online_fallback.RdImplements the online fallback procedure of Tian and Ramdas (2021), which guarantees strong FWER control under arbitrary dependence of the p-values.
online_fallback(
d,
alpha = 0.05,
gammai,
random = TRUE,
display_progress = FALSE,
date.format = "%Y-%m-%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 in sequence, arriving one at a time.
Overall significance level of the FDR procedure, the default is 0.05.
Optional vector of \(\gamma_i\). A default is provided as proposed by Javanmard and Montanari (2018), equation 31.
Logical. If TRUE (the default), then the order of the
p-values in each batch (i.e. those that have exactly the same date) is
randomised.
Logical. If TRUE prints out a progress bar for
the algorithm runtime.
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 in sequence, arriving one at a time. Given an overall significance level \(\alpha\), we choose a sequence of non-negative non-increasing numbers \(\gamma_i\) that sum to 1.
The online fallback procedure provides a uniformly more powerful method than Alpha-spending, by saving the significance level of a previous rejection. More specifically, the procedure 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 online fallback procedure can be found in Tian and Ramdas (2021).
Tian, J. and Ramdas, A. (2021). Online control of the familywise error rate. Statistical Methods for Medical Research, 30(4):976–993.
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))
online_fallback(sample.df, random=FALSE)
#> pval alphai R
#> 1 2.9000e-08 0.0026758385 1
#> 2 6.7430e-02 0.0032577488 0
#> 3 1.5140e-02 0.0004956249 0
#> 4 8.1740e-02 0.0004121803 0
#> 5 1.7100e-03 0.0003494435 0
#> 6 3.6000e-05 0.0003022950 1
#> 7 7.9149e-01 0.0005682672 0
#> 8 2.7201e-01 0.0002372613 0
#> 9 2.8295e-01 0.0002140474 0
#> 10 7.5900e-08 0.0001949126 1
#> 11 6.9274e-01 0.0003737922 0
#> 12 3.0443e-01 0.0001652568 0
#> 13 1.3600e-03 0.0001535420 0
#> 14 7.2342e-01 0.0001433627 0
#> 15 5.4757e-01 0.0001344368 0
set.seed(1); online_fallback(sample.df)
#> pval alphai R
#> 1 2.9000e-08 0.0026758385 1
#> 2 6.7430e-02 0.0032577488 0
#> 3 1.5140e-02 0.0004956249 0
#> 4 8.1740e-02 0.0004121803 0
#> 5 1.7100e-03 0.0003494435 0
#> 6 2.7201e-01 0.0003022950 0
#> 7 3.6000e-05 0.0002659722 1
#> 8 7.9149e-01 0.0005032335 0
#> 9 7.5900e-08 0.0002140474 1
#> 10 2.8295e-01 0.0004089600 0
#> 11 6.9274e-01 0.0001788796 0
#> 12 7.2342e-01 0.0001652568 0
#> 13 3.0443e-01 0.0001535420 0
#> 14 5.4757e-01 0.0001433627 0
#> 15 1.3600e-03 0.0001344368 0
set.seed(1); online_fallback(sample.df, alpha=0.1)
#> pval alphai R
#> 1 2.9000e-08 0.0053516771 1
#> 2 6.7430e-02 0.0065154977 0
#> 3 1.5140e-02 0.0009912499 0
#> 4 8.1740e-02 0.0008243606 0
#> 5 1.7100e-03 0.0006988870 0
#> 6 2.7201e-01 0.0006045900 0
#> 7 3.6000e-05 0.0005319444 1
#> 8 7.9149e-01 0.0010064670 0
#> 9 7.5900e-08 0.0004280949 1
#> 10 2.8295e-01 0.0008179201 0
#> 11 6.9274e-01 0.0003577593 0
#> 12 7.2342e-01 0.0003305137 0
#> 13 3.0443e-01 0.0003070841 0
#> 14 5.4757e-01 0.0002867254 0
#> 15 1.3600e-03 0.0002688736 0