## Brief Background of the onlineFDR algorithms

Javanmard and Montanari proposed two procedures, LOND and LORD, to control the FDR in an online manner (Javanmard and Montanari (2015, 2018)), with the latter extended by Ramdas et al. (2017). The LOND procedure sets the adjusted significance thresholds based on the number of discoveries made so far, while LORD sets them according to the time of the most recent discovery. Ramdas et al. (2018) then proposed the SAFFRON procedure, which provides an adaptive method of online FDR control. They also proposed a variant of the Alpha-investing algorithm of Foster and Stine (2008) that guarantees FDR control, using SAFFRON’s update rule.

Subsequently, Zrnic et al. (2018) proposed procedures to control the modified FDR (mFDR) in the context of asynchronous testing, i.e. where each hypothesis test can itself be a sequential process and the tests can overlap in time. They presented asynchronous versions of the LOND, LORD and SAFFRON procedures for a variety of trial settings. For both synchronous and asynchronous testing, Tian & Ramdas (2019a) proposed the ADDIS algorithms which compensate for the loss in power in the presence of conservative nulls by adaptively ‘discarding’ these p-values.

Finally, Tian & Ramdas (2019b) proposed procedures that provide online control of the FWER. One procedure, online fallback, gives a uniform improvement to the naive Alpha-spending procedure (see below). The ADDIS-spending procedure compensates for the power loss of these procedures by including both adapativity in the fraction of null hypotheses and the conservativeness of nulls.

## Variations to the default options

In the following section, we consider the arguments that a typical user might consider amending for their analysis.

### Common arguments

As a default, the alpha argument is set to 0.05, where alpha sets the overall significance level of the FDR of FWER controlling procedure. By convention, the standard significance level utilised is the 5%. However, there are applications where an alternate threshold could be considered. For example, a more stringent threshold might be appropriate when there are limited resources to follow up significant findings. A less stringent threshold might be appropriate when the downstream analysis is a global analysis which can tolerate a higher proportion of false positives.

To ensure correct interpretation of the dates provided there is a date.format argument. As a default, the date format is set to receive dates as year-month(00-12)-day(number). The following website provides clear guidance on symbols used to interpret the date information: https://www.statmethods.net/input/dates.html

As a default, the random argument is set to TRUE. In this situation, the order of p-values in each batch (i.e. with the same date) are randomised. This is to avoid the risk of p-values being ordered post-hoc, which can lead to an inflation of the FDR. As the dataset grows the data is reprocessed. To ensure the consistency of the output (with the randomisation within the previous batches remaining the same), it is necessary to set the same seed for all analyses.

The user also has the option to turn off the randomisation step, by setting the random argument to FALSE. This approach would be appropriate if the user has both a date and a time stamp for the p-values, in which case the data should be ordered by date and time beforehand and then passed to a wrapper function. Another scenario would be when p-values within the batches are ordered using independent side information, so that hypotheses most likely to be rejected come first, which would potentially increase the power of the procedure (see Javanmard and Montanari (2018) and Li and Barber (2017)).

### LOND

As a default, the dep argument is set to FALSE. Alternatively, this can be set to TRUE and will implement the LOND procedure to guarantee FDR control for arbitrarily dependent p-values. This method will in general be more conservative.

set.seed(1); results.indep <- LOND(sample.df)    # for independent p-values
set.seed(1); results.dep <- LOND(sample.df, dep=TRUE)   # for dependent p-values

cbind(independent = results.indep$alphai, dependent = results.dep$alphai)
#>        independent    dependent
#>  [1,] 0.0026758385 0.0026758385
#>  [2,] 0.0011638206 0.0007758804
#>  [3,] 0.0009912499 0.0005406818
#>  [4,] 0.0008243606 0.0003956931
#>  [5,] 0.0006988870 0.0003060819
#>  [6,] 0.0006045900 0.0002467714
#>  [7,] 0.0005319444 0.0002051576
#>  [8,] 0.0007117838 0.0002618915
#>  [9,] 0.0006421423 0.0002269882
#> [10,] 0.0007796504 0.0002661860
#> [11,] 0.0007155186 0.0002369363
#> [12,] 0.0006610273 0.0002130140
#> [13,] 0.0006141682 0.0001931265
#> [14,] 0.0005734509 0.0001763616
#> [15,] 0.0005377472 0.0001620585

The vector betai is supplied by default, but can optionally be specified by the user (as described above, see the formula for $$\beta_j$$ here).

### LORD

The default version of LORD used is version ‘++’, but the user can optionally specify versions 3, ‘discard’ and ‘dep’ using the version argument (see here for further details about the different versions).

set.seed(1); results.LORD.plus <- LORD(sample.df)
set.seed(1); results.LORD3 <- LORD(sample.df, version=3)
set.seed(1); results.LORD.dep <- LORD(sample.df, version='dep')

cbind(LORD.plus = results.LORD.plus$alphai, LORD3 = results.LORD3$alphai,
LORD.discard  = results.LORD.discard$alphai, LORD.dep = results.LORD.dep$alphai)
#>  [1,] 0.0002675839 0.0002675839 0.0002675839 2.091542e-03
#>  [2,] 0.0024664457 0.0026615183 0.0011285264 1.002025e-02
#>  [3,] 0.0005732818 0.0005787961 0.0002823266 1.677763e-03
#>  [4,] 0.0004872805 0.0004929725 0.0002394680 6.262659e-04
#>  [5,] 0.0004059066 0.0004099744 0.0001998165 3.201787e-04
#>  [6,] 0.0003447286 0.0003475734 0.0001700069 1.933725e-04
#>  [7,] 0.0002986627 0.0003006772 0.0001475152 1.293954e-04
#>  [8,] 0.0029389397 0.0072216015 0.0014680343 1.548152e-04
#>  [9,] 0.0008168502 0.0015704700 0.0014680343 1.166482e-04
#> [10,] 0.0033835974 0.0091593329 0.0017451837 1.422614e-04
#> [11,] 0.0011873999 0.0019918653 0.0006438778 1.145119e-04
#> [12,] 0.0010225858 0.0016965126 0.0006438778 9.432408e-05
#> [13,] 0.0008785607 0.0014108836 0.0006438778 7.916885e-05
#> [14,] 0.0007679398 0.0011961369 0.0005497556 6.749313e-05
#> [15,] 0.0006820264 0.0010347488 0.0005497556 5.830055e-05

By default $$w_0 = \alpha/10$$ and (for LORD 3 and LORD dep) $$b0 = alpha - w0$$, but these parameters can optionally be specified by the user subject to the requirements that $$0 \leq w_0 \leq \alpha$$, $$b_0 > 0$$ and $$w_0+b_0 \leq \alpha$$.

The value of gammai is also supplied by default, but can optionally be specified by the user (as described above, see the formula for $$\gamma_j$$ here for version=‘dep’ and here for all other versions of LORD).

### SAFFRON

By default $$w_0 = \alpha/2$$ and $$\lambda = 0.5$$, but these parameters can optionally be specified by the user subject to the requirements that $$0 \leq w_0 < \alpha$$ and $$0 < \lambda < 1$$. The values of gammai are also supplied by default, but can optionally be specified by the user (as described above, see the formula for $$\gamma_j$$ here).

By default $$w_0 = \tau \lambda \alpha/2$$ and $$\lambda = \tau = 0.5$$, but these parameters can optionally be specified by the user subject to the requirements that $$0 \leq w_0 < \tau \lambda \alpha$$, $$0 < \lambda < 1$$ and $$0 < \tau < 1$$. The values of gammai are also supplied by default, but can optionally be specified by the user.

### Alpha-spending and online fallback

The values of gammai are supplied by default, but can optionally be specified by the user.

By default $$\lambda = 0.25$$ and $$\tau = 0.5$$, but these parameters can optionally be specified by the user subject to the requirements that $$\lambda < \tau$$, $$0 < \lambda < 1$$ and $$0 < \tau < 1$$. The values of gammai are also supplied by default, but can optionally be specified by the user.
2: Online mFDR control under local dependence: 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’.