---
title: "Reproduce Results From Fujikawa et al. (2020)"
author: "Lukas Baumann"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Reproduce Results From Fujikawa et al. (2020)}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

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

In this vignette it is shown how the results from Fujikawa et al. (2020) can be reproduced using `baskexact`. At first, we have to set up a a design object, using either `setupOneStageBasket()` for a single-stage design or `setupTwoStageBasket()`for a two-stage design with one interim analysis.

```{r setup, eval=FALSE}
library(baskexact)
design1 <- setupOneStageBasket(k = 3, shape1 = 1, shape2 = 1, p0 = 0.2)
```

To get the results shown in Figure 1, `basket_test()` can be used, which computes posterior probabilities for a given results-vector and also shows the pairwise weights and the parameters of the Beta posterior distribution.

```{r, eval=FALSE}
basket_test(design1, n = 15, r = c(1, 5, 7), lambda = 0.99, 
  weight_fun = weights_fujikawa, weight_params = list(epsilon = 2, tau = 0.5,
    logbase = exp(1)))

# $weights
#          Basket 1  Basket 2  Basket 3
# Basket 1        1 0.0000000 0.0000000
# Basket 2        0 1.0000000 0.7832585
# Basket 3        0 0.7832585 1.0000000
# 
# $post_dist_noborrow
#        Basket 1 Basket 2 Basket 3
# shape1        2        6        8
# shape2       15       11        9
# 
# $post_dist_borrow
#        Basket 1 Basket 2 Basket 3
# shape1        2 12.26607 12.69955
# shape2       15 18.04933 17.61584
# 
# $post_prob_noborrow
#  Basket 1  Basket 2  Basket 3 
# 0.1407375 0.9183121 0.9929964 
# 
# $post_prob_borrow
#  Basket 1  Basket 2  Basket 3 
# 0.1407375 0.9942795 0.9965258 
```

Note that at the moment it's not possible to reproduce the results from Table 1, as `baskexact` currently doesn't support baskets with unequal sample sizes.

To reproduce the results from Table 2 we can use `toer()` and `pow()`. First, the results from the single-stage design with two different choices of tuning parameter values. Fujikawa et al.'s "Proposed design (i)" uses $\varepsilon = 2$ and $\tau = 0$, "Proposed design (ii)" uses $\varepsilon = 2$ and $\tau = 0.5$.

Note that the default value for the logbase parameter is 2, such that the weights are always bounded between 0 and 1, but Fujikawa et al. use the natural logarithm, which leads to a lower limit for the weights that is strictly greater than 0.

```{r, eval=FALSE}
## p = (0.2, 0.2, 0.2)
# Proposed design (i)
toer(
  design = design1, n = 24, lambda = 0.99,
  weight_fun = weights_fujikawa, weight_params = list(epsilon = 2, tau = 0,
  logbase = exp(1)), results = "group"
)

# $rejection_probabilities
# [1] 0.02158174 0.02158174 0.02158174
# 
# $fwer
# [1] 0.03600149

# Proposed design (ii)
toer(
  design = design1, n = 24, lambda = 0.99,
  weight_fun = weights_fujikawa, weight_params = list(epsilon = 2, tau = 0.5,
  logbase = exp(1)), results = "group"
)

# $rejection_probabilities
# [1] 0.03239555 0.03239555 0.03239555
# 
# $fwer
# [1] 0.06315308
```

For the results from the two-stage design, we have to set up a different design object at first:

```{r, eval=FALSE}
design2 <- setupTwoStageBasket(k = 3, shape1 = 1, shape2 = 1, p0 = 0.2)
```

Fujikawa et al. use an interim analysis that allows stopping of individual baskets based on the posterior predictive probability.

```{r, eval = FALSE}
## p = (0.2, 0.2, 0.2)
# Proposed design (i)
toer(
  design = design2, n = 24, n1 = 15, lambda = 0.99, 
  interim_fun = interim_postpred, interim_params = list(prob_futstop = 0.1,
    prob_effstop = 0.9), weight_fun = weights_fujikawa,
  weight_params = list(logbase = exp(1), tau = 0, epsilon = 2),
  results = "group"
)

# $rejection_probabilities
# [1] 0.01703198 0.01703198 0.01703198
# 
# $fwer
# [1] 0.03722851

ess(
  design = design2, n = 24, n1 = 15, lambda = 0.99, 
  interim_fun = interim_postpred, interim_params = list(prob_futstop = 0.1,
    prob_effstop = 0.9), weight_fun = weights_fujikawa,
  weight_params = list(logbase = exp(1), tau = 0, epsilon = 2)
)

# [1] 16.06847 16.06847 16.06847

# Proposed design (ii)
toer(
  design = design2, n = 24, n1 = 15, lambda = 0.99, 
  interim_fun = interim_postpred, interim_params = list(prob_futstop = 0.1,
    prob_effstop = 0.9), weight_fun = weights_fujikawa,
  weight_params = list(logbase = exp(1), tau = 0.5, epsilon = 2),
  results = "group"
)

# $rejection_probabilities
# [1] 0.02175429 0.02175429 0.02175429
# 
# $fwer
# [1] 0.04955128

ess(
  design = design2, n = 24, n1 = 15, lambda = 0.99, 
  interim_fun = interim_postpred, interim_params = list(prob_futstop = 0.1,
    prob_effstop = 0.9), weight_fun = weights_fujikawa,
  weight_params = list(logbase = exp(1), tau = 0.5, epsilon = 2)
)

# [1] 16.22526 16.22526 16.22526
```

To reproduce the rest of Table 2, `p1` has to be changed accordingly.
Note that the results are slightly different, as Fujikawa et al.'s results are based on simulation with $n_{\text{sim}} = 5000$. `baskexact` calculates the results analytically.

## References
Fujikawa, K., Teramukai, S., Yokota, I., & Daimon, T. (2020). A Bayesian basket trial design that borrows information across strata based on the similarity between the posterior distributions of the response probability. Biometrical Journal, 62(2), 330-338.