Skip to contents

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 correlation where \(\Phi^{-1}\) denotes the inverse of the standard normal distribution function.


  alpha = 0.05,
  adjPValues = TRUE,
  verbose = FALSE,



A numeric vector specifying the p-values.


A numeric vector of weights.


A numeric specifying the maximal allowed type one error rate. If adjPValues==TRUE (default) the parameter alpha is not used.


Logical scalar. If TRUE (the default) an adjusted p-value for the weighted parametric test is returned. Otherwise if adjPValues==FALSE a logical value is returned whether the null hypothesis can be rejected.


Logical scalar. If TRUE verbose output is generated.


Correlation matrix. For parametric tests the p-values must arise from one-sided tests with multivariate normal distributed test statistics for which the correlation is (partially) known. In that case a weighted parametric closed test is performed (also see generatePvals). Unknown values can be set to NA. (See details for more information)


Further arguments possibly passed by gMCP which will be used by other test procedures but not this one.


adjusted p-value or decision of rejection


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: \(correlation[i,i] = 1\) for diagonal elements, \(correlation[i,j] = \rho_{ij}\), 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 correlation[1,2]=0 indicates that the first and second test statistic are uncorrelated, whereas correlation[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,j') for i!=j!=j' are specified then cor(j,j') has to be specified as well) otherwise the corresponding intersection null hypotheses tests are not uniquely defined and an error is returned.

For further details see the given references.


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. Biometrical Journal 53 (6), pages 894-913, Wiley. doi:10.1002/bimj.201000239


parametric.test(pvalues=c(0.1,0.2,0.05), weights=c(0.5,0.5,0), correlation = diag(3))
#> [1] 0.19
parametric.test(pvalues=c(0.1,0.2,0.05), weights=c(0.5,0.5,0), correlation = diag(3),
adjPValues = FALSE)
#> [1] FALSE