# 1 Introduction

With SummarizedBenchmark, a complete benchmarking workflow is comprised of three primary components:

1. data,
2. methods, and
3. performance metrics.

The first two (data and methods) are necessary for carrying out the benchmark experiment, and the last (performance metrics) is essential for evaluating the results of the experiment. Following this approach, the SummarizedBenchmark package defines two primary types of objects: BenchDesign objects and SummarizedBenchmark objects. BenchDesign objects contain only the design of the benchmark experiment, namely the data and methods, where a method is defined as the combination of a function or algorithm and parameter settings. After constructing a BenchDesign, the experiment is executed to create a SummarizedBenchmark containing the results of applying the methods to the data. SummarizedBenchmark objects extend the Bioconductor SummarizedExperiment class, with the additional capability of working with performance metrics.

The basic framework is illustrated in the figure below. A BenchDesign is created with methods and combined with data to create a SummarizedBenchmark, which contains the output of applying the methods to the data. This output is then paired with performance metrics specified by the user. Note that the same BenchDesign can be combined with several data sets to produce several SummarizedBenchmark objects with the corresponding outputs. For convenience, several default performance metrics are implemented in the package, and can be added to SummarizedBenchmark objects using simple commands.

In this vignette, we first illustrate the basic use of both the BenchDesign and SummarizedBenchmark classes with a simple comparison of methods for p-value correction in the context of multiple hypothesis testing. Then, we describe more advanced features of the package with a case study comparing three methods for differential expression analysis.

# 2 Quickstart Case Study

library("SummarizedBenchmark")
library("magrittr")

To illustrate the basic use of the BenchDesign class, we use the tdat data set included with this package.

data(tdat)

The data set is a data.frame containing the results of 50 two-sample t-tests. The tests were performed using independently simulated sets of 20 observations drawn from a single standard Normal distribution (when H = 0) or two mean-shifted Normal distributions (when H = 1).

head(tdat)
##   H test_statistic effect_size         pval        SE
## 1 1     -3.2083437 -1.17151466 4.872282e-03 0.3651463
## 2 0      0.1692236  0.07321912 8.675080e-01 0.4326768
## 3 1     -5.7077940 -1.81715381 2.061521e-05 0.3183636
## 4 1     -1.9805856 -1.09107836 6.313031e-02 0.5508867
## 5 1     -1.0014358 -0.37726058 3.298895e-01 0.3767197
## 6 1     -0.9190433 -0.47583669 3.702252e-01 0.5177522

Several approaches have been proposed and implemented to compute adjusted p-values and q-values with the goal of controlling the total number of false discoveries across a collection of tests. In this example, we compare three such methods:

1. Bonferroni correction (p.adjust w/ method = "bonferroni") (Dunn 1961),
2. Benjamini-Hochberg (p.adjust w/ method = "BH") (Benjamini and Hochberg 1995), and
3. Storey’s FDR q-value (qvalue::qvalue) (Storey 2002).

First, consider how benchmarking the three methods might look without the SummarizedBenchmark framework.

To compare methods, each is applied to tdat, and the results are stored in separate variables.

adj_bonf <- p.adjust(p = tdat$pval, method = "bonferroni") adj_bh <- p.adjust(p = tdat$pval, method = "BH")

qv <- qvalue::qvalue(p = tdat$pval) adj_qv <- qv$qvalues

Since the values of interest are available from the ouput of each method as a vector of length 50 (the number of hypotheses tested), to keep things clean, they can be combined into a single data.frame.

adj <- cbind.data.frame(adj_bonf, adj_bh, adj_qv)
head(adj)
##     adj_bonf     adj_bh       adj_qv
## 1 0.24361409 0.02191765 0.0079813228
## 2 1.00000000 0.90365415 0.3290660527
## 3 0.00103076 0.00103076 0.0003753518
## 4 1.00000000 0.12140444 0.0442094809
## 5 1.00000000 0.47127065 0.1716134119
## 6 1.00000000 0.48250104 0.1757029652

The data.frame of adjusted p-values and q-values can be used to compare the methods, either by directly parsing the table or using a framework like iCOBRA. Additionally, the data.frame can be saved as a RDS or Rdata object for future reference, eliminating the need for recomputing on the original data.

While this approach can work well for smaller comparisons, it can quickly become overwhelming and unweildy as the number of methods and parameters increases. Furthermore, once each method is applied and the final data.frame (adj) is constructed, there is no way to determine how each value was calculated. While an informative name can be used to “label” each method (as done above), this does not capture the full complexity, e.g. parameters and context, where the function was evaluated. One solution might involve manually recording function calls and parameters in a separate data.frame with the hope of maintaining synchrony with the output data.frame. However, this is prone to errors, e.g. during fast “copy and paste” operations or additions and deletions of parameter combinations. An alternative (and hopefully better) solution, is to use the framework of the SummarizedBenchmark package.

In the SummarizedBenchmark approach, a BenchDesign is constructed with the data and any number of methods. Optionally, a BenchDesign can also be constructed without any data or method inputs. Methods and data can be added or removed from the object modularly in a few different ways, as will be described in the following section. For simplicity, we first show how to construct the BenchDesign with just the data set as input. The data object, here tdat, must be passed explicitly to the data = parameter.

b <- BenchDesign(data = tdat)

Then, each method of interest can be added to the BenchDesign using addMethod().

b <- addMethod(bd = b, label = "bonf", func = p.adjust,
params = rlang::quos(p = pval, method = "bonferroni"))

At a minimum, addMethod() requires three parameters:

1. bd: the BenchDesign object to modify,
2. label: a character name for the method, and
3. func: the function to be called.

After the minimum parameters are specified, any parameters needed by the func method should be passed as named parameters, e.g. p = pval, method = "bonferroni", to params = as a list of quosures using rlang::quos(..). Notice here that the pval wrapped in rlang::quos(..) does not need to be specified as tdat$pval for the function to access the column in the data.frame. For readers familiar with the ggplot2 package, the use of params = rlang::quos(..) here should be viewed similar to the use of aes = aes(..) in ggplot2 for mapping between the data and plotting (or benchmarking) parameters. The process of adding methods can be written more concisely using the pipe operators from the magrittr package. b <- b %>% addMethod(label = "BH", func = p.adjust, params = rlang::quos(p = pval, method = "BH")) %>% addMethod(label = "qv", func = qvalue::qvalue, params = rlang::quos(p = pval), post = function(x) { x$qvalues })

For some methods, such as the q-value approach above, it may be necessary to call a “post-processing” function on the primary method to extract the desired output (here, the q-values). This should be specified using the optional post = parameter.

Now, the BenchDesign object contains three methods. This can be verified either by calling on the object.

b
## BenchDesign ------------------------------------------------
##   benchmark data:
##     type: data
##     names: H, test_statistic, effect_size, pval, SE
##   benchmark methods:
##     method:    qv; func: qvalue::qvalue

More details about each method can be seen by using the printMethods() function.

printMethods(b)
## bonf -------------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##     function (p, method = p.adjust.methods, n = length(p)...
##   parameters:
##     p : pval
##     method : "bonferroni"
##   post:
##     none
##   meta:
##     none
## BH ---------------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##     function (p, method = p.adjust.methods, n = length(p)...
##   parameters:
##     p : pval
##     method : "BH"
##   post:
##     none
##   meta:
##     none
## qv ---------------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##   method: qvalue::qvalue
##     function (p, fdr.level = NULL, pfdr = FALSE, lfdr.out...
##   parameters:
##     p : pval
##   post:
##     default : function (x) { ...
##   meta:
##     none

While the bench now includes all the information necessary for performing the benchmarking study, the actual adjusted p-values and q-values have not yet been calculated. To do this, we simply call buildBench(). While buildBench() does not require any inputs other than the BenchDesign object, when the corresponding ground truth is known, the truthCols = parameter should be specified. In this example, the H column of the tdat data.frame contains the true null or alternative status of each simulated hypothesis test. Note that if any of the methods are defined in a separate package, they must be installed and loaded before running the experiment.

sb <- buildBench(b, truthCols = "H")

The returned object is a SummarizedBenchmark class. The SummarizedBenchmark object is an extension of a SummarizedExperiment object. The table of adjusted p-values and q-values is contained in a single “assay” of the object with each method added using addMethod() as a column with the corresponding label as the name.

head(assay(sb))
##            bonf         BH           qv
## [1,] 0.24361409 0.02191765 0.0079813228
## [2,] 1.00000000 0.90365415 0.3290660527
## [3,] 0.00103076 0.00103076 0.0003753518
## [4,] 1.00000000 0.12140444 0.0442094809
## [5,] 1.00000000 0.47127065 0.1716134119
## [6,] 1.00000000 0.48250104 0.1757029652

Metadata for the methods is contained in the colData() of the same object, with each row corresponding to one method in the comparison.

colData(sb)
## DataFrame with 3 rows and 6 columns
##         func.pkg func.pkg.vers func.pkg.manual     param.p param.method
##      <character>   <character>       <logical> <character>  <character>
## bonf       stats         3.6.0           FALSE        pval "bonferroni"
## BH         stats         3.6.0           FALSE        pval         "BH"
## qv        qvalue        2.16.0           FALSE        pval           NA
##      session.idx
##        <numeric>
## bonf           1
## BH             1
## qv             1

In addition to columns for the functions and parameters specified with addMethod() (func, post, label, param.*), the colData() includes several other columns added during the buildBench() process. Most notably, columns for the package name and version of func if available (func.pkg, func.pkg.vers).

When available, ground truth data is contained in the rowData() of the SummarizedBenchmark object.

rowData(sb)
## DataFrame with 50 rows and 1 column
##             H
##     <numeric>
## 1           1
## 2           0
## 3           1
## 4           1
## 5           1
## ...       ...
## 46          0
## 47          1
## 48          0
## 49          1
## 50          1

In addition, the SummarizedBenchmark class contains an additional slot where users can define performance metrics to evaluate the different methods. Since different benchmarking experiments may require the use of different metrics to evaluate the performance of the methods, the SummarizedBenchmark class provides a flexible way to define performance metrics. We can define performance metrics using the function addPerformanceMetric() by providing a SummarizedBenchmark object, a name of the metric, an assay name, and the function that defines it. Importantly, the function must contain the following two arguments: query (referring to a vector of values being evaluated, i.e. the output of one method) and truth (referring to the vector of ground truths). If further arguments are provided to the performance function, these must contain default values.

For our example, we define the performance metric “TPR” (True Positive Rate) that calculates the fraction of true positives recovered given an alpha value. This performance metric uses the H assay of our SummarizedBenchmark example object.

sb <- addPerformanceMetric(
object = sb,
assay = "H",
evalMetric = "TPR",
evalFunction = function(query, truth, alpha = 0.1) {
goodHits <- sum((query < alpha) & truth == 1)
goodHits / sum(truth == 1)
}
)

performanceMetrics(sb)[["H"]]
## $TPR ## function(query, truth, alpha = 0.1) { ## goodHits <- sum((query < alpha) & truth == 1) ## goodHits / sum(truth == 1) ## } Having defined all the desired performance metrics, the function estimatePerformanceMetrics() calculates these for each method. Parameters for the performance functions can be passed here. In the case below, we specify several alpha = values to be used for calculating the performance metrics with each function. resWide <- estimatePerformanceMetrics(sb, alpha = c(0.05, 0.1, 0.2)) resWide ## DataFrame with 3 rows and 9 columns ## func.pkg func.pkg.vers func.pkg.manual param.p param.method ## <character> <character> <logical> <character> <character> ## bonf stats 3.6.0 FALSE pval "bonferroni" ## BH stats 3.6.0 FALSE pval "BH" ## qv qvalue 2.16.0 FALSE pval NA ## session.idx TPR.1 TPR.2 TPR.3 ## <numeric> <numeric> <numeric> <numeric> ## bonf 1 0.125 0.2 0.225 ## BH 1 0.375 0.55 0.625 ## qv 1 0.6 0.7 0.9 By default, the function above returns a DataFrame, where the parameters of the performance function are stored in its elementMetadata(). elementMetadata(resWide) ## DataFrame with 9 rows and 4 columns ## colType assay performanceMetric alpha ## <character> <character> <character> <numeric> ## func.pkg methodInformation NA NA NA ## func.pkg.vers methodInformation NA NA NA ## func.pkg.manual methodInformation NA NA NA ## param.p methodInformation NA NA NA ## param.method methodInformation NA NA NA ## session.idx methodInformation NA NA NA ## TPR.1 performanceMetric H TPR 0.05 ## TPR.2 performanceMetric H TPR 0.1 ## TPR.3 performanceMetric H TPR 0.2 A second possibility is to set the parameter addColData = TRUE for these results to be stored in the colData() of the SummarizedBenchmark object. sb <- estimatePerformanceMetrics(sb, alpha = c(0.05, 0.1, 0.2), addColData = TRUE) colData(sb) ## DataFrame with 3 rows and 10 columns ## func.pkg func.pkg.vers func.pkg.manual param.p param.method ## <character> <character> <logical> <character> <character> ## bonf stats 3.6.0 FALSE pval "bonferroni" ## BH stats 3.6.0 FALSE pval "BH" ## qv qvalue 2.16.0 FALSE pval NA ## session.idx TPR.1 TPR.2 TPR.3 pm.session ## <numeric> <numeric> <numeric> <numeric> <numeric> ## bonf 1 0.125 0.2 0.225 1 ## BH 1 0.375 0.55 0.625 1 ## qv 1 0.6 0.7 0.9 1 elementMetadata(colData(sb)) ## DataFrame with 10 rows and 4 columns ## colType assay performanceMetric ## <character> <character> <character> ## func.pkg methodInformation NA NA ## func.pkg.vers methodInformation NA NA ## func.pkg.manual methodInformation NA NA ## param.p methodInformation NA NA ## param.method methodInformation NA NA ## session.idx methodInformation NA NA ## TPR.1 performanceMetric H TPR ## TPR.2 performanceMetric H TPR ## TPR.3 performanceMetric H TPR ## pm.session performanceMetricSession NA NA ## alpha ## <numeric> ## func.pkg NA ## func.pkg.vers NA ## func.pkg.manual NA ## param.p NA ## param.method NA ## session.idx NA ## TPR.1 0.05 ## TPR.2 0.1 ## TPR.3 0.2 ## pm.session NA Finally, if the user prefers tidier formats, by setting the parameter tidy = TRUE the function returns a long-formated version of the results. estimatePerformanceMetrics(sb, alpha = c(0.05, 0.1, 0.2), tidy = TRUE) ## func.pkg func.pkg.vers func.pkg.manual param.p param.method session.idx ## 1 stats 3.6.0 FALSE pval "bonferroni" 1 ## 2 stats 3.6.0 FALSE pval "BH" 1 ## 3 qvalue 2.16.0 FALSE pval <NA> 1 ## 4 stats 3.6.0 FALSE pval "bonferroni" 1 ## 5 stats 3.6.0 FALSE pval "BH" 1 ## 6 qvalue 2.16.0 FALSE pval <NA> 1 ## 7 stats 3.6.0 FALSE pval "bonferroni" 1 ## 8 stats 3.6.0 FALSE pval "BH" 1 ## 9 qvalue 2.16.0 FALSE pval <NA> 1 ## pm.session label key value assay performanceMetric alpha ## 1 1 bonf TPR.1 0.125 H TPR 0.05 ## 2 1 BH TPR.1 0.375 H TPR 0.05 ## 3 1 qv TPR.1 0.600 H TPR 0.05 ## 4 1 bonf TPR.2 0.200 H TPR 0.10 ## 5 1 BH TPR.2 0.550 H TPR 0.10 ## 6 1 qv TPR.2 0.700 H TPR 0.10 ## 7 1 bonf TPR.3 0.225 H TPR 0.20 ## 8 1 BH TPR.3 0.625 H TPR 0.20 ## 9 1 qv TPR.3 0.900 H TPR 0.20 As an alternative to get the same data.frame as the previous chunk, we can call the function tidyUpMetrics() on the saved results from a SummarizedBenchmark object. head(tidyUpMetrics(sb)) ## func.pkg func.pkg.vers func.pkg.manual param.p param.method session.idx ## 1 stats 3.6.0 FALSE pval "bonferroni" 1 ## 2 stats 3.6.0 FALSE pval "BH" 1 ## 3 qvalue 2.16.0 FALSE pval <NA> 1 ## 4 stats 3.6.0 FALSE pval "bonferroni" 1 ## 5 stats 3.6.0 FALSE pval "BH" 1 ## 6 qvalue 2.16.0 FALSE pval <NA> 1 ## pm.session label key value assay performanceMetric alpha ## 1 1 bonf TPR.1 0.125 H TPR 0.05 ## 2 1 BH TPR.1 0.375 H TPR 0.05 ## 3 1 qv TPR.1 0.600 H TPR 0.05 ## 4 1 bonf TPR.2 0.200 H TPR 0.10 ## 5 1 BH TPR.2 0.550 H TPR 0.10 ## 6 1 qv TPR.2 0.700 H TPR 0.10 For example, the code below extracts the TPR for an alpha of 0.1 for the Bonferroni method. tidyUpMetrics(sb) %>% dplyr:::filter(label == "bonf", alpha == 0.1, performanceMetric == "TPR") %>% dplyr:::select(value) ## value ## 1 0.2 # 3 Class Details Before moving to the second case study, we describe the anatomy of the BenchDesign and SummarizedBenchmark classes in a little more detail. ## 3.1 BenchDesign As noted in the previous section, BenchDesign objects are composed of a data set and methods. Formally, the methods are stored in the BenchDesign as a BDMethodList and the data as a BDData object. As would be expected, the BDMethodList is a list (List) of BDMethod objects, each containing the definition of a method to be compared in the benchmark experiment. The general structure is illustrated in the figure below. Continuing with the BenchDesign constructed in the example above, we can access the list of methods and the data set stored in the object using the BDMethodList() and BDData() functions. BDMethodList(b) ## BenchDesign Method List (BDMethodList) --------------------- ## list of 3 methods ## method: bonf; func: p.adjust ## method: BH; func: p.adjust ## method: qv; func: qvalue::qvalue BDData(b) ## BenchDesign Data (BDData) ---------------------------------- ## type: data ## data: ## H test_statistic effect_size pval SE ## 1 1 -3.2083437 -1.17151466 4.872282e-03 0.3651463 ## 2 0 0.1692236 0.07321912 8.675080e-01 0.4326768 ## 3 1 -5.7077940 -1.81715381 2.061521e-05 0.3183636 ## 4 1 -1.9805856 -1.09107836 6.313031e-02 0.5508867 ## 5 1 -1.0014358 -0.37726058 3.298895e-01 0.3767197 ## 6 1 -0.9190433 -0.47583669 3.702252e-01 0.5177522 ## 7 1 -1.3184486 -0.72750221 2.038846e-01 0.5517866 ## 8 1 -3.8492346 -1.47787871 1.175116e-03 0.3839409 ## 9 1 -3.4836213 -1.35474550 2.651401e-03 0.3888900 ## 10 0 0.6238195 0.25470364 5.405719e-01 0.4082970 ## ... with 30 more rows. The list of methods inherits from the SimpleList class, and supports several useful accessor and setter features. BDMethodList(b)[["bonf"]] ## BenchDesign Method (BDMethod) ------------------------------ ## method: p.adjust ## function (p, method = p.adjust.methods, n = length(p)... ## parameters: ## p : pval ## method : "bonferroni" ## post: ## none ## meta: ## none This interface allows for adding new methods by creating a new named entry in the BDMethodList of the BenchDesign object. BDMethodList(b)[["bonf2"]] <- BDMethodList(b)[["bonf"]] b ## BenchDesign ------------------------------------------------ ## benchmark data: ## type: data ## names: H, test_statistic, effect_size, pval, SE ## benchmark methods: ## method: bonf; func: p.adjust ## method: BH; func: p.adjust ## method: qv; func: qvalue::qvalue ## method: bonf2; func: p.adjust Methods can also be removed by setting the list value to NULL. An equivalent convenience function, dropMethod() also exists for removing methods using syntax similar to addMethod(). BDMethodList(b)[["bonf"]] <- NULL b ## BenchDesign ------------------------------------------------ ## benchmark data: ## type: data ## names: H, test_statistic, effect_size, pval, SE ## benchmark methods: ## method: BH; func: p.adjust ## method: qv; func: qvalue::qvalue ## method: bonf2; func: p.adjust Each BDMethod object in a BDMethodList encapsulates a method to be benchmarked. The contents of the object are all of the arguments passed through the addMethod() convenience function used in the example above. New BDMethod objects can be constructed with the BDMethod() constructor, which takes the same input parameters as addMethod(). bdm_bonf <- BDMethod(x = p.adjust, params = rlang::quos(p = pval, method = "bonferroni")) bdm_bonf ## BenchDesign Method (BDMethod) ------------------------------ ## method: p.adjust ## function (p, method = p.adjust.methods, n = length(p)... ## parameters: ## p : pval ## method : "bonferroni" ## post: ## none ## meta: ## none Directly modifying the BDMethodList object provides an alternative approach (aside form using addMethod()) to adding methods to a BenchDesign object. BDMethodList(b)[["bonf"]] <- bdm_bonf b ## BenchDesign ------------------------------------------------ ## benchmark data: ## type: data ## names: H, test_statistic, effect_size, pval, SE ## benchmark methods: ## method: BH; func: p.adjust ## method: qv; func: qvalue::qvalue ## method: bonf2; func: p.adjust ## method: bonf; func: p.adjust The BDData object is a simple object which only contains two slots, @type and @data. The @type of a BDData object can be either data or hash. If the @type is data, the @data slot is a standard list or data.frame object of raw data. However, if the @type is hash, then the @data slot is just a MD5 hash of some raw data object computed using digest::digest(). As described next in Section 3.2, SummarizedBenchmark objects also contain the BenchDesign object used to generate the benchmark results. Often, the original raw data can be large, and saving the full data set as part of the SummarizedBenchmark object can be undesirable and unnecessary. While the raw data set is needed during the benchmark experiment, by default, the BDData is converted to a MD5 hash beforing stored the BenchDesign as part of a SummarizedBenchmark object. Using this approach, it is still possible to confirm whether a certain data set was used for a benchmarking experiment without having to store copies of the raw data with every SummarizedBenchmark. Returning to the BenchDesign example from above, the BDData object of a BenchDesign can also be similarly extracted and modified. As with BDMethod objects, the data set may be removed by setting the value to NULL. BDData(b) <- NULL b ## BenchDesign ------------------------------------------------ ## benchmark data: ## NULL ## benchmark methods: ## method: BH; func: p.adjust ## method: qv; func: qvalue::qvalue ## method: bonf2; func: p.adjust ## method: bonf; func: p.adjust A new data set can be constructed using the BDData() function and used to replace the data set in a BenchDesign object. bdd <- BDData(data = tdat) BDData(b) <- bdd b ## BenchDesign ------------------------------------------------ ## benchmark data: ## type: data ## names: H, test_statistic, effect_size, pval, SE ## benchmark methods: ## method: BH; func: p.adjust ## method: qv; func: qvalue::qvalue ## method: bonf2; func: p.adjust ## method: bonf; func: p.adjust The helper function hashBDData() can be called on either BDData or BenchDesign objects to convert the data to a MD5 hash. By default, this function is called on the BenchDesign object during buildBench() before storing the object in the resulting SummarizedBenchmark. b <- hashBDData(b) b ## BenchDesign ------------------------------------------------ ## benchmark data: ## type: md5hash ## MD5 hash: 09867f2696d3178a8f3bc038917042e1 ## benchmark methods: ## method: BH; func: p.adjust ## method: qv; func: qvalue::qvalue ## method: bonf2; func: p.adjust ## method: bonf; func: p.adjust We drop the "bonf2" method from the object to return to the same set of methods used to generate the SummarizedBenchmark object. BDMethodList(b)[["bonf2"]] <- NULL ## 3.2 SummarizedBenchmark As described above, the SummarizedBenchmark class builds on the existing SummarizedExperiment class and Bioconductor infrastructure. By doing so, saved results are tightly linked to metadata. Thus, it is possible, for example, to easily subset the results without losing references to the respective metadata. For example, the code below extracts the data for only the first two methods. sbSub <- sb[, 1:2] colData(sbSub) ## DataFrame with 2 rows and 10 columns ## func.pkg func.pkg.vers func.pkg.manual param.p param.method ## <character> <character> <logical> <character> <character> ## bonf stats 3.6.0 FALSE pval "bonferroni" ## BH stats 3.6.0 FALSE pval "BH" ## session.idx TPR.1 TPR.2 TPR.3 pm.session ## <numeric> <numeric> <numeric> <numeric> <numeric> ## bonf 1 0.125 0.2 0.225 1 ## BH 1 0.375 0.55 0.625 1 Building on the SummarizedExperiment class, in addition to the slot for performanceMetrics described above, the SummarizedBenchmark class also includes a slot which stores a copy of the BenchDesign object used to generate the results with buildBench(). The BenchDesign can be accessed by simply passing the object to BenchDesign(). BenchDesign(sb) ## BenchDesign ------------------------------------------------ ## benchmark data: ## type: md5hash ## MD5 hash: 09867f2696d3178a8f3bc038917042e1 ## benchmark methods: ## method: bonf; func: p.adjust ## method: BH; func: p.adjust ## method: qv; func: qvalue::qvalue Notice, however, unlike the original BenchDesign above, the data is stored as a MD5 hash. To prevent bloat when the data set is large, by default, only a MD5 hash computed using digest::digest() is stored with the benchmarking results. This behavior can be changed by setting keepData = TRUE when calling buildBench(), in which case, the complete data set will be stored as part of the BenchDesign and SummarizedBenchmark objects. A simple helper function, compareBDData(), can be used to verify that while the data type may be different between the data originally used to construct the SummarizedBenchmark and the data stored with the SummarizedBenchmark, the referenced data sets are the same (i.e. have matching MD5 hash values). compareBDData(BDData(tdat), BenchDesign(sb)) ##$data
## [1] TRUE
##
## $type ## [1] FALSE Finally, information about the R session where the methods were executed is stored in the metadata of the SummarizedBenchmark as an entry in a list called sessions. The package supports rerunning benchmark experiments, and for this purpose, each new session is stored as a new entry in the sessions list. Since this experiment has only been executed once, the list only has a single entry in the sessions list. In addition to storing the session info and list of methods evaluated during the session, the list also keeps track of all parameters in the buildBench() call, i.e. truthCols = "H". metadata(sb)$sessions[[1]]
## $methods ## [1] "bonf" "BH" "qv" ## ##$results
## $results$bonf
## $results$bonf$default ## [1] "success" ## ## ##$results$BH ##$results$BH$default
## [1] "success"
##
##
## $results$qv
## $results$qv$default ## [1] "success" ## ## ## ##$parameters
## $parameters$truthCols
## [1] "H"
##
## $parameters$ftCols
## NULL
##
## $parameters$sortIDs
## [1] FALSE
##
## $parameters$keepData
## [1] FALSE
##
## $parameters$catchErrors
## [1] TRUE
##
## $parameters$parallel
## [1] FALSE
##
## $parameters$BPPARAM
## class: MulticoreParam
##   bpisup: FALSE; bpnworkers: 4; bptasks: 0; bpjobname: BPJOB
##   bplog: FALSE; bpthreshold: INFO; bpstopOnError: TRUE
##   bpRNGseed: ; bptimeout: 2592000; bpprogressbar: FALSE
##   bpexportglobals: TRUE
##   bplogdir: NA
##   bpresultdir: NA
##   cluster type: FORK
##
##
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yaml 2.2.0 2018-07-25 [2] CRAN (R 3.6.0) ## zlibbioc 1.30.0 2019-06-14 [2] Bioconductor ## ## [1] /tmp/Rtmp4IP0pJ/Rinst368840ec693e ## [2] /home/biocbuild/bbs-3.9-bioc/R/library ## 3.3 Reproducibility and Replicability While the BenchDesign and SummarizedBenchmark classes are designed to enable easier reproduction and replication of benchmark experiments, they are not meant to completely replace properly written and documented analyses. If any functions or packages are needed for the methods defined in the BenchDesign, they must be available and sourced or loaded when running (or re-running) the analyses. As an example, suppose we have the following locally defined method, mymethod(), which depends on another locally defined, but unavailable, function, mysubmethod(). mymethod <- function(x) { mysubmethod(x) } mybd <- BenchDesign(data = list(vals = 1:5)) mybd <- addMethod(mybd, "method1", mymethod, params = rlang::quos(x = vals)) The top level mymethod() is captured in the corresponding BDMethod and BenchDesign. No errors are thrown since including mymethod() in a BenchDesign object does not require evaluating the function. BDMethod(mybd, "method1")@f ## function(x) { ## mysubmethod(x) ## } However, when buildBench() is called, the method naturally fails because the necessary mysubmethod() is not available in the current session. tryCatch({ buildBench(mybd) }, error = function(e) print(e)) ## !! error caught in buildBench !! ## !! error in main function of method: 'method1' ## !! original message: ## !! could not find function "mysubmethod" ## <simpleError in buildBench(mybd): ## No method returned valid values for any assays! ## > Therefore, if any locally defined functions are necessary for a method, they should be available along with the BenchDesign object or defined at the top level. Additionally, for reproducibility and clarity, we generally recommend keeping functions as “self-contained” as possible, and never relying on variables defined externally, e.g. in the global enviroment. Scoping with functions can be confusing, and it may not always be clear where constants are being evaluated. As an example, consider the following simple multiplying function. m <- 5 mymult <- function(x) { m * x } m <- 2 mybd <- BenchDesign(data = list(vals = 1:5, m = 10)) mybd <- addMethod(mybd, "methodr", mymult, params = rlang::quos(x = vals)) While experienced R programmers may know which value of m (5, 2 or 10) will be used when buildBench() is called, for many users, this is less obvious. assay(buildBench(mybd)) ## methodr ## [1,] 2 ## [2,] 4 ## [3,] 6 ## [4,] 8 ## [5,] 10 The answer is the value assigned most recently in the environment where mymult() was defined, R_GlobalEnv, the global environment (m = 2). Note, however, that if the BenchDesign is saved and reloaded in a new session, whatever value of m is defined in the global environment of the new session will be used. If m is not defined in the new session, an error will be thrown. In this case, m should either be explicitly defined within the function or passed as a second input variable of mymult() and defined with params = in the BDMethod. While it may be possible to aggressively capture all code and environment variables defined with each method in a BenchDesign, it is not within the scope of this package, and furthermore, may not be the best appraoch to constructing a reproducible benchmark. While redundant, for emphasis, we reiterate that the SummarizedBenchmark framework is a solution for structuring benchmark experiments which complements, but does not replace, well documented and organized code. # 4 Differential Expression Case Study library("limma") library("edgeR") library("DESeq2") library("tximport") In this more advanced case study, we use a simulated data set from (Soneson and Robinson 2016) to demonstrate how the SummarizedBenchmark package can be used to benchmark methods for differential expression analysis. Namely, we compare the methods implemented in the DESeq2, edgeR, and limma packages. The simulated data set includes 6 samples of three replicates each from two conditions. For each sample, transcript-level expression is provided as transcripts per-million (TPM) values for 15,677 transcripts from human chromosome 1 (Ensembl GRCh37.71). A more complete description of the data, including code for how the data ws generated, is available in the Supplementary Materials of (Soneson and Robinson 2016) here. We provide precomputed objects containing these count and ground truth data. data("soneson2016") head(txi$counts)
##                         A1          A2         A3         B1        B2
## ENST00000367770   1.670571   0.8064829   5.472561  58.700418  32.89297
## ENST00000367771 558.834722 458.8887676 662.352695   5.299743  20.73813
## ENST00000367772 155.881534 110.6033685 183.417201   0.000000   0.00000
## ENST00000423670  11.809207  16.4752934  10.426669  20.392491   1.26733
## ENST00000470238  96.489863  34.2755231  32.489730 785.514128 614.71261
## ENST00000286031  12.442872  13.4797855   4.781290   1.152118  20.50770
##                         B3
## ENST00000367770  16.648100
## ENST00000367771  14.862318
## ENST00000367772   0.000000
## ENST00000423670   7.546371
## ENST00000470238 815.123229
## ENST00000286031   5.760588
head(truthdat)
## # A tibble: 6 x 13
##   transcript status logFC_cat   logFC avetpm length eff_length   tpm1  tpm2
##   <chr>       <int> <chr>       <dbl>  <dbl>  <int>      <dbl>  <dbl> <dbl>
## 1 ENST00000…      1 [ 3.38,3…   4.17   0.225   2916      2771. 0.0237 0.427
## 2 ENST00000…      1 [ 3.38,3…   5.25   3.20    2921      2776. 6.23   0.164
## 3 ENST00000…      1 Inf       Inf      0.817   3477      3332. 1.63   0
## 4 ENST00000…      1 [ 0.00, …   0.322  0.203   2077      1932. 0.181  0.226
## 5 ENST00000…      1 [ 3.38,3…   4.07   4.08    1538      1393. 0.460  7.71
## 6 ENST00000…      0 [ 0.00, …   0      0.154   4355      4210. 0.154  0.154
## # … with 4 more variables: isopct1 <dbl>, isopct2 <dbl>, gene <chr>,
## #   avetpm_cat <chr>

## 4.1 Benchmark Set-Up and Execution

We begin the benchmarking process by creating our BenchDesign object with the data set. The BenchDesign can be initialized with a data.frame (as in the case study above), or more generally, with a list object. In this case study, since methods for differential expression require more than just the expression counts, e.g. the experimental design, we construct a list containing each of these inputs as a named entry.

The scaled TPM values are rounded before passing to the differential expression methods.

mycounts <- round(txi$counts) Here, we simply use the the conditions for each sample to define the experimental design. The design matrix is stored as a data.frame, mycoldat. mycoldat <- data.frame(condition = factor(rep(c(1, 2), each = 3))) rownames(mycoldat) <- colnames(mycounts) The data object for the benchmark experiment is now constructed with both the counts and the design matrix, along with some ground truth information (“status”: the true presence or absence of differential expression between conditions, and “lfc”: the expected log-fold change between conditions). mydat <- list(coldat = mycoldat, cntdat = mycounts, status = truthdat$status,
lfc = truthdat$logFC) As before, the BenchDesign is constructed with the data as the sole input. bd <- BenchDesign(data = mydat) For simplicity, we focus on comparing only the p-values returned by each method after testing for differential expression between the two conditions. However, later in this vignette, we also show how multiple metrics (p-values and log-fold change) can be compared in a single BenchDesign object. Since each method requires running multiple steps, we write wrapper functions which return only the vector of p-values for each method. deseq2_pvals <- function(countData, colData, design, contrast) { dds <- DESeqDataSetFromMatrix(countData, colData = colData, design = design) dds <- DESeq(dds) res <- results(dds, contrast = contrast) res$pvalue
}

edgeR_pvals <- function(countData, group, design) {
y <- DGEList(countData, group = group)
y <- calcNormFactors(y)
des <- model.matrix(design)
y <- estimateDisp(y, des)
fit <- glmFit(y, des)
lrt <- glmLRT(fit, coef=2)
lrt$table$PValue
}

voom_pvals <- function(countData, group, design) {
y <- DGEList(countData, group = group)
y <- calcNormFactors(y)
des <- model.matrix(design)
y <- voom(y, des)
eb <- eBayes(lmFit(y, des))
eb$p.value[, 2] } Next, each method is added to the BenchDesign using addMethod(), and the corresponding wrapper function passed as func. (For a review of the basic usage of addMethod(), revisit Section 2.) We again use the pipe notation for compactness. bd <- bd %>% addMethod(label = "deseq2", func = deseq2_pvals, params = rlang::quos(countData = cntdat, colData = coldat, design = ~condition, contrast = c("condition", "2", "1")) ) %>% addMethod(label = "edgeR", func = edgeR_pvals, params = rlang::quos(countData = cntdat, group = coldat$condition,
design = ~coldat$condition) ) %>% addMethod(label = "voom", func = voom_pvals, params = rlang::quos(countData = cntdat, group = coldat$condition,
design = ~coldat$condition) ) So far, none of the methods have been executed. The BenchDesign object simply serves as a container describing how the methods should be executed. The methods are applied by a simple call to buildBench(). Since the ground truth is known and available in mydat$status, this is specified to truthCols=.

sb <- buildBench(bd, truthCols = "status")

We can inspect the results.

sb
## class: SummarizedBenchmark
## dim: 15677 3
## assays(1): status
## rownames: NULL
## rowData names(1): status
## colnames(3): deseq2 edgeR voom
## colData names(9): func.pkg func.pkg.vers ... param.group session.idx

## 4.2 Benchmark Evaluation

By running the code above, the results of three differential expression methods (edgeR, limma-voom and DESeq2 will be stored in a SummarizedBenchmark container. The next step is to define metrics to evaluate the performance of these three methods. This can be done by using the function addPerformanceMetric(), as described before in Section 2. However, in this package there are implementations for several ‘default’ metrics that are commonly used to evaluate methods. The function availableMetrics() returns a data.frame of these metrics.

availableMetrics()
##            functions                                   description
## 1         rejections                          Number of rejections
## 2                TPR                            True Positive Rate
## 3                TNR                            True Negative Rate
## 4                FDR              False Discovery Rate (estimated)
## 5                FNR                           False Negative Rate
## 6        correlation                           Pearson correlation
## 7               sdad Standard Deviation of the Absolute Difference
## 8            hamming                              Hamming distance
## 9             LPnorm                                    L_{p} norm
##    requiresTruth
## 1          FALSE
## 2           TRUE
## 3           TRUE
## 4           TRUE
## 5           TRUE
## 6           TRUE
## 7           TRUE
## 8           TRUE
## 9           TRUE
## 10          TRUE

For example, the predefined metrics rejections, TPR, TNR, FDR and FNR can be added to the assay H of our object using the following code,

sb <- addPerformanceMetric( sb,
evalMetric=c("rejections", "TPR", "TNR", "FDR", "FNR"),
assay="status" )
names(performanceMetrics(sb)[["status"]])
## [1] "rejections" "TPR"        "TNR"        "FDR"        "FNR"

Having defined the desired performance metrics, the function estimatePerformanceMetrics() will calculate these metrics for each of the three methods.

estimatePerformanceMetrics(
sb,
alpha = c(0.01, 0.05, 0.1, 0.2),
tidy = TRUE) %>%
dplyr:::select(label, value, performanceMetric, alpha) %>%
tail()
##     label      value performanceMetric alpha
## 55 deseq2 0.10949477               FNR   0.1
## 56  edgeR 0.09381113               FNR   0.1
## 57   voom 0.11381970               FNR   0.1
## 58 deseq2 0.09181118               FNR   0.2
## 59  edgeR 0.07986767               FNR   0.2
## 60   voom 0.08741018               FNR   0.2

Furthermore, the functions plotMethodsOverlap() and plotROC() are helpful to visualize the performance of the different methods, in case these methods output q-values.

plotMethodsOverlap() is a wrapper for the function upset() from the UpSetR package that is helpful to visualize the overlaps between hits of different methods for a given alpha value.

plotMethodsOverlap( sb, assay="status", alpha=0.1, order.by="freq")

From the plot above, it is evident that there is a large number of transcripts that are detected to be differentially expressed by all three methods. There are also smallers sets of transcripts that are detected uniquely by a single method or subsets of methods. Another typical way to compare the performance of different methods are Receiver Operating Characteristic (ROC) curves. The function plotROC() inputs a SummarizeBenchmark object and draws the ROC curves for all methods contained in it.

SummarizedBenchmark::plotROC(sb, assay="status")

Here, we describe several additional features implemented in SummarizedBenchmark for building on the standard workflow described in the previous sections. The features are illustrated using the same differential expression example from above.

## 5.1 Reusing a BenchDesign Across Data Sets

When benchmarking several methods, it is generally considered good practice to apply the methods to more than just a single data set. Under the SummarizedBenchmark framework, this naturally translates to recycling the same set of methods defined in a single BenchDesign object across multiple data sets. As mentioned early on, a BenchDesign can be intialized without any methods or data set.

bdnull <- BenchDesign()
bdnull
## BenchDesign ------------------------------------------------
##   benchmark data:
##     NULL
##   benchmark methods:
##     none

As before, methods can be added to the BenchDesign with addMethod(), and the benchmark experiment run using buildBench().

bdnull <- bdnull %>%
params = rlang::quos(p = pval,
method = "bonferroni")) %>%
params = rlang::quos(p = pval,
method = "BH"))

The buildBench() method accepts an optional data = parameter. When specified, this data set is used to run the experiment, taking precedence over the data set specified in (or missing from) the BenchDesign object.

buildBench(bdnull, data = tdat)
## class: SummarizedBenchmark
## dim: 50 2
## assays(1): default
## rownames: NULL
## rowData names(1): default
## colnames(2): bonf BH
## colData names(6): func.pkg func.pkg.vers ... param.method
##   session.idx

By specifying data during the buildBench() step, the exact same benchmark comparison, as defined in the common BenchDesign object, can be carried out consistently across multiple data sets. While this approach works even if the common BenchDesign object contains a default data set, it is recommended that the BenchDesign be created without any data to avoid errors if the design is going to be reused across data sets.

## 5.2 Storing Multiple Outputs

The differential expression case study described above has assumed that we are interested in a single numeric vector for each method, namely, a vector of p-values. These p-values are stored as the sole assay in the SummarizedBenchmark object returned by buildBench(). However, in many cases, there are multiple values of interest to be compared across methods. For example, looking at the estimated log-fold changes in addition to p-values may be informative when comparing methods for differential expression.

The BenchDesign framework supports multiple assays with the post = parameter of the addMethod() call. When zero or one function is specified to post = for all methods, as in the examples above, the results are stored as a single assay. However, if post = is passed a named list of functions, separate assays will be created using the names and functions in each list. Since the assay names are taken from post =, all entries in the list must be named. Furthermore, since results are matched across methods using the assay names, if a method is missing a post = function for a given assay name, the corresponding column in the assays will be set to NA.

To track both p-values and log-fold change values for each method, we write new wrapper functions. Separate wrapper functions are written for first returning the primary analysis results, and separate accessor functions for extracting p-values and log-fold changes from the results.

deseq2_run <- function(countData, colData, design, contrast) {
dds <- DESeqDataSetFromMatrix(countData,
colData = colData,
design = design)
dds <- DESeq(dds)
results(dds, contrast = contrast)
}
deseq2_pv <- function(x) {
x$pvalue } deseq2_lfc <- function(x) { x$log2FoldChange
}

edgeR_run <- function(countData, group, design) {
y <- DGEList(countData, group = group)
y <- calcNormFactors(y)
des <- model.matrix(design)
y <- estimateDisp(y, des)
fit <- glmFit(y, des)
glmLRT(fit, coef=2)
}
edgeR_pv <- function(x) {
x$table$PValue
}
edgeR_lfc <- function(x) {
x$table$logFC
}

voom_run <- function(countData, group, design) {
y <- DGEList(countData, group = group)
y <- calcNormFactors(y)
des <- model.matrix(design)
y <- voom(y, des)
eBayes(lmFit(y, des))
}
voom_pv <- function(x) {
x$p.value[, 2] } voom_lfc <- function(x) { x$coefficients[, 2]
}

The primary wrapper function and a list of accessor functions are passed to func = and post = respectively.

bd <- BenchDesign(data = mydat) %>%
func = deseq2_run,
post = list(pv = deseq2_pv,
lfc = deseq2_lfc),
params = rlang::quos(countData = cntdat,
colData = coldat,
design = ~condition,
contrast = c("condition", "2", "1"))
) %>%
func = edgeR_run,
post = list(pv = edgeR_pv,
lfc = edgeR_lfc),
params = rlang::quos(countData = cntdat,
group = coldat$condition, design = ~coldat$condition)
) %>%
func = voom_run,
post = list(pv = voom_pv,
lfc = voom_lfc),
params = rlang::quos(countData = cntdat,
group = coldat$condition, design = ~coldat$condition)
)

When the BenchDesign is evaluated using buildBench(), the resulting SummarizedBenchmark will be generated with two assays: "pv" and "lfc". As before, the ground truth can be specified using the truthCols = parameter. When multiple assays are used, truthCols = expects a named vector of assay-name = "column-name" pairs.

sb <- buildBench(bd = bd, truthCols = c(pv = "status", lfc = "lfc"))
sb
## class: SummarizedBenchmark
## dim: 15677 3
## assays(2): pv lfc
## rownames: NULL
## rowData names(2): pv lfc
## colnames(3): deseq2 edgeR voom
## colData names(9): func.pkg func.pkg.vers ... param.group session.idx

We can verify that the two assays contain the expected values.

head(assay(sb, "pv"))
##            deseq2        edgeR         voom
## [1,] 2.875541e-04 2.074073e-04 6.103023e-03
## [2,] 2.799371e-23 2.121422e-16 1.239445e-04
## [3,] 8.450676e-14 3.109331e-18 5.221157e-06
## [4,] 6.930834e-01 6.621236e-01 3.357040e-01
## [5,] 2.479616e-11 2.468951e-09 3.978989e-04
## [6,] 9.057130e-01 8.947438e-01 5.199404e-01
head(assay(sb, "lfc"))
##          deseq2       edgeR       voom
## [1,]  3.7329019   3.6767779  3.5232524
## [2,] -5.3905346  -5.3773940 -5.5543949
## [3,] -9.7125453 -10.2467488 -8.2533275
## [4,] -0.4706410  -0.4547095 -1.0510456
## [5,]  3.7048724   3.7024426  3.9084940
## [6,] -0.1554938  -0.1504299 -0.7607029

## 5.3 Parallelizing with BiocParallel

The simple examples considered in this vignette were constructed to be computational manageable with only one core. However, when working with larger data sets, running each method in serial with a single machine is often undesirable. Similarly, when replicating benchmark experiments across multiple data sets, running each experiment in serial may be inefficent. In this section, we describe how to use the BiocStyle::Biocpkg("BiocParallel") package to parallelize benchmarking across methods and data sets. More details on how to specify the parallelization back-end can be found in the Introduction to BiocParallel vignette for the BiocParallel package.

### 5.3.1 Across Methods

Since constructing a BenchDesign object requires no computation, the bottleneck only appears at the buildBench() step of the process. Parallelization of this step is enabled using the BiocParallel package. By default, parallel evaluation is disabled, but can easily be enabled by setting parallel = TRUE and optionally specifying the BPPARAM = parameter. If BPPARAM = is not specified, the default back-end will be used. The default back-end can be checked with bpparam().

Parallelization of buildBench() is carried out across the set of methods specified with addMethod(). Thus, there is no benefit to specifying more cores than the number of methods.

bpparam()
## class: MulticoreParam
##   bpisup: FALSE; bpnworkers: 4; bptasks: 0; bpjobname: BPJOB
##   bplog: FALSE; bpthreshold: INFO; bpstopOnError: TRUE
##   bpRNGseed: ; bptimeout: 2592000; bpprogressbar: FALSE
##   bpexportglobals: TRUE
##   bplogdir: NA
##   bpresultdir: NA
##   cluster type: FORK
sbp <- buildBench(bd, parallel = TRUE,
BPPARAM = BiocParallel::SerialParam())
sbp
## class: SummarizedBenchmark
## dim: 15677 3
## assays(2): pv lfc
## rownames: NULL
## rowData names(2): pv lfc
## colnames(3): deseq2 edgeR voom
## colData names(9): func.pkg func.pkg.vers ... param.group session.idx

The results, as expected, are the same as when buildBench() was called without parallelization.

all(assay(sbp) == assay(sb), na.rm = TRUE)
## [1] TRUE

### 5.3.2 Across Datasets

Typically, benchmark studies have more than a single dataset. For this cases, users can create a BenchDesign object and execute this design on many datasets. Again, running the step of executing the benchmark design on every dataset using a single core might take to long. However, parallelization across datasets is possible using the BiocParallel package.

To demonstrate this, we split the count data and ground truths of the (Soneson and Robinson 2016) dataset, as if they were three different datasets.

#mydat <- as.list( bd@data@data )
ndat <- length(mydat$status) spIndexes <- split( seq_len( ndat ), rep( 1:3, length.out=ndat ) ) datasetList <- lapply( spIndexes, function(x){ list( coldat=mydat$coldat,
cntdat=mydat$cntdat[x,], status=mydat$status[x],
lfc=mydat$lfc[x] ) } ) names( datasetList ) <- c("dataset1", "dataset2", "dataset3") Then, using a call to bplapply() function, we can execute the BenchDesign object for each dataset. Note that with the BPPARAM = parameter, the execution of the BenchDesign is done across many computing instances. In the example below, we use 3 cores. sbL <- bplapply( datasetList, function( x ){ buildBench( bd, data=x ) }, BPPARAM = MulticoreParam( 3 ) ) sbL ##$dataset1
## class: SummarizedBenchmark
## dim: 5226 3
## assays(2): pv lfc
## rownames: NULL
## rowData names(2): pv lfc
## colnames(3): deseq2 edgeR voom
## colData names(9): func.pkg func.pkg.vers ... param.group session.idx
##
## $dataset2 ## class: SummarizedBenchmark ## dim: 5226 3 ## metadata(1): sessions ## assays(2): pv lfc ## rownames: NULL ## rowData names(2): pv lfc ## colnames(3): deseq2 edgeR voom ## colData names(9): func.pkg func.pkg.vers ... param.group session.idx ## ##$dataset3
## class: SummarizedBenchmark
## dim: 5225 3
## assays(2): pv lfc
## rownames: NULL
## rowData names(2): pv lfc
## colnames(3): deseq2 edgeR voom
## colData names(9): func.pkg func.pkg.vers ... param.group session.idx

## 5.4 Updating Analyses

Often, benchmarking analyses are performed in multiple steps, iteratively, with methods added or updated based on previous results. Additionally, over the course of a benchmarking study, developers may update methods to improve the implementation or add new features. Ideally, whenever a methods are updated or added, the benchmark results should also be updated. Naively, all methods in the benchmark could be re-computed by calling buildBench() on the original BenchDesign whenever the benchmarker notices that a method has been changed. However, not only is this computational inefficient, but requires the benchmarker to manually check if any methods have been verified.

To make this process easier, the package includes the updateBench() function which can be used to both check if any results in a SummarizedBenchmark object need to be updated, and to update these results with a call to buildBench().

To demonstrate the use of updateBench(), we continue with the example of comparing methods for differential expression. First, the function can be called with just a single SummarizedBenchmark object to see if any of the methods have become outdated. By default, the function will not actually run any methods.

updateBench(sb = sb)
## Update SummarizedBenchmark (dryrun) ----------------------------------
##   benchmark data: unchanged (full data missing)
##     MD5 hash: 3f41eeab0a157bf127e2b7ffb98ab685
##     names: ??
##   benchmark methods:
##     |              | Need to  |           Outdated            |
##     | Method       | (Re)Run  |  Func Param  Meta  Post  Vers |
##     | deseq2       |       N  |     N     N     N     N     N |
##     | edgeR        |       N  |     N     N     N     N     N |
##     | voom         |       N  |     N     N     N     N     N |

The output is a table showing whether any methods need to be re-evaluated, as well as which components of the methods are now outdated. Here, the headings correspond to:

• Func: the main method function,
• Param: the method parameters,
• Meta: the manually specified method metadata,
• Post: the post-processing functions,
• Vers: the corresponding package version for the main method.

As can be seen by the Ns under Need to (Re)Run, all results in the SummarizedBenchmark are still valid, and no method needs to be re-run. Also note that the benchmark data is listed as full data missing. This is because the SummarizedBenchmark object only contains the data set as a MD5 hash. If any of the methods needed to be updated, the original data would have to be specified to updateBench() using the data = parameter.

BDData(BenchDesign(sb))
## BenchDesign Data (BDData) ----------------------------------
##   type: md5hash
##   MD5 hash: 3f41eeab0a157bf127e2b7ffb98ab685

The second way to call updateBench() is to specify both a SummarizedBenchmark object and a BenchDesign object. Intuitively, this will result in updating the results of the SummarizedBenchmark object to include any new or modified methods in the BenchDesign. Methods between the two objects are matched by the method label, e.g. deseq2. As an example, suppose we modify one of the methods in the BenchDesign object and want to rerun the benchmarking experiment. We first create a modified BenchDesign object, where we modify a single method’s manually specified meta data. Note that this is just a trivial update and normally would not warrant updating the results.

bd2 <- modifyMethod(bd, "deseq2", rlang::quos(bd.meta = list(note = "minor update")))

Next, we pass the object to updateBench() along with the original SummarizedBenchmark object.

updateBench(sb = sb, bd = bd2)
## Update SummarizedBenchmark (dryrun) ----------------------------------
##   benchmark data: unchanged
##     MD5 hash: 3f41eeab0a157bf127e2b7ffb98ab685
##     names: coldat cntdat status lfc
##   benchmark methods:
##     |              | Need to  |           Outdated            |
##     | Method       | (Re)Run  |  Func Param  Meta  Post  Vers |
##     | deseq2       |       Y  |     N     N     Y     N     N |
##     | edgeR        |       N  |     N     N     N     N     N |
##     | voom         |       N  |     N     N     N     N     N |

Notice that now the Need to (Re)Run and Meta columns are now set to Y for the deseq2 method, indicating that the metadata of the method has been updated, and the method now needs to be rerun. Next, suppose we also decide to add a new method to our comparison. Here, we add an alternate version of DESeq2 to the comparison based on adaptive shrinkage (ASH) (Stephens 2017).

deseq2_ashr <- function(countData, colData, design, contrast) {
dds <- DESeqDataSetFromMatrix(countData,
colData = colData,
design = design)
dds <- DESeq(dds)
lfcShrink(dds, contrast = contrast, type = "ashr")
}

deseq2_ashr,
post = list(pv = deseq2_pv,
lfc = deseq2_lfc),
params = rlang::quos(countData = cntdat,
colData = coldat,
design = ~condition,
contrast = c("condition", "2", "1")))

Now we see that bo the modified deseq2 and deseq2_ashr methods are listed as needing to be rerun.

updateBench(sb = sb, bd = bd2)
## Update SummarizedBenchmark (dryrun) ----------------------------------
##   benchmark data: unchanged
##     MD5 hash: 3f41eeab0a157bf127e2b7ffb98ab685
##     names: coldat cntdat status lfc
##   benchmark methods:
##     |              | Need to  |           Outdated            |
##     | Method       | (Re)Run  |  Func Param  Meta  Post  Vers |
##     | deseq2       |       Y  |     N     N     Y     N     N |
##     | edgeR        |       N  |     N     N     N     N     N |
##     | voom         |       N  |     N     N     N     N     N |
##     | deseq2_ashr  |       Y  |     -     -     -     -     - |

To run the update, we simply modify the call to updateBench() with dryrun = FALSE. When we do this, only the two methods needing to be rerun are evaluated.

sb2 <- updateBench(sb = sb, bd = bd2, dryrun = FALSE)
sb2
## class: SummarizedBenchmark
## dim: 15677 4
## assays(2): pv lfc
## rownames: NULL
## rowData names(2): pv lfc
## colnames(4): edgeR voom deseq2_ashr deseq2
## colData names(10): func.pkg func.pkg.vers ... session.idx meta.note

We can check the output to see that the results of the deseq2_ashr method have been added along with the updated deseq2 results.

head(assay(sb2, "lfc"))
##            edgeR       voom deseq2_ashr     deseq2
## [1,]   3.6767779  3.5232524  3.40956554  3.7329019
## [2,]  -5.3773940 -5.5543949 -5.32293234 -5.3905346
## [3,] -10.2467488 -8.2533275 -9.20816381 -9.7125453
## [4,]  -0.4547095 -1.0510456 -0.08956306 -0.4706410
## [5,]   3.7024426  3.9084940  3.61844604  3.7048724
## [6,]  -0.1504299 -0.7607029 -0.02921225 -0.1554938

Additionally, notice that a session index is also stored in the column data of each method. This can be used to map methods quickly to the corresponding entry of the sessions list stored in the object metadata.

colData(sb2)[, "session.idx", drop = FALSE]
## DataFrame with 4 rows and 1 column
##             session.idx
##               <numeric>
## edgeR                 1
## voom                  1
## deseq2_ashr           2
## deseq2                2

Recall, sessions is a list of session information stored in the metadata of the SummarizedBenchmark object. Each entry of the sessions list includes the names of methods for which the results were generated during that session, the corresponding method results, the run parameters, and the session info. We can check the list of methods for each session and see that it matches the session.idx values above.

lapply(metadata(sb2)$sessions, [[, "methods") ## [[1]] ## [1] "edgeR" "voom" ## ## [[2]] ## [1] "deseq2_ashr" "deseq2" As another example, we can also compare the new SummarizedBenchmark object against the original BenchDesign. By default, updateBench() will keep all methods in the SummarizedBenchmark object even if a corresponding method is not found in the new BenchDesign object. To avoid this behavior, we can specify keepAll = FALSE. updateBench(sb2, bd, keepAll = FALSE) ## Update SummarizedBenchmark (dryrun) ---------------------------------- ## benchmark data: unchanged ## MD5 hash: 3f41eeab0a157bf127e2b7ffb98ab685 ## names: coldat cntdat status lfc ## benchmark methods: ## | | Need to | Outdated | ## | Method | (Re)Run | Func Param Meta Post Vers | ## | deseq2 | Y | N N Y N N | ## | edgeR | N | N N N N N | ## | voom | N | N N N N N | ## | deseq2_ashr | Drop | - - - - - | As expected, the deseq2 method must again be updated to return to match the original method definition, and the newer deseq2_ashr method must be dropped. If users update the BenchDesign, it is possible to calculate the performance metrics only for methods that have been added or modified. This feature is useful, for example, when performance metrics are computationally expensive to calculate. In the example below, we calculate the number of rejections at a FDR of 10%. sb <- addPerformanceMetric( sb, assay="pv", evalMetric="rejections", evalFunction=function( query, truth ){ sum( p.adjust( query, method="BH" ) < 0.1, na.rm=TRUE) } ) sb <- estimatePerformanceMetrics(sb, addColData=TRUE) Then, we update the BenchDesign and rerun the function estimatePerformanceMetrics() with the parameter rerun=FALSE. Setting this parameter to FALSE will detect which performance metrics have been calculated before and will only recalculate metrics for methods that have been added or modified. sb2 <- updateBench( sb, bd2, dryrun=FALSE ) estimatePerformanceMetrics( sb2, rerun=FALSE, addColData=TRUE ) ## ## Option rerun is set to FALSE: ## Rerunning performance metrics only for the following methods: deseq2_ashr, deseq2 ## class: SummarizedBenchmark ## dim: 15677 4 ## metadata(1): sessions ## assays(2): pv lfc ## rownames: NULL ## rowData names(2): pv lfc ## colnames(4): edgeR voom deseq2_ashr deseq2 ## colData names(12): func.pkg func.pkg.vers ... pm.session meta.note ## 5.5 Error Handling When running a large benchmark study, not uncommonly, a single or a small subset of methods may fail during execution. This may be the result of misspecified parameters, an underlying bug in the software, or any number of other reasons. By default, errors thrown by methods which fail during buildBench() or updateBench() are caught and handled in a user-friendly way. As long as a single method executes without any errors, a SummarizedBenchmark object is returned as usual, with the assay columns of failed methods set to NA. Additionally, the corresponding error messages are stored in the metadata of the object for reference. As an example, consider the following example where we run case where we benchmark two simple methods. The first, slowMethod draws 5 random normal samples after waiting 5 seconds, and the second, fastMethod draws 5 random normal samples immediately. Each method is then passed through two post-processing functions, keepSlow and makeSlower, and keepFast and makeSlower, respectively. This results in three partially overlapping assays, keepSlow, keepFast and makeSlower. With this example, we also demonstrate how mismatched assays are handled across methods. bdslow <- BenchDesign(data = tdat) %>% addMethod("slowMethod", function() { Sys.sleep(5); rnorm(5) }, post = list(keepSlow = identity, makeSlower = function(x) { Sys.sleep(5); x })) %>% addMethod("fastMethod", function() { rnorm(5) }, post = list(keepFast = identity, makeSlower = function(x) { Sys.sleep(5); x })) We run these methods in parallel using parallel = TRUE and specify a timeout limit of 1 second for the BPPARAM. Naturally, slowMethod will fail, and fastMethod will fail during the makeSlower post-processing function. bpp <- SerialParam() bptimeout(bpp) <- 1 sbep <- buildBench(bdslow, parallel = TRUE, BPPARAM = bpp) ## !! error caught in buildBench !! ## !! error in main function of method: 'slowMethod' ## !! original message: ## !! reached elapsed time limit ## !! error caught in buildBench !! ## !! error in method: 'fastMethod', post: 'makeSlower' ## !! original message: ## !! reached elapsed time limit Notice that during the execution process, errors caught by buildBench() are printed to the console along with the name of the failed method and post-processing function when appropriate. We can verify that a valid SummarizedBenchmark object is still returned with the the remaining results. sbep ## class: SummarizedBenchmark ## dim: 5 2 ## metadata(1): sessions ## assays(3): keepFast keepSlow makeSlower ## rownames: NULL ## rowData names(3): keepFast keepSlow makeSlower ## colnames(2): slowMethod fastMethod ## colData names(4): func.pkg func.pkg.vers func.pkg.manual session.idx We can also check the values of the assays. sapply(assayNames(sbep), assay, x = sbep, simplify = FALSE) ##$keepFast
##      slowMethod fastMethod
## [1,]         NA -1.4920954
## [2,]         NA -0.5808546
## [3,]         NA -1.4016588
## [4,]         NA -0.6005302
## [5,]         NA -0.2767537
##
## $keepSlow ## slowMethod fastMethod ## [1,] NA NA ## [2,] NA NA ## [3,] NA NA ## [4,] NA NA ## [5,] NA NA ## ##$makeSlower
##      slowMethod fastMethod
## [1,]         NA         NA
## [2,]         NA         NA
## [3,]         NA         NA
## [4,]         NA         NA
## [5,]         NA         NA

Notice that most columns contain only NA values. These columns correspond to both methods which returned errors, as well as methods missing post-processing functions, e.g. no keepSlow function was defined for the fastMethod method. While the NA values cannot be used to distinguish the sources of the NA values, this is documented in the sessions list of the SummarizedBenchmark metadata. While the sessions object is a list containing information for all previous sessions, we are only interested in the current, first session. (For more details on why multiple sessions may be run, see Section 5.4.)

names(metadata(sbep)$sessions[[1]]) ## [1] "methods" "results" "parameters" "sessionInfo" In sessions, there is a "results" entry which includes a summary of the results for each combination of method and post-processing function (assay). The entries of results can take one of three values: "success", "missing", or an error message of class buildbench-error. The easiest way to view these resultsis by passing the results to the base R function, simplify2array(). sess1res <- metadata(sbep)$sessions[[1]]$results simplify2array(sess1res) ## slowMethod fastMethod ## keepSlow "reached elapsed time limit" "reached elapsed time limit" ## makeSlower "reached elapsed time limit" "missing" ## keepFast "missing" "success" In the returned table, columns correspond to methods, and rows correspond to assays. We clearly see that many of the methods failed due to exceeding the specified time limit. If we check one of these entries more closesly, we see that it is indeed a buildbench-error object that occurred ("origin") during the "main" function. sess1res$slowMethod$keepSlow ## [1] "reached elapsed time limit" ## attr(,"class") ## [1] "buildbench-error" ## attr(,"origin") ## [1] "main" If this error handling is not wanted, and the user would like the benchmark experiment to terminate when an error is thrown, then the option catchErrors = FALSE can be specified to eiher buildBench() or updateBench(). ## 5.6 Specifying Method Metadata Metadata for methods are stored in the colData() of SummarizedBenchmark objects. As metioned above, several default metadata columns are populated in the colData() of the SummarizedBenchmark object generated by a call to buildBench(). Sometimes it may be useful to include additional metadata columns beyond just the default columns. While this can be accomplished manually by modifying the colData() of the SummarizedBenchmark object post hoc, method metadata can also be specified at the addMethod() step using the meta = optional parameter. The meta = parameter accepts a named list of metadata information. Each list entry will be added to the colData() as a new column. To avoid collisions between metadata columns specified with meta = and the default set of columns, metadata specified using meta = will be added to colData() with meta. prefixed to the column name. As an example, we construct a BenchDesign object again using the differential expression example. The BenchDesign is created with two methods, DESeq2 and edgeR. Each method is specified with the optional meta = parameter. We can verify that the manually defined metadata column (meta.reason) is available in the colData() of the generated SummarizedBenchmark. tempbd <- BenchDesign(data = mydat) %>% addMethod(label = "deseq2", func = deseq2_pvals, meta = list(reason = "recommended by friend X"), params = rlang::quos(countData = cntdat, colData = coldat, design = ~condition, contrast = c("condition", "2", "1")) ) %>% addMethod(label = "edgeR", func = edgeR_pvals, meta = list(reason = "recommended by friend Y"), params = rlang::quos(countData = cntdat, group = coldat$condition,
design = ~coldat$condition) ) buildBench(tempbd) %>% colData() ## DataFrame with 2 rows and 10 columns ## func.pkg func.pkg.vers func.pkg.manual param.countData ## <character> <character> <logical> <character> ## deseq2 NA NA FALSE cntdat ## edgeR NA NA FALSE cntdat ## param.colData param.design param.contrast ## <character> <character> <character> ## deseq2 coldat ~condition c("condition", "2", "1") ## edgeR NA ~coldat$condition                       NA
##                    meta.reason      param.group session.idx
##                    <character>      <character>   <numeric>
## deseq2 recommended by friend X               NA           1
## edgeR  recommended by friend Y coldat$condition 1 While all methods in this example had the meta = option specified, this is not necessary. It is completely acceptable to specify the meta = parameter for only a subset of methods. ## 5.7 Specifying Version Metadata Arguably, two of the most important pieces of metadata stored in the colData() of the SummarizedBenchmark returned by buildBench() are the relevant package name and version (pkg_name, pkg_vers). Determining the package name and version requires the primary “workhorse” function of the method be directly specified as func = in the addMethod() call. In some cases, this may not be possible, e.g. if the “workhorse” function is a wrapper as in the differential expression example above. However, there still might exist an important function for which we would like to track the package name and version. The meta parameter can help. The meta = parameter will handle the following named list entries as special values: pkg_name, pkg_vers, pkg_func. First, if values are specified for pkg_name and pkg_vers in meta =, these will overwrite the values determined from func =. To trace the source of pkg_name and pkg_vers information, the vers_src column of the colData will be set to "meta_manual" (the default value is "func"). Alternatively, a function can be passed to meta = as pkg_func. This function will be used to determine both pkg_name and pkg_vers, and will take precendence over manually specified pkg_name and pkg_vers values. If pkg_func is specified, it will be included in the colData() as a new column with the same name, and the vers_src column will be set to "meta_func". **Note: the function should be wrapped in rlang::quo to be properly parsed. The following example illustrates the behavior when using either pkg_func or pkg_name and pkg_vers with the meta optional parameter. tempbd <- BenchDesign(data = mydat) %>% addMethod(label = "deseq2", func = deseq2_pvals, meta = list(pkg_func = rlang::quo(DESeq2::DESeq)), params = rlang::quos(countData = cntdat, colData = coldat, design = ~condition, contrast = c("condition", "2", "1")) ) %>% addMethod(label = "edgeR", func = edgeR_pvals, meta = list(pkg_name = "edgeR", pkg_vers = as.character(packageVersion("edgeR"))), params = rlang::quos(countData = cntdat, group = coldat$condition,
design = ~coldat$condition) ) buildBench(tempbd) %>% colData() ## DataFrame with 2 rows and 9 columns ## func.pkg func.pkg.vers func.pkg.manual param.countData ## <character> <character> <logical> <character> ## deseq2 DESeq2 1.24.0 TRUE cntdat ## edgeR edgeR 3.26.4 TRUE cntdat ## param.colData param.design param.contrast ## <character> <character> <character> ## deseq2 coldat ~condition c("condition", "2", "1") ## edgeR NA ~coldat$condition                       NA
##             param.group session.idx
##             <character>   <numeric>
## deseq2               NA           1
## edgeR  coldat\$condition           1

## 5.8 Modifying Methods in a BenchDesign

Modifying the defintion of a method after it has been added to a BenchDesign is supported by the modifyMethod() function. The BenchDesign object created in the differential expression above includes a method called DESeq2. We can check the definition of this method using printMethod().

printMethod(bd, "deseq2")
## deseq2 -----------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##   method: deseq2_run
##     function (countData, colData, design, contrast) { ...
##   parameters:
##     countData : cntdat
##     colData : coldat
##     design : ~condition
##     contrast : c("condition", "2", "1")
##   post:
##     pv : function (x) { ...
##     lfc : function (x) { ...
##   meta:
##     none

Suppose we wish to both flip the order of the contrast, and add a metadata tag. This can be easily accomplished by passing both new parameters to modifyMethod() as a rlang::quos(..) list, similar to how they would be passed to addMethod(). If the func, post, or meta valuesshould be modified, they should be included in the list using the special keywords, bd.func =, bd.post =, or bd.meta =, as shown in the example below.

modifyMethod(bd, label = "deseq2",
params = rlang::quos(contrast = c("condition", "1", "2"),
bd.meta = list(note = "modified post hoc"))
) %>%
printMethod("deseq2")
## deseq2 -----------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##   method: deseq2_run
##     function (countData, colData, design, contrast) { ...
##   parameters:
##     countData : cntdat
##     colData : coldat
##     design : ~condition
##     contrast : c("condition", "1", "2")
##   post:
##     pv : function (x) { ...
##     lfc : function (x) { ...
##   meta:
##     note : "modified post hoc"

Sometimes it may be desirable to completely overwrite all function parameters for a method, e.g. countData, colData, design, and contrast in the case of DESeq2. This may occur if some parameters were optional and originally specified, but no longer necessary. All function parameters can be overwritten by specifying .overwrite = TRUE.

modifyMethod(bd, label = "deseq2",
params = rlang::quos(contrast = c("condition", "1", "2"),
bd.meta = list(note = "modified post hoc")),
.overwrite = TRUE) %>%
printMethod("deseq2")
## deseq2 -----------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##   method: deseq2_run
##     function (countData, colData, design, contrast) { ...
##   parameters:
##     contrast : c("condition", "1", "2")
##   post:
##     pv : function (x) { ...
##     lfc : function (x) { ...
##   meta:
##     note : "modified post hoc"

Notice that all parameters other than contrast = c("condition", "1", "2") have been dropped.

## 5.9 Duplicating Methods in a BenchDesign

In addition to comparing multiple methods, a benchmark study may also involve comparing a single method across several parameter settings. The expandMethod() function provides the capability to take a method already defined in the BenchDesign, and expand it to multiple methods with differing parameter values in the BenchDesign object. In the following example, expandMethod() is used to duplicate the DESeq2 method with only the "contrast" parameter modified.

bde <- expandMethod(bd, label = "deseq2",
onlyone = "contrast",
params = rlang::quos(deseq2_v1 = c("condition", "1", "2"),
deseq2_v2 = c("condition", "2", "2"))
)
printMethod(bde, "deseq2_v1")
## deseq2_v1 --------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##   method: deseq2_run
##     function (countData, colData, design, contrast) { ...
##   parameters:
##     countData : cntdat
##     colData : coldat
##     design : ~condition
##     contrast : c("condition", "1", "2")
##   post:
##     pv : function (x) { ...
##     lfc : function (x) { ...
##   meta:
##     none
printMethod(bde, "deseq2_v2")
## deseq2_v2 --------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##   method: deseq2_run
##     function (countData, colData, design, contrast) { ...
##   parameters:
##     countData : cntdat
##     colData : coldat
##     design : ~condition
##     contrast : c("condition", "2", "2")
##   post:
##     pv : function (x) { ...
##     lfc : function (x) { ...
##   meta:
##     none

Notice that the parameter to be modified is specified with onlyone = (denoting “only one” parameter will be changed), and that method names are taken from names of the quosure list passed to params = in the expandMethod() call. To modify more than a single parameter in the duplicated methods, the onlyone = parameter can be ignored, and the new parameter values specified to params = as a list of quosure lists. Below, both the "contrast" and meta parameters are modified in the expanded methods.

bde <- expandMethod(bd, label = "deseq2",
params = list(deseq2_v1 = rlang::quos(contrast = c("condition", "1", "2"),
meta = list(note = "filp order")),
deseq2_v2 = rlang::quos(contrast = c("condition", "2", "2"),
meta = list(note = "nonsensical order"))
)
)
printMethod(bde, "deseq2_v1")
## deseq2_v1 --------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##   method: deseq2_run
##     function (countData, colData, design, contrast) { ...
##   parameters:
##     countData : cntdat
##     colData : coldat
##     design : ~condition
##     contrast : c("condition", "1", "2")
##     meta : list(note = "filp order")
##   post:
##     pv : function (x) { ...
##     lfc : function (x) { ...
##   meta:
##     none
printMethod(bde, "deseq2_v2")
## deseq2_v2 --------------------------------------------------
## BenchDesign Method (BDMethod) ------------------------------
##   method: deseq2_run
##     function (countData, colData, design, contrast) { ...
##   parameters:
##     countData : cntdat
##     colData : coldat
##     design : ~condition
##     contrast : c("condition", "2", "2")
##     meta : list(note = "nonsensical order")
##   post:
##     pv : function (x) { ...
##     lfc : function (x) { ...
##   meta:
##     none

# References

Benjamini, Yoav, and Yosef Hochberg. 1995. “Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing.” Journal of the Royal Statistical Society. Series B (Methodological), 289–300.

Dunn, Olive Jean. 1961. “Multiple Comparisons Among Means.” Journal of the American Statistical Association 56 (293):52–64.

Soneson, Charlotte, and Mark D Robinson. 2016. “iCOBRA: Open, Reproducible, Standardized and Live Method Benchmarking.” Nature Methods 13 (4):283–83. https://doi.org/10.1038/nmeth.3805.

Stephens, Matthew. 2017. “False discovery rates: a new deal.” Biostatistics 18 (2):275–94. https://doi.org/10.1093/biostatistics/kxw041.

Storey, John D. 2002. “A Direct Approach to False Discovery Rates.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64 (3):479–98.