## 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 FDR in an online manner (Javanmard and Montanari (2015, 2018)), with the latter extended by Ramdas et al. (2017). The LOND procedure sets the significance level $$\alpha_i$$ based on the number of discoveries made so far, while LORD sets them according to the time of the most recent discovery. This package, onlineFDR, implements these procedures and provides wrapper functions to apply them to a historic dataset or a growing dataset. As a comparison, we have also provided a wrapper function for implementation of a method 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.

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 four 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 p-values, with an option (dep = TRUE) which guarantees control for 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) and Ramdas et al. (2017). The function provides different versions of the procedure for independent p-values, see ‘Theory behind onlineFDR’.

3: LORDdep Implements the LORD procedure for online FDR control for dependent p-values, where LORD stands for (significance) Levels based On Recent Discovery, as presented by Javanmard and Montanari (2018).

4: BonfInfinite Implements online FDR control using a Bonferroni-like test.

### Quick start

Here we show the steps to achieve online FDR control of a growing dataset. First you 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 0.0026615183 0
#> 3  C18705 2014-12-01 1.5140e-02 0.0005787961 0
#> 4  B49731 2015-09-21 8.1740e-02 0.0004929725 0
#> 5  E99902 2015-09-21 1.7100e-03 0.0004099744 0
#> 6  D46627 2015-09-21 2.7201e-01 0.0003475734 0
#> 7  C38292 2015-09-21 3.6100e-05 0.0003006772 1
#> 8  A30619 2015-09-21 7.9149e-01 0.0048133468 0
#> 9  A41418 2016-05-19 7.5900e-08 0.0010467508 1
#> 10 E29198 2016-05-19 2.8295e-01 0.0069079880 0
#> 11 D51456 2016-11-12 6.9274e-01 0.0015022690 0
#> 12 A63155 2017-03-27 7.2342e-01 0.0012795133 0
#> 13 C88669 2017-03-27 3.0443e-01 0.0010640913 0
#> 14 B66033 2017-03-27 5.4757e-01 0.0009021289 0
#> 15 E03673 2017-03-27 4.8700e-04 0.0007804097 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).

### 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).

### LOND procedure

The LOND procedure controls the FDR for independent $$p$$-values. Given an overall significance level $$\alpha$$, we choose any infinite 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 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 similat 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 procedures

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 $$i$$: $\tau_i = \max \left\{ l \in \{1, \ldots, i-1\} : R_l = 1\right\}$ Also let $$T(i)$$ be the set of discovery times up to time $$i$$: $T(i) = \left\{ l \in \{1, \ldots, i-1 \} : R_l = 1 \right\}$

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 signifcance thresholds $$\alpha_i$$.

• LORD 1: The significance thresholds are based on the time of the last discovery, with $\alpha_i = \begin{cases} \gamma_i w_0 & \text{ if } i \leq t_1 \\ \gamma_{i - \tau_i} b_0 & \text{ if } i > t_1 \end{cases}$ where $$t_1$$ denotes the time of the first discovery.

• LORD 2: The significance thresholds are based on all previous discovery times, with $\alpha_i = \gamma_i w_0 + \left(\sum_{l \in T(i)} \gamma_{i-l} \right) b_0$

• 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} where $$w_0$$ represents the initial ‘wealth’, and $$b_0$$ represents the amount earned for rejecting a hypothesis.

• LORD++: Ramdas et al. (2017) derived a modified version of LORD 2, called LORD++, which has at least as high power. The significance thresholds for LORD++ are chosen as follows: $\alpha_i = \gamma_i w_0 + (\alpha - w_0) \gamma_{i-\tau_1} + \alpha \sum_{j : \tau_j < i, \tau_j \neq \tau_1} \gamma_{i - \tau_j}$ Hence the significance thresholds for LORD++ will always be equal or greater than those for LORD 2.

LORD 1 and LORD 2 are instances of monotone generalised alpha investing rules, and hence provably control the FDR for independent p-values provided $$w_0 + b_0 \leq \alpha$$. LORD++ also provable controls the FDR for independent p-values provided $$w_0 \leq \alpha$$. In Javanmard and Montanari (2018), the authors demonstrate empirically that LORD 3 controls the FDR.

Note that LORD 1 will always have a lower power than LORD 2 (and hence LORD++) as well as LORD 3, and is included for completeness. In many scenarios with large $$N$$, LORD 3 will have a slightly higher power than LORD++ (see Robertson et al., 2018). Hence we would recommend using LORD 3 (the default) unless there is a need for a method with a provable guarantee of FDR control in which case we would recommend using LORD++.

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.

#### LORDdep

Javanmard and Montanari (2018) also presented an adjusted version of LORD that is valid for 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.

### Bonferroni-like procedure (BonfInfinite)

The procedure controls the FDR (as well as 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 $$\gamma_i \geq 0$$. Note that the procedure is also valid for dependent p-values.

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

### Accounting for dependent p-values

As noted above, the LOND procedure and LORD procedures (1, 2 and 3) assumes independent p-values, with adjusted versions of LOND and LORD available for dependent p-values (where the dependencies may be arbitrary). The Bonferroni-like procedure also controls the FDR for dependent p-values.

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

## 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 procedure. As a community 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 could be set to TRUE and will implement the LOND procedure to guarantee FDR control for 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 beta 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 3, but the user can optionally specify versions 1, 2 and ++ using the version argument (see here for further details about the different versions).

set.seed(1); results.LORD1 <- LORD(sample.df, version=1)
set.seed(1); results.LORD2 <- LORD(sample.df, version=2)
set.seed(1); results.LORD.plus <- LORD(sample.df, version='++')
set.seed(1); results.LORD3 <- LORD(sample.df)   # default version = 3

cbind(LORD1 = results.LORD1$alphai, LORD2 = results.LORD2$alphai,
LORD.plus = results.LORD.plus$alphai, LORD3 = results.LORD3$alphai)
#>              LORD1        LORD2    LORD.plus        LORD3
#>  [1,] 0.0002675839 0.0002675839 0.0002675839 0.0002675839
#>  [2,] 0.0024082547 0.0024664457 0.0024664457 0.0026615183
#>  [3,] 0.0005237193 0.0005732818 0.0005732818 0.0005787961
#>  [4,] 0.0004460624 0.0004872805 0.0004872805 0.0004929725
#>  [5,] 0.0003709623 0.0004059066 0.0004059066 0.0004099744
#>  [6,] 0.0003144991 0.0003447286 0.0003447286 0.0003475734
#>  [7,] 0.0002720655 0.0002986627 0.0002986627 0.0003006772
#>  [8,] 0.0024082547 0.0026713558 0.0029389397 0.0048133468
#>  [9,] 0.0005237193 0.0007586591 0.0008168502 0.0010467508
#> [10,] 0.0024082547 0.0030664511 0.0033835974 0.0069079880
#> [11,] 0.0005237193 0.0010879908 0.0011873999 0.0015022690
#> [12,] 0.0004460624 0.0009380789 0.0010225858 0.0012795133
#> [13,] 0.0003709623 0.0008071131 0.0008785607 0.0010640913
#> [14,] 0.0003144991 0.0007063982 0.0007679398 0.0009021289
#> [15,] 0.0002720655 0.0006280708 0.0006820264 0.0007804097

By default w0 = alpha/10 and (for LORD 1/2/3) 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).

### LORDdep

By default w0 = alpha/10 and b0 = alpha - w0, but can optionally be specified by the user subject to the requirements that $$w_0 \geq 0$$, $$b_0 > 0$$ and $$w_0+b_0 \leq \alpha$$.

The value of xi is also supplied by default, but can optionally be specified by the user (as described above, see the formula for $$\xi_j$$ here).

### BonfInfinite

The significance thresholds alphai are supplied by default, but can optionally be specified by the user (as described above, see here).

## Acknowledgements

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

## References

Benjamini, Y., and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. The Annals of Statistics, 29(4):1165-1188.

Bourgon, R., Gentleman, R., and Huber, W. (2010). Independent filtering increases detection power for high-throughput experiments. Proceedings of the National Academy of Sciences, 107(21), 9546-9551.

Ioannidis, J.P.A. (2005). Why most published research findings are false. PLoS Medicine, 2.8:e124.

Javanmard, A., and Montanari, A. (2015). On Online Control of False Discovery Rate. arXiv preprint, https://arxiv.org/abs/1502.06197.

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.

Koscielny, G., et al. (2013). The International Mouse Phenotyping Consortium Web Portal, a unified point of access for knockout mice and related phenotyping data. Nucleic Acids Research, 42.D1:D802-D809.

Li, A., and Barber, F.G. (2017). Accumulation Tests for FDR Control in Ordered Hypothesis Testing. Journal of the American Statistical Association, 112(518):837-849.

Ramdas, A. et al. (2017). Online control of the false discovery rate with decaying memory. Advances in Neural Information Processing Systems 30, 5650-5659.

Robertson, D.S. and Wason, J.M.S. (2018). Online control of the false discovery rate in biomedical research. arXiv preprint, https://arxiv.org/abs/1809.07292.