LORD (dep): Online FDR control based on recent discovery for dependent p-values

This funcion is deprecated, please use LORD instead with version = 'dep'.

LORDdep(
  d,
  alpha = 0.05,
  xi,
  w0 = alpha/10,
  b0 = alpha - w0,
  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 in sequence, arriving one at a time.

alpha

Overall significance level of the FDR procedure, the default is 0.05.

xi

Optional vector of \(\xi_i\). A default is provided to satisfy the condition given in Javanmard and Montanari (2018), example 3.7.

w0

Initial `wealth' of the procedure. Defaults to \(\alpha/10\).

b0

The `payout' for rejecting a hypothesis. Defaults to \(\alpha - w_0\).

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.

Value

d.out

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).

Details

LORDdep implements the LORD procedure for online FDR control for dependent p-values, where LORD stands for (significance) Levels based On Recent Discovery, as presented by Javanmard and Montanari (2018).

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.

This modified LORD procedure controls FDR for dependent p-values. Given an overall significance level \(\alpha\), we choose a sequence of non-negative numbers \(\xi_i\) such that they satisfy a condition given in Javanmard and Montanari (2018), example 3.8.

The procedure depends on constants \(w_0\) and \(b_0\), where \(w_0 \ge 0\) represents the intial `wealth' and \(b_0 > 0\) represents the `payout' for rejecting a hypothesis. We require \(w_0+b_0 \le \alpha\) for FDR control to hold.

Further details of the modified LORD procedure can be found in Javanmard and Montanari (2018).

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.