online_fallback_code.md

title	author	date	output	html_document
Online_fallback_code	Lathan Liou	12/16/2020		self_contained

Online fallback procedure for FWER control

Description

Implements the online fallback procedure of Tian and Ramdas (2021), which guarantees strong FWER control under arbitrary dependence of the p-values.

Usage

 online_fallback(   d,   alpha = 0.05,   gammai,   random = TRUE,   display_progress = FALSE,   date.format = "%Y-%m-%d" ) 

Arguments

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 γ_i. A default is provided as proposed by Javanmard and Montanari (2018), equation 31.
random	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.
display_progress	Logical. If TRUE prints out a progress bar for the algorithm runtime.
date.format	Optional string giving the format that is used for dates.

Details

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 α, we choose a sequence of non-negative non-increasing numbers γ_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

α_i = α γ_i + R_{i-1} α_{i-1}

where R_i = 1\{p_i ≤q α_i\} denotes a rejected hypothesis.

Further details of the online fallback procedure can be found in Tian and Ramdas (2021).

Value

out

A dataframe with the original data d (which will be reordered if there are batches and random = TRUE), the LORD-adjusted significance thresholds α_i and the indicator function of discoveries R. Hypothesis i is rejected if the i-th p-value is less than or equal to α_i, in which case R[i] = 1 (otherwise R[i] = 0).

References

Tian, J. and Ramdas, A. (2021). Online control of the familywise error rate. Statistical Methods for Medical Research (to appear), https://arxiv.org/abs/1910.04900.

Examples

 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)  set.seed(1); online_fallback(sample.df)  set.seed(1); online_fallback(sample.df, alpha=0.1)