LOND: Online FDR control based on number of discoveries

Implements the LOND algorithm for online FDR control, where LOND stands for (significance) Levels based On Number of Discoveries, as presented by Javanmard and Montanari (2015).

LOND(
  d,
  alpha = 0.05,
  betai,
  dep = FALSE,
  random = TRUE,
  display_progress = FALSE,
  date.format = "%Y-%m-%d",
  original = TRUE
)

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.
betai: Optional vector of $\beta_i$. A default is provided as proposed by Javanmard and Montanari (2018), equation 31.
dep: Logical. If TRUE, runs the modified LOND algorithm which guarantees FDR control for dependent p-values. Defaults to FALSE.
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.
original: Logical. If TRUE, runs the original LOND algorithm of Javanmard and Montanari (2015), otherwise runs the modified algorithm of Zrnic et al. (2018). Defaults to TRUE.

Value

out: A dataframe with the original data d (which will be reordered if there are batches and random = TRUE), the LOND-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

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.

The LOND algorithm controls the FDR for independent p-values (see below for the modification for dependent p-values). Given an overall significance level $\alpha$, we choose a sequence of non-negative numbers $\beta_i$ such that they sum to $\alpha$. The values of the adjusted significance thresholds $\alpha_i$ are chosen as follows: $$\alpha_i = (D(i-1) + 1)\beta_i$$ where $D(n)$ denotes the number of discoveries in the first $n$ hypotheses.

A slightly modified version of LOND with thresholds $\alpha_i = max(D(i-1), 1)\beta_i$ provably controls the FDR under positive dependence (PRDS condition), see Zrnic et al. (2021).

For arbitrarily dependent p-values, LOND controls the FDR if it is modified with $\beta_i / H(i)$ in place of $\beta_i$, where $H(j)$ is the i-th harmonic number.

Further details of the LOND algorithm can be found in Javanmard and Montanari (2015).

References

Javanmard, A. and Montanari, A. (2015) On Online Control of False Discovery Rate. arXiv preprint, https://arxiv.org/abs/1502.06197.

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.

Zrnic, T., Ramdas, A. and Jordan, M.I. (2021). Asynchronous Online Testing of Multiple Hypotheses. Journal of Machine Learning Research (to appear), https://arxiv.org/abs/1812.05068.

Examples