AlphaSaving.RdImplements the Alpha-saving algorithm of Tian and Ramdas (2019b), which guarantees strong FWER control under arbitrary dependence of the p-values.
AlphaSaving(d, alpha = 0.05, gammai, random = TRUE, date.format = "%Y-%m-%d")
| d | Either a vector of p-values, or a dataframe with three columns: an identifier (`id'), date (`date') and p-value (`pval'). If no column of dates is provided, then the p-values are treated as being ordered sequentially with no batches. |
|---|---|
| alpha | Overall significance level of the FDR procedure, the default is 0.05. |
| gammai | Optional vector of \(\gamma_i\). A default is provided with \(\gamma_j\) proportional to \(1/j^(1.6)\). |
| random | Logical. If |
| date.format | Optional string giving the format that is used for dates. |
A dataframe with the original data d (which will
be reordered if there are batches and random = TRUE), the
LORD-adjusted significance thresholds \(\alpha_i\) 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 either a vector of p-values or a dataframe with three columns: an identifier (`id'), date (`date') and p-value (`pval'). The case where p-values arrive in batches corresponds to multiple instances of the same date. If no column of dates is provided, then the p-values are treated as being ordered sequentially with no batches. Given an overall significance level \(\alpha\), we choose a sequence of non-negative non-increasing numbers \(\gamma_i\) that sum to 1.
The Alpha-saving procedure provides a uniformly more powerful method than Alpha-spending, by saving the significance level of a previous rejection. More specifically, Alpha-saving tests hypothesis \(H_i\) at level $$\alpha_i = \alpha \gamma_i + R_{i-1} \alpha_{i-1}$$ where \(R_i = 1\{p_i \leq \alpha_i\}\) denotes a rejected hypothesis.
Further details of the Alpha-saving procedure 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.
sample.df <- data.frame( id = c('A15432', 'B90969', 'C18705', 'B49731', 'E99902', 'C38292', 'A30619', 'D46627', 'E29198', 'A41418', 'D51456', 'C88669', 'E03673', 'A63155', 'B66033'), date = as.Date(c(rep("2014-12-01",3), rep("2015-09-21",5), rep("2016-05-19",2), "2016-11-12", rep("2017-03-27",4))), pval = c(2.90e-08, 0.06743, 0.01514, 0.08174, 0.00171, 3.60e-05, 0.79149, 0.27201, 0.28295, 7.59e-08, 0.69274, 0.30443, 0.00136, 0.72342, 0.54757)) AlphaSaving(sample.df, random=FALSE)#> id date pval alphai R #> 1 A15432 2014-12-01 2.9000e-08 0.0026758385 1 #> 2 B90969 2014-12-01 6.7430e-02 0.0032577488 0 #> 3 C18705 2014-12-01 1.5140e-02 0.0004956249 0 #> 4 B49731 2015-09-21 8.1740e-02 0.0004121803 0 #> 5 E99902 2015-09-21 1.7100e-03 0.0003494435 0 #> 6 C38292 2015-09-21 3.6000e-05 0.0003022950 1 #> 7 A30619 2015-09-21 7.9149e-01 0.0005682672 0 #> 8 D46627 2015-09-21 2.7201e-01 0.0002372613 0 #> 9 E29198 2016-05-19 2.8295e-01 0.0002140474 0 #> 10 A41418 2016-05-19 7.5900e-08 0.0001949126 1 #> 11 D51456 2016-11-12 6.9274e-01 0.0003737922 0 #> 12 C88669 2017-03-27 3.0443e-01 0.0001652568 0 #> 13 E03673 2017-03-27 1.3600e-03 0.0001535420 0 #> 14 A63155 2017-03-27 7.2342e-01 0.0001433627 0 #> 15 B66033 2017-03-27 5.4757e-01 0.0001344368 0#> id date pval alphai R #> 1 A15432 2014-12-01 2.9000e-08 0.0026758385 1 #> 2 B90969 2014-12-01 6.7430e-02 0.0032577488 0 #> 3 C18705 2014-12-01 1.5140e-02 0.0004956249 0 #> 4 B49731 2015-09-21 8.1740e-02 0.0004121803 0 #> 5 E99902 2015-09-21 1.7100e-03 0.0003494435 0 #> 6 D46627 2015-09-21 2.7201e-01 0.0003022950 0 #> 7 C38292 2015-09-21 3.6000e-05 0.0002659722 1 #> 8 A30619 2015-09-21 7.9149e-01 0.0005032335 0 #> 9 A41418 2016-05-19 7.5900e-08 0.0002140474 1 #> 10 E29198 2016-05-19 2.8295e-01 0.0004089600 0 #> 11 D51456 2016-11-12 6.9274e-01 0.0001788796 0 #> 12 A63155 2017-03-27 7.2342e-01 0.0001652568 0 #> 13 C88669 2017-03-27 3.0443e-01 0.0001535420 0 #> 14 B66033 2017-03-27 5.4757e-01 0.0001433627 0 #> 15 E03673 2017-03-27 1.3600e-03 0.0001344368 0#> id date pval alphai R #> 1 A15432 2014-12-01 2.9000e-08 0.0053516771 1 #> 2 B90969 2014-12-01 6.7430e-02 0.0065154977 0 #> 3 C18705 2014-12-01 1.5140e-02 0.0009912499 0 #> 4 B49731 2015-09-21 8.1740e-02 0.0008243606 0 #> 5 E99902 2015-09-21 1.7100e-03 0.0006988870 0 #> 6 D46627 2015-09-21 2.7201e-01 0.0006045900 0 #> 7 C38292 2015-09-21 3.6000e-05 0.0005319444 1 #> 8 A30619 2015-09-21 7.9149e-01 0.0010064670 0 #> 9 A41418 2016-05-19 7.5900e-08 0.0004280949 1 #> 10 E29198 2016-05-19 2.8295e-01 0.0008179201 0 #> 11 D51456 2016-11-12 6.9274e-01 0.0003577593 0 #> 12 A63155 2017-03-27 7.2342e-01 0.0003305137 0 #> 13 C88669 2017-03-27 3.0443e-01 0.0003070841 0 #> 14 B66033 2017-03-27 5.4757e-01 0.0002867254 0 #> 15 E03673 2017-03-27 1.3600e-03 0.0002688736 0