Calculate sample size and bounds given targeted power and Type I error in group sequential design using average hazard ratio under non-proportional hazards

Usage

gs_design_ahr(
  enroll_rate = define_enroll_rate(duration = c(2, 2, 10), rate = c(3, 6, 9)),
  fail_rate = define_fail_rate(duration = c(3, 100), fail_rate = log(2)/c(9, 18), hr =
    c(0.9, 0.6), dropout_rate = 0.001),
  alpha = 0.025,
  beta = 0.1,
  info_frac = NULL,
  analysis_time = 36,
  ratio = 1,
  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),
  h1_spending = TRUE,
  test_upper = TRUE,
  test_lower = TRUE,
  info_scale = c("h0_h1_info", "h0_info", "h1_info"),
  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().

alpha

One-sided Type I error.

beta

Type II error.

info_frac

Targeted information fraction for analyses. See details.

analysis_time

Targeted calendar timing of analyses. See details.

ratio

Experimental:Control randomization ratio.

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.

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.

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.

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.

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.

The $input is a list including alpha, beta, ratio, etc.
The $enroll_rate is a table showing the enrollment for arriving the targeted power (1 - beta).
The $fail_rate is a table showing the failure and dropout rates, which is the same as input.
The $bound is a table summarizing the efficacy and futility bound per analysis.
The analysis is a table summarizing the analysis time, sample size, events, average HR, treatment effect and statistical information per analysis.

Details

The parameters info_frac and analysis_time are used to determine the timing for interim and final analyses.

If the interim analysis is determined by targeted information fraction and the study duration is known, then info_frac is a numerical vector where each element (greater than 0 and less than or equal to 1) represents the information fraction for each analysis. The analysis_time, which defaults to 36, indicates the time for the final analysis.
If interim analyses are determined solely by the targeted calendar analysis timing from start of study, then analysis_time will be a vector specifying the time for each analysis.
If both the targeted analysis time and the targeted information fraction are utilized for a given analysis, then timing will be the maximum of the two with both info_frac and analysis_time provided as vectors.

Specification

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

Examples

library(gsDesign)
#> 
#> Attaching package: ‘gsDesign’
#> The following objects are masked from ‘package:gsDesign2’:
#> 
#>     as_gt, as_rtf
library(gsDesign2)

# Example 1 ----
# call with defaults
gs_design_ahr()
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  13.2
#> 2 All            2  26.4
#> 3 All           10  39.7
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 1 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <dbl> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper         0.9        0.025  1.96          0.795      0.0250
#> 
#> $analysis
#> # A tibble: 1 × 10
#>   analysis  time     n event   ahr theta  info info0 info_frac info_frac0
#>      <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>     <dbl>      <dbl>
#> 1        1    36  476.  292. 0.683 0.381  71.7  73.0         1          1
#> 

# Example 2 ----
# Single analysis
gs_design_ahr(analysis_time = 40)
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  11.9
#> 2 All            2  23.8
#> 3 All           10  35.6
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 1 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <dbl> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper         0.9        0.025  1.96          0.791      0.0250
#> 
#> $analysis
#> # A tibble: 1 × 10
#>   analysis  time     n event   ahr theta  info info0 info_frac info_frac0
#>      <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>     <dbl>      <dbl>
#> 1        1    40  428.  280. 0.678 0.389  68.8  69.9         1          1
#> 

# Example 3 ----
# Multiple analysis_time
gs_design_ahr(analysis_time = c(12, 24, 36))
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  14.5
#> 2 All            2  29.1
#> 3 All           10  43.6
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 6 × 7
#>   analysis bound probability probability0      z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>  <dbl>          <dbl>       <dbl>
#> 1        1 upper     0.00248    0.0000538  3.87           0.459   0.0000538
#> 2        1 lower     0.00321    0.0443    -1.70           1.41    0.956    
#> 3        2 upper     0.579      0.00921    2.36           0.736   0.00919  
#> 4        2 lower     0.0556     0.830      0.953          0.884   0.170    
#> 5        3 upper     0.900      0.0244     2.01           0.799   0.0222   
#> 6        3 lower     0.100      0.976      2.01           0.799   0.0223   
#> 
#> $analysis
#> # A tibble: 3 × 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    12  436.  98.8 0.811 0.210  24.4  24.7     0.309      0.308
#> 2        2    24  523. 238.  0.715 0.335  58.1  59.4     0.738      0.741
#> 3        3    36  523. 321.  0.683 0.381  78.8  80.2     1          1    
#> 

# Example 4 ----
# Specified information fraction
# \donttest{
gs_design_ahr(info_frac = c(.25, .75, 1), analysis_time = 36)
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  14.6
#> 2 All            2  29.1
#> 3 All           10  43.7
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 6 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper    0.000295   0.00000737  4.33          0.380  0.00000737
#> 2        1 lower    0.00108    0.0135     -2.21          1.64   0.987     
#> 3        2 upper    0.599      0.00965     2.34          0.740  0.00965   
#> 4        2 lower    0.0570     0.843       1.01          0.878  0.157     
#> 5        3 upper    0.900      0.0244      2.01          0.799  0.0221    
#> 6        3 lower    0.100      0.976       2.01          0.799  0.0221    
#> 
#> $analysis
#> # A tibble: 3 × 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  10.7  382.  80.4 0.823 0.195  19.8  20.1     0.251      0.250
#> 2        2  24.4  524. 241.  0.714 0.337  59.0  60.3     0.747      0.750
#> 3        3  36    524. 322.  0.683 0.381  79.0  80.4     1          1    
#> 
# }

# Example 5 ----
# multiple analysis times & info_frac
# driven by times
gs_design_ahr(info_frac = c(.25, .75, 1), analysis_time = c(12, 25, 36))
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  14.6
#> 2 All            2  29.3
#> 3 All           10  43.9
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 6 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper     0.00251    0.0000538  3.87          0.460   0.0000538
#> 2        1 lower     0.00321    0.0446    -1.70          1.41    0.955    
#> 3        2 upper     0.635      0.0105     2.31          0.746   0.0104   
#> 4        2 lower     0.0599     0.862      1.09          0.871   0.138    
#> 5        3 upper     0.900      0.0243     2.02          0.799   0.0219   
#> 6        3 lower     0.100      0.976      2.01          0.799   0.0220   
#> 
#> $analysis
#> # A tibble: 3 × 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    12  439.  99.5 0.811 0.210  24.5  24.9     0.309      0.308
#> 2        2    25  527. 248.  0.711 0.341  60.5  61.9     0.763      0.766
#> 3        3    36  527. 323.  0.683 0.381  79.3  80.7     1          1    
#> 
# driven by info_frac
# \donttest{
gs_design_ahr(info_frac = c(1 / 3, .8, 1), analysis_time = c(12, 25, 36))
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  14.7
#> 2 All            2  29.5
#> 3 All           10  44.2
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 6 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper     0.00510     0.000104  3.71          0.490    0.000104
#> 2        1 lower     0.00459     0.0665   -1.50          1.33     0.934   
#> 3        2 upper     0.701       0.0122    2.25          0.756    0.0122  
#> 4        2 lower     0.0655      0.896     1.26          0.856    0.104   
#> 5        3 upper     0.900       0.0241    2.03          0.799    0.0214  
#> 6        3 lower     0.100       0.976     2.02          0.799    0.0216  
#> 
#> $analysis
#> # A tibble: 3 × 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  12.5  465.  108. 0.806 0.216  26.7  27.1     0.334      0.333
#> 2        2  26.4  530.  260. 0.706 0.348  63.7  65.1     0.797      0.800
#> 3        3  36    530.  325. 0.683 0.381  79.9  81.3     1          1    
#> 
# }

# Example 6 ----
# 2-sided symmetric design with O'Brien-Fleming spending
# \donttest{
gs_design_ahr(
  analysis_time = c(12, 24, 36),
  binding = TRUE,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  h1_spending = FALSE
)
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  13.7
#> 2 All            2  27.5
#> 3 All           10  41.2
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 6 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper 0.00226        0.0000538  3.87          0.449   0.0000538
#> 2        1 lower 0.000000613    0.0000538 -3.87          2.23    1.000    
#> 3        2 upper 0.550          0.00921    2.36          0.730   0.00919  
#> 4        2 lower 0.00000125     0.00921   -2.36          1.37    0.991    
#> 5        3 upper 0.900          0.0250     2.01          0.794   0.0222   
#> 6        3 lower 0.00000128     0.0250    -2.01          1.26    0.978    
#> 
#> $analysis
#> # A tibble: 3 × 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    12  412.  93.4 0.811 0.210  23.0  23.3     0.309      0.308
#> 2        2    24  494. 224.  0.715 0.335  54.9  56.1     0.738      0.741
#> 3        3    36  494. 303.  0.683 0.381  74.4  75.8     1          1    
#> 
# }
# 2-sided asymmetric design with O'Brien-Fleming upper spending
# Pocock lower spending under H1 (NPH)
# \donttest{
gs_design_ahr(
  analysis_time = c(12, 24, 36),
  binding = TRUE,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDPocock, total_spend = 0.1, param = NULL, timing = NULL),
  h1_spending = TRUE
)
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  16.5
#> 2 All            2  32.9
#> 3 All           10  49.4
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 6 × 7
#>   analysis bound probability probability0      z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>  <dbl>          <dbl>       <dbl>
#> 1        1 upper     0.00305    0.0000538  3.87           0.481   0.0000538
#> 2        1 lower     0.0430     0.268     -0.619          1.12    0.732    
#> 3        2 upper     0.638      0.00921    2.36           0.750   0.00920  
#> 4        2 lower     0.0823     0.874      1.13           0.871   0.129    
#> 5        3 upper     0.900      0.0250     1.98           0.813   0.0240   
#> 6        3 lower     0.100      0.975      1.97           0.813   0.0243   
#> 
#> $analysis
#> # A tibble: 3 × 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    12  494.  112. 0.811 0.210  27.6  28.0     0.309      0.308
#> 2        2    24  593.  269. 0.715 0.335  65.9  67.3     0.738      0.741
#> 3        3    36  593.  364. 0.683 0.381  89.3  90.9     1          1    
#> 
# }

# Example 7 ----
# \donttest{
gs_design_ahr(
  alpha = 0.0125,
  analysis_time = c(12, 24, 36),
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.0125, param = NULL, timing = NULL),
  lower = gs_b,
  lpar = rep(-Inf, 3)
)
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  16.1
#> 2 All            2  32.2
#> 3 All           10  48.3
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 3 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper    0.000619   0.00000679  4.35          0.435  0.00000679
#> 2        2 upper    0.505      0.00371     2.68          0.719  0.00371   
#> 3        3 upper    0.900      0.0125      2.28          0.785  0.0114    
#> 
#> $analysis
#> # A tibble: 3 × 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    12  483.  109. 0.811 0.210  27.0  27.4     0.309      0.308
#> 2        2    24  579.  263. 0.715 0.335  64.3  65.8     0.738      0.741
#> 3        3    36  579.  355. 0.683 0.381  87.2  88.8     1          1    
#> 

gs_design_ahr(
  alpha = 0.0125,
  analysis_time = c(12, 24, 36),
  upper = gs_b,
  upar = gsDesign::gsDesign(
    k = 3, test.type = 1, n.I = c(.25, .75, 1),
    sfu = sfLDOF, sfupar = NULL, alpha = 0.0125
  )$upper$bound,
  lower = gs_b,
  lpar = rep(-Inf, 3)
)
#> $design
#> [1] "ahr"
#> 
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  16.1
#> 2 All            2  32.2
#> 3 All           10  48.3
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770        0.001   0.9
#> 2 All          100    0.0385        0.001   0.6
#> 
#> $bound
#> # A tibble: 3 × 7
#>   analysis bound probability probability0     z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl> <dbl>          <dbl>       <dbl>
#> 1        1 upper   0.0000938  0.000000587  4.86          0.395 0.000000587
#> 2        2 upper   0.513      0.00393      2.66          0.721 0.00393    
#> 3        3 upper   0.900      0.0125       2.28          0.785 0.0113     
#> 
#> $analysis
#> # A tibble: 3 × 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    12  483.  110. 0.811 0.210  27.0  27.4     0.309      0.308
#> 2        2    24  580.  263. 0.715 0.335  64.4  65.9     0.738      0.741
#> 3        3    36  580.  356. 0.683 0.381  87.3  88.9     1          1    
#> 
# }