Hypothesis Tests for Quantiles and Quantile-Based Measures

Functions to conduct hypothesis tests and derive confidence intervals for quantiles, linear combinations of quantiles, ratios of dependent linear combinations and differences and ratios of all of the above for comparisons between independent samples. Additionally, quantile-based measures of inequality are also considered.

rquest

Overview

The rquest package provides convenient functionality for researchers to carry out hypothesis tests and obtain confidence intervals for measures based on quantiles. This includes single quantiles (e.g., the median), linear combinations of quantiles (such as the interquartile range), ratios of linear combinations commonly found in skewness and kurtosis measures, and quantile-based inequality measures. Importantly, the package also allows users to define their own measures based on quantiles and carry out hypothesis tests and construct confidence intervals for these measures.

The package also provides functionality for quantile density estimation, computation of quantile optimality ratios (QOR), covariance estimation for quantiles and ratios of linear combinations of quantiles, and inference for robust versions of the coefficient of variation.

Key functionality

rquest provides a flexible framework for quantile-based statistical inference. Key functionality includes:

• Quantile inference: hypothesis tests and confidence intervals for individual quantiles, linear combinations of quantiles, and ratios of linear combinations of quantiles.

• User-defined measures: specification of custom quantile-based measures, allowing users to carry out hypothesis tests and construct confidence intervals beyond the measures built into the package.

• One- and two-sample inference: comparison of quantiles and quantile-based measures within a single population or between two independent populations.

• Robust measures of variation: inference for robust versions of the coefficient of variation based on the MAD and IQR.

• Quantile-based inequality measures: inference for built-in inequality measures as well as user-defined quantile-based inequality measures.

• Quantile density estimation: estimation of quantile densities using QOR-based or density-based methods.

• Flexible QOR computation: computation of quantile optimality ratios using the generalized lambda distribution, lognormal, normal, exponential, or a user-supplied quantile function.

• Covariance estimation: covariance matrices for quantile estimators and for ratios of linear combinations of quantiles.

Main functions

The main functions in the package are:

• q.test() carries out hypothesis tests and obtains associated confidence intervals for quantiles, linear combinations of quantiles, and ratios of linear combinations of quantiles.

• qineq() carries out hypothesis tests and obtains associated confidence intervals for quantile-based inequality measures, including user-defined measures.

• qcov() computes covariance matrices containing variances and covariances of quantile estimators.

• qden() estimates the quantile density function using either a QOR-based method or a density-based method.

• qor() computes quantile optimality ratios for supported distributions or user-supplied quantile functions.

• qrcov() computes approximate covariance matrices for ratios of linear combinations of quantile estimators.

• rcv.test() carries out one- and two-sample inference for robust versions of the coefficient of variation based on the MAD or IQR.

Installation

You can install the released version of rquest from CRAN with:

install.packages("rquest")

You can install the development version of rquest from GitHub with:

# install.packages("pak")
pak::pak("shenal-dkumara/rquest")

Usage

library(rquest)

Quantile-based inference with q.test()

# Create some data
x <- c(8.43, 7.08, 8.79, 8.88, 7.87,
       5.94, 8.79, 5.46, 8.11, 7.08)

y <- c(13.44, 13.65, 14.77, 9.51, 14.07,
       10.92, 11.59, 13.42, 8.93, 10.88)

# One-sample hypothesis test for the IQR
q.test(x, measure = "iqr")
#> 
#>  One sample test of the interquartile range (IQR)
#> 
#> data:  x
#> Z = 1.4037, p-value = 0.1604
#> alternative hypothesis: true IQR  is not equal to 0
#> 95 percent confidence interval:
#>  -0.7153314  4.3253314
#> sample estimates:
#>  IQR  
#> 1.805

# Two-sample hypothesis test for robust coefficients of variation
# (0.75 * IQR / median), with log transformation and
# back-transformation to the ratio scale
q.test(
  x, y,
  measure = "rCViqr",
  log.transf = TRUE,
  back.transf = TRUE
)
#> 
#>  Two sample test of the robust coefficient of variation
#>  (0.75*IQR/median)
#> 
#> data:  x and y
#> Z = -0.032603, p-value = 0.974
#> alternative hypothesis: true ratio of Robust CVs  is not equal to 1
#> 95 percent confidence interval:
#>  0.1347132 6.9518490
#> sample estimates:
#> ratio of Robust CVs  
#>            0.9677323

The same measure can also be specified directly in terms of the required quantiles and coefficients:

u <- c(0.25, 0.75)
coef <- 0.75 * c(-1, 1)
u2 <- 0.5
coef2 <- 1

q.test(
  x, y,
  u = u,
  u2 = u2,
  coef = coef,
  coef2 = coef2,
  log.transf = TRUE,
  back.transf = TRUE
)
#> 
#>  Two sample test of a ratio of two linear combinations of quantiles
#>  (LCQs)
#> 
#> data:  x and y
#> Z = -0.032603, p-value = 0.974
#> alternative hypothesis: true ratio of Ratio of LCQs  is not equal to 1
#> 95 percent confidence interval:
#>  0.1347132 6.9518490
#> sample estimates:
#> ratio of Ratio of LCQs  
#>               0.9677323

Alternatively, the numerator and denominator can be defined using a coefficient matrix:

u <- c(0.25, 0.5, 0.75)
num <- 0.75 * c(-1, 0, 1)
den <- c(0, 1, 0)
coef <- rbind(num, den)

q.test(
  x, y,
  u = u,
  coef = coef,
  log.transf = TRUE,
  back.transf = TRUE
)
#> 
#>  Two sample test of a ratio of two linear combinations of quantiles
#>  (LCQs)
#> 
#> data:  x and y
#> Z = -0.032603, p-value = 0.974
#> alternative hypothesis: true ratio of Ratio of LCQs  is not equal to 1
#> 95 percent confidence interval:
#>  0.1347132 6.9518490
#> sample estimates:
#> ratio of Ratio of LCQs  
#>               0.9677323

Covariance estimation with qcov()

set.seed(1234)
xq <- rnorm(100)

# Covariance matrix for sample quartiles
qcov(xq, c(0.25, 0.5, 0.75))
#>             0.25         0.5        0.75
#> 0.25 0.014141467 0.008979251 0.008130512
#> 0.5  0.008979251 0.017104367 0.015487625
#> 0.75 0.008130512 0.015487625 0.042071102

Quantile density estimation with qden()

set.seed(1234)
xd <- rnorm(100)

# QOR-based quantile density estimation using
# the flexible GLD distribution
qden(
  xd,
  c(0.25, 0.5, 0.75),
  method = "qor"
)
#> [1] 2.746291 2.615673 4.736868

# QOR-based quantile density estimation using
# the normal distribution
qden(
  xd,
  c(0.25, 0.5, 0.75),
  dist = "norm",
  method = "qor"
)
#> [1] 2.409224 2.638314 4.776829

# Density-based quantile density estimation
qden(
  xd,
  c(0.25, 0.5, 0.75),
  method = "density"
)
#> [1] 2.542878 2.386256 4.299268

Quantile optimality ratios with qor()

set.seed(1234)
xqor <- rlnorm(100)

# QOR using the flexible FKML GLD distribution
# with parameters estimated from the data
qor(
  seq(0.1, 0.9, by = 0.2),
  x = xqor
)
#> $qor
#> [1] -0.458044160  0.214678367  0.088987175  0.027832449  0.002757499
#> 
#> $params
#> NULL

# QOR using the lognormal distribution
qor(
  seq(0.1, 0.9, by = 0.2),
  dist = "lnorm",
  x = xqor
)
#> $qor
#> [1] 0.021508337 0.123522437 0.079227681 0.029207901 0.003364183
#> 
#> $params
#> NULL

# QOR for the exponential distribution
qor(
  seq(0.1, 0.9, by = 0.2),
  dist = "exp"
)
#> $qor
#> [1] 0.405 0.245 0.125 0.045 0.005
#> 
#> $params
#> NULL

# QOR for a user-supplied quantile function
qor(
  c(0.2, 0.5, 0.8),
  dist = qbeta,
  params = list(shape1 = 2, shape2 = 5)
)
#> $qor
#> [1] 0.07645133 0.19841722 0.03361339
#> 
#> $params
#> $params$shape1
#> [1] 2
#> 
#> $params$shape2
#> [1] 5

Covariance estimation for ratios with qrcov()

set.seed(1234)
xr <- rnorm(100)

# Covariance matrix for the ratio of the third quartile
# to the median and the ratio of the first quartile
# to the median
coef1 <- matrix(
  c(0, 0, 1,
    1, 0, 0),
  nrow = 2,
  byrow = TRUE
)

coef2 <- matrix(
  c(0, 1, 0,
    0, 1, 0),
  nrow = 2,
  byrow = TRUE
)

qrcov(
  xr,
  c(0.25, 0.5, 0.75),
  coef1 = coef1,
  coef2 = coef2
)
#> $ratios
#> [1] -1.265365  2.346111
#> 
#> $cov
#>            R1         R2
#> R1  0.7344446 -0.4570857
#> R2 -0.4570857  0.4471806

Quantile-based inequality measures with qineq()

# Create some data
x <- c(8.43, 7.08, 8.79, 8.88, 7.87,
       5.94, 8.79, 5.46, 8.11, 7.08)

y <- c(13.44, 13.65, 14.77, 9.51, 14.07,
       10.92, 11.59, 13.42, 8.93, 10.88)

# Test equality of the QRI measure between two groups
qineq(x, y, measure = "QRI")
#> 
#>  Two sample test of the QRI statistic
#> 
#> data:  x and y
#> Z = -0.21439, p-value = 0.8302
#> alternative hypothesis: true difference in QRI statistic is not equal to 0
#> 95 percent confidence interval:
#>  -0.2522859  0.2025363
#> sample estimates:
#> difference in QRI statistic 
#>                 -0.02487481

A user-defined inequality measure can be specified directly using a formula. For example, the QRI can equivalently be defined as:

qineq(
  x, y,
  measure = ~ Q(p/2) / Q(1 - p/2)
)
#> 
#>  Two sample test of the user defined statistic
#> 
#> data:  x and y
#> Z = -0.21439, p-value = 0.8302
#> alternative hypothesis: true difference in user defined statistic is not equal to 0
#> 95 percent confidence interval:
#>  -0.2522859  0.2025363
#> sample estimates:
#> difference in user defined statistic 
#>                          -0.02487481

More generally, a user-defined measure can also be specified directly using the required quantiles and coefficient matrices:

J <- 100
p <- (1:J - 0.5) / J
u <- sort(c(p/2, 1 - p/2))

num <- cbind(
  diag(rep(1, J)),
  matrix(0, J, J)
)

den <- cbind(
  matrix(0, J, J),
  diag(J)[, J:1]
)

qineq(
  x, y,
  measure = list(
    u = u,
    coef1 = num,
    coef2 = den
  )
)
#> 
#>  Two sample test of the user defined statistic
#> 
#> data:  x and y
#> Z = -0.21439, p-value = 0.8302
#> alternative hypothesis: true difference in user defined statistic is not equal to 0
#> 95 percent confidence interval:
#>  -0.2522859  0.2025363
#> sample estimates:
#> difference in user defined statistic 
#>                          -0.02487481

Robust coefficients of variation with rcv.test()

set.seed(123)
x1 <- rnorm(100, 8)
x2 <- rnorm(120, 10)

# One-sample inference using the MAD
rcv.test(x1)
#> 
#>  One sample test of the robust coefficient of variation (MAD/median)
#> 
#> data:  x1
#> Z = -18.082, p-value < 2.2e-16
#> alternative hypothesis: true Robust CV  is not equal to 1
#> 95 percent confidence interval:
#>  0.08691013 0.14014523
#> sample estimates:
#> Robust CV  
#>  0.1103632

# One-sample inference using the IQR
rcv.test(x1, numerator = "iqr")
#> 
#>  One sample test of the robust coefficient of variation
#>  (0.75*IQR/median)
#> 
#> data:  x
#> Z = -16.613, p-value < 2.2e-16
#> alternative hypothesis: true Robust CV  is not equal to 1
#> 95 percent confidence interval:
#>  0.0856860 0.1439217
#> sample estimates:
#> Robust CV  
#>  0.1110499

# Two-sample inference using the MAD
rcv.test(x1, x2)
#> 
#>  Two sample test of the robust coefficient of variation (MAD/median)
#> 
#> data:  x1 and x2
#> Z = 0.97732, p-value = 0.3284
#> alternative hypothesis: true ratio of Robust CVs  is not equal to 1
#> 95 percent confidence interval:
#>  0.8489523 1.6314679
#> sample estimates:
#> ratio of Robust CVs  
#>             1.176877

# Two-sample inference using the IQR
rcv.test(x1, x2, numerator = "iqr")
#> 
#>  Two sample test of the robust coefficient of variation
#>  (0.75*IQR/median)
#> 
#> data:  x and y
#> Z = 0.65049, p-value = 0.5154
#> alternative hypothesis: true ratio of Robust CVs  is not equal to 1
#> 95 percent confidence interval:
#>  0.7898182 1.6005955
#> sample estimates:
#> ratio of Robust CVs  
#>             1.124357