Group sequential design for negative binomial outcomes

Creates a group sequential design for negative binomial outcomes based on sample size calculations from sample_size_nbinom().

Usage

gsNBCalendar(
  x,
  k = 3,
  test.type = 4,
  alpha = 0.025,
  beta = 0.1,
  astar = 0,
  delta = 0,
  sfu = gsDesign::sfHSD,
  sfupar = -4,
  sfl = gsDesign::sfHSD,
  sflpar = -2,
  sfharm = gsDesign::sfHSD,
  sfharmparam = -2,
  testUpper = TRUE,
  testLower = TRUE,
  testHarm = TRUE,
  tol = 1e-06,
  r = 18,
  usTime = NULL,
  lsTime = NULL,
  analysis_times = NULL
)

Arguments

x

An object of class sample_size_nbinom_result from sample_size_nbinom().

k

Number of analyses (interim + final). Default is 3.

test.type

Test type as in gsDesign::gsDesign():

1: One-sided
2: Two-sided symmetric
3: Two-sided, asymmetric, binding futility bound, beta-spending
4: Two-sided, asymmetric, non-binding futility bound, beta-spending
5: Two-sided, asymmetric, binding futility bound, lower spending
6: Two-sided, asymmetric, non-binding futility bound, lower spending
7: Two-sided, asymmetric, binding futility and binding harm bounds
8: Two-sided, asymmetric, non-binding futility and non-binding harm bounds

Default is 4.

alpha

Type I error (one-sided). Default is 0.025.

beta

Type II error (1 - power). Default is 0.1.

astar

For test.type 5 or 6, allocated Type I error for the lower bound. For test.type 7 or 8, total probability of crossing the harm bound under the null. Default is 0 (set to 1 - alpha internally when needed).

delta

Standardized effect size. Default is 0 (computed from design).

sfu

Spending function for upper bound. Default is gsDesign::sfHSD.

sfupar

Parameter for upper spending function. Default is -4.

sfl

Spending function for lower bound. Default is gsDesign::sfHSD.

sflpar

Parameter for lower spending function. Default is -2.

sfharm

Spending function for the harm bound (test.type 7 or 8). Default is gsDesign::sfHSD.

sfharmparam

Parameter for the harm spending function. Default is -2.

testUpper

Logical scalar or vector of length k specifying which analyses include an upper (efficacy) bound. TRUE (default) means all analyses. Where FALSE, the upper bound is set to +20 (effectively Inf) and displayed as NA. Must be TRUE at the final analysis.

testLower

Logical scalar or vector of length k specifying which analyses include a lower (futility) bound. TRUE (default) means all analyses. Where FALSE, the lower bound is set to -20 (effectively -Inf) and displayed as NA. Ignored for test.type 1.

testHarm

Logical scalar or vector of length k specifying which analyses include a harm bound (test.type 7 or 8 only). TRUE (default) means all analyses. Where FALSE, the harm bound is set to -20 and displayed as NA.

tol

Tolerance for convergence. Default is 1e-06.

r

Integer controlling grid size for numerical integration. Default is 18.

usTime

Spending time for upper bound (optional).

lsTime

Spending time for lower bound (optional).

analysis_times

Vector of calendar times for each analysis. Must have length k. These times are stored in the T element and displayed by gsDesign::gsBoundSummary().

Value

An object of class gsNB which inherits from gsDesign and sample_size_nbinom_result. While the final sample size would be planned total enrollment, interim analysis sample sizes are the expected number enrolled at the times specified in analysis_times. Output value contains all elements from gsDesign::gsDesign() plus:

nb_design: The original sample_size_nbinom_result object
n1: A vector with sample size per analysis for group 1
n2: A vector with sample size per analysis for group 2
n_total: A vector with total sample size per analysis
events: A vector with expected total events per analysis
events1: A vector with expected events per analysis for group 1
events2: A vector with expected events per analysis for group 2
exposure: A vector with expected average calendar exposure per analysis
exposure_at_risk1: A vector with expected at-risk exposure per analysis for group 1
exposure_at_risk2: A vector with expected at-risk exposure per analysis for group 2
variance: A vector with variance of log rate ratio per analysis
T: Calendar time at each analysis (if analysis_times provided)
testUpper: Logical vector indicating which analyses have an efficacy bound
testLower: Logical vector indicating which analyses have a futility bound
testHarm: Logical vector indicating which analyses have a harm bound (test.type 7 or 8 only)

Note that n.I in the returned object represents the statistical information at each analysis.

References

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

Examples

# First create a sample size calculation
nb_ss <- sample_size_nbinom(
  lambda1 = 0.5, lambda2 = 0.3, dispersion = 0.1, power = 0.9,
  accrual_rate = 10, accrual_duration = 20, trial_duration = 24
)

# Then create a group sequential design with analysis times
gs_design <- gsNBCalendar(nb_ss,
  k = 3, test.type = 4,
  analysis_times = c(10, 18, 24)
)

# Selective bound testing: futility only at IA1, efficacy deferred to IA2+
gs_selective <- gsNBCalendar(nb_ss,
  k = 3, test.type = 4,
  analysis_times = c(10, 18, 24),
  testUpper = c(FALSE, TRUE, TRUE),
  testLower = c(TRUE, TRUE, FALSE)
)
gs_selective
#> 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.
#> 
#>                 ----Lower bounds----  ----Upper bounds-----
#>   Analysis N    Z   Nominal p Spend+  Z   Nominal p Spend++
#>          1 11 -0.24    0.4057 0.0148 3.01    0.0013  0.0013
#>          2 29  0.94    0.8267 0.0289 2.55    0.0054  0.0049
#>          3 44  2.00    0.9772 0.0563 2.00    0.0228  0.0188
#>      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.
#> 
#> 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  Total E{N}
#>   0.0000 0.0013 0.0049 0.0171 0.0233 25.4
#>   0.5084 0.1412 0.4403 0.3185 0.9000 32.2
#> 
#> Lower boundary (futility or Type II Error)
#>           Analysis
#>    Theta      1      2      3  Total
#>   0.0000 0.4057 0.4290 0.1420 0.9767
#>   0.5084 0.0148 0.0289 0.0563 0.1000