Group sequential design using weighted log-rank test under non-proportional hazards

Usage

gs_design_wlr(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9)),
  fail_rate = tibble(stratum = "All", duration = c(3, 100), fail_rate = log(2)/c(9, 18),
    hr = c(0.9, 0.6), dropout_rate = rep(0.001, 2)),
  weight = "logrank",
  approx = "asymptotic",
  alpha = 0.025,
  beta = 0.1,
  ratio = 1,
  info_frac = NULL,
  info_scale = c("h0_h1_info", "h0_info", "h1_info"),
  analysis_time = 36,
  binding = FALSE,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = alpha),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = beta),
  test_upper = TRUE,
  test_lower = TRUE,
  h1_spending = TRUE,
  r = 18,
  tol = 1e-06,
  interval = c(0.01, 1000)
)

Arguments

enroll_rate

Enrollment rates defined by define_enroll_rate().

fail_rate

Failure and dropout rates defined by define_fail_rate().

weight

Weight of weighted log rank test:

"logrank" = regular logrank test.
list(method = "fh", param = list(rho = ..., gamma = ...)) = Fleming-Harrington weighting functions.
list(method = "mb", param = list(tau = ..., w_max = ...)) = Magirr and Burman weighting functions.

approx

Approximate estimation method for Z statistics.

"event_driven" = only work under proportional hazard model with log rank test.
"asymptotic".

alpha

One-sided Type I error.

beta

Type II error.

ratio

Experimental:Control randomization ratio.

info_frac

Targeted information fraction for analyses. See details.

info_scale

Information scale for calculation. Options are:

"h0_h1_info" (default): variance under both null and alternative hypotheses is used.
"h0_info": variance under null hypothesis is used.
"h1_info": variance under alternative hypothesis is used.

analysis_time

Targeted calendar timing of analyses. See details.

binding

Indicator of whether futility bound is binding; default of FALSE is recommended.

upper

Function to compute upper bound.

gs_spending_bound(): alpha-spending efficacy bounds.
gs_b(): fixed efficacy bounds.

upar

Parameters passed to upper.

If upper = gs_b, then upar is a numerical vector specifying the fixed efficacy bounds per analysis.
If upper = gs_spending_bound, then upar is a list including
- sf for the spending function family.
- total_spend for total alpha spend.
- param for the parameter of the spending function.
- timing specifies spending time if different from information-based spending; see details.

lower

Function to compute lower bound, which can be set up similarly as upper. See this vignette.

lpar

Parameters passed to lower, which can be set up similarly as upar.

test_upper

Indicator of which analyses should include an upper (efficacy) bound; single value of TRUE (default) indicates all analyses; otherwise, a logical vector of the same length as info should indicate which analyses will have an efficacy bound.

test_lower

Indicator of which analyses should include an lower bound; single value of TRUE (default) indicates all analyses; single value FALSE indicated no lower bound; otherwise, a logical vector of the same length as info should indicate which analyses will have a lower bound.

h1_spending

Indicator that lower bound to be set by spending under alternate hypothesis (input fail_rate) if spending is used for lower bound. If this is FALSE, then the lower bound spending is under the null hypothesis. This is for two-sided symmetric or asymmetric testing under the null hypothesis; See this vignette.

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.

tol

Tolerance parameter for boundary convergence (on Z-scale); normally not changed by the user.

interval

An interval presumed to include the times at which expected event count is equal to targeted event. Normally, this can be ignored by the user as it is set to c(.01, 1000).

Value

A list with input parameters, enrollment rate, analysis, and bound.

Specification

The contents of this section are shown in PDF user manual only.

Examples

library(mvtnorm)
library(gsDesign)
library(gsDesign2)

# set enrollment rates
enroll_rate <- define_enroll_rate(duration = 12, rate = 1)

# set failure rates
fail_rate <- define_fail_rate(
  duration = c(4, 100),
  fail_rate = log(2) / 15, # median survival 15 month
  hr = c(1, .6),
  dropout_rate = 0.001
)

# Example 1 ----
# Information fraction driven design
gs_design_wlr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  ratio = 1,
  alpha = 0.025, beta = 0.2,
  weight = list(method = "mb", param = list(tau = Inf, w_max = 2)),
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.2),
  analysis_time = 36,
  info_frac = c(0.6, 1)
)
#> $design
#> [1] "wlr"
#> 
#> $enroll_rate
#> # A tibble: 1 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All           12  22.3
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            4    0.0462        0.001   1  
#> 2 All          100    0.0462        0.001   0.6
#> 
#> $bounds
#> # A tibble: 4 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper      0.648       0.00381  2.67          0.629     0.00381
#> 2        1 lower      0.0976      0.959    1.74          0.738     0.0406 
#> 3        2 upper      0.800       0.0141   1.98          0.742     0.0238 
#> 4        2 lower      0.197       0.985    1.91          0.750     0.0280 
#> 
#> $analysis
#> # A tibble: 2 × 10
#>   analysis  time     n event   ahr theta  info info0 info_frac info_frac0
#>      <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>     <dbl>      <dbl>
#> 1        1  24.1  267.  132. 0.690 0.371  66.3  67.8     0.610      0.600
#> 2        2  36    267.  177. 0.657 0.259 109.  113.      1          1    
#> 

# Example 2 ----
# Calendar time driven design
gs_design_wlr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  ratio = 1,
  alpha = 0.025, beta = 0.2,
  weight = list(method = "mb", param = list(tau = Inf, w_max = 2)),
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.2),
  analysis_time = c(24, 36),
  info_frac = NULL
)
#> $design
#> [1] "wlr"
#> 
#> $enroll_rate
#> # A tibble: 1 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All           12  25.0
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            4    0.0462        0.001   1  
#> 2 All          100    0.0462        0.001   0.6
#> 
#> $bounds
#> # A tibble: 4 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper      0.326       0.00366 2.68           0.643     0.00366
#> 2        1 lower      0.0977      0.821   0.918          0.860     0.179  
#> 3        2 upper      0.800       0.0228  1.98           0.755     0.0238 
#> 4        2 lower      0.198       0.975   1.93           0.760     0.0267 
#> 
#> $analysis
#> # A tibble: 2 × 10
#>   analysis  time     n event   ahr theta  info info0 info_frac info_frac0
#>      <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>     <dbl>      <dbl>
#> 1        1    24  300.  148. 0.690 0.256  73.9  75.5     0.605      0.595
#> 2        2    36  300.  199. 0.657 0.259 122.  127.      1          1    
#> 

# Example 3 ----
# Both calendar time and information fraction driven design
gs_design_wlr(
  enroll_rate = enroll_rate,
  fail_rate = fail_rate,
  ratio = 1,
  alpha = 0.025, beta = 0.2,
  weight = list(method = "mb", param = list(tau = Inf, w_max = 2)),
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.2),
  analysis_time = c(24, 36),
  info_frac = c(0.6, 1)
)
#> $design
#> [1] "wlr"
#> 
#> $enroll_rate
#> # A tibble: 1 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All           12  22.3
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            4    0.0462        0.001   1  
#> 2 All          100    0.0462        0.001   0.6
#> 
#> $bounds
#> # A tibble: 4 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper      0.648       0.00381  2.67          0.629     0.00381
#> 2        1 lower      0.0976      0.959    1.74          0.738     0.0406 
#> 3        2 upper      0.800       0.0141   1.98          0.742     0.0238 
#> 4        2 lower      0.197       0.985    1.91          0.750     0.0280 
#> 
#> $analysis
#> # A tibble: 2 × 10
#>   analysis  time     n event   ahr theta  info info0 info_frac info_frac0
#>      <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>     <dbl>      <dbl>
#> 1        1  24.1  267.  132. 0.690 0.371  66.3  67.8     0.610      0.600
#> 2        2  36    267.  177. 0.657 0.259 109.  113.      1          1    
#>