# 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.

True discovery guarantee using generic statistics: `sumStats`
Access a `sumObj` object: `discoveries`, `tdp`, `fdp`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```# generate matrix of p-values for 5 variables and 10 permutations G <- simData(prop = 0.6, m = 5, B = 10, alpha = 0.4, seed = 42) # subset of interest (variables 1 and 2) S <- c(1,2) # create object of class sumObj # combination: harmonic mean (Vovk and Wang with r = -1) res <- sumPvals(G, S, alpha = 0.4, r = -1) res summary(res) # lower confidence bound for the number of true discoveries in S discoveries(res) # lower confidence bound for the true discovery proportion in S tdp(res) # upper confidence bound for the false discovery proportion in S fdp(res) ```