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

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calcPower(weights, alpha, G, mean = rep(0, nrow(corr.sim)),
  corr.sim = diag(length(mean)), corr.test = NULL, n.sim = 10000,
  type = c("quasirandom", "pseudorandom"), f = list(), upscale = FALSE,
  graph, ...)

Arguments

weights

Initial weight levels for the test procedure (see graphTest function). Alternatively a graphMCP object can be given as parameter graph.

alpha

Overall alpha level of the procedure, see graphTest function. (For entangled graphs alpha should be a numeric vector of length equal to the number of graphs, each element specifying the partial alpha for the respective graph. The overall alpha level equals sum(alpha).)

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 elements. Alternatively a graphMCP object can be given as parameter graph.

mean

Mean under the alternative

corr.sim

Covariance matrix under the alternative.

corr.test

Correlation matrix that should be used for the parametric test. If corr.test==NULL the Bonferroni based test procedure is used. Can contain NAs.

n.sim

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 and at least 1024.

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.

f

List of user defined power functions (or just a single power function). If one is interested in the power to reject hypotheses 1 and 3 one could specify:
f=function(x) {x[1] && x[3]}.
If the power of rejecting hypotheses 1 and 2 is also of interest one would use a (optionally named) list:
f=list(power1and3=function(x) {x[1] && x[3]},
power1and2=function(x) {x[1] && x[2]}). If the list has no names, the functions will be referenced to as "func1", "func2", etc. in the output.

upscale

Logical. If upscale=FALSE then for each intersection of hypotheses (i.e. each subgraph) a weighted test is performed at the possibly reduced level alpha of sum(w)*alpha, where sum(w) is the sum of all node weights in this subset. If upscale=TRUE all weights are upscaled, so that sum(w)=1.

graph

A graph of class graphMCP.

...

For backwards compatibility. For example up to version 0.8-7 the parameters corr.model and corr.test were called sigma and cr. Also instead of supplying a graph object one could supply a parameter weights and a transition matrix G.

test

In the parametric case there is more than one way to handle subgraphs with less than the full alpha. If the parameter test is missing, the tests are performed as described by Bretz et al. (2011), i.e. tests of intersection null hypotheses always exhaust the full alpha level even if the sum of weights is strictly smaller than one. If test="simple-parametric" the tests are performed as defined in Equation (3) of Bretz et al. (2011).

Value

A list containg three elements

LocalPower

A numeric giving the local powers for the hypotheses

ExpRejections

The expected number of rejections

PowAtlst1

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

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## reproduce example from Stat Med paper (Bretz et al. 2010, Table I)
## first only consider line 2 of Table I
## significance levels
graph <- simpleSuccessiveII()
## 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(graph=graph, alpha=0.025, mean=theta, corr.sim=corMat, n.sim= 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(weights=al, alpha=0.025, G=G, mean=mean, corr.sim=corMat, n.sim = 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)

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