drmdel (version 1.1)

quantileCompWald: Wald test for pairwise comparison and linear hypotheses about quantiles under the DRM

Description

Suppose we have m+1 samples, labeled as $0, \, 1, \, \ldots, \, m$, whose population distributions satisfy the density ratio model (DRM) (see drmdel for the definition of DRM). We now want to test the linear hypothesis about a vector of quantiles $q = (q_1, q_2, \ldots, q_s)^T$ of probably different populations: $$H_0: \, Aq = b \ \ \mbox{against} \ \ H_1: \, Aq\neq b,$$ where A is a t by s, t $\le$ s, non-singular matrix and b is a vector. The quantileCompWald function performs a Wald-test for the above hypothesis and also pairwise comparisons of the population quantiles.

Usage

quantileCompWald(quantileDRMObject, n_total, pairwise=TRUE,
                 p_adj_method="none", A=NULL, b=NULL)

Arguments

quantileDRMObject

an output from the quantileDRM function. It must be a list containing a vector of quantile estimates (quantileDRMobject$est) and a estimated covariance matrix of the quantile estimates (

n_total

total sample size.

pairwise

a logical variable specifying whether to perform pairwise comparisons of the quantiles. The default is TRUE.

p_adj_method

when pairwise=TRUE, how should the p-values be adjusted for multiple comparisons. The available methods are: "holm", "hochberg", "hommel", "bonferroni", "BH", "BY", "fdr" and "none". See help(p.adjust) for details. The default

A

the left-hand side t by s, t $\le$ s, matrix in the linear hypothesis.

b

the right-hand side t-dimensional vector in the linear hypothesis.

Value

• p_val_pair
  p-values of pairwise comparisons, in the form of a lower triangular matrix. Available only if argument pairwise=TRUE
• p_val
  p-value of the linear hypothesis. Available only if argumen 'A' and 'b' are not NULL.

Details

Denote the EL quantile estimate of the $q$ vector as $\hat{q}$, and the estimate of the corresponding covariance matrix as $\hat{\Sigma}$. $\hat{q}$ and $\hat{\Sigma}$ can be calculated using function quantileDRM with 'cov=TRUE'. It is known that, $\sqrt{n_{total}}(\hat{q} - q)$ converges in distribution to a normal distribution with 0 mean and covariance matrix $\Sigma$. Also, $\hat{\Sigma}$ is a consistent estimator of $\Sigma$. Hence, under the null of the linear hypothesis, $$n_total (A \hat{q} - b)^T {(A \hat\Sigma A^T)}^{-1} (A \hat{q} - b)$$ has a chi-square limiting distribution with t (=ncol(A)) degrees of freedom.

References

J. Chen and Y. Liu (2013), Quantile and quantile-function estimations under density ratio model. To appear in The Annals of Statistics, 2013.

Examples

# Data generation
set.seed(25)
n_samples <- c(100, 200, 180, 150, 175)  # sample sizes
x0 <- rgamma(n_samples[1], shape=5, rate=1.8)
x1 <- rgamma(n_samples[2], shape=12, rate=1.2)
x2 <- rgamma(n_samples[3], shape=12, rate=1.2)
x3 <- rgamma(n_samples[4], shape=18, rate=5)
x4 <- rgamma(n_samples[5], shape=25, rate=2.6)
x <- c(x0, x1, x2, x3, x4)

# Fit a DRM with the basis function q(x) = (x, log(abs(x))), which
# is the basis function for gamma family. This basis function is
# the built-in basis function 6.
drmfit <- drmdel(x=x, n_samples=n_samples, basis_func=6)

# Quantile comparisons
# Compare the 5^th percentile of population 0, 1, 2 and 3.

# Estimate these quantiles first
qe <- quantileDRM(k=c(0, 1, 2, 3), p=0.05, drmfit=drmfit)

# Create a matrix A and a vector b for testing the equality of all
# these 5^th percentiles. Note that, for this test, the contrast
# matrix A is not unique.
A <- matrix(rep(0, 12), 3, 4)
A[1,] <- c(1, -1, 0, 0)
A[2,] <- c(0, 1, -1, 0)
A[3,] <- c(0, 0, 1, -1)
b <- rep(0, 3)

# Quantile comparisons
# No p-value adjustment for pairwise comparisons
(qComp <- quantileCompWald(qe, n_total=sum(n_samples), A=A, b=b))

# Adjust the p-values for pairwise comparisons using the "holm"
# method.
(qComp1 <- quantileCompWald(qe, n_total=sum(n_samples),
                            p_adj_method="holm", A=A, b=b))