Conditional Power at Interim Boundaries

gsBoundCP() computes the total probability of crossing future upper bounds given an interim test statistic at an interim bound. For each interim boundary, assumes an interim test statistic at the boundary and computes the probability of crossing any of the later upper boundaries.

See Conditional power section of manual for further clarification. See also Muller and Schaffer (2001) for background theory.

Usage

gsBoundCP(x, theta = "thetahat", r = 18)

Arguments

x: An object of type gsDesign or gsProbability
theta: if "thetahat" and class(x)!="gsDesign", conditional power computations for each boundary value are computed using estimated treatment effect assuming a test statistic at that boundary (zi/sqrt(x$n.I[i]) at analysis i, interim test statistic zi and interim sample size/statistical information of x$n.I[i]). Otherwise, conditional power is computed assuming the input scalar value theta.
r: Integer value controlling grid for numerical integration as in Jennison and Turnbull (2000); default is 18, range is 1 to 80. Larger values provide larger number of grid points and greater accuracy. Normally r will not be changed by the user.

Value

A list containing two vectors, CPlo and CPhi.

CPlo: A vector of length x$k-1 with conditional powers of crossing upper bounds given interim test statistics at each lower bound
CPhi: A vector of length x$k-1 with conditional powers of crossing upper bounds given interim test statistics at each upper bound.

Note

The gsDesign technical manual is available at https://keaven.github.io/gsd-tech-manual/.

References

Jennison C and Turnbull BW (2000), Group Sequential Methods with Applications to Clinical Trials. Boca Raton: Chapman and Hall.

Muller, Hans-Helge and Schaffer, Helmut (2001), Adaptive group sequential designs for clinical trials: combining the advantages of adaptive and classical group sequential approaches. Biometrics;57:886-891.

Author

Keaven Anderson keaven_anderson@merck.com

Examples


# set up a group sequential design
x <- gsDesign(k = 5)
x
#> Asymmetric two-sided group sequential design with
#> 90 % power and 2.5 % Type I Error.
#> Upper bound spending computations assume
#> trial continues if lower bound is crossed.
#> 
#>            Sample
#>             Size    ----Lower bounds----  ----Upper bounds-----
#>   Analysis Ratio*   Z   Nominal p Spend+  Z   Nominal p Spend++
#>          1  0.220 -0.90    0.1836 0.0077 3.25    0.0006  0.0006
#>          2  0.441 -0.04    0.4853 0.0115 2.99    0.0014  0.0013
#>          3  0.661  0.69    0.7563 0.0171 2.69    0.0036  0.0028
#>          4  0.881  1.36    0.9131 0.0256 2.37    0.0088  0.0063
#>          5  1.101  2.03    0.9786 0.0381 2.03    0.0214  0.0140
#>      Total                        0.1000                 0.0250 
#> + lower bound beta spending (under H1):
#>  Hwang-Shih-DeCani spending function with gamma = -2.
#> ++ alpha spending:
#>  Hwang-Shih-DeCani spending function with gamma = -4.
#> * Sample size ratio compared to fixed design with no interim
#> 
#> Boundary crossing probabilities and expected sample size
#> assume any cross stops the trial
#> 
#> Upper boundary (power or Type I Error)
#>           Analysis
#>    Theta      1      2      3      4      5  Total   E{N}
#>   0.0000 0.0006 0.0013 0.0028 0.0062 0.0117 0.0226 0.5726
#>   3.2415 0.0417 0.1679 0.2806 0.2654 0.1444 0.9000 0.7440
#> 
#> Lower boundary (futility or Type II Error)
#>           Analysis
#>    Theta      1      2      3      4      5  Total
#>   0.0000 0.1836 0.3201 0.2700 0.1477 0.0559 0.9774
#>   3.2415 0.0077 0.0115 0.0171 0.0256 0.0381 0.1000

# compute conditional power based on interim treatment effects
gsBoundCP(x)
#>              CPlo      CPhi
#> [1,] 2.294534e-06 1.0000001
#> [2,] 2.238566e-03 0.9998352
#> [3,] 2.669114e-02 0.9922459
#> [4,] 1.296705e-01 0.9200502

# compute conditional power based on original x$delta
gsBoundCP(x, theta = x$delta)
#>           CPlo      CPhi
#> [1,] 0.4936972 0.9940265
#> [2,] 0.3676577 0.9954019
#> [3,] 0.3331896 0.9912361
#> [4,] 0.3871332 0.9590607