emplikAUC (version 0.5)

el2test4auc: Testing one AUC value by Empirical likelihood.

Description

By calling upon a function from the package emplik2 (using EM), this function computes the two sample Log Empirical Likelihood ratio for testing \(H_0\): AUC = theta. The two samples are in the x-vector and y-vector inputs.

Usage

el2test4auc(theta, x, y, ind)

Value

A list that is the same as el2.cen.EMs() from emplik2 package. Which contains

lambda

The final tilting parameter.

"-2LLR"

The -2 log empirical likelihood ratio.

Pval

The p-value.

iterNum

The iteration number used in computing.

Arguments

theta

The "true" value of the AUC under \(H_0\), to be tested.

x

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

y

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

ind

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

Author

Mai Zhou <[e-mail]>.

Details

This function is similar to the function eltest4aucONE(), the difference is that we call the function emplik2::el2.cen.EMs() to do the heavy computation (instead of by our own code). So, the speed and convergence property may be slightly different. When they both converge the results should be the same.

The empirical likelihood we used here is defined as $$ EL = \prod_{i=1}^m v_i \prod_{j=1}^n \nu_j ~;~~~~~ \sum v_i =1 ~,~~ \sum \nu_j =1 ~. $$

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)
#### The estimation of AUC
sum(smooth3(x=x, y=y))/(length(x)*length(y))
#### This does not work in Rcmd check: (truncate at %*%) 
####   rep(1/length(x), length(x))%*%smooth3(x=x, y=y)%*%rep(1/length(y), length(y))
#### The result should be 0.75.
#### We may then test a hypothesis about the AUC value: H0: AUC= 0.7
el2test4auc(theta=0.7, x=x, y=y, ind=smooth3)
#### Two of the outputs should be '-2LLR'=0.1379561 and Pval=0.7103214