Reproduce Results From Fujikawa et al. (2020)

Lukas Baumann

2024-04-09

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.

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.

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.

## 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:

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.

## 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.