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Post hoc bound on the number of false positives

Source: R/tree-functions.R
V.star.Rd

Computes the post hoc upper bound \(V^*(S)\) on the number of false positives in a given selection set \(S\) of hypotheses, using a reference family \((R_k, \zeta_k)\) that possess the forest structure (see Reference).

Usage

V.star(S, C, ZL, leaf_list)

Arguments

S

An integer vector with the indices of the hypotheses of the selection set. Does not need to be ordered.

C

A list of list representing the forest structure. Each list of C represents a level of depth in the forest structure. So, C[[1]] lists the regions at depth 1, C[[2]] lists the regions at depth 2, and so on. We then use the fact that each region of the reference family is the union of some of its atom over an integer interval. So he elements of each list C[[i]] are integer vectors of length 2 representing this interval. For example, C[[1]][[1]] = c(1, 5) means that the first region at depth 1 is the union of the first 5 atoms, C[[2]][[3]] = c(4, 5) means that the third region at depth 2 is the union of the atoms 4 and 5, and C[[3]][[5]] = c(5, 5) means that the fifth region at depth 3 is simply the fifth atom.

ZL

A list of integer vectors representing the upper bound \(\zeta_k\) associated to a region \(R_k\) in the reference family. ZL[[h]][j] is the \(\zeta_k\) associated to the \(R_k\) described by C[[h]][[j]].

leaf_list

A list of vectors. Each vector is an integer array. The i-th vector contains the indices of the hypotheses in the i-th atom. Atoms form a partition of the set of hypotheses indices : there cannot be overlap, and each index has to be inside one of the atoms.

Value

An integer value, the post hoc upper bound \(V^*(S)\).

Details

For V.star, the forest structure doesn't need to be complete. That is, in C, some trivial intervals c(i,i) corresponding to regions that are atoms may be missing.

References

Durand, G., Blanchard, G., Neuvial, P., & Roquain, E. (2020). Post hoc false positive control for structured hypotheses. Scandinavian Journal of Statistics, 47(4), 1114-1148.

Examples

m <- 20
C <- list(
  list(c(2, 5), c(8, 15), c(16, 19)),
  list(c(3, 5), c(8, 10), c(12, 15), c(16, 16), c(17, 19)),
  list(c(4, 5), c(8, 9), c(10, 10), c(12, 12), c(13, 15), c(17, 17), c(18, 19)),
  list(c(8, 8), c(9, 9), c(13, 13), c(14, 15), c(18, 18), c(19, 19))
)
ZL <- list(
  c(4, 8, 4),
  c(3, 3, 4, 1, 3),
  c(2, 2, 1, 1, 2, 1, 2),
  c(1, 1, 1, 2, 1, 1)
)
leaf_list <- as.list(1:m)
V.star(1, C, ZL, leaf_list)
#> [1] 1

V.star(1:5, C, ZL, leaf_list)
#> [1] 5

V.star(13:15, C, ZL, leaf_list)
#> [1] 2

V.star(1:m, C, ZL, leaf_list)
#> [1] 19