gsDesign (version 2.4-01)

gsCP: 2.4: Conditional and Predictive Power, Overall and Conditional Probability of Success

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. gsPP() averages conditional power across a posterior distribution to compute predictive power. 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

gsCP(x, theta=NULL, i=1, zi=0, r=18)
gsPP(x, i=1, zi=0, theta=c(0,3), wgts=c(.5,.5), r=18, total=TRUE)
gsPOS(x, theta, wgts)
gsCPOS(i, x, theta, wgts)

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

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

zi

interim z-value at analysis i (scalar)

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 th

total

Needs documentation.

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. 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 .

Details

See Conditional power section of manual for further clarification. See also Muller and Schaffer (2001) for background theory.

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

gsDesign, gsProbability, gsBoundCP

Examples

# 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)))