---
title: "Asynchronous Control to A to A+B stepped-wedge designs"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Asynchronous Control to A to A+B stepped-wedge designs}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
set.seed(1)
```

```{r setup}
library(stepwedgepower)
```

## Design question

Some implementation studies add interventions in stages. A cluster first moves
from control to intervention A and later receives B in addition to A:

\[
\text{Control} \longrightarrow A \longrightarrow A+B.
\]

The sequences need not add B in the same calendar period. This distinction is
important: the design is not a conventional three-arm trial, and there is no
B-only condition. The B coefficient therefore represents the **incremental
add-on effect of B among clusters already receiving A**.

## An asynchronous schedule

The example below has eight sequences. A and B begin at different times, and
the duration of A-only exposure also varies across sequences.

```{r}
three_state <- sw_multistage_design(
  clusters_per_sequence = rep(6, 8),
  a_start = c(2, 3, 4, 5, 6, 7, 8, 9),
  b_start = c(6, 8, 7, 10, 9, 11, 12, 12),
  n_periods = 12,
  sequence_names = paste0("S", 1:8)
)
three_state
```

The varying A-to-B lags help distinguish the B effect from a treatment-A effect
that simply matures with exposure time. The schedule can also be entered as an
explicit matrix containing `0` (Control), `1` (A), and `2` (A+B).

## Delayed effects

```{r}
assumptions <- sw_multistage_assumptions(
  baseline_prob = 0.15,
  treatment_or_a = 1.40,
  incremental_or_b = 1.30,
  delay_a = 1,
  delay_b = 1,
  icc = 0.05,
  period_effects = log(seq(1.00, 1.12, length.out = 12)),
  n_per_cluster_period = 30
)
assumptions
```

A delay of one makes the first active period a wash-in period. The effect begins
in the second period after the corresponding intervention starts. On the logit
scale, the generating model is

\[
\operatorname{logit}(p_{ijt}) = \beta_{0,jt}+\gamma_t+b_i+
\beta_A A^*_{ijt}+\beta_B B^*_{ijt}.
\]

The estimands are:

* `exp(beta_A)`: A versus Control;
* `exp(beta_B)`: A+B versus A, the incremental B effect;
* `exp(beta_A + beta_B)`: A+B versus Control, the total effect.

## Audit before simulation

```{r}
audit_multistage_design(
  three_state,
  delay_a = assumptions$delay_a,
  delay_b = assumptions$delay_b
)
```

The audit checks fixed-effect rank and identifies whether calendar periods
contain concurrent Control/A and A/A+B comparisons. It warns when all sequences
add B in one calendar period or after the same duration of A exposure.

## Simulate and analyze one trial

```{r}
trial <- simulate_multistage_swcrt(three_state, assumptions, seed = 123)
head(trial)
```

The active rollout state and the delayed-effect indicators are both retained.
This makes the assumptions auditable rather than hiding wash-in periods inside
a single treatment code.

```{r, eval = FALSE}
fit <- fit_multistage_model(trial, multiplicity = "holm")
fit$contrasts
```

The default mixed model contains separate A and B indicators, categorical
calendar time, and a cluster random intercept. The total A+B contrast uses the
estimated covariance between the A and B coefficients.

## Power and failure-aware performance

```{r, eval = FALSE}
power_three <- power_multistage_swcrt(
  three_state,
  assumptions,
  nsim = 5000,
  multiplicity = "holm",
  n_cores = 4,
  seed = 2026
)
summary(power_three)
plot(power_three, type = "failure-aware")
```

The output distinguishes conditional power among usable fits from
failure-aware power, which counts a failed or non-estimable analysis as an
unsuccessful trial. It also reports Monte Carlo intervals, type-specific
contrasts, bias, standard-error calibration, coverage, convergence, and
singular fits.

## Compare Control to A with Control to A to A+B

Construct a two-state schedule with the same A rollout, clusters, periods, and
sample sizes, but no B transition.

```{r}
two_state <- sw_multistage_design(
  clusters_per_sequence = three_state$clusters_per_sequence,
  a_start = three_state$a_start,
  b_start = rep(Inf, three_state$n_sequences),
  n_periods = three_state$n_periods,
  sequence_names = three_state$sequence_names
)
```

```{r, eval = FALSE}
comparison <- compare_multistage_designs(
  design_two = two_state,
  design_three = three_state,
  assumptions_two = assumptions,
  nsim = 5000,
  multiplicity = "holm",
  n_cores = 4,
  seed = 2026
)
comparison
```

The A-versus-Control rows can be compared directly under equal resources. The
three-state design additionally reports incremental-B power, total A+B power,
and the probability that both multiplicity-adjusted component objectives
succeed.

## Check type I error first

```{r, eval = FALSE}
type1 <- type1_multistage_swcrt(
  three_state,
  assumptions,
  nsim = 5000,
  multiplicity = "holm",
  n_cores = 4,
  seed = 22026
)
type1
```

Power should be interpreted only after confirming that the intended tests have
acceptable null rejection rates for the proposed number of clusters and ICC.