---
title: "Missing data: reference-based and delta-adjusted imputation"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Missing data: reference-based and delta-adjusted imputation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 7, fig.height = 4)
```

An MMRM assumes the missing responses are missing at random. The two standard ways to test that assumption are
reference-based imputation, which replaces the assumption with a clinically stated one, and delta adjustment, which
asks how much worse the imputed values would have to be before the conclusion changes.

```{r spec}
library(ballast)
dat <- mmrm::fev_data[mmrm::fev_data$USUBJID %in% paste0("PT", 1:100), ]
spec <- stress_spec(dat, "FEV1", "USUBJID", "AVISIT", "ARMCD", "PBO", covariates = ~ FEV1_BL,
                    covariance = "us", method = "Satterthwaite", vcov = "Asymptotic")
stress_fit(spec)[, c("estimate", "stderr", "p", "n_subj")]
```

## Reference-based imputation

`stress_reference()` fits a multivariate-normal imputation model (treatment by visit, the subject-level covariates,
unstructured covariance) and draws the missing values under four assumptions: missing at random, jump to reference,
copy reference, and copy increments in reference. Parameter uncertainty comes from bootstrap refits of the imputation
model, one per imputation. The vignette uses few imputations; use 25 or more in practice.

```{r ref}
ref <- stress_reference(spec, m = 5, seed = 1)
ref[, c("label", "estimate", "stderr", "p", "note")]
```

The package's implementation was checked against rbmi on the antidepressant data that rbmi ships: with the same
imputation model, the same number of bootstrap draws and the same analysis model applied to both engines' completed
data sets, all four assumptions agree within Monte Carlo error. One caution for rbmi users: `rbmi::set_vars()` adds
only the main effects of group and visit, so the treatment-by-visit interaction has to be named in its `covariates`.

## Delta adjustment

`stress_tipping()` imputes by fully conditional specification (the regression method of SAS PROC MI), adds a shift to
the active arm's imputed values, refits and pools. `delta = 0` is the missing-at-random multiple-imputation analysis
and is counted as an alternative; the other deltas are searches. `tipping_point()` interpolates the crossing and
`plot_pcurve()` draws the curve.

```{r tip}
tp <- stress_tipping(spec, deltas = c(0, -5, -10, -15), m = 5, seed = 1)
tp[, c("label", "estimate", "p", "kind")]
tipping_point(tp)
plot_pcurve(tp, xlab = "Shift applied to imputed active-arm values")
```

## Intercurrent events

When an intercurrent event is the reason for the missing data, `stress_estimand()` derives the analysis data sets
implied by each handling strategy (treatment policy, hypothetical, while on treatment, composite) and refits each. A
hypothetical strategy sets the post-event responses to missing; the reference-based family can then be run on that
derived data set with `spec_update(spec, data = ...)`.