online_fallback {onlineFDR} | R Documentation

## 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)

[Package onlineFDR version 2.2.0 Index]