LOND {onlineFDR} | R Documentation

## LOND: Online FDR control based on number of discoveries

### Description

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).

### Usage

    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 sequentially with no batches.
`alpha` | Overall significance level of the FDR procedure, the default is 0.05.
`betai` | Optional vector of β_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`.

### 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.

The LOND algorithm controls the FDR for independent p-values (see below for the modification for dependent p-values). Given an overall significance level α, we choose a sequence of non-negative numbers β_i such that they sum to α. The values of the adjusted significance thresholds α_i are chosen as follows:

α_i = (D(i-1) + 1)β_i

where D(n) denotes the number of discoveries in the first n hypotheses.

A slightly modified version of LOND with thresholds α_i = max(D(i-1), 1)β_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 β_i / H(i) in place of β_i, where H(j) is the i-th harmonic number.

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

### 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 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

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.

### See Also

`LONDstar` presents versions of LORD for synchronous p-values, i.e. where each test can only start when the previous test has finished.

### 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))

    set.seed(1); LOND(sample.df)

    LOND(sample.df, random=FALSE)

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

[Package onlineFDR version 2.2.0 Index]