gsCP: 2.4: Conditional and Predictive Power, Overall and...
In gsDesign: Group Sequential Design
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
Description
gsCP() computes conditional boundary crossing probabilities at future planned analyses for a given group sequential design assuming an interim z-statistic at a specified interim analysis.
While gsCP() is designed toward computing conditional power for a variety of underlying parameter values, condPower is built to compute conditional power for a variety of interim test statistic values which is useful for sample size adaptation (see ssrCP).
gsPP() averages conditional power across a posterior distribution to compute predictive power.
gsPI() computes Bayesian prediction intervals for future analyses corresponding to results produced by gsPP().
gsPosterior() computes the posterior density for the group sequential design parameter of interest given a prior density and an interim outcome that is exact or in an interval.
gsPOS() computes the probability of success for a trial using a prior distribution to average power over a set of theta values of interest.
gsCPOS() assumes no boundary has been crossed before and including an interim analysis of interest, and computes the probability of success based on this event.
Note that gsCP() and gsPP() take only the interim test statistic into account in computing conditional probabilities, while gsCPOS() conditions on not crossing any bound through a specified interim analysis.
Usage
1
2
3
4
5
6
Arguments
x
An object of type gsDesign or gsProbability
theta
a vector with theta value(s) at which conditional power is to be computed; for gsCP() if NULL, an estimated value of theta based on the interim test statistic (zi/sqrt(x$n.I[i])) as well as at x$theta is computed.
For gsPosterior, this may be a scalar or an interval; for gsPP and gsCP, this must be a scalar.
wgts
Weights to be used with grid points in theta. Length can be one if weights are equal, otherwise should be the same length as theta. Values should be positive, but do not need to sum to 1.
i
analysis at which interim z-value is given; must be from 1 to x$k-1
prior
provides a prior distribution in the form produced by normalGrid
zi
interim z-value at analysis i (scalar)
j
specific analysis for which prediction is being made; must be >i and no more than x$k
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.
total
The default of total=TRUE produces the combined probability for all planned analyses after the interim analysis specified in i. Otherwise, information on each analysis is provided separately.
level
The level to be used for Bayes credible intervals (which approach confidence intervals for vague priors). The default level=.95 corresponds to a 95% credible interval. level=0 provides a point estimate rather than an interval.
Details
See Conditional power section of manual for further clarification. See also Muller and Schaffer (2001) for background theory.
For gsPP(), gsPI(), gsPOS() and gsCPOS(), the prior distribution for the standardized parameter theta () for a group sequential design specified through a gsDesign object is specified through the arguments theta and wgts. This can be a discrete or a continuous probability density function. For a discrete function, generally all weights would be 1. For a continuous density, the wgts would contain integration grid weights, such as those provided by normalGrid.
For gsPosterior, a prior distribution in prior must be composed of the vectors z density.
The vector z contains points where the prior is evaluated and density the corresponding density or, for a discrete distribution, the probabilities of each point in z.
Densities may be supplied as from normalGrid() where grid weights for numerical integration are supplied in gridwgts.
If gridwgts are not supplied, they are defaulted to 1 (equal weighting).
To ensure a proper prior distribution, you must have sum(gridwgts * density) equal to 1; this is NOT checked, however.
Value
gsCP() returns an object of the class gsProbability.
Based on the input design and the interim test statistic, the output gsDesign object has bounds for test statistics computed based on solely on observations after interim i.
Boundary crossing probabilities are computed for the input theta values.
See manual and examples.
gsPP() if total==TRUE, returns a real value indicating the predictive power of the trial conditional on the interim test statistic zi at analysis i; otherwise returns vector with predictive power for each future planned analysis.
gsPI() returns an interval (or point estimate if level=0) indicating 100level% credible interval for the z-statistic at analysis j conditional on the z-statistic at analysis i<j.
The interval does not consider intervending interim analyses.
The probability estimate is based on the predictive distribution used for gsPP() and requires a prior distribution for the group sequential parameter theta specified in theta and wgts.
gsPosterior() returns a posterior distribution containing the the vector z input in prior$z, the posterior density in density, grid weights for integrating the posterior density as input in prior$gridwgts or defaulted to a vector of ones, and the product of the output values in density and gridwgts in wgts.
gsPOS() returns a real value indicating the probability of a positive study weighted by the prior distribution input for theta.
gsCPOS() returns a real value indicating the probability of a positive study weighted by the posterior distribution derived from the interim test statistic and the prior distribution input for theta conditional on an interim test statistic.
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.
Proschan, Michael A., Lan, KK Gordon and Wittes, Janet Turk (2006), Statiscal Monitoring of Clinical Trials. NY: Springer.
Muller, Hans-Helge and Schaffer, Helmut (2001), Adaptive group sequential designs for clinical trials:
combining the advantages of adaptive and classical group sequential approaches. Biometrics;57:886-891.
See Also
normalGrid, gsDesign, gsProbability, gsBoundCP, ssrCP, condPower
Examples
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# set up a group sequential design
x <- gsDesign(k=5)
x
# set up a prior distribution for the treatment effect
# that is normal with mean .75*x$delta and standard deviation x$delta/2
mu0 <-.75*x$delta
sigma0 <- x$delta/2
prior <- normalGrid(mu=mu0, sigma=sigma0)
# compute POS for the design given the above prior distribution for theta
gsPOS(x=x, theta=prior$z, wgts=prior$wgts)
# assume POS should only count cases in prior where theta >= x$delta/2
gsPOS(x=x, theta=prior$z, wgts=prior$wgts*(prior$z>=x$delta/2))
# assuming a z-value at lower bound at analysis 2, what are conditional
# boundary crossing probabilities for future analyses
# assuming theta values from x as well as a value based on the interim
# observed z
CP <- gsCP(x, i=2, zi=x$lower$bound[2])
CP
# summing values for crossing future upper bounds gives overall
# conditional power for each theta value
CP$theta
t(CP$upper$prob)
# compute predictive probability based on above assumptions
gsPP(x, i=2, zi=x$lower$bound[2], theta=prior$z, wgts=prior$wgts)
# if it is known that boundary not crossed at interim 2, use
# gsCPOS to compute conditional POS based on this
gsCPOS(x=x, i=2, theta=prior$z, wgts=prior$wgts)
# 2-stage example to compare results to direct computation
x<-gsDesign(k=2)
z1<- 0.5
n1<-x$n.I[1]
n2<-x$n.I[2]-x$n.I[1]
thetahat<-z1/sqrt(n1)
theta<-c(thetahat, 0 , x$delta)
# conditional power direct computation - comparison w gsCP
pnorm((n2*theta+z1*sqrt(n1)-x$upper$bound[2]*sqrt(n1+n2))/sqrt(n2))
gsCP(x=x, zi=z1, i=1)$upper$prob
# predictive power direct computation - comparison w gsPP
# use same prior as above
mu0 <- .75 * x$delta * sqrt(x$n.I[2])
sigma2 <- (.5 * x$delta)^2 * x$n.I[2]
prior <- normalGrid(mu=.75 * x$delta, sigma=x$delta/2)
gsPP(x=x, zi=z1, i=1, theta=prior$z, wgts=prior$wgts)
t <- .5
z1 <- .5
b <- z1 * sqrt(t)
# direct from Proschan, Lan and Wittes eqn 3.10
# adjusted drift at n.I[2]
pnorm(((b - x$upper$bound[2]) * (1 + t * sigma2) +
(1 - t) * (mu0 + b * sigma2)) /
sqrt((1 - t) * (1 + sigma2) * (1 + t * sigma2)))
# plot prior then posterior distribution for unblinded analysis with i=1, zi=1
xp <- gsPosterior(x=x,i=1,zi=1,prior=prior)
plot(x=xp$z, y=xp$density, type="l", col=2, xlab=expression(theta), ylab="Density")
points(x=x$z, y=x$density, col=1)
# add posterior plot assuming only knowlede that interim bound has
# not been crossed at interim 1
xpb <- gsPosterior(x=x,i=1,zi=1,prior=prior)
lines(x=xpb$z,y=xpb$density,col=4)
# prediction interval based in interim 1 results
# start with point estimate, followed by 90% prediction interval
gsPI(x=x, i=1, zi=z1, j=2, theta=prior$z, wgts=prior$wgts, level=0)
gsPI(x=x, i=1, zi=z1, j=2, theta=prior$z, wgts=prior$wgts, level=.9)
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 0.220 -0.90 0.1836 0.0077 3.25 0.0006 0.0006
2 0.441 -0.04 0.4853 0.0115 2.99 0.0014 0.0013
3 0.661 0.69 0.7563 0.0171 2.69 0.0036 0.0028
4 0.881 1.36 0.9131 0.0256 2.37 0.0088 0.0063
5 1.101 2.03 0.9786 0.0381 2.03 0.0214 0.0140
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 4 5 Total E{N}
0.0000 0.0006 0.0013 0.0028 0.0062 0.0117 0.0226 0.5726
3.2415 0.0417 0.1679 0.2806 0.2654 0.1444 0.9000 0.7440
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 4 5 Total
0.0000 0.1836 0.3201 0.2700 0.1477 0.0559 0.9774
3.2415 0.0077 0.0115 0.0171 0.0256 0.0381 0.1000
[1] 0.5954771
[1] 0.5554313
Lower bounds Upper bounds
Analysis N Z Nominal p Z Nominal p
1 1 1.25 0.8952 4.71 0.0000
2 1 1.96 0.9750 3.39 0.0003
3 1 2.64 0.9959 2.64 0.0041
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}
-0.0554 0e+00 0.0003 0.0019 0.0022 0.2
0.0000 0e+00 0.0003 0.0022 0.0026 0.2
3.2415 7e-04 0.1038 0.2631 0.3677 0.4
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 Total
-0.0554 0.8999 0.0841 0.0138 0.9978
0.0000 0.8952 0.0872 0.0150 0.9974
3.2415 0.3950 0.1368 0.1006 0.6323
[1] -0.05536767 0.00000000 3.24151555
[,1] [,2] [,3]
[1,] 1.068365e-06 0.0002906764 0.001946821
[2,] 1.214159e-06 0.0003326578 0.002227698
[3,] 7.046361e-04 0.1038159665 0.263137090
[1] 0.06730167
[1] 0.6114033
[1] 0.03579292 0.01067483 0.51555676
[,1] [,2] [,3]
[1,] 0.03579292 0.01067483 0.5155568
[1] 0.1556447
[1] 0.1556447
[1] 1.081745
[1] -0.3793133 2.5428090
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 Usage Arguments Details Value Note Author(s) References See Also Examples
Description
gsCP() computes conditional boundary crossing probabilities at future planned analyses for a given group sequential design assuming an interim z-statistic at a specified interim analysis.
While gsCP() is designed toward computing conditional power for a variety of underlying parameter values, condPower is built to compute conditional power for a variety of interim test statistic values which is useful for sample size adaptation (see ssrCP).
gsPP() averages conditional power across a posterior distribution to compute predictive power.
gsPI() computes Bayesian prediction intervals for future analyses corresponding to results produced by gsPP().
gsPosterior() computes the posterior density for the group sequential design parameter of interest given a prior density and an interim outcome that is exact or in an interval.
gsPOS() computes the probability of success for a trial using a prior distribution to average power over a set of theta values of interest.
gsCPOS() assumes no boundary has been crossed before and including an interim analysis of interest, and computes the probability of success based on this event.
Note that gsCP() and gsPP() take only the interim test statistic into account in computing conditional probabilities, while gsCPOS() conditions on not crossing any bound through a specified interim analysis.
Usage
1 2 3 4 5 6 |
Arguments
x |
An object of type |
theta |
a vector with theta value(s) at which conditional power is to be computed; for |
wgts |
Weights to be used with grid points in |
i |
analysis at which interim z-value is given; must be from 1 to |
prior |
provides a prior distribution in the form produced by |
zi |
interim z-value at analysis i (scalar) |
j |
specific analysis for which prediction is being made; must be |
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 |
total |
The default of |
level |
The level to be used for Bayes credible intervals (which approach confidence intervals for vague priors). The default |
Details
See Conditional power section of manual for further clarification. See also Muller and Schaffer (2001) for background theory.
For gsPP(), gsPI(), gsPOS() and gsCPOS(), the prior distribution for the standardized parameter theta () for a group sequential design specified through a gsDesign object is specified through the arguments theta and wgts. This can be a discrete or a continuous probability density function. For a discrete function, generally all weights would be 1. For a continuous density, the wgts would contain integration grid weights, such as those provided by normalGrid.
For gsPosterior, a prior distribution in prior must be composed of the vectors z density.
The vector z contains points where the prior is evaluated and density the corresponding density or, for a discrete distribution, the probabilities of each point in z.
Densities may be supplied as from normalGrid() where grid weights for numerical integration are supplied in gridwgts.
If gridwgts are not supplied, they are defaulted to 1 (equal weighting).
To ensure a proper prior distribution, you must have sum(gridwgts * density) equal to 1; this is NOT checked, however.
Value
gsCP() returns an object of the class gsProbability.
Based on the input design and the interim test statistic, the output gsDesign object has bounds for test statistics computed based on solely on observations after interim i.
Boundary crossing probabilities are computed for the input theta values.
See manual and examples.
gsPP() if total==TRUE, returns a real value indicating the predictive power of the trial conditional on the interim test statistic zi at analysis i; otherwise returns vector with predictive power for each future planned analysis.
gsPI() returns an interval (or point estimate if level=0) indicating 100level% credible interval for the z-statistic at analysis j conditional on the z-statistic at analysis i<j.
The interval does not consider intervending interim analyses.
The probability estimate is based on the predictive distribution used for gsPP() and requires a prior distribution for the group sequential parameter theta specified in theta and wgts.
gsPosterior() returns a posterior distribution containing the the vector z input in prior$z, the posterior density in density, grid weights for integrating the posterior density as input in prior$gridwgts or defaulted to a vector of ones, and the product of the output values in density and gridwgts in wgts.
gsPOS() returns a real value indicating the probability of a positive study weighted by the prior distribution input for theta.
gsCPOS() returns a real value indicating the probability of a positive study weighted by the posterior distribution derived from the interim test statistic and the prior distribution input for theta conditional on an interim test statistic.
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.
Proschan, Michael A., Lan, KK Gordon and Wittes, Janet Turk (2006), Statiscal Monitoring of Clinical Trials. NY: Springer.
Muller, Hans-Helge and Schaffer, Helmut (2001), Adaptive group sequential designs for clinical trials: combining the advantages of adaptive and classical group sequential approaches. Biometrics;57:886-891.
See Also
normalGrid, gsDesign, gsProbability, gsBoundCP, ssrCP, condPower
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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | # set up a group sequential design
x <- gsDesign(k=5)
x
# set up a prior distribution for the treatment effect
# that is normal with mean .75*x$delta and standard deviation x$delta/2
mu0 <-.75*x$delta
sigma0 <- x$delta/2
prior <- normalGrid(mu=mu0, sigma=sigma0)
# compute POS for the design given the above prior distribution for theta
gsPOS(x=x, theta=prior$z, wgts=prior$wgts)
# assume POS should only count cases in prior where theta >= x$delta/2
gsPOS(x=x, theta=prior$z, wgts=prior$wgts*(prior$z>=x$delta/2))
# assuming a z-value at lower bound at analysis 2, what are conditional
# boundary crossing probabilities for future analyses
# assuming theta values from x as well as a value based on the interim
# observed z
CP <- gsCP(x, i=2, zi=x$lower$bound[2])
CP
# summing values for crossing future upper bounds gives overall
# conditional power for each theta value
CP$theta
t(CP$upper$prob)
# compute predictive probability based on above assumptions
gsPP(x, i=2, zi=x$lower$bound[2], theta=prior$z, wgts=prior$wgts)
# if it is known that boundary not crossed at interim 2, use
# gsCPOS to compute conditional POS based on this
gsCPOS(x=x, i=2, theta=prior$z, wgts=prior$wgts)
# 2-stage example to compare results to direct computation
x<-gsDesign(k=2)
z1<- 0.5
n1<-x$n.I[1]
n2<-x$n.I[2]-x$n.I[1]
thetahat<-z1/sqrt(n1)
theta<-c(thetahat, 0 , x$delta)
# conditional power direct computation - comparison w gsCP
pnorm((n2*theta+z1*sqrt(n1)-x$upper$bound[2]*sqrt(n1+n2))/sqrt(n2))
gsCP(x=x, zi=z1, i=1)$upper$prob
# predictive power direct computation - comparison w gsPP
# use same prior as above
mu0 <- .75 * x$delta * sqrt(x$n.I[2])
sigma2 <- (.5 * x$delta)^2 * x$n.I[2]
prior <- normalGrid(mu=.75 * x$delta, sigma=x$delta/2)
gsPP(x=x, zi=z1, i=1, theta=prior$z, wgts=prior$wgts)
t <- .5
z1 <- .5
b <- z1 * sqrt(t)
# direct from Proschan, Lan and Wittes eqn 3.10
# adjusted drift at n.I[2]
pnorm(((b - x$upper$bound[2]) * (1 + t * sigma2) +
(1 - t) * (mu0 + b * sigma2)) /
sqrt((1 - t) * (1 + sigma2) * (1 + t * sigma2)))
# plot prior then posterior distribution for unblinded analysis with i=1, zi=1
xp <- gsPosterior(x=x,i=1,zi=1,prior=prior)
plot(x=xp$z, y=xp$density, type="l", col=2, xlab=expression(theta), ylab="Density")
points(x=x$z, y=x$density, col=1)
# add posterior plot assuming only knowlede that interim bound has
# not been crossed at interim 1
xpb <- gsPosterior(x=x,i=1,zi=1,prior=prior)
lines(x=xpb$z,y=xpb$density,col=4)
# prediction interval based in interim 1 results
# start with point estimate, followed by 90% prediction interval
gsPI(x=x, i=1, zi=z1, j=2, theta=prior$z, wgts=prior$wgts, level=0)
gsPI(x=x, i=1, zi=z1, j=2, theta=prior$z, wgts=prior$wgts, level=.9)
|
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 0.220 -0.90 0.1836 0.0077 3.25 0.0006 0.0006
2 0.441 -0.04 0.4853 0.0115 2.99 0.0014 0.0013
3 0.661 0.69 0.7563 0.0171 2.69 0.0036 0.0028
4 0.881 1.36 0.9131 0.0256 2.37 0.0088 0.0063
5 1.101 2.03 0.9786 0.0381 2.03 0.0214 0.0140
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 4 5 Total E{N}
0.0000 0.0006 0.0013 0.0028 0.0062 0.0117 0.0226 0.5726
3.2415 0.0417 0.1679 0.2806 0.2654 0.1444 0.9000 0.7440
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 4 5 Total
0.0000 0.1836 0.3201 0.2700 0.1477 0.0559 0.9774
3.2415 0.0077 0.0115 0.0171 0.0256 0.0381 0.1000
[1] 0.5954771
[1] 0.5554313
Lower bounds Upper bounds
Analysis N Z Nominal p Z Nominal p
1 1 1.25 0.8952 4.71 0.0000
2 1 1.96 0.9750 3.39 0.0003
3 1 2.64 0.9959 2.64 0.0041
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}
-0.0554 0e+00 0.0003 0.0019 0.0022 0.2
0.0000 0e+00 0.0003 0.0022 0.0026 0.2
3.2415 7e-04 0.1038 0.2631 0.3677 0.4
Lower boundary (futility or Type II Error)
Analysis
Theta 1 2 3 Total
-0.0554 0.8999 0.0841 0.0138 0.9978
0.0000 0.8952 0.0872 0.0150 0.9974
3.2415 0.3950 0.1368 0.1006 0.6323
[1] -0.05536767 0.00000000 3.24151555
[,1] [,2] [,3]
[1,] 1.068365e-06 0.0002906764 0.001946821
[2,] 1.214159e-06 0.0003326578 0.002227698
[3,] 7.046361e-04 0.1038159665 0.263137090
[1] 0.06730167
[1] 0.6114033
[1] 0.03579292 0.01067483 0.51555676
[,1] [,2] [,3]
[1,] 0.03579292 0.01067483 0.5155568
[1] 0.1556447
[1] 0.1556447
[1] 1.081745
[1] -0.3793133 2.5428090
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