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compute adjusted p-values either for the closed test defined by the graph or for each elementary hypotheses within each intersection hypotheses


  adjusted = TRUE,
  hint = generateWeights(g, w),
  upscale = FALSE



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


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


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)


vector of observed unadjusted p-values, that belong to test-statistics with a joint multivariate normal null distribution with (partially) known correlation matrix cr


logical, if TRUE (default) adjusted p-values for the closed test are returned, else a matrix of p-values adjusted only for each intersection hypothesis is returned


if intersection hypotheses weights have already been computed (output of generateWeights) can be passed here otherwise will be computed during execution


if FALSE (default) the p-values are additionally adjusted for the case that non-exhaustive weights are specified. (See details)


If adjusted is set to true returns a vector of adjusted p-values. Any elementary null hypothesis is rejected if its corresponding adjusted p-value is below the predetermined alpha level. For adjusted set to false a matrix with p-values adjusted only within each intersection hypotheses is returned. The intersection corresponding to each line is given by conversion of the line number into binary (eg. 13 is binary 1101 and corresponds to (H1,H2,H4)). If any adjusted p-value within a given line falls below alpha, then the corresponding intersection hypotheses can be rejected.


It is assumed that under the global null hypothesis \((\Phi^{-1}(1-p_1),...,\Phi^{-1}(1-p_m))\) follow a multivariate normal distribution with correlation matrix cr where \(\Phi^{-1}\) 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 \(\Phi^{-1}(1-p_i)\) 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, \(cr[i,j] = \rho_{ij}\), where \(\rho_{ij}\) is the known value of the correlation between \(\Phi^{-1}(1-p_i)\) and \(\Phi^{-1}(1-p_j)\) 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 exhaust 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 and p-values adjusted accordingly such that test procedures with non-exhaustive weighting strategies may be implemented. If set to TRUE the tests are performed as defined in Equation (3) of (Bretz et al. (2011)).


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


Florian Klinglmueller


## 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
## p-values as Section 3 of Bretz et al. (2011),
p <- c(0.0121,0.0337,0.0084,0.0160)

## Boundaries for correlated test statistics at alpha level .05:
#> [1] 0.0242 0.0337 0.0242 0.0337

g <- Entangled2Maurer2012()
generatePvals(g=g, cr=diag(5), p=rep(0.1,5))
#> [1] 0.19 0.19 0.19 0.19 0.19