot outcome. The pilot outcome is assumed to have already been rescaled to
the unit interval. The goal is to choose the main-wave assignment share, not to
estimate a treatment effect.

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

## Simulate a pilot

The treatment arm has a slightly higher mean and lower variance in this
simulation. CMR uses the pilot only to choose the main-wave treatment
assignment probability.

```{r simulate-pilot}
set.seed(101)

n_pilot <- 160
d <- rbinom(n_pilot, size = 1, prob = 0.5)
y <- numeric(n_pilot)
y[d == 1] <- rbeta(sum(d == 1), shape1 = 5, shape2 = 3)
y[d == 0] <- rbeta(sum(d == 0), shape1 = 2.2, shape2 = 2.2)

pilot_summary <- aggregate(y, by = list(treatment = d), FUN = function(x) {
  c(mean = mean(x), variance = var(x), n = length(x))
})
pilot_summary
```

## Compute the CMR assignment

The default bounded-outcome confidence rectangle uses the
Maurer-Pontil-style empirical variance bounds for outcomes in `[0, 1]`.

```{r bounded-cmr}
fit_bounded <- cmr_two_arm(y, d, method = "bounded", alpha = 0.05)

fit_bounded$pi
round(fit_bounded$rectangle, 4)
fit_bounded$U_CMR
fit_bounded$binding
```

Here `$pi` is the main-wave probability assigned to treatment. The rectangle
contains lower and upper confidence bounds for the treatment and control
variances. The CMR certificate `$U_CMR` is the worst-case regret over the two
least favorable corners of that rectangle; it is not a treatment-effect
confidence interval.

```{r inspect-pilot-fields}
fit_bounded$pilot$n
round(fit_bounded$pilot$vhat, 4)
fit_bounded$joint_error_bound
```

## Compare confidence-set methods

The package exposes the bounded/MP and MTR variance confidence sets through the
same interface. The MTR option implements the Martinez-Taboada-Ramdas one-sided
variance bounds.

```{r compare-methods}
methods <- c("bounded", "mp", "mtr")

comparison <- do.call(rbind, lapply(methods, function(method) {
  fit <- cmr_two_arm(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)
```

`"bounded"` and `"mp"` are synonyms. In applications, use the method that
matches the confidence set you want to report, and record the method in the
analysis plan.

## Work from a precomputed rectangle

If the rectangle was computed elsewhere, the CMR rule can be applied directly.

```{r from-rectangle}
manual_fit <- cmr_two_arm_from_rectangle(fit_bounded$rectangle)

manual_fit$pi
manual_fit$U_CMR
```