# 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. ```r 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 (>= 1 and <= 80) controlling the number of numerical integration grid points. Default is 18, as recommended by Jennison and Turnbull (2000). Grid points are spread out in the tails for accurate probability calculations. Larger values provide more grid points and greater accuracy but slow down computation. Jennison and Turnbull (p. 350) note an accuracy of $10^{-6}$ with `r = 16`. This parameter is normally not changed by users. - `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` ## Returns `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 100`level`% credible interval for the z-statistic at analysis `j` conditional on the z-statistic at analysis `i= 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) %*% c(1, 1, 1) # 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) ``` ## 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. ## See Also `normalGrid`, `gsDesign`, `gsProbability`, `gsBoundCP`, `ssrCP`, `condPower` ## Author(s) Keaven Anderson keaven_anderson@merck.com

gsCP function