Conditional and Predictive Power, Overall and Conditional Probability of Success

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.

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.

Usage

gsCP(x, theta = NULL, i = 1, zi = 0, r = 18)

gsPP(
  x,
  i = 1,
  zi = 0,
  theta = c(0, 3),
  wgts = c(0.5, 0.5),
  r = 18,
  total = TRUE
)

gsPI(
  x,
  i = 1,
  zi = 0,
  j = 2,
  level = 0.95,
  theta = c(0, 3),
  wgts = c(0.5, 0.5)
)

gsPosterior(x = gsDesign(), i = 1, zi = NULL, prior = normalGrid(), r = 18)

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 is computed. For gsPosterior, this may be a scalar or an interval; for gsPP and gsCP, this must be a scalar.
i: analysis at which interim z-value is given; must be from 1 to x$k-1
zi: interim z-value at analysis i (scalar)
r: Integer value controlling grid for numerical integration as in Jennison and Turnbull (2000); default is 18, range is 1 to 80. Larger values provide larger number of grid points and greater accuracy. Normally r will not be changed by the user.
wgts: Weights to be used with grid points in theta. Length can be one if weights are equal, otherwise should be the same length as theta. Values should be positive, but do not need to sum to 1.
total: The default of total=TRUE produces the combined probability for all planned analyses after the interim analysis specified in i. Otherwise, information on each analysis is provided separately.
j: specific analysis for which prediction is being made; must be >i and no more than x$k
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.
prior: provides a prior distribution in the form produced by normalGrid

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 gsDesign technical manual is available at https://keaven.github.io/gsd-tech-manual/.

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), Statistical 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.

Author

Keaven Anderson keaven_anderson@merck.com

Examples

library(ggplot2)
# set up a group sequential design
x <- gsDesign(k = 5)
x
#> 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

# 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)
#> [1] 0.5954771

# 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))
#> [1] 0.5554313

# 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
#>               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
#>           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
#>           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

# summing values for crossing future upper bounds gives overall
# conditional power for each theta value
CP$theta
#> [1] -0.05536767  0.00000000  3.24151555
t(CP$upper$prob) %*% c(1, 1, 1)
#>             [,1]
#> [1,] 0.002238566
#> [2,] 0.002561570
#> [3,] 0.367657693

# compute predictive probability based on above assumptions
gsPP(x, i = 2, zi = x$lower$bound[2], theta = prior$z, wgts = prior$wgts)
#> [1] 0.06730167

# 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)
#> [1] 0.6114033

# 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))
#> [1] 0.03579292 0.01067483 0.51555676

gsCP(x = x, zi = z1, i = 1)$upper$prob
#>            [,1]       [,2]      [,3]
#> [1,] 0.03579292 0.01067483 0.5155568

# 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)
#> [1] 0.1556447
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)))
#> [1] 0.1556447

# 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)
#> [1] 1.081745
gsPI(x = x, i = 1, zi = z1, j = 2, theta = prior$z, wgts = prior$wgts, level = .9)
#> [1] -0.3793133  2.5428090