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

This vignette shows the proxy-outcome extension. The pilot observes a short-run
proxy, and the researcher supplies a bridge constant `zeta` that limits how far
the primary-outcome standard deviation can be from the proxy standard deviation
within each arm. The bridge constant is part of the design assumption, not a
quantity estimated by `cmrdesign`.

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

## Simulate proxy and primary outcomes

In a real delayed-outcome design, the pilot may only observe the proxy in time
for the main-wave assignment decision. Here the primary outcome is simulated
only so that we can compare the proxy design to an oracle calculation.

```{r simulate-proxy-pilot}
set.seed(404)

n_pilot <- 200
d <- rbinom(n_pilot, size = 1, prob = 0.5)

primary <- numeric(n_pilot)
primary[d == 1] <- rbeta(sum(d == 1), shape1 = 6, shape2 = 3)
primary[d == 0] <- rbeta(sum(d == 0), shape1 = 3.5, shape2 = 3.5)

proxy_noise <- ifelse(d == 1, 0.06, 0.08) * rnorm(n_pilot)
proxy_y <- pmin(1, pmax(0, primary + proxy_noise))

round(rbind(
  proxy_treatment = c(mean = mean(proxy_y[d == 1]), variance = var(proxy_y[d == 1])),
  proxy_control = c(mean = mean(proxy_y[d == 0]), variance = var(proxy_y[d == 0]))
), 4)
```

## Choose a bridge constant

The bridge constant is an assumption, not something estimated by the package.
It can be scalar or arm-specific. Arm-specific values can be named with
`"treatment"` and `"control"`.

```{r bridge}
zeta <- c(treatment = 0.04, control = 0.06)
```

## Proxy CMR

```{r proxy-cmr}
fit_proxy <- cmr_proxy(
  proxy_y = proxy_y,
  d = d,
  zeta = zeta,
  method = "bounded",
  alpha = 0.05
)

fit_proxy$pi
round(fit_proxy$rectangle, 4)
fit_proxy$zeta
fit_proxy$diagnostics$bridge
```

The reported rectangle is the widened primary-outcome rectangle. The proxy
rectangle remains available separately.

```{r proxy-vs-primary-rectangle}
round(fit_proxy$confidence_set$bridge$proxy_rectangle, 4)
round(fit_proxy$confidence_set$bridge$primary_rectangle, 4)
```

## Simulated oracle comparison

The following calculation uses the simulated primary outcome directly. This is
not available at the decision time in a delayed-outcome application; it is only
a diagnostic for this simulated example.

```{r oracle-comparison}
fit_oracle <- cmr_two_arm(primary, d, method = "bounded")

comparison <- rbind(
  proxy = c(pi = fit_proxy$pi, U_CMR = fit_proxy$U_CMR),
  oracle_primary = c(pi = fit_oracle$pi, U_CMR = fit_oracle$U_CMR)
)

round(comparison, 4)
```