Alpha_investing {onlineFDR} | R Documentation

## Alpha-investing for online FDR control

### Description

Implements a variant of the Alpha-investing algorithm of Foster and Stine (2008) that guarantees FDR control, as proposed by Ramdas et al. (2018). This procedure uses SAFFRON's update rule with the constant λ replaced by a sequence λ_i = α_i. This is also equivalent to using the ADDIS algorithm with τ = 1 and λ_i = α_i.

### Usage

    Alpha_investing(
      d,
      alpha = 0.05,
      gammai,
      w0,
      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 with γ_j proportional to 1/j^(1.6).
`w0` | Initial ‘wealth’ of the procedure, defaults to α/2. Must be between 0 and α.
`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.

The Alpha-investing procedure provably controls FDR for independent p-values. Given an overall significance level α, we choose a sequence of non-negative non-increasing numbers γ_i that sum to 1. Alpha-investing depends on a constant w_0, which satisfies 0 ≤ w_0 ≤ α and represents the initial ‘wealth’ of the procedure.

Further details of the Alpha-investing procedure and its modification can be found in Foster and Stine (2008) and Ramdas et al. (2018).

### 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

Foster, D. and Stine R. (2008). α-investing: a procedure for sequential control of expected false discoveries. Journal of the Royal Statistical Society (Series B), 29(4):429-444.

Ramdas, A., Zrnic, T., Wainwright M.J. and Jordan, M.I. (2018). SAFFRON: an adaptive algorithm for online control of the false discovery rate. Proceedings of the 35th International Conference in Machine Learning, 80:4286-4294.

### See Also

`SAFFRON` uses the update rule of Alpha-investing but with constant λ.

### 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))

    Alpha_investing(sample.df, random=FALSE)

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

    set.seed(1); Alpha_investing(sample.df, alpha=0.1, w0=0.025)

[Package onlineFDR version 2.2.0 Index]