Type: Package
Title: Valid Inference on Multiple Quantile Regressions
Version: 0.2.0
Date: 2026-01-09
Description: The approach is based on the closed testing procedure to control familywise error rate in a strong sense. The local tests implemented are Wald-type and rank-score. The method is described in De Santis, et al., (2026), <doi:10.48550/arXiv.2511.07999>.
Depends: quantreg, Matrix, MASS, pracma, methods, sn
License: GPL (≥ 3)
Encoding: UTF-8
RoxygenNote: 7.3.2
NeedsCompilation: no
Packaged: 2026-01-09 11:00:48 UTC; Andreella
Author: Angela Andreella [aut, cre], Anna Vesely [aut], Riccardo De Santis [aut]
Maintainer: Angela Andreella <angela.andreella@unive.it>
Repository: CRAN
Date/Publication: 2026-01-09 11:10:02 UTC

The Imhof method for x=0 and central variables only.

Description

The Imhof method for x=0 and central variables only.

Usage

.pImhof(lams, eps = c(1e-04, 1e-04), x)

Arguments

lams

eigenvalues

eps

tolerance vector (one for integrate, one for quadgk)

x

observed stat test

Assert that X is numeric or (if categorical) has at most two levels

Description

Works with rq/rqs even when model.frame or mod$x are not stored.

Usage

assert_binary_categorical_X(mod, X)

Arguments

mod

An "rq"/"rqs" object

X

Character scalar: name of the covariate of interest

Assert that the fitted model includes an intercept

Description

Assert that the fitted model includes an intercept

Usage

assert_intercept_present(mod)

Arguments

mod

An object of class "rq" or "rqs".

Compute weights from error densities

Description

Compute weights from error densities

Usage

build_B_from_dist(taus, error.distr, error.par = list())

Arguments

taus

Numeric vector of quantile levels in (0,1).

error.distr

Character. One of "normal", "skew-normal", "t".

error.par

List of parameters for the chosen distribution.

Closed testing for quantile regression

Description

Applies the closed testing procedure to strongly control the familywise error rate (FWER) when testing the effect of a covariate of interest across multiple quantile regression models.

Usage

closedTesting(mod, X, tau = NULL, test = "rank-score", ...)

Arguments

mod

An object of class rqs returned by rq, representing the fitted quantile regression models.

X

A string indicating the covariate of interest.

tau

A numeric vector of quantiles of interest used in mod. If NULL (default), all quantiles from the mod object are considered.

test

Character. Type of test to be used. Options are "rank-score" and "wald".

...

Additional arguments, see rankTest, waldTest.

Details

This procedure requires that the covariate of interest X is either numeric or, if categorical, has at most two levels. Multilevel categorical covariates are not supported and will trigger an error.

The weighting matrix used in the multivariate rank test is the identity matrix, i.e., it is currently the only one implemented explicitly as a shortcut within the closed testing procedure.

Value

An object of class quasar containing:

Author(s)

Angela Andreella

References

Marcus, R., Eric, P., & Gabriel, K. R. (1976). On closed testing procedures with special reference to ordered analysis of variance. Biometrika, 63(3), 655–660.

Goeman, J. J., Hemerik, J., & Solari, A. (2021). Only closed testing procedures are admissible for controlling false discovery proportions. The Annals of Statistics, 49(2), 1218–1238.

See Also

rq, rankTest, waldTest

Examples

# Simulate data
set.seed(1234)
D <- simulateData(n = 100, gamma = 0.5, sigma.y = "1 + 2 * pmax(X, 0)")

# Quantile regressions at different levels
tau <- c(0.1, 0.25, 0.5, 0.75, 0.9)
mod <- quantreg::rq(y ~ X + Z1, tau = tau, data=D)

# Closed testing
res <- closedTesting(mod, X = "X")
res

# Summary and plot
summary(res, alpha = 0.1)
plot(res, alpha = 0.1, legend.position = "bottomright")

Plot method for quasar objects

Description

Produces a plot of a quasar object, typically returned by the closedTesting function. It shows the estimated coefficients by quantile level, highlighting statistically significant coefficients based on adjusted p-values.

Usage

## S3 method for class 'quasar'
plot(
  x,
  alpha = 0.05,
  legend.position = "topright",
  main = NULL,
  xlab = "Quantile level",
  ylab = "Coefficient",
  col.line = "darkgrey",
  col.sig = "darkred",
  col.nonsig = "darkgrey",
  pch.sig = 19,
  pch.nonsig = 17,
  show.legend = TRUE,
  ...
)

Arguments

x

An object of class quasar.

alpha

Significance level.

legend.position

Position of the legend.

main

Main plot title.

xlab

Label for the x-axis.

ylab

Label for the y-axis.

col.line

Color of the connecting line.

col.sig

Color for significant points.

col.nonsig

Color for non-significant points.

pch.sig

Point character for significant points.

pch.nonsig

Point character for non-significant points.

show.legend

Logical; whether to display a legend.

...

Additional graphical parameters passed to plot().

Value

A base R plot.

Author(s)

Anna Vesely

See Also

closedTesting

Print and summary methods for quasar objects

Description

These methods provide basic information about objects of class quasar, typically returned by the closedTesting function.

Usage

## S3 method for class 'quasar'
print(x, ...)

## S3 method for class 'quasar'
summary(object, ..., alpha = 0.05)

Arguments

x, object

An object of class quasar.

...

Additional arguments passed to other methods.

alpha

Significance level.

Value

The input object invisibly.

Author(s)

Anna Vesely

Rank-score test for quantile regression

Description

Performs the rank-score test for the covariate of interest X, at the quantiles defined in tau, using a fitted quantile regression model. The test evaluates the null hypothesis that the coefficient of X is equal to zero against a two-sided alternative, at each specified quantile level. Testing equality to a non-zero value is not yet implemented.

Usage

rankTest(mod, X, tau = NULL, full = FALSE, h = NULL, alpha = 0.05,
eps = c(1e-04,1e-04), B = "identity", error.distr = NULL, error.par = NULL)

Arguments

mod

An object of class rqs returned by rq, representing the fitted quantile regression models.

X

A string indicating the covariate of interest.

tau

A numeric vector of quantiles of interest used in mod. If NULL (default), all quantiles from the mod object are considered.

full

Logical. If TRUE, the function returns the test statistics and corresponding p-values for all intersection hypotheses containing tau. If FALSE (default), only the results for the single hypotheses are returned.

h

Character string specifying the bandwidth selection method for sparsity estimation. One of "Hall-Sheather" or "Bofinger". The default, NULL, uses the Hall-Sheather (1988) rule.

alpha

A numeric value used for bandwidth estimation. Following Koenker (2005), it is typically set equal to the desired significance level.

eps

Numeric vector of length 2 specifying the relative accuracies requested for approximating the distribution of the test statistic using Imhof's (1961) method, "Computing the Distribution of Quadratic Forms in Normal Variables". The first component controls the accuracy of the initial (more accurate) numerical integration (using integrate), while the second component controls the accuracy of a subsequent, less accurate integration step (using quadgk).

B

Weight specification used in the computation of the test statistic. One of "identity" (default), "distribution", "inverse diagonal", or "inverse variance" (not recommended). Alternatively, the user can supply a numeric matrix of dimension length(mod$tau) x length(mod$tau). This argument is ignored when full = TRUE.

error.distr

A character string specifying the assumed distribution of the regression errors, used only when B = "distribution". Allowed values are "normal", "skew-normal", "t".

error.par

A named list of parameters associated with error.distr. Required elements depend on the chosen distribution:

  • For "normal": mean, sd.

  • For "skew-normal": xi, omega, alpha.

  • For "t": df.

Details

This procedure requires that the covariate of interest X is either numeric or, if categorical, has at most two levels. Multilevel categorical covariates are not supported and will trigger an error.

If full = TRUE, B is ignored and the identity weight is used. If B is user-defined (a matrix), it must be square, symmetric, positive definite, and numerically invertible.

Value

A data.frame containing:

Author(s)

Angela Andreella

References

Koenker, R. (2005). Quantile Regression. Cambridge University Press.

Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48(3–4), 419–426.

De Santis, R., Veselý, A., and Andreella, A. (2026). Inference on multiple quantiles in regression models by a rank-score approach. arXiv preprint <doi:10.48550/arXiv.2511.07999>.

See Also

rq, waldTest

Examples

set.seed(1234)
D <- simulateData(n = 100, gamma = 0.5, sigma.y = "1 + 2 * pmax(X, 0)")

#Quantile regressions at different levels
tau <- c(0.1, 0.25, 0.5, 0.75, 0.9)
mod <- quantreg::rq(y ~ X + Z1, tau = tau, data=D)

# Rank test
rankTest(mod, X = "X")

Simulate data

Description

Simulates a main covariate X, a vector of additional covariates Z, and a response y drawn from the chosen distribution.

Usage

simulateData(n, beta = 0, gamma = 0, mu = 0, Sigma = NULL,
             sigma.y = 1, distribution = "normal", df = 5,
             xi = -1.453, omega = 2, alpha = 2.2, seed = NULL)

Arguments

n

Integer. Number of observations.

beta

Numeric scalar. Effect of X.

gamma

Numeric vector. Effects of Z (length p - 1, where p = ncol(Sigma)).

mu

Numeric scalar. Intercept.

Sigma

Numeric p x p symmetric positive-definite covariance matrix for (X, Z). The first column corresponds to X, the remaining columns to Z1, Z2, ....

sigma.y

Either a numeric scalar or a one-sided expression/string (e.g., "0.3 * abs(X) + 0.1") defining the scale of y.

distribution

Character. One of "normal", "t", or "skew-normal". This is the distribution of y.

df

Numeric scalar > 0. Degrees of freedom for t-distribution.

xi

Numeric scalar. Location parameter for the skew-normal distribution. In particular, this will be mu + X * beta + Z %*% gamma + xi. Default -1.453.

omega

Numeric scalar > 0. Scale parameter for the skew-normal distribution. In particular, this will be sigma.y + omega. Default 2.

alpha

Numeric scalar. Slant parameter for the skew-normal distribution. Default 2.2.

seed

Numeric scalar > 0. Seed for random number generator.

Details

The response is generated as y = mu + X * beta + Z %*% gamma + error. The error term can be drawn from a normal distribution, scaled Student-t with df degrees of freedom, or a skew-normal. Its standard deviation is defined by sigma.y: if numeric, a fixed scale is used; if a character expression, the scale can vary with X and/or Z1.

Value

A data.frame with columns y, X, and Z1, ..., Zk.

Author(s)

Angela Andreella

Examples

set.seed(1)
p <- 3
Sigma <- diag(p)

# Normal
dat_n <- simulateData(n = 200, beta = 0.5, gamma = c(0.2,-0.1),
                      sigma.y = 0.5, distribution = "normal")

# Student-t
dat_t0 <- simulateData(n = 200, beta = 0.5, gamma = c(0.2,-0.1),
                       sigma.y = 0.5, distribution = "t", df = 7)
# Skew-normal
dat_sn <- simulateData(n = 200, beta = 0.5, gamma = c(0.2,-0.1),
                      sigma.y = "abs(Z1) + 1", distribution = "skew-normal")

Wald-type test for quantile regression

Description

Performs the Wald-type test for the covariate of interest X, at the quantiles defined in tau, using a fitted quantile regression model. The test evaluates the null hypothesis that the coefficient of X is equal to a given value beta against a two-sided alternative, at each specified quantile level.

Usage

waldTest(mod, X, tau = NULL, full = FALSE, h = NULL, beta = 0, alpha = 0.05)

Arguments

mod

An object of class rqs returned by rq, representing the fitted quantile regression models.

X

A string indicating the covariate of interest.

tau

A numeric vector of quantiles of interest used in mod. If NULL (default), all quantiles from the mod object are considered.

full

Logical. If TRUE, the function returns the test statistics and corresponding p-values for all intersection hypotheses containing tau. If FALSE (default), only the results for the single hypotheses are returned.

h

A numeric value for the bandwidth.

beta

Numeric value of the parameter of interest under the null hypothesis.

alpha

A numeric value used for bandwidth estimation. Following Koenker (2005), it is typically set equal to the desired significance level.

Details

This procedure requires that the covariate of interest X is either numeric or, if categorical, has at most two levels. Multilevel categorical covariates are not supported and will trigger an error.

Value

A data.frame containing:

Author(s)

Angela Andreella

References

Koenker, R. (2005). Quantile Regression. Cambridge University Press.

See Also

rq, rankTest

Examples

set.seed(1234)
D <- simulateData(n = 100, gamma = 0.5, sigma.y = "1 + 2 * pmax(X, 0)")

#Quantile regressions at different levels
tau <- c(0.1, 0.25, 0.5, 0.75, 0.9)
mod <- quantreg::rq(y ~ X + Z1, tau = tau, data=D)

# Wald test
waldTest(mod, X = "X")