ADDIS.spending.RdImplements the ADDIS algorithm for online FWER control, where ADDIS stands for an ADaptive algorithm that DIScards conservative nulls, as presented by Tian and Ramdas (2019b). The procedure compensates for the power loss of Alpha-spending, by including both adapativity in the fraction of null hypotheses and the conservativeness of nulls.
ADDIS.spending(pval, alpha = 0.05, gammai, lambda = 0.25, tau = 0.5, dep = FALSE, lags)
| pval | A vector of p-values. |
|---|---|
| alpha | Overall significance level of the procedure, the default is 0.05. |
| gammai | Optional vector of \(\gamma_i\). A default is provided with \(\gamma_j\) proportional to \(1/j^(1.6)\). |
| lambda | Optional parameter that sets the threshold for `candidate' hypotheses. Must be between 0 and 1, defaults to 0.25. |
| tau | Optional threshold for hypotheses to be selected for testing. Must be between 0 and 1, defaults to 0.5. |
| dep | Logical. If |
| lags | A vector of lags or the hypothesis tests, this is required if
|
A dataframe with the original p-values pval, the
adjusted testing levels \(\alpha_i\) and the indicator function of
discoveries R. Hypothesis \(i\) is rejected if the \(i\)-th
p-value is less than or equal to \(\alpha_i\), in which case R[i] =
1 (otherwise R[i] = 0).
The function takes as its input a vector of p-values. Given an overall significance level \(\alpha\), ADDIS depends on constants \(\lambda\) and \(\tau\), where \(\lambda < \tau\). 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-spending will never reject a p-value larger than \(\lambda\). The algorithms also require a sequence of non-negative non-increasing numbers \(\gamma_i\) that sum to 1.
The ADDIS-spending procedure provably controls the FWER in the strong sense for independent p-values. Note that the procedure also controls the generalised familywise error rate (k-FWER) for \(k > 1\) if \(\alpha\) is replaced by min(\(1, k\alpha\)).
Tian and Ramdas (2019b) also presented a version for handling local
dependence. More precisely, for any \(t>0\) we allow the p-value \(p_t\)
to have arbitrary dependence on the previous \(L_t\) p-values. The fixed
sequence \(L_t\) is referred to as `lags', and is given as the input
lags for this version of the ADDIS-spending algorithm.
Further details of the ADDIS-spending algorithms can be found in Tian and Ramdas (2019b).
Tian, J. and Ramdas, A. (2019b). Online control of the familywise error rate. arXiv preprint, https://arxiv.org/abs/1910.04900.
ADDIS provides online control of the FDR.
pval = c(2.90e-08, 0.06743, 3.51e-04, 0.0154, 0.04723, 3.60e-05, 0.79149, 0.27201, 0.28295, 7.59e-06, 0.69274, 0.30443, 0.00136, 0.82342, 5.4757e-04) ADDIS.spending(pval)#> pval alphai R #> 1 2.9000e-08 0.0054686271 1 #> 2 6.7430e-02 0.0054686271 0 #> 3 3.5100e-04 0.0054686271 1 #> 4 1.5400e-02 0.0054686271 0 #> 5 4.7230e-02 0.0054686271 0 #> 6 3.6000e-05 0.0054686271 1 #> 7 7.9149e-01 0.0054686271 0 #> 8 2.7201e-01 0.0054686271 0 #> 9 2.8295e-01 0.0018039742 0 #> 10 7.5900e-06 0.0009429405 1 #> 11 6.9274e-01 0.0009429405 0 #> 12 3.0443e-01 0.0009429405 0 #> 13 1.3600e-03 0.0005950895 0 #> 14 8.2342e-01 0.0005950895 0 #> 15 5.4757e-04 0.0005950895 1#> pval alphai R #> 1 2.9000e-08 0.0054686271 1 #> 2 6.7430e-02 0.0054686271 0 #> 3 3.5100e-04 0.0054686271 1 #> 4 1.5400e-02 0.0054686271 0 #> 5 4.7230e-02 0.0054686271 0 #> 6 3.6000e-05 0.0054686271 1 #> 7 7.9149e-01 0.0054686271 0 #> 8 2.7201e-01 0.0054686271 0 #> 9 2.8295e-01 0.0018039742 0 #> 10 7.5900e-06 0.0009429405 1 #> 11 6.9274e-01 0.0009429405 0 #> 12 3.0443e-01 0.0009429405 0 #> 13 1.3600e-03 0.0005950895 0 #> 14 8.2342e-01 0.0005950895 0 #> 15 5.4757e-04 0.0005950895 1#> pval alphai R #> 1 2.9000e-08 0.0054686271 1 #> 2 6.7430e-02 0.0018039742 0 #> 3 3.5100e-04 0.0018039742 1 #> 4 1.5400e-02 0.0018039742 0 #> 5 4.7230e-02 0.0018039742 0 #> 6 3.6000e-05 0.0018039742 1 #> 7 7.9149e-01 0.0018039742 0 #> 8 2.7201e-01 0.0018039742 0 #> 9 2.8295e-01 0.0018039742 0 #> 10 7.5900e-06 0.0009429405 1 #> 11 6.9274e-01 0.0005950895 0 #> 12 3.0443e-01 0.0005950895 0 #> 13 1.3600e-03 0.0005950895 0 #> 14 8.2342e-01 0.0004164149 0 #> 15 5.4757e-04 0.0004164149 0