LORDstar: LORDstar: Asynchronous online mFDR control based on recent...
In onlineFDR: Online error control

Description Usage Arguments Details Value References See Also Examples

Implements LORD algorithms for asynchronous online testing, as presented by Zrnic et al. (2018).

1	LORDstar(d, alpha = 0.05, version, gammai, w0, batch.sizes)

`d`	Either a vector of p-values, or a dataframe with three columns: an identifier (‘id’), p-value (‘pval’), and either ‘decision.times’, or ‘lags’, depending on which version you're using. See version for more details.
`alpha`	Overall significance level of the procedure, the default is 0.05.
`version`	Takes values 'async', 'dep' or 'batch'. This specifies the version of LORDstar to use. `version='async'` requires a column of decision times (‘decision.times’). `version='dep'` requires a column of lags (‘lags’). `version='batch'` requires a vector of batch sizes (‘batch.sizes’).
`gammai`	Optional vector of γ_i. A default is provided as proposed by Javanmard and Montanari (2018), equation 31.
`w0`	Initial ‘wealth’ of the procedure, defaults to α/10.
`batch.sizes`	A vector of batch sizes, this is required for `version='batch'`.

The function takes as its input either a vector of p-values, or a dataframe with three columns: an identifier (‘id’), p-value (‘pval’), or a column describing the conflict sets for the hypotheses. This takes the form of a vector of decision times or lags. Batch sizes can be specified as a separate argument (see below).

Zrnic et al. (2018) present explicit three versions of LORDstar:

1) version='async' is for an asynchronous testing process, consisting of tests that start and finish at (potentially) random times. The discretised finish times of the test correspond to the decision times. These decision times are given as the input decision.times for this version of the LORDstar algorithm.

2) version='dep' is for online testing under local dependence of the p-values. More precisely, for any t>0 we allow the p-value p_t to have arbitrary dependence on the previous L_t p-values. The fixed sequence L_t is referred to as ‘lags’, and is given as the input lags for this version of the LORDstar algorithm.

3) version='batch' is for controlling the mFDR in mini-batch testing, where a mini-batch represents a grouping of tests run asynchronously which result in dependent p-values. Once a mini-batch of tests is fully completed, a new one can start, testing hypotheses independent of the previous batch. The batch sizes are given as the input batch.sizes for this version of the LORDstar algorithm.

Given an overall significance level α, LORDstar depends on w_0 (where 0 ≤ w_0 ≤ α), which represents the intial ‘wealth’ of the procedure. The algorithms also require a sequence of non-negative non-increasing numbers γ_i that sum to 1.

Note that these LORDstar algorithms control the modified FDR (mFDR). The ‘async’ version also controls the usual FDR if the p-values are assumed to be independent.

Further details of the LORDstar algorithms can be found in Zrnic et al. (2018).

d.out

A dataframe with the original p-values pval, the adjusted testing levels 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).

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. (2018). Asynchronous Online Testing of Multiple Hypotheses. arXiv preprint, https://arxiv.org/abs/1812.05068.

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

sample.df <- data.frame(
id = c('A15432', 'B90969', 'C18705', 'B49731', 'E99902',
    'C38292', 'A30619', 'D46627', 'E29198', 'A41418',
    'D51456', 'C88669', 'E03673', 'A63155', 'B66033'),
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),
decision.times = seq_len(15) + 1)

LORDstar(sample.df, version='async')

sample.df2 <- data.frame(
id = c('A15432', 'B90969', 'C18705', 'B49731', 'E99902',
    'C38292', 'A30619', 'D46627', 'E29198', 'A41418',
    'D51456', 'C88669', 'E03673', 'A63155', 'B66033'),
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),
lags = rep(1,15))

LORDstar(sample.df2, version='dep')

sample.df3 <- data.frame(
id = c('A15432', 'B90969', 'C18705', 'B49731', 'E99902',
    'C38292', 'A30619', 'D46627', 'E29198', 'A41418',
    'D51456', 'C88669', 'E03673', 'A63155', 'B66033'),
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))

LORDstar(sample.df3, version='batch', batch.sizes = c(4,6,5))