## Introduction

Multiple hypothesis testing is a fundamental problem in statistical inference. The failure to manage the multiple testing problem has been highlighted as one of the elements contributing to the replicability crisis in science (Ioannidis 2015). Methodologies have been developed for a family of hypotheses to adjust the significance levels to manage the multiple testing situation by controlling error metrics such as the familywise error rate (FWER) or the false discovery rate (FDR).

Frequently, modern data analysis problems have a further complexity that the hypothesis arrive sequentially in a stream. This introduces the challenge that at each step the investigator must decide whether to reject the current null hypothesis without having access to the future p-values or the total number of hypothesis to be tested, but does have knowledge of the historic decisions to date. The International Mouse Phenotyping Consortium (Koscielny et al., 2013), provides a concrete example of such a scenario. Here the dataset is constantly growing as new knockout mice lines are generated and phenotyping data uploaded to a database.

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.

The onlineFDR package implements all these procedures and provides wrapper functions to apply them to a historic or growing dataset. As a comparison, we have also provided a function for implementation of the Alpha-spending procedure, which is based on the Bonferroni procedure adapted to the online scenario. This vignette explains the use of the package and demonstrates a typical workflow.

## Overview of the process

1. A dataset with three columns (an identifier (‘id’), date (‘date’) and p-value (‘pval’)) is passed to one of the onlineFDR wrapper functions. Alternatively, a vector of p-values can be provided, in which case step 2 is skipped.

2. The function orders the information by date. If there are multiple p-values with the same date (i.e. the same batch), the order of the p-values within each batch is randomised by default. In order for the randomisation of the p-values to be reproducible, it is necessary to set a seed (via the set.seed function) before calling the wrapper function (see also step 6).

3. For each hypothesis test, an adjusted significance threshold (alphai) is calculated, which gives the threshold at which the corresponding p-value would be declared significant.

4. Using the p-values provided and the alphai, an indicator of discoveries (R) is calculated, where R[i] = 1 corresponds to hypothesis i being rejected (and R[i] = 0 otherwise).

5. A dataframe is returned, reordered by batch, with the original data and the newly calculated alphai and R.

6. For simplicity, as the dataset grows the new larger dataset should be passed to the wrapper function and the values recalculated repeating steps 1 to 5. In order for the randomisation of the data within the previous batches to remain the same (and hence to allow for reproducibility of the results), the same seed should be used for all analyses.

### Outline of the functions available

1: LOND Implements the LOND procedure for online FDR control, where LOND stands for (significance) Levels based On Number of Discoveries, as presented by Javanmard and Montanari (2015). The procedure controls the FDR for independent or positively dependent (PRDS) p-values, with an option (dep = TRUE) which guarantees control for arbitrarily dependent p-values.

2: LORD Implements the LORD procedure for online FDR control, where LORD stands for (significance) Levels based On Recent Discovery, as presented by Javanmard and Montanari (2018), Ramdas et al. (2017) and Tian & Ramdas (2019). The function provides different versions of the procedure valid for independent p-values, see ‘Theory behind onlineFDR’. There is also a version (‘dep’) that guarantees control for dependent p-values.

3: SAFFRON Implements the SAFFRON procedure for online FDR control, where SAFFRON stands for Serial estimate of the Alpha Fraction that is Futilely Rationed On true Null hypotheses, as presented by Ramdas et al. (2018). The procedure provides an adaptive method of online FDR control.

4: ADDIS Implements the ADDIS algorithm for online FDR control, where ADDIS stands for an ADaptive algorithm that DIScards conservative nulls, as presented by Tian & Ramdas (2019). The algorithm compensates for the power loss of SAFFRON with conservative nulls, by including both adapativity in the fraction of null hypotheses (like SAFFRON) and the conservativeness of nulls (unlike SAFFRON). This procedure controls the FDR for independent p-values.

5: Alpha_investing 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 a variant of SAFFRON’s update rule. This procedure controls the FDR for independent p-values.

6: LONDstar Implements the LOND algorithm for asynchronous online testing, as presented by Zrnic et al. (2018). These algorithms control the mFDR.

7: LORDstar Implements LORD algorithms for asynchronous online testing, as presented by Zrnic et al. (2018). These algorithms control the mFDR.

8: SAFFRONstar Implements the SAFFRON algorithm for asynchronous online testing, as presented by Zrnic et al. (2018). These algorithms control the mFDR.

9: Alpha_spending Implements online FWER control using a Bonferroni-like test. Alpha-spending provides strong FWER control for arbitrarily dependent p-values.

10: online_fallback Implements the online fallback algorithm for FWER control, as proposed by Tian & Ramdas (2019b). Online fallback is a uniformly more powerful method than Alpha-spending, as it saves the significance level of a previous rejection. Online fallback strongly controls the FWER for arbitrarily dependent p-values.

11: ADDIS-spending Implements the ADDIS-spending algorithm for online FWER control, as proposed by Tian & Ramdas (2019b). The algorithm compensates for the power loss of Alpha-spending, by including both adapativity in the fraction of null hypotheses and the conservativeness of nulls. ADDIS-spending provides strong FWER control for independent p-values. Tian & Ramdas (2019b) also presented a version for handling local dependence.

### Quick start

Here we show the steps to achieve online FDR control of a growing dataset. First load a dataframe with the three columns: an identifier (‘id’), date (‘date’) and p-value (‘pval’), and then call the wrapper function of interest. In order for the results to be reproducible, we also set a seed using the set.seed function.

library(onlineFDR)

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-14, 0.06743, 0.01514, 0.08174, 0.00171,
3.61e-05, 0.79149, 0.27201, 0.28295, 7.59e-08,
0.69274, 0.30443, 0.000487, 0.72342, 0.54757))

set.seed(1)
results <- LORD(sample.df)
results
#>        id       date       pval       alphai R
#> 1  A15432 2014-12-01 2.9000e-14 0.0002675839 1
#> 2  B90969 2014-12-01 6.7430e-02 0.0024664457 0
#> 3  C18705 2014-12-01 1.5140e-02 0.0005732818 0
#> 4  B49731 2015-09-21 8.1740e-02 0.0004872805 0
#> 5  E99902 2015-09-21 1.7100e-03 0.0004059066 0
#> 6  D46627 2015-09-21 2.7201e-01 0.0003447286 0
#> 7  C38292 2015-09-21 3.6100e-05 0.0002986627 1
#> 8  A30619 2015-09-21 7.9149e-01 0.0029389397 0
#> 9  A41418 2016-05-19 7.5900e-08 0.0008168502 1
#> 10 E29198 2016-05-19 2.8295e-01 0.0033835974 0
#> 11 D51456 2016-11-12 6.9274e-01 0.0011873999 0
#> 12 A63155 2017-03-27 7.2342e-01 0.0010225858 0
#> 13 C88669 2017-03-27 3.0443e-01 0.0008785607 0
#> 14 B66033 2017-03-27 5.4757e-01 0.0007679398 0
#> 15 E03673 2017-03-27 4.8700e-04 0.0006820264 1

### Input data

A dataset with three columns (an identifier (‘id’), date (‘date’) and p-value (‘pval’)). All p-values generated should be passed to the function (and not just the significant p-values). An exception to this would be if you have implemented an orthogonal filter to reduce the dataset size, such as discussed in (Burgon et al., 2010).

Alternatively, just the vector of p-values can be passed to the function. In this case, the p-values are treated as being ordered sequentially with no batches.

### Understanding the output

For each hypothesis test, the functions calculate the adjusted significance thresholds (alphai) at which the corresponding p-value would be declared statistically significant.

Also calculated is an indicator function of discoveries (R), where R[i] = 1 corresponds to hypothesis i being rejected, otherwise R[i] = 0.

A dataframe is returned, reordered by batch, with the original data and the newly calculated alphai and R.

## How to get help for onlineFDR

All questions regarding onlineFDR should be posted to the Bioconductor support site, which serves as a searchable knowledge base of questions and answers:

https://support.bioconductor.org

Posting a question and tagging with “onlineFDR” will automatically send an alert to the package authors to respond on the support site.

## Theory behind onlineFDR

### Online hypothesis testing

Consider a sequence of hypotheses $$H_1, H_2, H_3, \ldots$$ that arrive sequentially in a stream, with corresponding $$p$$-values $$(p_1, p_2, p_3, \ldots)$$. A testing procedure provides a sequence of adjusted significance thresholds $$\alpha_i$$, with corresponding decision rule: $R_i = \begin{cases} 1 & \text{if } p_i \leq \alpha_i & (\text{reject } H_i)\\ 0 & \text{otherwise} & (\text{accept } H_i) \end{cases}$

In online testing, the significance thresholds can only be functions of the prior decisions, i.e. $$\alpha_i = \alpha_i(R_1, R_2, \ldots, R_{i-1})$$.

Javanmard and Montanari (2015, 2018) proposed two procedures for online control. The first is LOND, which stands for (significance) Levels based On Number of Discoveries. The second is LORD, which stands for (significance) Levels based On Recent Discovery. LORD was subsequently extended by Ramdas et al. (2017). Ramdas et al. (2018) also proposed the SAFFRON procedure, which provides an adaptive method of online FDR control, which includes a variant of Alpha-investing. Finally, Tian & Ramdas (2019) proposed the ADDIS procedure as an improvement of SAFFRON in the presence of conservative nulls.

### LOND

The LOND procedure controls the FDR for independent or positively dependent (PRDS) $$p$$-values. Given an overall significance level $$\alpha$$, we choose a sequence of non-negative numbers $$\beta = (\beta_i)_{i \in \mathbb{N}}$$ such that they sum to $$\alpha$$. The values of the adjusted significance thresholds $$\alpha_i$$ are chosen as follows: $\alpha_i = \beta_i (D(i-1) + 1)$ where $$D(n) = \sum_{i=1}^n R_i$$ denotes the number of discoveries (i.e. rejections) in the first $$n$$ hypotheses tested.

LOND can be adjusted to also control FDR under arbitrarily dependent $$p$$-values. To do so, it is modified with $$\tilde{\beta}_i = \beta_i/H(i)$$ in place of $$\beta_i$$, where $$H(i) = \sum_{j=1}^i \frac{1}{j}$$ is the $$i$$-th harmonic number. Note that this leads to a substantial loss in power compared to the unadjusted LOND procedure. The correction factor is similar to the classical one used by Benjamini and Yekutieli (2001), except that in this case the $$i$$-th hypothesis among $$N$$ is penalised by a factor of $$H(i)$$ to give consistent results across time (as compared to a factor $$H(N)$$ for the Benjamini and Yekutieli method).

The default sequence of $$\beta$$ is given by $\beta_j = C \alpha \frac{\log(\max(j, 2))}{j e^{\sqrt{\log j}}}$ where $$C \approx 0.07720838$$, as proposed by Javanmard and Montanari (2018) equation 31.

### LORD

The LORD procedure controls the FDR for independent $$p$$-values. We first fix a sequence of non-negative numbers $$\gamma = (\gamma_i)_{i \in \mathbb{N}}$$ such that $$\gamma_i \geq \gamma_j$$ for $$i \leq j$$ and $$\sum_{i=1}^{\infty} \gamma_i = 1$$. At each time $$i$$, let $$\tau_i$$ be the last time a discovery was made before $$i$$: $\tau_i = \max \left\{ l \in \{1, \ldots, i-1\} : R_l = 1\right\}$

LORD depends on constants $$w_0$$ and $$b_0$$, where $$w_0 \geq 0$$ represents the initial ‘wealth’ of the procedure and $$b_0 > 0$$ represents the ‘payout’ for rejecting a hypothesis. We require $$w_0+b_0 \leq \alpha$$ for FDR control to hold.

Javanmard and Montanari (2018) presented three different versions of LORD, which have different definitions of the adjusted significance thresholds $$\alpha_i$$. Versions 1 and 2 have since been superseded by the LORD++ procedure of Ramdas et al. (2017), so we do not describe them here.

• LORD++: The significance thresholds for LORD++ are chosen as follows: $\alpha_i = \gamma_i w_0 + (\alpha - w_0) \gamma_{i-\tau_1} + \alpha \sum_{j : \tau_j < i, \tau_j \neq \tau_1} \gamma_{i - \tau_j}$

• LORD 3: The significance thresholds depend on the time of the last discovery time and the wealth accumulated at that time, with $\alpha_i = \gamma_{i - \tau_i} W(\tau_i)$ where $$\tau_1 = 0$$. Here $$\{W(j)\}_{j \geq 0}$$ represents the ‘wealth’ available at time $$j$$, and is defined recursively: \begin{align} W(0) & = w_0 \nonumber \\ W(j) & = W(j-1) - \alpha_{j-1} + b_0 R_j \end{align}

• D-LORD: This is equivalent to the LORD++ procedure with discarding. Given a discarding threshold $$\tau \in (0,1)$$ and initial wealth $$w_0 \leq \tau\alpha$$ the significance thresholds are chosen as follows: $\alpha_t = \min\{\tau, \tilde{\alpha}_t\}$ where $\tilde{\alpha}_t = w_0 \gamma_{S^t} + (\tau\alpha - w_0)\gamma_{S^t - \kappa_1^*} + \tau\alpha \sum_{j \geq 2} \gamma_{S^t - \kappa_j^*}$ and $\kappa_j = \min\{i \in [t-1] : \sum_{k \leq i} 1 \{p_k \leq \alpha_k\} \geq j\}, \; \kappa_j^* = \sum_{i \leq \kappa_j} 1 \{p_i \leq \tau \}, \; S^t = \sum_{i < t} 1 \{p_i \leq \tau \}$

LORD++ is an instance of a monotone rule, and provably controls the FDR for independent p-values provided $$w_0 \leq \alpha$$. LORD 3 is a non-monotone rule, and FDR control is only demonstrated empirically. In some scenarios with large $$N$$, LORD 3 will have a slightly higher power than LORD++ (see Robertson et al., 2018), but since it is a non-monotone rule we would recommend using LORD++ (which is the default), especially since it also has a provable guarantee of FDR control.

In all versions, the default sequence of $$\gamma$$ is given by $\gamma_j = C \frac{\log(\max(j, 2))}{j e^{\sqrt{\log j}}}$ where $$C \approx 0.07720838$$, as proposed by Javanmard and Montanari (2018) equation 31.

Javanmard and Montanari (2018) also proposed an adjusted version of LORD that is valid for arbitrarily dependent p-values. Similarly to LORD 3, the adjusted significance thresholds are set equal to $\alpha_i = \xi_i W(\tau_i)$ where (assuming $$w_0 \leq b_0$$), $$\sum_{j=1}^{\infty} \xi_i (1 + \log(j)) \leq \alpha / b_0$$

The default sequence of $$\xi$$ is given by $\xi_j = \frac{C \alpha }{b_0 j \log(\max(j, 2))^3}$ where $$C \approx 0.139307$$.

Note that allowing for dependent p-values can lead to a substantial loss in power compared with the LORD procedures described above.

### SAFFRON

The SAFFRON procedure controls the FDR for independent p-values, and was proposed by Ramdas et al. (2018). The algorithm is based on an estimate of the proportion of true null hypotheses. More precisely, SAFFRON sets the adjusted test levels based on an estimate of the amount of alpha-wealth that is allocated to testing the true null hypotheses.

SAFFRON depends on constants $$w_0$$ and $$\lambda$$, where $$w_0$$ satisfies $$0 \leq w_0 < (1 - \lambda)\alpha$$ and represents the initial ‘wealth’ of the procedure, and $$\lambda \in (0,1)$$ represents the threshold for a ‘candidate’ hypothesis. A ‘candidate’ refers to p-values smaller than $$\lambda$$, since SAFFRON will never reject a p-value larger than $$\lambda$$. These candidates can be thought of as the hypotheses that are a-priori more likely to be non-null.

The SAFFRON procedure runs as follows:

1. Set the initial alpha-wealth $$w_0 < (1-\lambda)\alpha$$

2. At each time $$i$$, define the number of candidates after the $$k$$-th rejection as $C_{k+} = C_{k+}(i) = \sum_{j = \tau_k + 1}^{i-1} C_j$ where $$C_j = 1\{p_j \leq \lambda \}$$ is the indicator for candidacy.

3. SAFFRON starts with $$\alpha_1 = \min\{\gamma_1 w_0, \lambda\}$$. Subsequent test levels are chosen as $$\alpha_i = \min\{ \lambda, \tilde{\alpha}_i\}$$, where $\tilde{\alpha}_i = w_0 \gamma_{i-C_{0+}} + ((1-\lambda)\alpha - w_0)\gamma_{i-\tau_1-C_{1+}} + (1-\lambda)\alpha \sum_{j : \tau_j < i, \tau_j \neq \tau_1} \gamma_{i - \tau_j- C_{j+}}$

The default sequence of $$\gamma$$ for SAFFRON is given by $$\gamma_j \propto j^{-1.6}$$.

### Alpha-investing

Ramdas et al. (2018) proposed a variant of the Alpha-investing algorithm of Foster and Stine (2008) that guarantees FDR control for independent p-values. This procedure uses SAFFRON’s update rule with the constant replaced by a sequence $$\lambda_i = \alpha_i$$. This is also equivalent to using the ADDIS algorithm (see below) with $$\tau = 1$$ and $$\lambda_i = \alpha_i$$.

The ADDIS procedure controls the FDR for independent p-values, and was proposed by Tian & Ramdas (2019). The algorithm compensates for the power loss of SAFFRON with conservative nulls, by including both adapativity in the fraction of null hypotheses (like SAFFRON) and the conservativeness of nulls (unlike SAFFRON).

ADDIS depends on constants $$w_0, \lambda$$ and $$\tau$$. $$w_0$$ represents the initial wealth’ of the procedure and satisfies $$0 \leq w_0 \leq \tau \lambda \alpha$$. $$\tau \in (0,1)$$ represents the threshold for a hypothesis to be selected for testing: p-values greater than $$\tau$$ are implicitly ‘discarded’ by the procedure. Finally, $$\lambda \in (0,1)$$ sets the threshold for a p-value to be a candidate for rejection: ADDIS will never reject a p-value larger than $$\tau \lambda$$.

The significance thresholds for ADDIS are chosen as follows: $\alpha_t = \min\{\tau\lambda, \tilde{\alpha}_t\}$ where $\tilde{\alpha}_t = w_0 \gamma_{S^t-C_{0+}} + (\tau(1-\lambda)\alpha - w_0)\gamma_{S^t - \kappa_1^*-C_{1+}} + \tau(1-\lambda)\alpha \sum_{j \geq 2} \gamma_{S^t - \kappa_j^* - C_{j+}}$ and $\kappa_j = \min\{i \in [t-1] : \sum_{k \leq i} 1 \{p_k \leq \alpha_k\} \geq j\}, \; \kappa_j^* = \sum_{i \leq \kappa_j} 1 \{p_i \leq \tau \}, \; S^t = \sum_{i < t} 1 \{p_i \leq \tau \}, \; C_{j+} = \sum_{i = \tau_k + 1}^{t-1} 1\{p_i \leq \tau\}$

The default sequence of $$\gamma$$ for ADDIS is the same as for SAFFRON given here.

### Alpha-spending

The Alpha-spending procedure controls the FWER for a potentially infinite stream of p-values using a Bonferroni-like test. Given an overall significance level $$\alpha$$, the significance thresholds are chosen as $\alpha_i = \alpha \gamma_i$ where $$\sum_{i=1}^{\infty} \gamma_i = 1$$ and $$\gamma_i \geq 0$$. The procedure strongly controls the FWER for arbitrarily dependent p-values.

Note that the procedure also controls the generalised familywise error rate (k-FWER) for $$k > 1$$ if $$\alpha$$ is replaced by $$\min(1,k\alpha)$$.

The default sequence of $$\gamma$$ is the same as that for $$\xi$$ for LORD given here.

### Online Fallback

The online fallback procedure of Tian & Ramdas (2019b) provides a uniformly more powerful method than Alpha-spending, by saving the significance level of a previous rejection. More specifically, online fallback tests hypothesis $$H_i$$ at level $\alpha_i = \alpha \gamma_i + R_{i-1} \alpha_{i-1}$ where $$R_i = 1\{p_i \leq \alpha_i\}$$ denotes a rejected hypothesis. The procedure strongly controls the FWER for arbitrarily dependent p-values.

The default sequence of $$\gamma$$ is the same as that for $$\xi$$ for LORD given here.

The ADDIS-spending procedure strongly controls the FWER for independent p-values, and was proposed by Tian & Ramdas (2019b). The procedure compensates for the power loss of Alpha-spending, by including both adapativity in the fraction of null hypotheses and the conservativeness of nulls.

ADDIS depends on constants $$\lambda$$ and $$\tau$$, where $$\lambda < \tau$$. Here $$\tau \in (0,1)$$ represents the threshold for a hypothesis to be selected for testing: p-values greater than $$\tau$$ are implicitly discarded’ by the procedure, while $$\lambda \in (0,1)$$ sets the threshold for a p-value to be a candidate for rejection: ADDIS-spending will never reject a p-value larger than $$\lambda$$.

Note that the procedure controls the generalised familywise error rate (k-FWER) for $$k > 1$$ if $$\alpha$$ is replaced by $$\min(1,k\alpha)$$. Tian and Ramdas (2019b) also presented a version for handling local dependence, see the Section on Asynchronous testing below.

The default sequence of $$\gamma$$ for ADDIS-spending is the same as for SAFFRON given here.

### Accounting for dependent p-values

As noted above, the LORD, SAFFRON, ADDIS and ADDIS-spending procedures assume independent p-values, while the LOND procedure is also valid under positive dependencies (like the Benjamini-Hochberg method, see below). Adjusted versions of LOND and LORD available for arbitrarily dependent p-values. Alpha-spending and online fallback also control the FWER and FDR for arbitrarily dependent p-values.

By way of comparison, the usual Benjamini-Hochberg method for controlling the FDR assumes that the p-values are positively dependent (PRDS). As an example, the PRDS is satisfied for multivariate normal test statistics with a positive correlation matrix). See Benjamini & Yekutieli (2001) for further technical details.

## Asynchronous testing

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, including the following:

1: Asynchronous online mFDR control: This 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.

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’.

3: mFDR control in asynchronous mini-batch testing: 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.

## 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 2.323935e-03
#>  [2,] 0.0024664457 0.0026615183 0.0011285264 1.107961e-02
#>  [3,] 0.0005732818 0.0005787961 0.0002823266 1.855138e-03
#>  [4,] 0.0004872805 0.0004929725 0.0002394680 6.924756e-04
#>  [5,] 0.0004059066 0.0004099744 0.0001998165 3.540284e-04
#>  [6,] 0.0003447286 0.0003475734 0.0001700069 2.138161e-04
#>  [7,] 0.0002986627 0.0003006772 0.0001475152 1.430752e-04
#>  [8,] 0.0029389397 0.0048133468 0.0014680343 1.685669e-04
#>  [9,] 0.0008168502 0.0010467508 0.0014680343 1.270096e-04
#> [10,] 0.0033835974 0.0069079880 0.0017451837 1.560048e-04
#> [11,] 0.0011873999 0.0015022690 0.0006438778 1.255746e-04
#> [12,] 0.0010225858 0.0012795133 0.0006438778 1.034364e-04
#> [13,] 0.0008785607 0.0010640913 0.0006438778 8.681710e-05
#> [14,] 0.0007679398 0.0009021289 0.0005497556 7.401343e-05
#> [15,] 0.0006820264 0.0007804097 0.0005497556 6.393279e-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.

## Acknowledgements

We would like to thank the IMPC team (via Jeremy Mason and Hamed Haseli Mashhadi) for useful discussions during the development of the package.

## References

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