# sumPvals: True Discovery Guarantee for p-Value Combinations In sumSome: Permutation True Discovery Guarantee by Sum-Based Tests

## Description

This function determines confidence bounds for the number of true discoveries, the true discovery proportion and the false discovery proportion within a set of interest, when using p-values as test statistics. The bounds are simultaneous over all sets, and remain valid under post-hoc selection.

## Usage

 1 2 sumPvals(G, S = NULL, alpha = 0.05, truncFrom = NULL, truncTo = 0.5, type = "vovk.wang", r = 0, nMax = 50)

## Arguments

 G numeric matrix of p-values, where columns correspond to variables, and rows to data transformations (e.g. permutations). The first transformation is the identity. S vector of indices for the variables of interest (if not specified, all variables). alpha significance level. truncFrom truncation parameter: values greater than truncFrom are truncated. If NULL, it is set to alpha. truncTo truncation parameter: truncated values are set to truncTo. If NULL, p-values are not truncated. type p-value combination among edgington, fisher, pearson, liptak, cauchy, vovk.wang (see details). r parameter for Vovk and Wang's p-value combination. nMax maximum number of iterations.

## Details

A p-value p is transformed as following.

• Edgington: -p

• Fisher: -log(p)

• Pearson: log(1-p)

• Liptak: -qnorm(p)

• Cauchy: tan(0.5 - p)/p

• Vovk and Wang: - sign(r)p^r

An error message is returned if the transformation produces infinite values.

Truncation parameters should be such that truncTo is not smaller than truncFrom. As Pearson's and Liptak's transformations produce infinite values in 1, for such methods truncTo should be strictly smaller than 1.

The significance level alpha should be in the interval [1/B, 1), where B is the number of data transformations (rows in G).

## Value

sumPvals returns an object of class sumObj, containing

• total: total number of variables (columns in G)

• size: size of S

• alpha: significance level

• TD: lower (1-alpha)-confidence bound for the number of true discoveries in S

• maxTD: maximum value of TD that could be found under convergence of the algorithm

• iterations: number of iterations of the algorithm

Anna Vesely.

## References

Goeman, J. J. and Solari, A. (2011). Multiple testing for exploratory research. Statistical Science, 26(4):584-597.

Hemerik, J. and Goeman, J. J. (2018). False discovery proportion estimation by permutations: confidence for significance analysis of microarrays. JRSS B, 80(1):137-155.

Vesely, A., Finos, L., and Goeman, J. J. (2020). Permutation-based true discovery guarantee by sum tests. Pre-print arXiv:2102.11759.