AlphaSpending.RdImplements online FWER control using a Bonferroni-like test.
AlphaSpending(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 \(\gamma_i\), where hypothesis \(i\) is rejected if the \(i\)-th p-value is less than or equal to \(\alpha \gamma_i\). A default is provided as proposed by Javanmard and Montanari (2018), equation 31. |
| 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 adjusted
signifcance thresholds alphai and the indicator function of
discoveries R, where R[i] = 1 corresponds to hypothesis
\(i\) being rejected (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.
Alpha-spending provides strong FWER control for a potentially infinite stream of p-values by using a Bonferroni-like test. Given an overall significance level \(\alpha\), we choose a (potentially infinite) sequence of non-negative numbers \(\gamma_i\) such that they sum to 1. Hypothesis \(i\) is rejected if the \(i\)-th p-value is less than or equal to \(\alpha \gamma_i\).
Note that the procedure controls the generalised familywise error rate (k-FWER) for \(k > 1\) if \(\alpha\) is replaced by min(\(1, k\alpha\)).
Javanmard, A. and Montanari, A. (2018) Online Rules for Control of False Discovery Rate and False Discovery Exceedance. Annals of Statistics, 46(2):526-554.
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-17, 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)) set.seed(1); AlphaSpending(sample.df)#> id date pval alpha...gammai R #> 1 A15432 2014-12-01 2.9000e-17 0.0026758385 1 #> 2 B90969 2014-12-01 6.7430e-02 0.0005819103 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 0.0002372613 0 #> 9 A41418 2016-05-19 7.5900e-08 0.0002140474 1 #> 10 E29198 2016-05-19 2.8295e-01 0.0001949126 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 0AlphaSpending(sample.df, random=FALSE)#> id date pval alpha...gammai R #> 1 A15432 2014-12-01 2.9000e-17 0.0026758385 1 #> 2 B90969 2014-12-01 6.7430e-02 0.0005819103 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 0.0002659722 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 0.0001788796 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 alpha...gammai R #> 1 A15432 2014-12-01 2.9000e-17 0.0053516771 1 #> 2 B90969 2014-12-01 6.7430e-02 0.0011638206 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 0.0004745225 0 #> 9 A41418 2016-05-19 7.5900e-08 0.0004280949 1 #> 10 E29198 2016-05-19 2.8295e-01 0.0003898252 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