gsProbability: 2.2: Boundary Crossing Probabilities
In gsDesign: Group Sequential Design
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
Description
Computes power/Type I error and expected sample size for a group sequential design
across a selected set of parameter values for a given set of analyses and boundaries.
The print function has been extended using print.gsProbability to print gsProbability objects; see examples.
Usage
1
2
3
gsProbability(k=0, theta, n.I, a, b, r=18, d=NULL, overrun=0)
## S3 method for class 'gsProbability'
print(x,...)
Arguments
k
Number of analyses planned, including interim and final.
theta
Vector of standardized effect sizes for which boundary crossing probabilities are to be computed.
n.I
Sample size or relative sample size at analyses; vector of length k. See gsDesign and manual.
a
Lower bound cutoffs (z-values) for futility or harm at each analysis, vector of length k.
b
Upper bound cutoffs (z-values) for futility at each analysis; vector of length k.
r
Control for grid as in Jennison and Turnbull (2000); default is 18, range is 1 to 80.
Normally this will not be changed by the user.
d
If not NULL, this should be an object of type gsDesign returned by a call to gsDesign().
When this is specified, the values of k, n.I, a, b, and r will be obtained from d and
only theta needs to be specified by the user.
x
An item of class gsProbability.
overrun
Scalar or vector of length k-1 with patients enrolled that are not included in each interim analysis.
...
Not implemented (here for compatibility with generic print input).
Details
Depending on the calling sequence, an object of class gsProbability or class gsDesign is returned.
If it is of class gsDesign then the members of the object will be the same as described in gsDesign.
If d is input as NULL (the default), all other arguments (other than r) must be specified
and an object of class gsProbability is returned.
If d is passed as an object of class gsProbability or gsDesign the only other argument required is theta;
the object returned has the same class as the input d.
On output, the values of theta input to gsProbability will be the parameter values for which the
design is characterized.
Value
k
As input.
theta
As input.
n.I
As input.
lower
A list containing two elements: bound is as input in a and prob is a matrix of boundary
crossing probabilities. Element i,j contains the boundary crossing probability at analysis i for the j-th
element of theta input. All boundary crossing is assumed to be binding for this computation;
that is, the trial must stop if a boundary is crossed.
upper
A list of the same form as lower containing the upper bound and upper boundary crossing probabilities.
en
A vector of the same length as theta containing expected sample sizes for the trial design
corresponding to each value in the vector theta.
r
As input.
Note: print.gsProbability() returns the input x.
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
Plots for group sequential designs, gsDesign, gsDesign package overview
Examples
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# making a gsDesign object first may be easiest...
x <- gsDesign()
# take a look at it
x
# default plot for gsDesign object shows boundaries
plot(x)
# plottype=2 shows boundary crossing probabilities
plot(x, plottype=2)
# now add boundary crossing probabilities and
# expected sample size for more theta values
y <- gsProbability(d=x, theta=x$delta*seq(0, 2, .25))
class(y)
# note that "y" below is equivalent to print(y) and
# print.gsProbability(y)
y
# the plot does not change from before since this is a
# gsDesign object; note that theta/delta is on x axis
plot(y, plottype=2)
# now let's see what happens with a gsProbability object
z <- gsProbability(k=3, a=x$lower$bound, b=x$upper$bound,
n.I=x$n.I, theta=x$delta*seq(0, 2, .25))
# with the above form, the results is a gsProbability object
class(z)
z
# default plottype is now 2
# this is the same range for theta, but plot now has theta on x axis
plot(z)
Example output
Loading required package: xtable
Loading required package: ggplot2
Asymmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Upper bound spending computations assume
trial continues if lower bound is crossed.
Sample
Size ----Lower bounds---- ----Upper bounds-----
Analysis Ratio* Z Nominal p Spend+ Z Nominal p Spend++
1 0.357 -0.24 0.4057 0.0148 3.01 0.0013 0.0013
2 0.713 0.94 0.8267 0.0289 2.55 0.0054 0.0049
3 1.070 2.00 0.9772 0.0563 2.00 0.0228 0.0188
Total 0.1000 0.0250
+ lower bound beta spending (under H1):
Hwang-Shih-DeCani spending function with gamma = -2.
++ alpha spending:
Hwang-Shih-DeCani spending function with gamma = -4.
* 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 0.0013 0.0049 0.0171 0.0233 0.6249
3.2415 0.1412 0.4403 0.3185 0.9000 0.7913
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 Total
0.0000 0.4057 0.4290 0.1420 0.9767
3.2415 0.0148 0.0289 0.0563 0.1000
[1] "gsDesign"
Asymmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Upper bound spending computations assume
trial continues if lower bound is crossed.
Sample
Size ----Lower bounds---- ----Upper bounds-----
Analysis Ratio* Z Nominal p Spend+ Z Nominal p Spend++
1 0.357 -0.24 0.4057 0.0148 3.01 0.0013 0.0013
2 0.713 0.94 0.8267 0.0289 2.55 0.0054 0.0049
3 1.070 2.00 0.9772 0.0563 2.00 0.0228 0.0188
Total 0.1000 0.0250
+ lower bound beta spending (under H1):
Hwang-Shih-DeCani spending function with gamma = -2.
++ alpha spending:
Hwang-Shih-DeCani spending function with gamma = -4.
* 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 0.0013 0.0049 0.0171 0.0233 0.6249
0.8104 0.0058 0.0279 0.0872 0.1209 0.7523
1.6208 0.0205 0.1038 0.2393 0.3636 0.8520
2.4311 0.0595 0.2579 0.3636 0.6810 0.8668
3.2415 0.1412 0.4403 0.3185 0.9000 0.7913
4.0519 0.2773 0.5353 0.1684 0.9810 0.6765
4.8623 0.4574 0.4844 0.0559 0.9976 0.5701
5.6727 0.6469 0.3410 0.0119 0.9998 0.4868
6.4830 0.8053 0.1930 0.0016 1.0000 0.4266
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 Total
0.0000 0.4057 0.4290 0.1420 0.9767
0.8104 0.2349 0.3812 0.2630 0.8791
1.6208 0.1138 0.2385 0.2841 0.6364
2.4311 0.0455 0.1017 0.1718 0.3190
3.2415 0.0148 0.0289 0.0563 0.1000
4.0519 0.0039 0.0054 0.0097 0.0190
4.8623 0.0008 0.0006 0.0009 0.0024
5.6727 0.0001 0.0001 0.0000 0.0002
6.4830 0.0000 0.0000 0.0000 0.0000
[1] "gsProbability"
Lower bounds Upper bounds
Analysis N Z Nominal p Z Nominal p
1 1 -0.24 0.4057 3.01 0.0013
2 1 0.94 0.8267 2.55 0.0054
3 2 2.00 0.9772 2.00 0.0228
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 0.0013 0.0049 0.0171 0.0233 0.6
0.8104 0.0058 0.0279 0.0872 0.1209 0.8
1.6208 0.0205 0.1038 0.2393 0.3636 0.9
2.4311 0.0595 0.2579 0.3636 0.6810 0.9
3.2415 0.1412 0.4403 0.3185 0.9000 0.8
4.0519 0.2773 0.5353 0.1684 0.9810 0.7
4.8623 0.4574 0.4844 0.0559 0.9976 0.6
5.6727 0.6469 0.3410 0.0119 0.9998 0.5
6.4830 0.8053 0.1930 0.0016 1.0000 0.4
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 Total
0.0000 0.4057 0.4290 0.1420 0.9767
0.8104 0.2349 0.3812 0.2630 0.8791
1.6208 0.1138 0.2385 0.2841 0.6364
2.4311 0.0455 0.1017 0.1718 0.3190
3.2415 0.0148 0.0289 0.0563 0.1000
4.0519 0.0039 0.0054 0.0097 0.0190
4.8623 0.0008 0.0006 0.0009 0.0024
5.6727 0.0001 0.0001 0.0000 0.0002
6.4830 0.0000 0.0000 0.0000 0.0000
gsDesign documentation built on May 2, 2019, 4:49 p.m.
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Description Usage Arguments Details Value Note Author(s) References See Also Examples
Description
Computes power/Type I error and expected sample size for a group sequential design
across a selected set of parameter values for a given set of analyses and boundaries.
The print function has been extended using print.gsProbability to print gsProbability objects; see examples.
Usage
1 2 3 | gsProbability(k=0, theta, n.I, a, b, r=18, d=NULL, overrun=0)
## S3 method for class 'gsProbability'
print(x,...)
|
Arguments
k |
Number of analyses planned, including interim and final. |
theta |
Vector of standardized effect sizes for which boundary crossing probabilities are to be computed. |
n.I |
Sample size or relative sample size at analyses; vector of length k. See |
a |
Lower bound cutoffs (z-values) for futility or harm at each analysis, vector of length k. |
b |
Upper bound cutoffs (z-values) for futility at each analysis; vector of length k. |
r |
Control for grid as in Jennison and Turnbull (2000); default is 18, range is 1 to 80. Normally this will not be changed by the user. |
d |
If not |
x |
An item of class |
overrun |
Scalar or vector of length |
... |
Not implemented (here for compatibility with generic print input). |
Details
Depending on the calling sequence, an object of class gsProbability or class gsDesign is returned.
If it is of class gsDesign then the members of the object will be the same as described in gsDesign.
If d is input as NULL (the default), all other arguments (other than r) must be specified
and an object of class gsProbability is returned.
If d is passed as an object of class gsProbability or gsDesign the only other argument required is theta;
the object returned has the same class as the input d.
On output, the values of theta input to gsProbability will be the parameter values for which the
design is characterized.
Value
k |
As input. |
theta |
As input. |
n.I |
As input. |
lower |
A list containing two elements: |
upper |
A list of the same form as |
en |
A vector of the same length as |
r |
As input. |
Note: print.gsProbability() returns the input x.
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
Plots for group sequential designs, gsDesign, gsDesign package overview
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 26 27 28 29 30 31 32 33 34 35 36 | # making a gsDesign object first may be easiest...
x <- gsDesign()
# take a look at it
x
# default plot for gsDesign object shows boundaries
plot(x)
# plottype=2 shows boundary crossing probabilities
plot(x, plottype=2)
# now add boundary crossing probabilities and
# expected sample size for more theta values
y <- gsProbability(d=x, theta=x$delta*seq(0, 2, .25))
class(y)
# note that "y" below is equivalent to print(y) and
# print.gsProbability(y)
y
# the plot does not change from before since this is a
# gsDesign object; note that theta/delta is on x axis
plot(y, plottype=2)
# now let's see what happens with a gsProbability object
z <- gsProbability(k=3, a=x$lower$bound, b=x$upper$bound,
n.I=x$n.I, theta=x$delta*seq(0, 2, .25))
# with the above form, the results is a gsProbability object
class(z)
z
# default plottype is now 2
# this is the same range for theta, but plot now has theta on x axis
plot(z)
|
Example output
Loading required package: xtable
Loading required package: ggplot2
Asymmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Upper bound spending computations assume
trial continues if lower bound is crossed.
Sample
Size ----Lower bounds---- ----Upper bounds-----
Analysis Ratio* Z Nominal p Spend+ Z Nominal p Spend++
1 0.357 -0.24 0.4057 0.0148 3.01 0.0013 0.0013
2 0.713 0.94 0.8267 0.0289 2.55 0.0054 0.0049
3 1.070 2.00 0.9772 0.0563 2.00 0.0228 0.0188
Total 0.1000 0.0250
+ lower bound beta spending (under H1):
Hwang-Shih-DeCani spending function with gamma = -2.
++ alpha spending:
Hwang-Shih-DeCani spending function with gamma = -4.
* 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 0.0013 0.0049 0.0171 0.0233 0.6249
3.2415 0.1412 0.4403 0.3185 0.9000 0.7913
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 Total
0.0000 0.4057 0.4290 0.1420 0.9767
3.2415 0.0148 0.0289 0.0563 0.1000
[1] "gsDesign"
Asymmetric two-sided group sequential design with
90 % power and 2.5 % Type I Error.
Upper bound spending computations assume
trial continues if lower bound is crossed.
Sample
Size ----Lower bounds---- ----Upper bounds-----
Analysis Ratio* Z Nominal p Spend+ Z Nominal p Spend++
1 0.357 -0.24 0.4057 0.0148 3.01 0.0013 0.0013
2 0.713 0.94 0.8267 0.0289 2.55 0.0054 0.0049
3 1.070 2.00 0.9772 0.0563 2.00 0.0228 0.0188
Total 0.1000 0.0250
+ lower bound beta spending (under H1):
Hwang-Shih-DeCani spending function with gamma = -2.
++ alpha spending:
Hwang-Shih-DeCani spending function with gamma = -4.
* 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 0.0013 0.0049 0.0171 0.0233 0.6249
0.8104 0.0058 0.0279 0.0872 0.1209 0.7523
1.6208 0.0205 0.1038 0.2393 0.3636 0.8520
2.4311 0.0595 0.2579 0.3636 0.6810 0.8668
3.2415 0.1412 0.4403 0.3185 0.9000 0.7913
4.0519 0.2773 0.5353 0.1684 0.9810 0.6765
4.8623 0.4574 0.4844 0.0559 0.9976 0.5701
5.6727 0.6469 0.3410 0.0119 0.9998 0.4868
6.4830 0.8053 0.1930 0.0016 1.0000 0.4266
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 Total
0.0000 0.4057 0.4290 0.1420 0.9767
0.8104 0.2349 0.3812 0.2630 0.8791
1.6208 0.1138 0.2385 0.2841 0.6364
2.4311 0.0455 0.1017 0.1718 0.3190
3.2415 0.0148 0.0289 0.0563 0.1000
4.0519 0.0039 0.0054 0.0097 0.0190
4.8623 0.0008 0.0006 0.0009 0.0024
5.6727 0.0001 0.0001 0.0000 0.0002
6.4830 0.0000 0.0000 0.0000 0.0000
[1] "gsProbability"
Lower bounds Upper bounds
Analysis N Z Nominal p Z Nominal p
1 1 -0.24 0.4057 3.01 0.0013
2 1 0.94 0.8267 2.55 0.0054
3 2 2.00 0.9772 2.00 0.0228
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 0.0013 0.0049 0.0171 0.0233 0.6
0.8104 0.0058 0.0279 0.0872 0.1209 0.8
1.6208 0.0205 0.1038 0.2393 0.3636 0.9
2.4311 0.0595 0.2579 0.3636 0.6810 0.9
3.2415 0.1412 0.4403 0.3185 0.9000 0.8
4.0519 0.2773 0.5353 0.1684 0.9810 0.7
4.8623 0.4574 0.4844 0.0559 0.9976 0.6
5.6727 0.6469 0.3410 0.0119 0.9998 0.5
6.4830 0.8053 0.1930 0.0016 1.0000 0.4
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 Total
0.0000 0.4057 0.4290 0.1420 0.9767
0.8104 0.2349 0.3812 0.2630 0.8791
1.6208 0.1138 0.2385 0.2841 0.6364
2.4311 0.0455 0.1017 0.1718 0.3190
3.2415 0.0148 0.0289 0.0563 0.1000
4.0519 0.0039 0.0054 0.0097 0.0190
4.8623 0.0008 0.0006 0.0009 0.0024
5.6727 0.0001 0.0001 0.0000 0.0002
6.4830 0.0000 0.0000 0.0000 0.0000
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