ADDIS-exhaustive: Exhaustive ADDIS-spending procedure for online FWER control
ADDIS_exhaustive.RdImplements an exhaustive variant of the ADDIS-spending algorithm for online FWER control, as presented by Fischer et al. (2024). The procedure is a uniform improvement of ADDIS-spending, and, under the conditions of their Theorem 3.5, no other FWER controlling procedure can enlarge the event of rejecting any hypothesis.
Arguments
- d
Either a vector of p-values, or a dataframe with at least a
pvalcolumn (and optionallyid).- alpha
Overall significance level of the procedure, default 0.05.
- tau
Optional threshold for hypotheses to be selected for testing. Must be between 0 and 1, defaults to 0.5.
- lambda
Optional parameter that sets the threshold for `candidate' hypotheses. Must be at least alpha * tau and less than tau, defaults to 0.25.
- gamma
Optional vector of initial weights, with at least one more element than the number of p-values and summing to at most 1. If
NULL(the default), a decreasing sequence proportional to \(j^{-1.6}\) is used, as inADDIS.
Value
A dataframe with the original p-values pval, the
per-hypothesis testing levels alphai, and the indicator of
discoveries R.
Details
The function takes as its input either a vector of p-values, or a dataframe with a column of p-values (`pval') and optionally an identifier (`id'). Given an overall significance level \(\alpha\), ADDIS-exhaustive depends on constants \(\lambda\) and \(\tau\), where \(\alpha\tau \le \lambda < \tau\). Here \(\tau \in (0,1]\) represents the threshold for a hypothesis to be selected for testing: p-values greater than \(\tau\) are implicitly `discarded' by the procedure, while \(\lambda \in (0,1)\) sets the threshold for a p-value to be a candidate for rejection: ADDIS-exhaustive will never reject a p-value larger than \(\lambda\). The algorithm also requires a sequence of non-negative numbers \(\gamma_i\) that sum to at most 1.
The ADDIS-exhaustive procedure provably controls the FWER in the strong sense
for independent p-values. FWER control is not guaranteed for positively
dependent p-values; Alpha_spending and
online_fallback control the FWER under arbitrary dependence.