des_curtailed: Design a curtailed group sequential single-arm trial for a...
In mjg211/singlearm: Design and analysis of single-arm clinical trials

Description Usage Arguments Details Value See Also Examples

Determines curtailed group sequential single-arm clinical trial designs for a single binary primary endpoint.

des_curtailed(J = 2, pi0 = 0.1, pi1 = 0.3, alpha = 0.05,
  beta = 0.2, thetaF = rep(0, J), thetaE = rep(1, J), Nmin = 1,
  Nmax = 50, futility = T, efficacy = F, optimality = "null_ess",
  point_prior, beta_prior, equal_n = F, ensign = F, summary = F)

`J`	The maximal number of stages to allow.
`pi0`	The (undesirable) response probability used in the definition of the null hypothesis.
`pi1`	The (desirable) response probability at which the trial is powered.
`alpha`	The desired maximal type-I error-rate.
`beta`	The desired maximal type-II error-rate.
`thetaF`	The vector of futility curtailment probabilities.
`thetaE`	The vector of efficacy curtailment probabilities.
`Nmin`	The minimal total sample size to allow in considered designs.
`Nmax`	The maximal total sample size to allow in considered designs.
`futility`	A logical variable indicating whether early stopping for futility should be allowed.
`efficacy`	A logical variable indicating whether early stopping for efficacy should be allowed.
`optimality`	Choice of optimal design criteria. Must be one of `"null_ess"`, `"alt_ess"`, `"null_med"`, `"alt_med"`, `"minimax"` or `"prior"`.
`point_prior`	Value of the response probability to minimise the expected sample size at. Only (potentially) required if `optimality == "prior"`.
`beta_prior`	Shape parameters of the beta distribution to optimise the expected sample over. Only (potentially) required if `optimality == "prior"`.
`equal_n`	A logical variable indicating that the sample size of each stage should be equal.
`ensign`	A logical variable indicating that the design of Ensign et al. (1994) should be mimicked, and the first stage futility boundary forced to be 0.
`summary`	A logical variable indicating a summary of the function's progress should be printed to the console.

des_curtailed() supports the determination of a variety of (optimised) curtailed group sequential single-arm clinical trial designs for a single binary primary endpoint. For all supported designs, the following hypotheses are tested for the response probability π

H₀ : π ≤ π ₀, H₁ : π > π₀,

for π₀, specified using the argument pi0.

In each instance, the optimal design is required to meet the following operating characteristics

P(π₀) ≤ α, P(π₁) ≥ 1 - β,

where P(π) is the probability of rejecting H₀ when the true response probability is π, and the values of α and β are specified using the arguments alpha and beta respectively. Moreover, π₁, satisfying π₀ < π₁, is specified using the argument pi1.

A group sequential single-arm design for a single binary endpoint, with a maximum of J allowed stages (specifying J through the argument J) is then indexed by three vectors: a = (a₁,…,a_J), r = (r₁,…,r_J), and n = (n₁,…,n_J).

With these vectors, and denoting the number of responses after m patients have been observed by s_m, the stopping rules for the trial are then as follows

For j = 1,…,J - 1
- If s_{N_j} ≤ a_j, then stop the trial and do not reject H₀.
- Else if s_{N_j} ≥ r_j, then stop the trial and reject H₀.
- Else if a_j < s_{N_j} < r_j, then continute to stage j + 1.
For j = J
- If s_{N_j} ≤ a_j, then do not reject H₀.
- Else if s_{N_j} ≥ r_j, then reject H₀.

Here, N_j = n₁ + &ctdot; + n_j.

The purpose of this function is then to optimise a , r , and n , accounting for the chosen restrictions placed on these vectors, the chosen optimality criteria, and the chosen curtailment rule.

The arguments Nmin, Nmax, and equal_n allow restrictions to be placed on n. Precisely, Nmin and Nmax set an inclusive range of allowed values for N_J. While, if set to TRUE, equal_n enforces n₁ = &ctdot; = n_J.

The arguments futility, efficacy, and ensign allow restrictions to be placed on a and r. If futility is set to FALSE, early stopping for futility (to not reject H₀) is prevented by enforcing a₁ = &ctdot; = a_{J - 1} = -∞. Similarly, if efficacy is set to FALSE, early stopping for efficacy (to reject H₀) is prevented by enforcing r₁ = &ctdot; = r_{J - 1} = ∞. Finally, if set to TRUE, ensign enforces the restriction that a₁ = 0, as suggested in Ensign et al (1994) for 3-stage designs.

Note that to ensure a decision is made about H ₀, this function enforces a _J + 1 = r_J.

In addition, two vectors θ_F ) and θ_E ), each of length J, must be specified which determine the chosen curtailment rule.

To describe the supported optimality criteria, denote the expected sample size and median required sample size when the true response probability is π by ESS(π) and Med(π) respectively. Then, the following optimality criteria are currently supported:

"minimax": The design which minimises N_J.
"null_ess": The design which minimises ESS(π₀).
"alt_ess": The design which minimises ESS(π₁).
"null_med": The design which minimises Med(π₀).
"alt_med": The design which minimises Med(π₁).
"prior": Either the design which minimises ESS(π) for the value of π specified using "point_prior". Or, the design which minimises ∫ESS(π)Beta(π, a,b)dπ over [0,1], where Beta(π,x, y) is the PDF of a beta distribution with shape parameters x and y, specified through "beta_prior" as a vector, evaluated at point π.

Note that the optimal design is determined by an exhaustive search. This means that vast speed improvements can be made by carefully choosing the values of Nmin and Nmax.

A list of class "sa_des_gs" containing the following elements

A list in the slot $des containing details of the identified optimal design.
A tibble in the slot $feasible, consisting of the identified designs which met the required operating characteristics.
Each of the input variables as specified.

opchar_curtailed, and their associated plot family of functions.