Multiple testing procedures are important tools for identifying
statistically significant findings in massive and complex data while
controlling a specific error rate. An important focus has been given to
methods controlling the false discovery rate (FDR), i.e., the expected
proportion of falsely rejected hypotheses among all rejected hypotheses,
which has become the standard error rate for high dimensional data
analysis. Since the original procedure of Benjamini and
Hochberg (1995), much effort has been undertaken to design FDR
controlling procedures that adapt to various underlying structures of
the data, such as the quantity of signal, the signal strength and the
dependencies, among others.
1. A Toy Example
To give a first impression of how DiscreteFDR works,
we consider an artificial toy example. A more realistic example
involving pharmacovigilance data is given in Section 2.
Suppose we would like to compare two treatments in nine different
populations. For each population we do this by evaluating the responders
and non-responders for each treatment. This leads to categorical data
which can be represented, for each population \(i = 1, \ldots, 9\) in the following 2 \(\times\) 2 table:
| Treatment 1 |
\(x_{1i}\) |
\(y_{1i}\) |
\(n_{1i}\) |
| Treatment 2 |
\(x_{2i}\) |
\(y_{2i}\) |
\(n_{2i}\) |
| Total |
\(x_{1i} +
x_{2i}\) |
\(y_{1i} +
y_{2i}\) |
\(n = n_{1i} +
n_{2i}\) |
Denoting the responder probabilities for population \(i\) by \(\pi_{1i}\) and \(\pi_{2i}\) we can test e.g.
\[H_{0i}: \pi_{1i} = \pi_{2i} \qquad
\text{vs.} \qquad H_{1i}: \pi_{1i} \neq \pi_{2i}\]
by using Fisher’s (two-sided) exact test (see Lehmann and Romano
(2006)), which is implemented in the R function
fisher.test. Suppose the data in the nine populations are
independent and we observe the following data frame df
library(knitr)
X1 <- c(4, 2, 2, 14, 6, 9, 4, 0, 1)
X2 <- c(0, 0, 1, 3, 2, 1, 2, 2, 2)
N1 <- rep(148, 9)
N2 <- rep(132, 9)
Y1 <- N1 - X1
Y2 <- N2 - X2
df <- data.frame(X1, Y1, X2, Y2)
kable(df, caption = "Toy Example")
Toy Example
| 4 |
144 |
0 |
132 |
| 2 |
146 |
0 |
132 |
| 2 |
146 |
1 |
131 |
| 14 |
134 |
3 |
129 |
| 6 |
142 |
2 |
130 |
| 9 |
139 |
1 |
131 |
| 4 |
144 |
2 |
130 |
| 0 |
148 |
2 |
130 |
| 1 |
147 |
2 |
130 |
In this data frame each of the 9 rows represents the data of an
observed 2 \(\times\) 2 table: e.g.,
the third row of the data corresponds to \(x_{13} = 2, y_{13} = 146, x_{23} = 1, y_{23} =
131\). Even though in this example, the total number of tested
hypotheses \(m = 9\) is very small, for
illustrative purposes we deal with the multiplicity problem here by
controlling FDR at level \(\alpha =
5\%\). The DBH step-down procedure can be applied directly to the
data frame object df and perform Fisher’s exact test
in-between. This yields an S3 object of class DiscreteFDR,
for which we provide both print and summary
methods:
library(DiscreteFDR)
DBH.sd.fast <- direct.discrete.BH(df, "fisher", direction = "sd")
print(DBH.sd.fast)
#>
#> Discrete Benjamini-Hochberg procedure (step-down)
#>
#> Data: df
#> Number of tests = 9
#> Number of rejections = 2 at global FDR level 0.05
#> (Original BH rejections = 0)
#> Largest rejected p value: 0.02126871
summary(DBH.sd.fast)
#>
#> Discrete Benjamini-Hochberg procedure (step-down)
#>
#> Data: df
#> Number of tests = 9
#> Number of rejections = 2 at global FDR level 0.05
#> (Original BH rejections = 0)
#> Largest rejected p value: 0.02126871
#>
#> Index P.value Adjusted Rejected
#> 1 4 0.01243145 0.03819796 TRUE
#> 2 6 0.02126871 0.03819796 TRUE
#> 3 1 0.12476691 0.25630985 FALSE
#> 4 8 0.22135177 0.47895996 FALSE
#> 5 5 0.28849298 0.51482782 FALSE
#> 6 2 0.49984639 1.00000000 FALSE
#> 7 9 0.60329543 1.00000000 FALSE
#> 8 7 0.68723229 1.00000000 FALSE
#> 9 3 1.00000000 1.00000000 FALSE
The output of the summary function contains the same
output as the print method, but adds a table that lists the
raw \(p\)-values, their adjusted
counterparts and their respective rejection decisions. It is sorted by
raw \(p\)-values in ascending order.
Our summary method actually creates a
summary.DiscreteFDR object, which includes all contents of
an DiscreteFDR object plus the aforementioned table. This
table can be accessed directly by the $Table command.
DBH.sd.fast.summary <- summary(DBH.sd.fast)
summary.table <- DBH.sd.fast.summary$Table
kable(summary.table, caption = "Summary table")
Summary table
| 4 |
0.0124314 |
0.0381980 |
TRUE |
| 6 |
0.0212687 |
0.0381980 |
TRUE |
| 1 |
0.1247669 |
0.2563098 |
FALSE |
| 8 |
0.2213518 |
0.4789600 |
FALSE |
| 5 |
0.2884930 |
0.5148278 |
FALSE |
| 2 |
0.4998464 |
1.0000000 |
FALSE |
| 9 |
0.6032954 |
1.0000000 |
FALSE |
| 7 |
0.6872323 |
1.0000000 |
FALSE |
| 3 |
1.0000000 |
1.0000000 |
FALSE |
Thus we can reject two hypotheses at FDR-level \(\alpha = 5\%\). Note, that our
print method also gives the number of rejections of the
usual (continuous) BH procedure. In order to compare its adjusted \(p\)-values with ours, we have to determine
the raw \(p\)-values first. This would
be possible by applying the fisher.test function to all
nine 2 \(\times\) 2 tables.
Alternatively, we may use the more convenient function
generate.pvalues, which is included in our package, for
accessing the raw \(p\)-values. Since
it only accept hypothesis test functions from the package
DiscreteTests (either as a function object or a character
string that can be abbreviated), we could also use such a function
directly, e.g. fisher.test.pv. An even more simple way is
to extract them from the DiscreteFDR object that we
obtained before and contains the results:
# compute results of Fisher's exact test for each row of 'df'
fisher.p <- generate.pvalues(df, "fisher", list(alternative = "two.sided"))
# extract raw observed p-values
raw.pvalues <- fisher.p$get_pvalues()
# or
library(DiscreteTests)
fisher.p.2 <- fisher.test.pv(df, "two.sided")
#> Warning: `fisher.test.pv()` was deprecated in DiscreteTests 0.2.
#> ℹ Please use `fisher_test_pv()` instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.
raw.pvalues.2 <- fisher.p.2$get_pvalues()
# or:
raw.pvalues.3 <- DBH.sd.fast$Data$raw.pvalues
all(raw.pvalues == raw.pvalues.2) && all(raw.pvalues == raw.pvalues.3)
#> [1] TRUE
p.adjust(raw.pvalues, method = "BH")
#> [1] 0.37430072 0.74976959 1.00000000 0.09570921 0.51928737 0.09570921 0.77313633
#> [8] 0.49804147 0.77313633
Applying the continuous BH procedure from the stats
package in the last line of code, we find that we cannot reject any
hypotheses at FDR-level \(\alpha =
5\%\). Another approach of determining this is to compare the
critical values of the discrete and continuous BH procedures. In the
discrete case, we need the observed \(p\)-values and their distributions. Both
were computed by our generate.pvalues function above. We
can either extract them and pass them to the DBH function,
or directly apply the function to the test results object itself:
# extract raw observed p-values
raw.pvalues <- fisher.p$get_pvalues()
# extract p-value support sets
pCDFlist <- fisher.p$get_pvalue_supports()
# use raw pvalues and list of supports:
DBH.sd.crit <- DBH(raw.pvalues, pCDFlist, 0.05, "sd", TRUE)
crit.vals.BH.disc <- DBH.sd.crit$Critical.values
# use test results object directly
DBH.sd.crit.2 <- DBH(fisher.p, 0.05, "sd", TRUE)
crit.vals.BH.disc.2 <- DBH.sd.crit.2$Critical.values
# compare
all(crit.vals.BH.disc == crit.vals.BH.disc.2)
#> [1] TRUE
The latter way enables the use of a pipe:
df |>
fisher.test.pv(alternative = "two.sided") |>
DBH(0.05, "sd", TRUE) |>
with(Critical.values)
#> [1] 0.01243145 0.02832448 0.03109596 0.04839433 0.05014119 0.07657062 0.07657062
#> [8] 0.10328523 0.10328523
Now, we can compare the critical values:
crit.vals.BH.cont <- 1:9 * 0.05/9
tab <- data.frame(raw.pvalues = sort(raw.pvalues), crit.vals.BH.disc, crit.vals.BH.cont)
kable(tab, caption = "Observed p-values and critical values")
Observed p-values and critical values
| 0.0124314 |
0.0124314 |
0.0055556 |
| 0.0212687 |
0.0283245 |
0.0111111 |
| 0.1247669 |
0.0310960 |
0.0166667 |
| 0.2213518 |
0.0483943 |
0.0222222 |
| 0.2884930 |
0.0501412 |
0.0277778 |
| 0.4998464 |
0.0765706 |
0.0333333 |
| 0.6032954 |
0.0765706 |
0.0388889 |
| 0.6872323 |
0.1032852 |
0.0444444 |
| 1.0000000 |
0.1032852 |
0.0500000 |
Obviously, the critical values of the discrete approach are
considerably larger than those of the continuous method. As a result,
the two smallest \(p\)-values are
rejected by the discrete BH procedure, because they are smaller than or
equal to the respective critical values. The continuous BH approach does
not reject any hypothesis, because all its critical values are smaller
than the observed \(p\)-values.
For illustration, our package includes a plot method for
DiscreteFDR S3 class objects. It allows to define colors,
line types and point characters separately for accepted and rejected
\(p\)-values and for critical values
(if present in the object).
plot(DBH.sd.crit, col = c("red", "blue", "green"), pch = c(4, 2, 19), lwd = 2, type.crit = 'o',
legend = "topleft", cex = 1.3)
Now, it is visible which observed \(p\)-values are rejected. A comparison with
the continuous BH procedure could be done as follows:
plot(DBH.sd.crit, col = c("red", "blue", "green"), pch = c(4, 2, 19), lwd = 2, type.crit = 'o',
cex = 1.3, ylim = c(0, 0.25), main = "Comparison of discrete and continuous BH procedures")
points(crit.vals.BH.cont, pch = 19, cex = 1.3, lwd = 2)
legend("topright", c("Rejected", "Accepted", "Critical Values (disc.)", "Critical Values (cont.)"),
col = c("red", "blue", "green", "black"), pch = c(4, 2, 19, 19), lwd = 2, lty = 0)
As this example illustrates, the discrete approach can potentially
yield a large increase in power. The gain depends on the involved
distribution functions and the raw \(p\)-values. To appreciate where this comes
from, it is instructive to consider the distribution functions \(F_1, \ldots, F_9\) of the \(p\)-values under the null in more detail.
Take for instance the first 2 \(\times\) 2 table:
| Treatment 1 |
4 |
144 |
148 |
| Treatment 2 |
0 |
132 |
132 |
| Total |
4 |
276 |
280 |
Fisher’s exact test works by determining the probability of observing
this (or a more ‘extreme’) table, given that the margins are fixed. So
each \(F_i\) is determined by the
margins of table \(i\). Since \(x_{11} + x_{21} = 4\), the only potentially
observable tables are given by \(x_{11} = 0,
\ldots, 4\). For each one of these 5 values a \(p\)-value can be determined using the
hypergeometric distribution. Therefore, the \(p\)-value of any 2 \(\times\) 2 table with margins given by the
above table can take (at most) 5 distinct values, say \(x_1, \ldots, x_5\). Combining these 5
values into a set, we obtain the support \(\mathcal{A}_1 = \{x_1, \ldots, x_5\}\) of
distribution \(F_1\). Now we can
continue in this vein for the remaining 2 \(\times\) 2 tables to obtain the supports
\(\mathcal{A}_1, \ldots,
\mathcal{A}_9\) for the distribution functions \(F_1, \ldots, F_{9}\). The supports can be
accessed via the $get_pvalue_supports() function of the
results object, e.g.
# extract p-value support sets
pCDFlist <- fisher.p$get_pvalue_supports()
# extract 1st and 5th support set
pCDFlist[c(1,5)]
#> [[1]]
#> [1] 0.04820493 0.12476691 0.34598645 0.62477763 1.00000000
#>
#> [[2]]
#> [1] 0.002173856 0.007733719 0.028324482 0.069964309 0.154043258 0.288492981
#> [7] 0.481808361 0.726262402 1.000000000
Here, this returns \(\mathcal{A}_1\)
and \(\mathcal{A}_5\). Panel (a) in the
following figure shows a graph of the distribution functions \(F_1, \ldots, F_9\). Each \(F_i\) is a step-function with \(F_i(0) = 0\), the jumps occurring only on
the support \(\mathcal{A}_i\) and \(F_i(x) = x\) only for \(x \in \mathcal{A}_i\). In particular, all
distributions are stochastically larger than the uniform distribution
(i.e., \(F_i(x) \le x\)), but in a
heterogeneous manner. This heterogeneity can be exploited e.g., by
transforming the raw \(p\)-values from
the exact Fisher’s test using the function \[\displaystyle \xi_{\text{SD}}(x) = \sum_{i =
1}^{9} \frac{F_i(x)}{1 - F_i(x)}.\] Panel (b) of the following
plot shows that \(\xi_{\text{SD}}\) is
a step function. Its jumps occur on the joint support \(\mathcal{A}= \mathcal{A}_1 \cup \ldots \cup
\mathcal{A}_9\). Panel (b) also shows that \(\displaystyle \xi_{\text{SD}}(x) \ll x\),
at least for small values of \(x\). It
turns out that the critical values of our new DBH step-down procedure
are essentially given by inverting \(\xi_{\text{SD}}\) at the critical values of
the [BH] procedure \(1 \cdot \alpha / 9, 2
\cdot \alpha / 9, \ldots, \alpha\), so that these values are
considerably larger than the [BH] critical values. This is illustrated
in panel (c) along with the ordered \(p\)~values. In particular, all asterisks
are located above the green [BH] dots, therefore this procedure can not
reject any hypothesis. In contrast, the two smallest \(p\)~values are located below red DBH
step-down dots, so that this procedure rejects two hypotheses as we had
already seen earlier.
stepf <- lapply(pCDFlist, 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(pCDFlist)))
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 and raw p-values
DBH.sd <- DBH(fisher.p, 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 1", 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 \(\xi_{\text{SD}}\). 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).
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.
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")
#> Warning: `poisson.test.pv()` was deprecated in DiscreteTests 0.2.
#> ℹ Please use `poisson_test_pv()` instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.
raw.pvalues.2 <- poisson.p$get_pvalues()
print(raw.pvalues.2)
#> [1] 0.023115288 0.120512901 0.503414696 0.373176876 0.080301397 0.001148481
#> [7] 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).