Alpha_spending_code.md

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

Alpha-spending for online FWER control

Description

Implements online FWER control using a Bonferroni-like test.

Usage

 Alpha_spending(   d,   alpha = 0.05,   gammai,   random = TRUE,   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, where hypothesis i is rejected if the i-th p-value is less than or equal to α 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.
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.

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 α, we choose a (potentially infinite) sequence of non-negative numbers γ_i such that they sum to 1. Hypothesis i is rejected if the i-th p-value is less than or equal to α γ_i.

Note that the procedure controls the generalised familywise error rate (k-FWER) for k > 1 if α is replaced by min(1, kα).

Value

out	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).

References

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. (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-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); Alpha_spending(sample.df)  Alpha_spending(sample.df, random=FALSE)  set.seed(1); Alpha_spending(sample.df, alpha=0.1)