des_gs: Design a group sequential single-arm trial for a single...

In mjg211/singlearm: Design and analysis of single-arm clinical trials

Description

Determines group sequential single-arm clinical trial designs for a single binary primary endpoint. In particular, this allows Simon's two-stage designs (Simon, 1989) to be identified.

Usage

des_gs(J = 2, pi0 = 0.1, pi1 = 0.3, alpha = 0.05, beta = 0.2,
  Nmin = 1, Nmax = 30, futility = T, efficacy = F,
  optimality = "null_ess", point_prior, beta_prior, equal_n = F,
  ensign = F, summary = F)

Arguments

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.
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.

Details

des_gs() supports the determination of a variety of (optimised) 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 > π ₀,

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_Nj ≤ a_j, then stop the trial and do not reject H₀.

  • Else if s_Nj ≥ r_j, then stop the trial and reject H₀.

  • Else if a_j < s_Nj < r_j , then continute to stage j + 1.

• For j = J

  • If s_Nj ≤ a_j, then do not reject H₀.

  • Else if s_Nj ≥ 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, and the chosen optimality criteria.

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.

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 when J ≤ 3, 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. In contrast, if J > 3, simulated annealing is employed to stochastically search for the optimal design, as proposed by Chen and Lee (2013).

Value

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.

References

Chen K, Shan M (2008) Optimal and minimax three-stage designs for phase II oncology trials. Contemporary Clinical Trials 29:32-41.

Chen N, Lee JJ (2013) Optimal continuous-monitoring design of single-arm phase II trial based on the simulated annealing method Contemporary Clinical Trials 35:170-8.

Chen TT (1997) Optimal three-stage designs for phase II cancer clinical trials. Statistics in Medicine 16:2701-11.

Ensign LG et al. (1994) An optimal three-stage design for phase II clinical trials. Statistics in Medicine 13:1727-36.

Hanfelt JJ et al. (1999) A modification of Simon's optimal design for phase II trials when the criterion is median sample size. Controlled Clinical Trials 20:555-66.

Mander AP, Thompson SG (2010) Two-stage designs optimal under the alternative hypothesis for phase II cancer clinical trials. Contemporary Clinical Trials 31:572-8.

Simon R (1989) Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials 10:1-10.

Shuster J (2002) Optimal two-stage designs for single arm phase II cancer trials. Journal of Biopharmaceutical Statistics 12: 39-51.

See Also

opchar_gs, est_gs, pval_gs, ci_gs, and their associated plot family of functions. Note that similar functionality is available through ph2simon and getSolutions.

Examples

# The minimax design for the default parameters
minimax   <- des_gs(optimality = "minimax")
# The corresponding design minimising the expected
# sample size under the null hypothesis
null_ess  <- des_gs(optimality = "null_ess")
# The corresponding 3-stage minimax design
minimax_3 <- des_gs(J = 3)

mjg211/singlearm documentation built on May 8, 2021, 3:17 a.m.