Wang-Tsiatis-bounds: 5.0: Wang-Tsiatis Bounds
In gsDesign: Group Sequential Design
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\_anderson@merck.
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 2, 2019, 4:49 p.m.
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Note that we can't provide technical support on individual packages. You should contact the package authors for that.
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\_anderson@merck.
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | # 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
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