Stopping rule for toxicity/futility monitoring

Computes a stopping rule and its operating characteristics for toxicity monitoring based repeated significance testing.

Usage

toxbdry(pLo, pHi, n, cP0=0.1, cP1=0.9, ngrid=6, niter=10, delta=0,
          priority=c("null","alt"))
futilbdry(rLo, rHi, n, size=0.1, power=0.9, ngrid=3, niter=10, delta=0.5)
bdrycross.prob(n, r, ptox)
# S3 method for toxbdry
print(x, ...)
# S3 method for futilbdry
print(x, ...)

Arguments

pLo: the toxicity rate that is acceptable.
rLo: baseline (null) response rate.
pHi: the toxicity rate that is too high and hence unacceptable.
rHi: desirable response rate. Stop when it is too unlikely.
n: vector of times (sample size) when toxicty/response is monitored.
r: vector of maximum acceptable toxicities (non-responders for futility) corresponding to n.
ptox: the toxicity rates for which the operating characteristics are calculated. For futility this is the non-response rate.
cP0: boundary crossing probability under pLo i.e. type I error or the probability of declaring a treatment with toxicity rate pLo unacceptable.
cP1: boundary crossing probability under pHi i.e. power or the probability of declaring a treatment with toxicity rate pHi unacceptable.
size: probability of calling drug effective if response rate is rLo.
power: probability of calling drug effective if response rate is rHi.
ngrid: the number of toxicity rates from pLo to pHi for which the operating characteristics are computed.
niter: the number of iterations run to obtain the boundary.
delta: power determining the shape of the boundary. Should be between 0 (default) and 0.5.
priority: the error threshold to prioritize when the max sample size is too small to have both error thresholds satisfied. Default is the null i.e. error under pLo.
x: object returned by the function toxbdry.
...: additional arguments to print.

Value

the function returns a list with:

looks: when toxicty is monitored - same as input n.
lo.bdry: lower boundary is a vector of maximum acceptable number of toxicities corresponding the number of subjects in n. The boundary crossing probability for this is slightly above cP0.
hi.bdry: upper boundary is a vector of maximum acceptable number of toxicities corresponding the number of subjects in n. The boundary crossing probability for this is slightly below cP0.
bdry.oc: the operating characteristics i.e the toxicity rate, the probability of crossing, stopping (i.e. cross before the last observation) and the expected sample size for both the low (lo) and high (hi) boundaries.
bdry.alpha: the alpha levels for testing at each look for the two boundaries.

stopping for toxicity is done when the number of toxicities exceeda the boundary i.e. the boundary gives the maximum acceptable number.

Details

Default value of boundary shape corresponds to the Pocock boundary where the same significance level is used for all looks. For a more conservative stopping rule use delta greater than 0 where 0.5 corresponds to the O'Brien-Fleming boundary which is extremely conservative in the early looks. Value between 0.1 and 0.2 is a reasonable compromise.

The exact calculations in this function are done along the lines of the method in Chapter 12 of Jennison and Turnbull (2000). Ivanova, Qaqish and Schell (2005) have an illustrative paper.

References

Jennison C and Turnbull BW. (2000). Group Sequential Methods with Applications to Clinical Trials. Chapman and Hall/CRC

Ivanova A, Qaqish BF and Schell MJ. (2005). Continuous Toxicity Monitoring in Phase II Trials in Oncology. Biometrics 61, 540-545.

Examples

  toxbdry(0.2, 0.35, c(20,40,60,75))
#> 
#>  Toxicity boundary based on repeated significance testing 
#>     Boundary shape parameter delta = 0 
#> 
#>  ******************************************************************
#>  * Stop if the number of toxicities exceeds (i.e. >) the boundary *
#>  ******************************************************************
#> 
#>   Low boundary:   7/20   12/40   17/60   20/75   
#>  High boundary:   7/20   12/40   17/60   21/75   
#> 
#>  Operating Characteristics: 
#> 
#>      ptox pcross.lo pstop.lo ess.lo pcross.hi pstop.hi ess.hi
#> [1,] 0.20     0.101    0.079   72.0     0.088    0.079   72.0
#> [2,] 0.23     0.247    0.188   68.1     0.214    0.188   68.1
#> [3,] 0.26     0.456    0.354   62.1     0.406    0.354   62.1
#> [4,] 0.29     0.672    0.548   54.7     0.619    0.548   54.7
#> [5,] 0.32     0.838    0.727   47.0     0.797    0.727   47.0
#> [6,] 0.35     0.935    0.859   39.9     0.911    0.859   39.9
  toxbdry(0.2, 0.3, c(20,40,60,75), cP0=0.15, cP1=0.8)
#> Warning: Max sample size 75 may be small for the specified stopping probabilities
#> 
#>  Toxicity boundary based on repeated significance testing 
#>     Boundary shape parameter delta = 0 
#> 
#>  ******************************************************************
#>  * Stop if the number of toxicities exceeds (i.e. >) the boundary *
#>  ******************************************************************
#> 
#>   Low boundary:   6/20   12/40   16/60   20/75   
#>  High boundary:   7/20   12/40   16/60   20/75   
#> 
#>  Operating Characteristics: 
#> 
#>      ptox pcross.lo pstop.lo ess.lo pcross.hi pstop.hi ess.hi
#> [1,] 0.20     0.152    0.140   69.1     0.117    0.103   71.6
#> [2,] 0.22     0.253    0.229   65.7     0.213    0.186   69.0
#> [3,] 0.24     0.381    0.342   61.4     0.341    0.297   65.4
#> [4,] 0.26     0.522    0.469   56.5     0.487    0.427   61.0
#> [5,] 0.28     0.659    0.598   51.2     0.632    0.562   56.1
#> [6,] 0.30     0.777    0.715   45.9     0.758    0.688   51.0
  # continuous monitoring
  toxbdry(0.1, 0.3, 2:30)
#> 
#>  Toxicity boundary based on repeated significance testing 
#>     Boundary shape parameter delta = 0 
#> 
#>  ******************************************************************
#>  * Stop if the number of toxicities exceeds (i.e. >) the boundary *
#>  ******************************************************************
#> 
#>   Low boundary:   1/3   2/8   3/13   4/20   5/26   6/30   
#>  High boundary:   1/3   2/8   3/13   4/19   5/26   6/30   
#> 
#>  Operating Characteristics: 
#> 
#>      ptox pcross.lo pstop.lo ess.lo pcross.hi pstop.hi ess.hi
#> [1,] 0.10     0.100    0.099   28.1     0.097    0.096   28.1
#> [2,] 0.14     0.250    0.245   25.7     0.245    0.239   25.8
#> [3,] 0.18     0.448    0.437   22.4     0.441    0.430   22.5
#> [4,] 0.22     0.644    0.631   18.8     0.638    0.624   19.0
#> [5,] 0.26     0.799    0.786   15.5     0.795    0.781   15.6
#> [6,] 0.30     0.901    0.891   12.6     0.898    0.888   12.7
  # prioritize cP1 error threshold
  toxbdry(0.1, 0.3, 2:25, priority="alt")
#> Warning: Max sample size 25 may be small for the specified stopping probabilities
#> 
#>  Toxicity boundary based on repeated significance testing 
#>     Boundary shape parameter delta = 0 
#> 
#>  ******************************************************************
#>  * Stop if the number of toxicities exceeds (i.e. >) the boundary *
#>  ******************************************************************
#> 
#>   Low boundary:   1/4   2/10   3/16   4/23   5/25   
#>  High boundary:   1/4   2/10   3/16   4/22   5/25   
#> 
#>  Operating Characteristics: 
#> 
#>      ptox pcross.lo pstop.lo ess.lo pcross.hi pstop.hi ess.hi
#> [1,] 0.10     0.146    0.145   22.8     0.140    0.139   22.8
#> [2,] 0.14     0.312    0.310   20.5     0.300    0.296   20.5
#> [3,] 0.18     0.504    0.499   17.8     0.488    0.480   17.8
#> [4,] 0.22     0.679    0.673   15.0     0.663    0.652   15.1
#> [5,] 0.26     0.813    0.807   12.5     0.800    0.790   12.5
#> [6,] 0.30     0.902    0.897   10.3     0.893    0.885   10.3