Title: Query Composite Hypotheses
Version: 2.1.0
Maintainer: Tristan Mary-Huard <tristan.mary-huard@agroparistech.fr>
Description: Provides functions for the joint analysis of Q sets of p-values obtained for the same list of items. This joint analysis is performed by querying a composite hypothesis, i.e. an arbitrary complex combination of simple hypotheses, as described in Mary-Huard et al. (2021) <doi:10.1093/bioinformatics/btab592> and De Walsche et al.(2023) <doi:10.1101/2024.03.17.585412>. In this approach, the Q-uplet of p-values associated with each item is distributed as a multivariate mixture, where each of the 2^Q components corresponds to a specific combination of simple hypotheses. The dependence between the p-value series is considered using a Gaussian copula function. A p-value for the composite hypothesis test is derived from the posterior probabilities.
License: GPL-3
Depends: R (≥ 2.10)
Imports: copula, dplyr, graphics, ks, purrr, qvalue, Rcpp, stats, stringr, utils
LinkingTo: Rcpp, RcppArmadillo
Encoding: UTF-8
LazyData: true
RoxygenNote: 7.3.2
NeedsCompilation: yes
Packaged: 2025-07-04 11:22:16 UTC; Annaig
Author: Tristan Mary-Huard ORCID iD [aut, cre], Annaig De Walsche ORCID iD [aut], Franck Gauthier ORCID iD [ctb]
Repository: CRAN
Date/Publication: 2025-07-04 12:50:02 UTC

qch: Query Composite Hypotheses

Description

Provides functions for the joint analysis of Q sets of p-values obtained for the same list of items. This joint analysis is performed by querying a composite hypothesis, i.e. an arbitrary complex combination of simple hypotheses, as described in Mary-Huard et al. (2021) doi:10.1093/bioinformatics/btab592 and De Walsche et al.(2023) doi:10.1101/2024.03.17.585412. In this approach, the Q-uplet of p-values associated with each item is distributed as a multivariate mixture, where each of the 2^Q components corresponds to a specific combination of simple hypotheses. The dependence between the p-value series is considered using a Gaussian copula function. A p-value for the composite hypothesis test is derived from the posterior probabilities.

Details

The main functions of the package GetHconfig, GetH1AtLeast, GetH1Equal, qch.fit and qch.test correspond to the 4 steps for querying a composite hypothesis:

Author(s)

Maintainer: Tristan Mary-Huard tristan.mary-huard@agroparistech.fr (ORCID)

Authors:

Other contributors:

Gaussian copula density for each H-configuration.

Description

Gaussian copula density for each H-configuration.

Usage

Copula.Hconfig_gaussian_density(Hconfig, F0Mat, F1Mat, R)

Arguments

Hconfig

A list of all possible combination of H_0 and H_1 hypotheses generated by the GetHconfig() function.

F0Mat

a matrix containing the evaluation of the marginal cdf under H_0 at each items, each column corresponding to a p-value serie.

F1Mat

a matrix containing the evaluation of the marginal cdf under H_1 at each items, each column corresponding to a p-value serie.

R

the correlation matrix.

Value

A matrix containing the evaluation of the Gaussian density function for each H-configuration in columns.

EM calibration in the case of the Gaussian copula (unsigned)

Description

EM calibration in the case of the Gaussian copula (unsigned)

Usage

EM_calibration_gaussian(
  Hconfig,
  F0Mat,
  F1Mat,
  fHconfig,
  R.init,
  Prior.init,
  Precision = 1e-06
)

Arguments

Hconfig

A list of all possible combination of H_0 and H_1 hypotheses generated by the GetHconfig() function.

F0Mat

a matrix containing the evaluation of the marginal cdf under H_0 at each items, each column corresponding to a p-value serie.

F1Mat

a matrix containing the evaluation of the marginal cdf under H_1 at each items, each column corresponding to a p-value serie.

fHconfig

a matrix containing H-config densities evaluated at each items, each column corresponding to a configurations.

R.init

the initialization of the correlation matrix of the Gaussian copula parameter.

Prior.init

the initialization of prior probabilities for each of the H-configurations.

Precision

Precision for the stop criterion. (Default is 1e-6)

Value

A list with the following elements:

priorHconfig vector of estimated prior probabilities for each of the H-configurations.
Rcopula the estimated correlation matrix of the Gaussian copula.

EM calibration in the case of the Gaussian copula (unsigned) with memory management

Description

EM calibration in the case of the Gaussian copula (unsigned) with memory management

Usage

EM_calibration_gaussian_memory(
  Logf0Mat,
  Logf1Mat,
  F0Mat,
  F1Mat,
  Prior.init,
  R.init,
  Hconfig,
  Precision = 1e-06,
  threads_nb
)

Arguments

Logf0Mat

a matrix containing the \log(f_0(x^i_q))

Logf1Mat

a matrix containing the \log(f_1(x^i_q))

F0Mat

a matrix containing the evaluation of the marginal cdf under H_0 at each items, each column corresponding to a p-value serie.

F1Mat

a matrix containing the evaluation of the marginal cdf under H_1 at each items, each column corresponding to a p-value serie.

Prior.init

the initialization of prior probabilities for each of the H-configurations.

R.init

the initialization of the correlation matrix of the gaussian copula parameter.

Hconfig

A list of all possible combination of H_0 and H_1 hypotheses generated by the GetHconfig() function.

Precision

Precision for the stop criterion. (Default is 1e-6)

threads_nb

The number of threads to use.

Value

A list with the following elements:

priorHconfig vector of estimated prior probabilities for each of the H-configurations.
Rcopula the estimated correlation matrix of the Gaussian copula.

EM calibration in the case of conditional independence

Description

EM calibration in the case of conditional independence

Usage

EM_calibration_indep(fHconfig, Prior.init, Precision = 1e-06)

Arguments

fHconfig

a matrix containing config densities evaluated at each items, each column corresponding to a configurations.

Prior.init

the initialization of prior probabilities for each of the H-configurations.

Precision

Precision for the stop criterion. (Default is 1e-6)

Value

a vector of estimated prior probabilities for each of the H-configurations.

EM calibration in the case of conditional independence with memory management (unsigned)

Description

EM calibration in the case of conditional independence with memory management (unsigned)

Usage

EM_calibration_indep_memory(
  Logf0Mat,
  Logf1Mat,
  Prior.init,
  Hconfig,
  Precision = 1e-06,
  threads_nb
)

Arguments

Logf0Mat

a matrix containing the \log(f_0(x^i_q))

Logf1Mat

a matrix containing the \log(f_1(x^i_q))

Prior.init

the initialization of prior probabilities for each of the H-configurations.

Hconfig

A list of all possible combination of H_0 and H_1 hypotheses generated by the GetHconfig() function.

Precision

Precision for the stop criterion. (Default is 1e-6)

threads_nb

The number of threads to use.

Value

a vector of estimated prior probabilities for each of the H-configurations.

FastKerFdr signed

Description

Kernel estimation of the density in a two-components mixture model where one component are a standard Gaussian density.

Usage

FastKerFdr_signed(
  X,
  p0 = NULL,
  plotting = FALSE,
  NbKnot = 1e+05,
  tol = 1e-05,
  max_iter = 10000
)

Arguments

X

a vector of probit-transformed p-values (corresponding to a p-value serie).

p0

a priori proportion of H_0 hypotheses.

plotting

boolean, should some diagnostic graphs be plotted. (Default is FALSE.)

NbKnot

The (maximum) number of knot for the kde procedure. (Default is 1e5.)

tol

a tolerance value for convergence. (Default is 1e-5.)

max_iter

the maximum number of iterations allowed for the algorithm to converge or complete its process.(Default is 1e4.)

Value

A list with the following elements:

p0 vector of the estimated proportions of H_0 hypotheses for each of p-value serie.
tau the vector of H_1 posteriors.
f1 a numeric vector, each coordinate i corresponding to the evaluation of the H_1 density at point x_i, where x_i is the ith item in X.
F1 a numeric vector, each coordinate i corresponding to the evaluation of the H_1 cdf at point x_i, where x_i is the ith item in X.

FastKerFdr unsigned

Description

Kernel estimation of the density in a two-components mixture model where one component are a standard Gaussian density. Here we suppose that the density to estimate lives in R^+.

Usage

FastKerFdr_unsigned(
  X,
  p0 = NULL,
  plotting = FALSE,
  NbKnot = 1e+05,
  tol = 1e-05,
  max_iter = 10000
)

Arguments

X

a vector of probit-transformed p-values (corresponding to a p-value serie)

p0

a priori proportion of H_0 hypotheses

plotting

boolean, should some diagnostic graphs be plotted. (Default is FALSE.)

NbKnot

The (maximum) number of knot for the kde procedure. (Default is 1e5.)

tol

a tolerance value for convergence. (Default is 1e-5.)

max_iter

the maximum number of iterations allowed for the algorithm to converge or complete its process.(Default is 1e4.)

Value

A list with the following elements:

p0 vector of the estimated proportions of H_0 hypotheses for each of p-value serie.
tau the vector of H_1 posteriors.
f1 a numeric vector, each coordinate i corresponding to the evaluation of the H_1 density at point x_i, where x_i is the ith item in X.
F1 a numeric vector, each coordinate i corresponding to the evaluation of the H_1 cdf at point x_i, where x_i is the ith item in X.

Specify the configurations corresponding to the composite H_1 test "AtLeast".

Description

Specify which configurations among Hconfig correspond to the composite alternative hypothesis : {at least "AtLeast" H_1 hypotheses are of interest }

Usage

GetH1AtLeast(Hconfig, AtLeast, Consecutive = FALSE, SameSign = FALSE)

Arguments

Hconfig

A list of all possible combination of H_0 and H_1 hypotheses generated by the GetHconfig() function.

AtLeast

How many H_1 hypotheses at least for the item to be of interest ? (an integer or a vector).

Consecutive

Should the significant test series be consecutive ? (optional, default is FALSE).

SameSign

Should the significant test series have the same sign ? (optional, default is FALSE).

Value

A vector 'Hconfig.H1' of components of Hconfig that correspond to the 'AtLeast' specification.

See Also

GetH1Equal()

Examples

GetH1AtLeast(GetHconfig(4), 2)

Specify the configurations corresponding to the composite H_1 test "Equal".

Description

Specify which configurations among Hconfig correspond to the composite alternative hypothesis :{Exactly "Equal" H_1 hypotheses are of interest }

Usage

GetH1Equal(Hconfig, Equal, Consecutive = FALSE, SameSign = FALSE)

Arguments

Hconfig

A list of all possible combination of H0 and H1 hypotheses generated by the GetHconfig() function.

Equal

What is the exact number of H_1 hypotheses for the item to be of interest? (an integer or a vector).

Consecutive

Should the significant test series be consecutive ? (optional, default is FALSE).

SameSign

Should the significant test series have the same sign ? (optional, default is FALSE).

Value

A vector 'Hconfig.H1' of components of Hconfig that correspond to the 'Equal' specification.

See Also

GetH1AtLeast()

Examples

GetH1Equal(GetHconfig(4), 2)

Generate the H_0/H_1 configurations.

Description

Generate all possible combination of simple hypotheses H_0/H_1.

Usage

GetHconfig(Q, Signed = FALSE)

Arguments

Q

The number of test series to be combined.

Signed

Should the sign of the effect be taken into account? (optional, default is FALSE).

Value

A list 'Hconfig' of all possible combination of H_0 and H_1 hypotheses among Q hypotheses tested.

Examples

GetHconfig(4)

Synthetic example to illustrate the main qch functions

Description

PvalSets is a data.frame with 10,000 rows and 3 columns. Each row corresponds to an item, columns 'Pval1' and 'Pval2' each correspond to a test serie over the items, and column 'Class' provides the truth, i.e. if item i belongs to class 1 then the H0 hypothesis is true for the 2 tests, if item i belongs to class 2 (resp. 3) then the H0 hypothesis is true for the first (resp. second) test only, and if item i belongs to class 4 then both H0 hypotheses are false (for the first and the second test).

Usage

PvalSets

Format

A data.frame

Synthetic example to illustrate the main qch functions using Gaussian copula

Description

PvalSets_cor is a data.frame with 10,000 rows and 3 columns. Each row corresponds to an item, columns Pval1 and Pval2 each correspond to a test serie over the items, and column 'Class' provides the truth, i.e. if item i belongs to class 1 then the H_0 hypothesis is true for the 2 tests, if item i belongs to class 2 (resp. 3) then the H_0 hypothesis is true for the first (resp. second) test only, and if item i belongs to class 4 then both H0 hypotheses are false (for the first and the second test). The correlation between the two pvalues series within each class is 0.3.

Usage

PvalSets_cor

Format

A data.frame

Gaussian copula correlation matrix Maximum Likelihood estimator.

Description

Gaussian copula correlation matrix Maximum Likelihood estimator.

Usage

R.MLE(Hconfig, zeta0, zeta1, Tau)

Arguments

Hconfig

A list of all possible combination of H_0 and H_1 hypotheses generated by the GetHconfig() function.

zeta0

a matrix containing the \Phi(F_0(Z_{iq})), each column corresponding to a p-value serie.

zeta1

a matrix containing the \Phi(F_1(Z_{iq})), each column corresponding to a p-value serie.

Tau

a matrix providing for each item (in row) its posterior probability to belong to each of the H-configurations (in columns).

Value

Estimate of the correlation matrix.

Check the Gaussian copula correlation matrix Maximum Likelihood estimator

Description

Check the Gaussian copula correlation matrix Maximum Likelihood estimator

Usage

R.MLE.check(R)

Arguments

R

Estimate of the correlation matrix.

Value

Estimate of the correlation matrix.

Gaussian copula correlation matrix Maximum Likelihood estimator (memory handling)

Description

Gaussian copula correlation matrix Maximum Likelihood estimator (memory handling)

Usage

R.MLE.memory(
  Hconfig,
  fHconfig_sum,
  OldPrior,
  Logf0Mat,
  Logf1Mat,
  zeta0,
  zeta1,
  OldR,
  OldRinv
)

Arguments

Hconfig

A list of all possible combination of H_0 and H_1 hypotheses generated by the GetHconfig() function.

fHconfig_sum

a vector containing \sum_c(w_c*psi_c) for each items.

OldPrior

a vector containing the prior probabilities for each of the H-configurations.

Logf0Mat

a matrix containing \log(f_0Mat), each column corresponding to a p-value serie.

Logf1Mat

a matrix containing \log(f_1Mat), each column corresponding to a p-value serie.

zeta0

a matrix containing qnorm(F0Mat), each column corresponding to a p-value serie.

zeta1

a matrix containing qnorm(F1Mat), each column corresponding to a p-value serie.

OldR

the copula correlation matrix.

OldRinv

the inverse of copula correlation matrix.

Value

Estimate of the correlation matrix.

Update the estimate of R correlation matrix of the gaussian copula, parallelized version

Description

Update the estimate of R correlation matrix of the gaussian copula, parallelized version

Usage

R_MLE_update_gaussian_copula_ptr_parallel(
  Hconfig,
  fHconfig_sum,
  OldPrior,
  Logf0Mat,
  Logf1Mat,
  zeta0,
  zeta1,
  OldR,
  OldRinv,
  RhoIndex,
  threads_nb = 0L
)

Arguments

Hconfig

list of vector of 0 and 1, corresponding to the configurations

fHconfig_sum

a double vector containing sum_c(w_c*psi_c), obtained by fHconfig_sum_update_ptr_parallel()

OldPrior

a double vector containing the prior w_c

Logf0Mat

a double matrix containing the log(f0(xi_q))

Logf1Mat

a double matrix containing the log(f1(xi_q))

zeta0

a double matrix containing the qnorm(F0(x_iq))

zeta1

a double matrix containing the qnorm(F1(x_iq))

OldR

a double matrix corresponding to the copula parameter

OldRinv

a double matrix corresponding to the inverse copula parameter

RhoIndex

a int matrix containing the index of lower triangular part of a matrix

threads_nb

an int the number of threads

Value

a double vector containing the lower triangular part of the MLE of R

Signed case function: Separate f1 into f+ and f-

Description

Signed case function: Separate f1 into f+ and f-

Usage

f1_separation_signed(XMat, f0Mat, f1Mat, p0, plotting = FALSE)

Arguments

XMat

a matrix of probit-transformed p-values, each column corresponding to a p-value serie.

f0Mat

a matrix containing the evaluation of the marginal density functions under H_0 at each items, each column corresponding to a p-value serie.

f1Mat

a matrix containing the evaluation of the marginal density functions under H_1 at each items, each column corresponding to a p-value serie.

p0

the proportions of H_0 items for each series.

plotting

boolean, should some diagnostic graphs be plotted. (Default is FALSE.)

Value

A list with the following elements:

f1plusMat a matrix containing the evaluation of the marginal density functions under H_1^+ at each items, each column corresponding to a p-value serie.
f1minusMat a matrix containing the evaluation of the marginal density functions under H_1^- at each items, each column corresponding to a p-value serie.
p1plus an estimate of the proportions of H_1^+ items for each series.
p1minus an estimate of the proportions of H_1^- items for each series.

Computation of the sum sum_c(w_c*psi_c) using Gaussian copula parallelized version

Description

Computation of the sum sum_c(w_c*psi_c) using Gaussian copula parallelized version

Usage

fHconfig_sum_update_gaussian_copula_ptr_parallel(
  Hconfig,
  NewPrior,
  Logf0Mat,
  Logf1Mat,
  zeta0,
  zeta1,
  R,
  Rinv,
  threads_nb = 0L
)

Arguments

Hconfig

list of vector of 0 and 1, corresponding to the configurations

NewPrior

a double vector containing the prior w_c

Logf0Mat

a double matrix containing the log(f0(xi_q))

Logf1Mat

a double matrix containing the log(f1(xi_q))

zeta0

a double matrix containing the qnorm(F0(x_iq))

zeta1

a double matrix containing the qnorm(F1(x_iq))

R

a double matrix corresponding to the copula parameter

Rinv

a double matrix corresponding to the inverse copula parameter

threads_nb

an int the number of threads

Value

a double vector containing sum_c(w_c*psi_c)

Computation of the sum sum_c(w_c*psi_c) parallelized version

Description

Computation of the sum sum_c(w_c*psi_c) parallelized version

Usage

fHconfig_sum_update_ptr_parallel(
  Hconfig,
  NewPrior,
  Logf0Mat,
  Logf1Mat,
  threads_nb = 0L
)

Arguments

Hconfig

list of vector of 0 and 1, corresponding to the configurations

NewPrior

a double vector containing the prior w_c

Logf0Mat

a double matrix containing the log(f0(xi_q))

Logf1Mat

a double matrix containing the log(f1(xi_q))

threads_nb

an int the number of threads

Value

a double vector containing sum_c(w_c*psi_c)

Gaussian copula density

Description

Gaussian copula density

Usage

gaussian_copula_density(zeta, R, Rinv)

Arguments

zeta

the matrix of probit-transformed observations.

R

the correlation matrix.

Rinv

the inverse correlation matrix.

Value

A numeric vector, each coordinate i corresponding to the evaluation of the Gaussian copula density function at observation \code{zeta}_i.

Update of the prior estimate in EM algo parallelized version

Description

Update of the prior estimate in EM algo parallelized version

Usage

prior_update_arma_ptr_parallel(
  Hconfig,
  fHconfig_sum,
  OldPrior,
  Logf0Mat,
  Logf1Mat,
  threads_nb = 0L
)

Arguments

Hconfig

list of vector of 0 and 1, corresponding to the configurations

fHconfig_sum

a double vector containing sum_c(w_c*psi_c), obtained by fHconfig_sum_update_ptr_parallel()

OldPrior

a double vector containing the prior w_c

Logf0Mat

a double matrix containing the log(f0(xi_q))

Logf1Mat

a double matrix containing the log(f1(xi_q))

threads_nb

an int the number of threads

Value

a double vector containing the new estimate of prior w_c

Update of the prior estimate in EM algo using Gaussian copula, parallelized version

Description

Update of the prior estimate in EM algo using Gaussian copula, parallelized version

Usage

prior_update_gaussian_copula_ptr_parallel(
  Hconfig,
  fHconfig_sum,
  OldPrior,
  Logf0Mat,
  Logf1Mat,
  zeta0,
  zeta1,
  R,
  Rinv,
  threads_nb = 0L
)

Arguments

Hconfig

list of vector of 0 and 1, corresponding to the configurations

fHconfig_sum

a double vector containing sum_c(w_c*psi_c), obtained by fHconfig_sum_update_ptr_parallel()

OldPrior

a double vector containing the prior w_c

Logf0Mat

a double matrix containing the log(f0(xi_q))

Logf1Mat

a double matrix containing the log(f1(xi_q))

zeta0

a double matrix containing the qnorm(F0(x_iq))

zeta1

a double matrix containing the qnorm(F1(x_iq))

R

a double matrix corresponding to the copula parameter

Rinv

a double matrix corresponding to the inverse copula parameter

threads_nb

an int the number of threads

Value

a double vector containing the new estimate of prior w_c

Infer posterior probabilities of H_0/H_1 configurations.

Description

For each item, estimate the posterior probability for each configuration. This function use either the model accounting for the dependence structure through a Gaussian copula function (copula=="gaussian") or assuming the conditional independence (copula=="indep"). Utilizes parallel computing, when available. For package documentation, see qch-package.

Usage

qch.fit(
  pValMat,
  EffectMat = NULL,
  Hconfig,
  copula = "indep",
  threads_nb = 0,
  plotting = FALSE,
  Precision = 1e-06
)

Arguments

pValMat

A matrix of p-values, each column corresponding to a p-value serie.

EffectMat

A matrix of estimated effects corresponding to the p-values contained in pValMat. If specified, the procedure will account for the direction of the effect. (optional, default is NULL)

Hconfig

A list of all possible combination of H_0 and H_1 hypotheses generated by the GetHconfig() function.

copula

A string specifying the form of copula to use. Possible values are "indep"and "gaussian". Default is "indep" corresponding to the independent case.

threads_nb

The number of threads to use. The number of thread will set to the number of cores available by default.

plotting

A boolean. Should some diagnostic graphs be plotted ? Default is FALSE.

Precision

The precision for EM algorithm to infer the parameters. Default is 1e-6.

Value

A list with the following elements:

prior vector of estimated prior probabilities for each of the H-configurations.
Rcopula the estimated correlation matrix of the Gaussian copula. (if applicable)
Hconfig the list of all configurations.
null_prop the estimation of items under the null for each test series.

The elements of interest are the posterior probabilities matrix, posterior, the estimated proportion of observations belonging to each configuration, prior, and the estimated correlation matrix of the Gaussian copula, Rcopula. The remaining elements are returned primarily for use by other functions.

Examples

data(PvalSets_cor)
PvalMat <- as.matrix(PvalSets_cor[, -3])
## Build the Hconfig objects
Q <- 2
Hconfig <- GetHconfig(Q)

## Run the function
res.fit <- qch.fit(pValMat = PvalMat, Hconfig = Hconfig, copula = "gaussian")

## Display the prior of each class of items
res.fit$prior

## Display the correlation estimate of the gaussian copula
res.fit$Rcopula

## Display the first posteriors
head(res.fit$posterior)

Perform composite hypothesis testing.

Description

Perform any composite hypothesis test by specifying the configurations 'Hconfig.H1' corresponding to the composite alternative hypothesis among all configurations 'Hconfig'.

Usage

qch.test(res.qch.fit, Hconfig, Hconfig.H1 = NULL, Alpha = 0.05, threads_nb = 0)

Arguments

res.qch.fit

The result provided by the qch.fit() function.

Hconfig

A list of all possible combination of H_0 and H_1 hypotheses generated by the GetHconfig() function.

Hconfig.H1

An integer vector (or a list of such vector) of the Hconfig index corresponding to the composite alternative hypothesis configuration(s). Can be generated by the GetH1AtLeast() or GetH1Equal() functions. If NULL, the composite hypothesis tests of being associated with "at least q" simple tests, for q=1,..Q are performed.

Alpha

the nominal Type I error rate for FDR control. Default is 0.05.

threads_nb

The number of threads to use. The number of thread will set to the number of cores available by default.

Details

By default, the function performs the composite hypothesis test of being associated with "at least q" simple tests, for q=1,..Q.

Value

A list with the following elements:

Rejection a matrix providing for each item the result of the composite hypothesis test, after adaptive Benjamin-Höchberg multiple testing correction.
lFDR a matrix providing for each item its local FDR estimate.
Pvalues a matrix providing for each item its p-value of the composite hypothesis test.

See Also

qch.fit(), GetH1AtLeast(),GetH1Equal()

Examples

data(PvalSets_cor)
PvalMat <- as.matrix(PvalSets_cor[, -3])
Truth <- PvalSets[, 3]

## Build the Hconfig objects
Q <- 2
Hconfig <- GetHconfig(Q)

## Infer the posteriors
res.fit <- qch.fit(pValMat = PvalMat, Hconfig = Hconfig, copula = "gaussian")

## Run the test procedure with FDR control
H1config <- GetH1AtLeast(Hconfig, 2)
res.test <- qch.test(res.qch.fit = res.fit, Hconfig = Hconfig, Hconfig.H1 = H1config)
table(res.test$Rejection$AtLeast_2, Truth == 4)