# Further Analyses

# 2. Further Analyses

## 2.1. Analysis of Pharmacovigilance Data

To illustrate how the procedures in DiscreteFDR can be used for real data, we revisit the analysis of the pharmacovigilance data from Heller and Gur (2011) performed in [DDR]. This data set is obtained from a database for reporting, investigating and monitoring adverse drug reactions due to the Medicines and Healthcare products Regulatory Agency in the United Kingdom. It contains the number of reported cases of amnesia as well as the total number of adverse events reported for each of the \(m = 2446\) drugs in the database. For more details we refer to Heller and Gur (2011) and to the accompanying R-package discreteMTP (Heller et al. (2012)) (no longer available on CRAN), which also contains the data. Heller and Gur (2011) investigate the association between reports of amnesia and suspected drugs by performing for each drug a Fisher’s exact test (one-sided) for testing association between the drug and amnesia while adjusting for multiplicity by using several (discrete) FDR procedures. In what follows we present code that reproduces parts of Figure 2 and Table 1 in [DDR].

We proceed as in the example in Section 1. Since we need to access the critical values, we first determine the \(p\)-values and their support for the data set `amnesia` contained for convenience in the package DiscreteFDR. For this, we use `generate.pvalues` in conjunction with the preprocessing function `reconstruct_two` from package `DiscreteDatasets`, which rebuilds \(2 \times 2\) tables from single columns or rows by using additional knowledge of the marginals.

    library(DiscreteFDR)
    library(DiscreteDatasets)
    data(amnesia)
    amnesia.formatted <- generate.pvalues(amnesia, "fisher", list(alternative = "greater"), reconstruct_two)

A more comprehensible way is the use of a pipe:

    library(DiscreteDatasets)
    library(DiscreteTests)
    amnesia.formatted <- amnesia |>
      reconstruct_two() |>
      fisher_test_pv(alternative = "greater")

Then we perform the FDR analysis with functions `DBH` and `ADBH` (SU and SD) and `DBR` at level \(\alpha = 0.05\) including critical values.

    DBH.su  <-  DBH(amnesia.formatted, ret.crit.consts = TRUE)
    DBH.sd  <-  DBH(amnesia.formatted, direction = "sd", ret.crit.consts = TRUE)
    ADBH.su <- ADBH(amnesia.formatted, ret.crit.consts = TRUE)
    ADBH.sd <- ADBH(amnesia.formatted, direction = "sd", ret.crit.consts = TRUE)
    DBR     <-  DBR(amnesia.formatted, ret.crit.consts = TRUE)

It is helpful to have a histogram of the observed \(p\)-values. For this, this package provides a `hist` method for `DiscreteFDR` class objects, too.

    hist(DBH.sd)

This histogram indicates a highly discrete \(p\)-value distribution, which strongly suggests the use of discrete methods.

By accessing the critical values we can now generate a plot similar to Figure 2 from [DDR]. Note that both [DBH-SU] and [DBH-SD] are visually indistinguishable from their adaptive counterparts.

    raw.pvalues <- amnesia.formatted$get_pvalues()
    m <- length(raw.pvalues)
    crit.values.BH <- 0.05 * seq_len(m) / m
    scale.points <- 0.7

    plot(DBH.su, col = c("black", "black", "orange"), pch = NA, type.crit = 'p', xlim = c(1, 100),
         ylim = c(0, DBH.su$Critical.values[100]), ylab = "critical values", cex = scale.points, main = "")

    points(crit.values.BH[1:105],          col = "green",  pch = 19, cex = scale.points)
    points(DBH.sd$Critical.values[1:105],  col = "red",    pch = 19, cex = scale.points)
    points(ADBH.su$Critical.values[1:105], col = "blue",   pch = 19, cex = scale.points)
    points(ADBH.sd$Critical.values[1:105], col = "purple", pch = 19, cex = scale.points)
    points(DBR$Critical.values[1:105],     col = "yellow", pch = 19, cex = scale.points)
    points(sort(raw.pvalues),                              pch = 4,  cex = scale.points)
    mtext("Figure 2", 1, outer = TRUE, line = -1)

Critical values for [BH] (green dots), [DBH-SU] (orange dots), [DBH-SD] (red dots), [A-DBH-SU] (blue dots), [A-DBH-SD] (purple dots), [DBR] (yellow dots). The sorted raw \(p\)-values are represented by asterisks.

The rejected hypotheses can be accessed via the command `$Indices`. The following code yields some of the values from Table 1 in [DDR]:

    rej.BH      <- length(which(p.adjust(raw.pvalues, method = "BH") <= 0.05))
    rej.DBH.su  <- length(DBH.su$Indices)
    rej.DBH.sd  <- length(DBH.sd$Indices)
    rej.ADBH.su <- length(ADBH.su$Indices)
    rej.ADBH.sd <- length(ADBH.sd$Indices)
    rej.DBR     <- length(DBR$Indices)
    c(rej.BH, rej.DBH.su, rej.DBH.sd, rej.ADBH.su, rej.ADBH.sd, rej.DBR)

    ## [1] 24 27 27 27 27 27

The (continuous) BH rejects only 24 hypotheses whereas all the discrete procedures implemented in DiscreteFDR are able to identify three additional drug candidates potentially associated with amnesia.

## 2.2. Other Types of Discrete Tests

In this section we sketch how can be used to analyze arbitrary multiple discrete tests. Jiménez-Otero et al. (2018) used DiscreteFDR to detect disorder in NGS experiments based on one-sample tests of the Poisson mean. Rather than reproducing their analysis in detail, we illustrate the general approach by using a toy example similar to the one in Section 1 and show how the test of the Poisson mean can be accommodated by DiscreteFDR.

To fix ideas, suppose we observe \(m = 9\) independent Poisson distributed counts \(N_1, \ldots, N_9\) (Jiménez-Otero et al. (2018) used this to model the read counts of different DNA bases). We assume that \(N_i \sim \text{Pois}(\lambda_i)\) and the goal is to identify cases where \(\lambda_i\) is larger than some prespecified value \(\lambda^0_i\), i.e., we have the (one-sided) multiple testing problem \[H_{0i}: \lambda_i = \lambda^0_i \qquad \text{vs.} \qquad H_{1i}: \lambda_i > \lambda^0_i.\] As in Section 1, the goal is to adjust for multiple testing by using the [DBH-SD] procedure at FDR-level \(\alpha = 5\%\). In our example the observations \(n_1,\ldots, n_9\) and parameters \(\lambda^0_1, \ldots, \lambda^0_9\) are given as follows:

    lambda.vector <- c(0.6, 1.2, 0.7, 1.3, 1.0, 0.2, 0.8, 1.3, 0.9)
    observations <- c(3, 3, 1, 2, 3, 3, 1, 2, 4)
    configuration <- cbind(observations, lambda.vector)
    alpha <- 0.05
    m <- length(observations)
    print(configuration)

    ##       observations lambda.vector
    ##  [1,]            3           0.6
    ##  [2,]            3           1.2
    ##  [3,]            1           0.7
    ##  [4,]            2           1.3
    ##  [5,]            3           1.0
    ##  [6,]            3           0.2
    ##  [7,]            1           0.8
    ##  [8,]            2           1.3
    ##  [9,]            4           0.9

Denote by \(G_i\) the distribution of \(N_i\) under \(H_{0i}\) i.e., \(G_i(x) = P(N_i \le x)\). For observations \(n_1,\ldots, n_9\) of \(N_1, \ldots, N_9\) the \(p\)-values for the above one-sided test are given by \[p_i = P(N_i \ge n_i) = P(N_i > n_i - 1) = \overline{G_i}(n_i - 1),\] where \(\overline{G_i}(x) = P(N_i > x) = 1 - G_i(x)\) denotes the survival function of the Poisson distribution with parameter \(\lambda^0_i\). Thus the raw \(p\)-values are determined by the following R code:

    raw.pvalues <- ppois(observations - 1, lambda.vector, lower.tail = FALSE)
    poisson.p <- poisson_test_pv(observations, lambda.vector, "greater")
    raw.pvalues.2 <- poisson.p$get_pvalues()
    print(raw.pvalues.2)

    ##           1           2           3           4           5           6
    ## 0.023115288 0.120512901 0.503414696 0.373176876 0.080301397 0.001148481
    ##           7           8           9
    ## 0.550671036 0.373176876 0.013458721

Following the definition of the function in R we define the inverse function of \(\overline{G_i}\) by \[\left(\overline{G_i}\right)^{-1}(p) = \min\{n \in \mathbb{N}: \overline{G_i}(n) \le p\}\] and obtain for the distribution function of the \(i\)-th \(p\)-value under the null \[F_i(x) = \overline{G_i}\left(\left(\overline{G_i}\right)^{-1}(x)\right).\] Each function \(F_i\) is a step function with \(F_i(0) = 0\), \(F_i(1) = 1\) and there exists an infinite sequence of jumps at locations \(1 = x_1 > x_2 > \ldots > x_n > x_{n + 1} > \ldots > 0\) such that \(F(x_j) = x_j\) for \(j \in \mathbb{N}\).

Initially it seems that we run into a problem if we want to determine the critical values of [DBH-SD] since the supports of \(F_1, \ldots, F_9\) are no longer finite (but still discrete). We can deal with this problem by using the observation that it is sufficient to consider new, restricted supports \(\mathcal{A}_i \cap [s^{\tiny \mbox{min}},1]\) where the lower threshold satisfies \[\begin{align} s^{\tiny \mbox{min}} &\le \tau^{\tiny \mbox{min}}_1 = \max \left\{t \in \mathcal{A}\::\: t \leq y^{\tiny \mbox{min}} \right\} \qquad \text{where} \qquad y^{\tiny \mbox{min}} = \frac{\alpha}{m} \cdot \left(1 + \frac{\alpha}{m} \right)^{-1}. \end{align}\] To determine such an \(s^{\tiny \mbox{min}}\) we proceed as follows. Define \(n^{\tiny \mbox{max}}_i = \left(\overline{G_i}\right)^{-1}(y^{\tiny \mbox{min}}) + 1\), \(t^{\tiny \mbox{min}}_i = \overline{G_i}(n^{\tiny \mbox{max}}_i - 1)\) and set \(s^{\tiny \mbox{min}} = \min\left(t^{\tiny \mbox{min}}_1, \ldots, t^{\tiny \mbox{min}}_9 \right)\). It is easily checked that this choice of \(s^{\tiny \mbox{min}}\) satisfies the above equation. We can determine \(s^{\tiny \mbox{min}}\) by the following code

    y.min <- alpha/m * (1 + alpha/m)^(-1)
    n.max <- qpois(y.min,     lambda.vector, lower.tail = FALSE) + 1
    t.min <- ppois(n.max - 1, lambda.vector, lower.tail = FALSE)
    s.min <- min(t.min)
    print(s.min)

    ## [1] 0.0007855354

The `poisson_test_pv` function from package `DiscreteTests` computes the support with \(y^{\tiny \mbox{min}}\) being the smallest observable p-value which can be represented by double precision, i.e. the smallest one that is not rounded to 0.

    sapply(poisson.p$get_pvalue_supports(), min)

    ## [1] 1.482197e-323 7.905050e-323 2.519735e-322 5.434722e-323 9.881313e-324
    ## [6] 8.893182e-323 4.397184e-322 5.434722e-323 1.333977e-322

For determining the restricted supports it is actually more convenient to work with \(n^{\tiny \mbox{max}}_i\) than \(s^{\tiny \mbox{min}}\). We can subsequently use these supports as the `pCDFlist` argument in the usual way when calling the `DBH` function:

    supports <- lapply(1:m, function(w){sort(ppois(0:n.max[w] - 1, lambda.vector[w], lower.tail = FALSE))})
    DBH.sd <- DBH(raw.pvalues, supports, direction = "sd", ret.crit.consts = TRUE)
    print(DBH.sd)

    ##
    ##   Discrete Benjamini-Hochberg procedure (step-down)
    ##
    ## Data:  raw.pvalues and supports
    ## Number of tests = 9
    ## Number of rejections = 3 at global FDR level 0.05
    ## (Original BH rejections = 1)
    ## Largest rejected p-value:  0.02311529

We can also use the results object of `poisson_test_pv`:

    DBH.sd.2 <- DBH(poisson.p, direction = "sd", ret.crit.consts = TRUE)
    print(DBH.sd.2)

    ##
    ##   Discrete Benjamini-Hochberg procedure (step-down)
    ##
    ## Data:  poisson.p
    ## Number of tests = 9
    ## Number of rejections = 3 at global FDR level 0.05
    ## (Original BH rejections = 1)
    ## Largest rejected p-value:  0.02311529

Figure 3 shows a summary similar to Figure 1. Applying the continuous BH procedure

    p.adjust(raw.pvalues, method = "BH")

    ## [1] 0.06934586 0.21692322 0.55067104 0.47979884 0.18067814 0.01033633 0.55067104
    ## [8] 0.47979884 0.06056424

results in one rejection at FDR-level \(\alpha = 5\%\), whereas the DBH step-down procedure can reject three hypotheses:

    DBH.sd$Adjusted

    ## [1] 0.039602625 0.101622881 0.580898946 0.522450788 0.101509307 0.001935955
    ## [7] 0.626257875 0.522450788 0.033073393

This information can also be obtained by our `print` or `summary` methods:

    print(DBH.sd)

    ##
    ##   Discrete Benjamini-Hochberg procedure (step-down)
    ##
    ## Data:  raw.pvalues and supports
    ## Number of tests = 9
    ## Number of rejections = 3 at global FDR level 0.05
    ## (Original BH rejections = 1)
    ## Largest rejected p-value:  0.02311529

    summary(DBH.sd)

    ##
    ##   Discrete Benjamini-Hochberg procedure (step-down)
    ##
    ## Data:  raw.pvalues and supports
    ## Number of tests = 9
    ## Number of rejections = 3 at global FDR level 0.05
    ## (Original BH rejections = 1)
    ## Largest rejected p-value:  0.02311529
    ##
    ##   Index     P.value Critical.value    Adjusted Rejected
    ## 1     6 0.001148481    0.009079858 0.001935955     TRUE
    ## 2     9 0.013458721    0.018988157 0.033073393     TRUE
    ## 3     1 0.023115288    0.033768968 0.039602625     TRUE
    ## 4     5 0.080301397    0.034141584 0.101509307    FALSE
    ## 5     2 0.120512901    0.043095453 0.101622881    FALSE
    ## 6     4 0.373176876    0.047422596 0.522450788    FALSE
    ## 7     8 0.373176876    0.062856934 0.522450788    FALSE
    ## 8     3 0.503414696    0.062856934 0.580898946    FALSE
    ## 9     7 0.550671036    0.080301397 0.626257875    FALSE

As in Figure 1, Panel (c) presents a graphical comparison between the two procedures applied to the \(p\)-values.

    stepf <- lapply(supports, function(x) stepfun(x, c(0, x)))
    par(mfcol = c(1, 3), mai = c(1, 0.5, 0.3, 0.1))

    plot(stepf[[1]], xlim = c(0,1), ylim = c(0,1), do.points = FALSE, lwd = 1, lty = 1, ylab = "F(x)",
         main = "(a)")
    for(i in (2:9)){
      plot(stepf[[i]], add = TRUE, do.points = FALSE, lwd = 1, col = i)
    }
    segments(0, 0, 1, 1, col = "grey", lty = 2)

    #   Plot xi
    support <- sort(unique(unlist(supports)))
    components <- lapply(stepf, function(s){s(support) / (1 - s(support))})
    xi.values <- 1/9 * Reduce('+', components)
    xi <- stepfun(support, c(0, xi.values))
    plot(xi, xlim = c(0, 0.10), ylim = c(0, 0.10), do.points = FALSE, ylab = expression(xi), main = "(b)")
    segments(0, 0, 0.1, 0.1, col = "grey", lty = 2)

    #   Plot discrete critical values as well a BH constants
    DBH.sd <- DBH(raw.pvalues, supports, direction = "sd", ret.crit.consts = TRUE)
    plot(DBH.sd, col = c("black", "black", "red"), pch = c(4, 4, 19), type.crit = 'p', ylim = c(0, 0.15),
         cex = 1.3, main = "(c)", ylab = "Critical Values")
    points(1:9, 0.05 * (1:9) / 9, col = "green", pch = 19, cex = 1.3)

    mtext("Figure 3", 1, outer = TRUE, line = -2)

Panel (a) depicts the distribution functions \(F_1, \ldots, F_9\) in various colors, (b) is a graph of the transformation function \(\xi_{\text{SD}}\). The uniform distribution function is shown in light gray in (a) and (b). Panel (c) shows the [BH] critical values (green dots), the DBH step-down critical values (red dots) and the sorted raw \(p\)-values (asterisks).