eltest4paucT {emplikAUC}

Testing one pAUC and one quantile together by Empirical Likelihood.

Description

This function computes the two sample Log Empirical Likelihood ratio for testing H_0: pAUC(0,p) = theta and F(tau) = 1-p. The two samples are in the x-vector and y-vector.

Usage

eltest4paucT(tau, x, y, true, ind, epsxy, epsT, tol.u, tol.v, tol.H0, p)

Arguments

tau

The "true" value of the (1-p)th quantile of X, under H_0, to be tested.

x

a vector of observations, length m, for the first sample. Test-results with healthy subjects.

y

a vector of observations, length n, for the second sample. Test-results with desease subjects.

true

The H_0 value of the pAUC(0, p) to be tested.

ind

A smoothed indicator function, to generate a Matrix of (smoothed) indicator values: I[x[i] < y[j]].

epsxy

Window width for smoother (ind) when compare x-y.

epsT

Window width for smoother (ind) when find quantile.

tol.u

Error tol for final u probability vector. Must > 0.

tol.v

Error tol for final v probability vector. Must > 0.

tol.H0

The error bound for checking if the constrained NPMLE satisfy H0, must >0.

p

The probability p in pAUC(0, p), and also in F(tau) = 1-p.

Details

This function is similar to el2testPaucT( ). Just a different algorithm (not EM). Speed and convergence may be slightly different.

This function is called by eltest4paucONE. It is listed here because the user may find it useful elsewhere.

The empirical likelihood we used here is defined as

EL = \prod_{i=1}^m v_i \prod_{j=1}^n \nu_j ~;~~~~~~ \sum v_i =1 ~,~~ \sum \nu_j =1 ~.

Value

A list containing

lambda

The final tilting parameter.

u

the new u vector.

v

The new v vector.

"-2LLR"

The -2 log empirical likelihood ratio.

iterNum

The iteration number used in computing.

Author(s)

Mai Zhou <[e-mail]>.

References

Zhao, Y., Ding, X. and Zhou (2021). Confidence Intervals of AUC and pAUC by Empirical Likelihood. Tech Report. https://www.ms.uky.edu/~mai/research/eAUC1.pdf

Examples

 
y <- c(10, 209, 273, 279, 324, 391, 566, 785)
x <- c(21, 38, 39, 51, 77, 185, 240, 289, 524)