This function computes the two sample Log Empirical Likelihood ratio for testing \(H_0\): pAUC(0, p) = theta. The two samples are in the x-vector and y-vector inputs.
el2testPauc(theta, x, y, ind, nuilow, nuiup, partial, epsxy, epsT)A list containing
The -2 log empirical likelihood ratio.
The nuisance parameter value that achieved the minimum.
The p-value, by using chi square distribution with 1 df.
The "true" value of the pAUC(0, p) under \(H_0\), to be tested.
a vector of observations, length m, for the first sample, test-results with the healthy subjects.
a vector of observations, length n, for the second sample, test-results with the desease subjects.
The (smoothed) indicator function for compare x-y.
Lower bound for the nuisamce parameter (1-p)-th quantile of X) search.
Upper bound for nuisance parameter search.
The probability p in pAUC(0, p).
The smoothing parameter when compare x-y.
The smoothing parameter when calculating quantile.
Mai Zhou <maizhou@gmail.com>.
This function will call another function: el2testPaucT( ).
We then use optimize( ) to profile out the nuisance
parameter tau: the (1-p)-th quantile of X distribution.
Can be used by findUnew( ) etc.
The empirical likelihood we used here is defined as $$ EL = \prod_{i=1}^m v_i \prod_{j=1}^n \nu_j ~; ~~~~s.t. ~~~ \sum v_i =1 ~,~~ \sum \nu_j =1 ~. $$
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
y <- c(10, 209, 273, 279, 324, 391, 566, 785)
x <- c(21, 38, 39, 51, 77, 185, 240, 289, 524)
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