Description Usage Arguments Details Value Note Author(s) References See Also Examples
Given an interim analysis i
of a group sequential design and a vector of real values zi
, gsDensity()
computes an interim density function at analysis i
at the values in zi
.
For each value in zi
, this interim density is the derivative of the probability that the group sequential trial does not cross a boundary prior to the i
-th analysis and at the i
-th analysis the interim Z-statistic is less than that value.
When integrated over the real line, this density computes the probability of not crossing a bound at a previous analysis.
It corresponds to the subdistribution function at analysis i
that excludes the probability of crossing a bound at an earlier analysis.
The initial purpose of this routine was as a component needed to compute the predictive power for a trial given an interim result; see gsPP
.
1 | gsDensity(x, theta=0, i=1, zi=0, r=18)
|
x |
An object of type |
theta |
a vector with theta value(s) at which the interim density function is to be computed. |
i |
analysis at which interim z-values are given; must be from 1 to |
zi |
interim z-value at analysis |
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 |
See Jennison and Turnbull (2000) for details on how these computations are performed.
zi |
The input vector |
theta |
The input vector |
density |
A matrix with |
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.
Keaven Anderson keaven\_anderson@merck.
Jennison C and Turnbull BW (2000), Group Sequential Methods with Applications to Clinical Trials. Boca Raton: Chapman and Hall.
gsDesign
, gsProbability
, gsBoundCP
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 | # set up a group sequential design
x <- gsDesign()
# set theta values where density is to be evaluated
theta <- x$theta[2] * c(0, .5, 1, 1.5)
# set zi values from -1 to 7 where density is to be evaluated
zi <- seq(-3, 7, .05)
# compute subdensity values at analysis 2
y <- gsDensity(x, theta=theta, i=2, zi=zi)
# plot sub-density function for each theta value
plot(y$zi, y$density[,3], type="l", xlab="Z",
ylab="Interim 2 density", lty=3, lwd=2)
lines(y$zi, y$density[,2], lty=2, lwd=2)
lines(y$zi, y$density[,1], lwd=2)
lines(y$zi, y$density[,4], lty=4, lwd=2)
title("Sub-density functions at interim analysis 2")
legend(x=c(3.85,7.2), y = c(.27,.385), lty=1:5, lwd=2, cex=1.5,
legend=c(
expression(paste(theta,"=0.0")),
expression(paste(theta,"=0.5", delta)),
expression(paste(theta,"=1.0", delta)),
expression(paste(theta,"=1.5", delta))))
# add vertical lines with lower and upper bounds at analysis 2
# to demonstrate how likely it is to continue, stop for futility
# or stop for efficacy at analysis 2 by treatment effect
lines(array(x$upper$bound[2],2), c(0,.4),col=2)
lines(array(x$lower$bound[2],2), c(0,.4), lty=2, col=2)
# Replicate part of figures 8.1 and 8.2 of Jennison and Turnbull text book
# O'Brien-Fleming design with four analyses
x <- gsDesign(k=4, test.type=2, sfu="OF", alpha=.1, beta=.2)
z <- seq(-4.2, 4.2, .05)
d <- gsDensity(x=x, theta=x$theta, i=4, zi=z)
plot(z, d$density[,1], type="l", lwd=2, ylab=expression(paste(p[4],"(z,",theta,")")))
lines(z, d$density[,2], lty=2, lwd=2)
u <- x$upper$bound[4]
text(expression(paste(theta,"=",delta)),x=2.2, y=.2, cex=1.5)
text(expression(paste(theta,"=0")),x=.55, y=.4, cex=1.5)
|
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