generateTest

Description

generates a test function for the multiple comparison procedure with correlated test statistics defined by a graph

Usage

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generateTest(g, w, cr, al, upscale = FALSE)

Arguments

g

graph defined as a matrix, each element defines how much of the local alpha reserved for the hypothesis corresponding to its row index is passed on to the hypothesis corresponding to its column index

w

vector of weights, defines how much of the overall alpha is initially reserved for each elementary hypothesis

cr

correlation matrix if p-values arise from one-sided tests with multivariate normal distributed test statistics for which the correlation is partially known. Unknown values can be set to NA. (See details for more information)

al

overall alpha level at which the family error is controlled

upscale

if FALSE (default) the parametric tests are performed at a reduced level alpha of sum(w) * alpha. (See details)

Details

It is assumed that under the global null hypothesis (Φ^{-1}(1-p_1),...,Φ^{-1}(1-p_m)) follow a multivariate normal distribution with correlation matrix cr where Φ^{-1} denotes the inverse of the standard normal distribution function.

For example, this is the case if p_1,..., p_m are the raw p-values from one-sided z-tests for each of the elementary hypotheses where the correlation between z-test statistics is generated by an overlap in the observations (e.g. comparison with a common control, group-sequential analyses etc.). An application of the transformation Φ^{-1}(1-p_i) to raw p-values from a two-sided test will not in general lead to a multivariate normal distribution. Partial knowledge of the correlation matrix is supported. The correlation matrix has to be passed as a numeric matrix with elements of the form: cr[i,i] = 1 for diagonal elements, cr[i,j] = ρ_{ij}, where ρ_{ij} is the known value of the correlation between Φ^{-1}(1-p_i) and Φ^{-1}(1-p_j) or NA if the corresponding correlation is unknown. For example cr[1,2]=0 indicates that the first and second test statistic are uncorrelated, whereas cr[2,3] = NA means that the true correlation between statistics two and three is unknown and may take values between -1 and 1. The correlation has to be specified for complete blocks (ie.: if cor(i,j), and cor(i,k) for i!=j!=k are specified then cor(j,k) has to be specified as well) otherwise the corresponding intersection null hypotheses tests are not uniquely defined and an error is returned.

The parametric tests in (Bretz et al. (2011)) are defined such that the tests of intersection null hypotheses always upscale the full alpha level even if the sum of weights is strictly smaller than one. This has the consequence that certain test procedures that do not test each intersection null hypothesis at the full level alpha may not be implemented (e.g., a single step Dunnett test). If upscale is set to FALSE (default) the parametric tests are performed at a reduced level alpha of sum(w) * alpha. If set to TRUE the tests are performed as defined in Equation (3) of (Bretz et al. (2011)).

Value

Returns a function that will take a vector of z-scores to which the test will be applied. This function in turn will return a boolean vector with elements false if the particular elementary hypothesis can not be rejected and true otherwise.

Author(s)

Florian Klinglmueller

References

Bretz F, Maurer W, Brannath W, Posch M; (2008) - A graphical approach to sequentially rejective multiple testing procedures. - Stat Med - 28/4, 586-604 Bretz F, Posch M, Glimm E, Klinglmueller F, Maurer W, Rohmeyer K; (2011) - Graphical approaches for multiple endpoint problems using weighted Bonferroni, Simes or parametric tests - to appear

Examples

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## Define some graph as matrix
 g <- matrix(c(0,0,1,0,
               0,0,0,1,
               0,1,0,0,
               1,0,0,0), nrow = 4,byrow=TRUE)
 ## Choose weights
 w <- c(.5,.5,0,0)
 ## Some correlation (upper and lower first diagonal 1/2)
 c <- diag(4)
 c[1:2,3:4] <- NA
 c[3:4,1:2] <- NA
 c[1,2] <- 1/2
 c[2,1] <- 1/2
 c[3,4] <- 1/2
 c[4,3] <- 1/2

 ## Test function for further use:
 myTest <- generateTest(g,w,c,.05)
 myTest(c(3,2,1,2))

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