Wang-Tsiatis-bounds: 5.0: Wang-Tsiatis Bounds

Description Details Note Author(s) References See Also Examples

Description

gsDesign offers the option of using Wang-Tsiatis bounds as an alternative to the spending function approach to group sequential design. Wang-Tsiatis bounds include both Pocock and O'Brien-Fleming designs. Wang-Tsiatis bounds are currently only available for 1-sided and symmetric 2-sided designs. Wang-Tsiatis bounds are typically used with equally spaced timing between analyses, but the option is available to use them with unequal spacing.

Details

Wang-Tsiatis bounds are defined as follows. Assume k analyses and let Z_i represent the upper bound and t_i the proportion of the total planned sample size for the i-th analysis, i=1,2,…,k. Let Delta be a real-value. Typically Delta will range from 0 (O'Brien-Fleming design) to 0.5 (Pocock design). The upper boundary is defined by

ct_i^{Δ-0.5}

for i= 1,2,…,k where c depends on the other parameters. The parameter Delta is supplied to gsDesign() in the parameter sfupar. For O'Brien-Fleming and Pocock designs there is also a calling sequence that does not require a parameter. See examples.

Note

The manual is not linked to this help file, but is available in library/gsdesign/doc/gsDesignManual.pdf in the directory where R is installed.

Author(s)

Keaven Anderson keaven\[email protected]

References

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

See Also

Spending function overview, gsDesign, gsProbability

Examples

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# Pocock design
gsDesign(test.type=2, sfu="Pocock")

# alternate call to get Pocock design specified using 
# Wang-Tsiatis option and Delta=0.5
gsDesign(test.type=2, sfu="WT", sfupar=0.5)

# this is how this might work with a spending function approach
# Hwang-Shih-DeCani spending function with gamma=1 is often used 
# to approximate Pocock design
gsDesign(test.type=2, sfu=sfHSD, sfupar=1)

# unequal spacing works,  but may not be desirable 
gsDesign(test.type=2, sfu="Pocock", timing=c(.1, .2))

# spending function approximation to Pocock with unequal spacing 
# is quite different from this
gsDesign(test.type=2, sfu=sfHSD, sfupar=1, timing=c(.1, .2))

# One-sided O'Brien-Fleming design
gsDesign(test.type=1, sfu="OF")

# alternate call to get O'Brien-Fleming design specified using 
# Wang-Tsiatis option and Delta=0
gsDesign(test.type=1, sfu="WT", sfupar=0)

Example output

Loading required package: xtable
Loading required package: ggplot2
Symmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Spending computations assume trial stops
if a bound is crossed.

           Sample
            Size 
  Analysis Ratio*  Z   Nominal p  Spend
         1  0.384 2.29     0.011 0.0110
         2  0.767 2.29     0.011 0.0079
         3  1.151 2.29     0.011 0.0060
     Total                       0.0250 

++ alpha spending:
 Pocock boundary.
* 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 Total   E{N}
  0.0000 0.011 0.0079 0.0060 0.025 1.1276
  3.2415 0.389 0.3421 0.1689 0.900 0.7210

Lower boundary (futility or Type II Error)
          Analysis
   Theta     1      2     3 Total
  0.0000 0.011 0.0079 0.006 0.025
  3.2415 0.000 0.0000 0.000 0.000
Symmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Spending computations assume trial stops
if a bound is crossed.

           Sample
            Size 
  Analysis Ratio*  Z   Nominal p  Spend
         1  0.384 2.29     0.011 0.0110
         2  0.767 2.29     0.011 0.0079
         3  1.151 2.29     0.011 0.0060
     Total                       0.0250 

++ alpha spending:
 Wang-Tsiatis boundary with Delta = 0.5.
* 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 Total   E{N}
  0.0000 0.011 0.0079 0.0060 0.025 1.1276
  3.2415 0.389 0.3421 0.1689 0.900 0.7210

Lower boundary (futility or Type II Error)
          Analysis
   Theta     1      2     3 Total
  0.0000 0.011 0.0079 0.006 0.025
  3.2415 0.000 0.0000 0.000 0.000
Symmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Spending computations assume trial stops
if a bound is crossed.

           Sample
            Size 
  Analysis Ratio*  Z   Nominal p  Spend
         1  0.385 2.28    0.0112 0.0112
         2  0.771 2.28    0.0112 0.0080
         3  1.156 2.30    0.0107 0.0058
     Total                       0.0250 

++ alpha spending:
 Hwang-Shih-DeCani spending function with gamma = 1.
* 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 Total   E{N}
  0.0000 0.0112 0.0080 0.0058 0.025 1.1327
  3.2415 0.3933 0.3418 0.1649 0.900 0.7213

Lower boundary (futility or Type II Error)
          Analysis
   Theta      1     2      3 Total
  0.0000 0.0112 0.008 0.0058 0.025
  3.2415 0.0000 0.000 0.0000 0.000
Symmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Spending computations assume trial stops
if a bound is crossed.

           Sample
            Size 
  Analysis Ratio*  Z   Nominal p  Spend
         1  0.122 2.34    0.0096 0.0096
         2  0.244 2.34    0.0096 0.0070
         3  1.221 2.34    0.0096 0.0084
     Total                       0.0250 

++ alpha spending:
 Pocock boundary.
* 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 Total   E{N}
  0.0000 0.0096 0.0070 0.0084 0.025 1.1864
  3.2415 0.1135 0.1479 0.6386 0.900 0.9517

Lower boundary (futility or Type II Error)
          Analysis
   Theta      1     2      3  Total
  0.0000 0.0096 0.007 0.0084 0.0250
  3.2415 0.0003 0.000 0.0000 0.0003
Symmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Spending computations assume trial stops
if a bound is crossed.

           Sample
            Size 
  Analysis Ratio*  Z   Nominal p  Spend
         1  0.106 2.67    0.0038 0.0038
         2  0.213 2.63    0.0043 0.0034
         3  1.064 2.08    0.0189 0.0178
     Total                       0.0250 

++ alpha spending:
 Hwang-Shih-DeCani spending function with gamma = 1.
* 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 Total   E{N}
  0.0000 0.0038 0.0034 0.0178 0.025 1.0507
  3.2415 0.0531 0.0939 0.7530 0.900 0.9328

Lower boundary (futility or Type II Error)
          Analysis
   Theta      1      2      3  Total
  0.0000 0.0038 0.0034 0.0178 0.0250
  3.2415 0.0001 0.0000 0.0000 0.0001
One-sided group sequential design with
90 % power and 2.5 % Type I Error.
           Sample
            Size 
  Analysis Ratio*  Z   Nominal p  Spend
         1  0.339 3.47    0.0003 0.0003
         2  0.677 2.45    0.0071 0.0069
         3  1.016 2.00    0.0225 0.0178
     Total                       0.0250 

++ alpha spending:
 O'Brien-Fleming boundary.
* 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 Total   E{N}
  0.0000 0.0003 0.0069 0.0178 0.025 1.0136
  3.2415 0.0565 0.5288 0.3147 0.900 0.7987
One-sided group sequential design with
90 % power and 2.5 % Type I Error.
           Sample
            Size 
  Analysis Ratio*  Z   Nominal p  Spend
         1  0.339 3.47    0.0003 0.0003
         2  0.677 2.45    0.0071 0.0069
         3  1.016 2.00    0.0225 0.0178
     Total                       0.0250 

++ alpha spending:
 Wang-Tsiatis boundary with Delta = 0.
* 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 Total   E{N}
  0.0000 0.0003 0.0069 0.0178 0.025 1.0136
  3.2415 0.0565 0.5288 0.3147 0.900 0.7987

gsDesign documentation built on May 31, 2017, 2:15 a.m.