gMCP (version 0.7-8)

calcPower: Calculate power values

Description

Given the distribution under the alternative (assumed to be multivariate normal), this function calculates the power to reject at least one hypothesis, the local power for the hypotheses as well as the expected number of rejections.

Usage

calcPower(weights, alpha, G, mean = rep(0, nrow(sigma)),
            sigma = diag(length(mean)),corr = NULL,
            nSim = 10000, seed = 4711,
            type = c("quasirandom", "pseudorandom"), f=list())

Arguments

weights

Initial weight levels for the test procedure, see graphTest function.

alpha

Overall alpha level of the procedure, see graphTest function.

G

Matrix determining the graph underlying the test procedure. Note that the diagonal need to contain only 0s, while the rows need to sum to 1. When multiple graphs should be used this needs to be a list containing the different graphs as element

mean

Mean under the alternative

sigma, corr

Covariance or correlation matrix under the alternative.

seed

Seed for pseudo random numbers and the random shift of the lattice rule.

type

What type of random numbers to use. quasirandom uses a randomized Lattice rule, and should be more efficient than pseudorandom that uses ordinary (pseudo) random numbers.

nSim

Monte Carlo sample size. If type = "quasirandom" this number is rounded up to the next power of 2, e.g. 1000 is rounded up to $1024=2^10$.

f

List of user defined power functions. If one is interested in the power to reject hypotheses 1 and 3 one could specify f=list(function(x) {x[1] && x[3]})

Value

• A list containg three elements
• LocPower
  A numeric giving the local powers for the hypotheses
• ExpRejections
  The expected number of rejections
• PowerAtlst1
  The power to reject at least one hypothesis

References

Bretz, F., Maurer, W., Brannath, W. and Posch, M. (2009) A graphical approach to sequentially rejective multiple test procedures. Statistics in Medicine, 28, 586--604

Bretz, F., Maurer, W. and Hommel, G. (2010) Test and power considerations for multiple endpoint analyses using sequentially rejective graphical procedures, to appear in Statistics in Medicine

Examples

## reproduce example from Stat Med paper (Bretz et al. 2010, Table I)
## first only consider line 2 of Table I
## significance levels
weights <- c(1/2, 1/2, 0, 0)
## graph
G <- rbind(c(0, 0.5, 0.5, 0),
           c(0.5, 0, 0, 0.5),
           c(0, 1, 0, 0),
           c(1, 0, 0, 0))
## or equivalent:
G <- simpleSuccessiveII()@m
## alternative (mvn distribution)
corMat <- rbind(c(1, 0.5, 0.5, 0.5/2),
                c(0.5,1,0.5/2,0.5),
                c(0.5,0.5/2,1,0.5),
                c(0.5/2,0.5,0.5,1))
theta <- c(3, 0, 0, 0)
calcPower(weights, alpha=0.025, G, theta, corMat, seed = 4711, nSim = 100000)


## now reproduce all 14 simulation scenarios
## different graphs
weights1 <- c(rep(1/2, 12), 1, 1)
weights2 <- c(rep(1/2, 12), 0, 0)
eps <- 0.01
gam1 <- c(rep(0.5, 10), 1-eps, 0, 0, 0)
gam2 <- gam1
## different multivariate normal alternatives
rho <- c(rep(0.5, 8), 0, 0.99, rep(0.5,4))
th1 <- c(0, 3, 3, 3, 2, 1, rep(3, 7), 0)
th2 <- c(rep(0, 6), 3, 3, 3, 3, 0, 0, 0, 3)
th3 <- c(0, 0, 3, 3, 3, 3, 0, 2, 2, 2, 3, 3, 3, 3)
th4 <- c(0,0,0,3,3,3,0,2,2,2,0,0,0,0)

## function that calculates power values for one scenario
simfunc <- function(nSim, a1, a2, g1, g2, rh, t1, t2, t3, t4, Gr){
  al <- c(a1, a2, 0, 0)
  G <- rbind(c(0, g1, 1-g1, 0), c(g2, 0, 0, 1-g2), c(0, 1, 0, 0), c(1, 0, 0, 0))
  corMat <- rbind(c(1, 0.5, rh, rh/2), c(0.5,1,rh/2,rh), c(rh,rh/2,1,0.5), c(rh/2,rh,0.5,1))
  mean <- c(t1, t2, t3, t4)
  calcPower(al, alpha=0.025, G, mean, corr = corMat, seed = 4711, nSim = nSim)
}

## calculate power for all 14 scenarios
outList <- list()
for(i in 1:14){
  outList[[i]] <- simfunc(10000, weights1[i], weights2[i], gam1[i], gam2[i], rho[i], th1[i], th2[i], th3[i], th4[i])
}

## summarize data as in Stat Med paper Table I 
atlst1 <- as.numeric(lapply(outList, function(x) x$PowAtlst1))
locpow <- do.call("rbind", lapply(outList, function(x) x$LocalPower))

round(cbind(atlst1, locpow), 5)