Alpha_investing.RdImplements a variant of the Alpha-investing algorithm of Foster and Stine (2008) that guarantees FDR control, as proposed by Ramdas et al. (2018). This procedure uses SAFFRON's update rule with the constant \(\lambda\) replaced by a sequence \(\lambda_i = \alpha_i\). This is also equivalent to using the ADDIS algorithm with \(\tau = 1\) and \(\lambda_i = \alpha_i\).
Alpha_investing(
d,
alpha = 0.05,
gammai,
w0,
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.
Overall significance level of the FDR procedure, the default is 0.05.
Optional vector of \(\gamma_i\). A default is provided with \(\gamma_j\) proportional to \(1/j^(1.6)\).
Initial `wealth' of the procedure, defaults to \(\alpha/2\). Must be between 0 and \(\alpha\).
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.
The Alpha-investing procedure provably controls FDR for independent p-values. Given an overall significance level \(\alpha\), we choose a sequence of non-negative non-increasing numbers \(\gamma_i\) that sum to 1. Alpha-investing depends on a constant \(w_0\), which satisfies \(0 \le w_0 \le \alpha\) and represents the initial `wealth' of the procedure.
Further details of the Alpha-investing procedure and its modification can be found in Foster and Stine (2008) and Ramdas et al. (2018).
Foster, D. and Stine R. (2008). \(\alpha\)-investing: a procedure for sequential control of expected false discoveries. Journal of the Royal Statistical Society (Series B), 29(4):429-444.
Ramdas, A., Zrnic, T., Wainwright M.J. and Jordan, M.I. (2018). SAFFRON: an adaptive algorithm for online control of the false discovery rate. Proceedings of the 35th International Conference in Machine Learning, 80:4286-4294.
SAFFRON uses the update rule of Alpha-investing but with
constant \(\lambda\).
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))
Alpha_investing(sample.df, random=FALSE)
#> pval alphai R
#> 1 2.9000e-08 0.010818925 1
#> 2 6.7430e-02 0.021406257 0
#> 3 1.5140e-02 0.007164201 0
#> 4 8.1740e-02 0.003757589 0
#> 5 1.7100e-03 0.002374706 1
#> 6 3.6000e-05 0.023680499 1
#> 7 7.9149e-01 0.044095287 0
#> 8 2.7201e-01 0.015842430 0
#> 9 2.8295e-01 0.008711190 0
#> 10 7.5900e-08 0.005700295 1
#> 11 6.9274e-01 0.026865786 0
#> 12 3.0443e-01 0.011205456 0
#> 13 1.3600e-03 0.006863124 1
#> 14 7.2342e-01 0.027979664 0
#> 15 5.4757e-01 0.011945806 0
set.seed(1); Alpha_investing(sample.df)
#> pval alphai R
#> 1 2.9000e-08 0.010818925 1
#> 2 6.7430e-02 0.021406257 0
#> 3 1.5140e-02 0.007164201 0
#> 4 8.1740e-02 0.003757589 0
#> 5 1.7100e-03 0.002374706 1
#> 6 2.7201e-01 0.023680499 0
#> 7 3.6000e-05 0.008803369 1
#> 8 7.9149e-01 0.029838354 0
#> 9 7.5900e-08 0.012084065 1
#> 10 2.8295e-01 0.032981505 0
#> 11 6.9274e-01 0.014137539 0
#> 12 7.2342e-01 0.008529625 0
#> 13 3.0443e-01 0.005905508 0
#> 14 5.4757e-01 0.004412045 0
#> 15 1.3600e-03 0.003461425 1
set.seed(1); Alpha_investing(sample.df, alpha=0.1, w0=0.025)
#> pval alphai R
#> 1 2.9000e-08 0.010818925 1
#> 2 6.7430e-02 0.041915265 0
#> 3 1.5140e-02 0.014226480 0
#> 4 8.1740e-02 0.007487045 0
#> 5 1.7100e-03 0.004738159 1
#> 6 2.7201e-01 0.046265410 0
#> 7 3.6000e-05 0.017453092 1
#> 8 7.9149e-01 0.057947646 0
#> 9 7.5900e-08 0.023879569 1
#> 10 2.8295e-01 0.063856912 0
#> 11 6.9274e-01 0.027880910 0
#> 12 7.2342e-01 0.016914972 0
#> 13 3.0443e-01 0.011741675 0
#> 14 5.4757e-01 0.008785330 0
#> 15 1.3600e-03 0.006898971 1