gsd.dens {PwrGSD}

A function for computing the probability density for the group sequentially monitored test statistic.

Description

A function for computing the probability density for a sequentially monitored test. This is the joint density, in the rejection region, of (X_K, K), where X_K is the observed value of the test statistic upon efficacy boundary crossing, and K is the analysis number at which the efficacy boundary was crossed.

Usage

gsd.dens(x, frac = NULL, scale="Standard")

Arguments

x

The main argument, x, is either a object of class “boundaries” or a numeric vector. If it is of class “boundaries” then no other arguments are required. If it is a numeric vector then the frac argument must be specified. See below. In this case, x will be the observed values of the statistic at the current and all prior analyses, either on the standard normal scale (the default) or on the “Brownian” scale. For “Brownian” scale, set argument scale to “Brownian”.

frac

Required only when the main argument, x, is a numeric vector, and must be a vector of the same length. In this case, frac will be the information at the current and all prior interim analyses.

scale

Required only when the main argument, x, is a numeric vector. A switch indicating whether the elements of the numeric vector, x, are specified on the standard normal scale, x=“Standard”, or on the Brownian scale, x=“Brownian”.

Value

A list with elements x, dF, x1c, and dF1c:

x

Node points used in Gaussian quadrature. See examples below.

dF

Probability mass at each node point. See examples below.

x1c

Node points in the continuation region at the first analysis.

dF1c

Probability mass at each node point in the continuation region at the first analysis.

Note

Also used in computation of Rao-Blackwell-ized bias adjusted point estimate for statistic observed to cross the efficacy boundary.

Author(s)

Grant Izmirlian <[e-mail]>

References

Emerson, S. S. (1993). Computation of the uniform minimum variance unibiased estimator of a normal mean following a group sequential trialdiscrete sequential boundaries for clinical trials. Computers and Biomedical Research 26 68–73.

Izmirlian, G. (2014). Estimation of the relative risk following group sequential procedure based upon the weighted log-rank statistic. Statistics and its Interface 00 00–00

See Also

EX1gXK

Examples

  # Information fraction
  frac <- c(0.15, 0.37, 0.64, 0.76)

  # Efficacy Boundary 
  gsb <- GrpSeqBnds(frac=frac, EfficacyBoundary=LanDemets(spending=ObrienFleming, alpha=0.05))

  # To compute the p-value under the stagewise ordering, for an observed
  #  value of the monitoring statistic 2.1, crossing the efficacy
  #  boundary at the 4th analysis, we do the following

  be <- gsb$table[,"b.e"] 
  be[4] <- 2.1

  sum(gsd.dens(be, frac, scale="Standard")$dF)