| Version: | 0.4 |
| Title: | Empirical Likelihood Ratio Test/Confidence Interval for AUC or pAUC |
| Maintainer: | Mai Zhou <maizhou@gmail.com> |
| Depends: | R (≥ 3.2.5), emplik2, rootSolve |
| Imports: | stats |
| Description: | Test hypotheses and construct confidence intervals for AUC (area under Receiver Operating Characteristic curve) and pAUC (partial area under ROC curve), from the given two samples of test data with disease/healthy subjects. The method used is based on TWO SAMPLE empirical likelihood and PROFILE empirical likelihood, as described in https://www.ms.uky.edu/~mai/research/eAUC1.pdf. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| NeedsCompilation: | no |
| Author: | Mai Zhou [aut, cre, cph], Xue Ding [aut], Yumin Zhao [aut] |
| Packaged: | 2024-05-29 08:53:36 UTC; maizh |
| Repository: | CRAN |
| Date/Publication: | 2024-05-30 03:20:06 UTC |
Testing one pAUC(p1, p2) and two quantile values together by Empirical Likelihood.
Description
This function computes the two sample Log Empirical Likelihood ratio
for testing H_0: pAUC(p1, p2) = theta; F(tau1)=1-p2; F(tau2)= 1-p1.
The two samples are in the x-vector and y-vector.
Usage
EL2paucT1T2(tauVec, pauc, partial1, partial2, x, y, epsxy)
Arguments
tauVec |
The vector (tau1, tau2), the two quantile values, to be tested. |
pauc |
The value of the pAUC(p1, p2) under |
partial1 |
The probability that define the quantile 1 |
partial2 |
The probability that define the quantile 2. Must satisfy partial 1 < partial 2 . |
x |
a vector of observations, length m, for the first sample. |
y |
a vector of observations, length n, for the second sample. |
epsxy |
The parameter for smoothing when compare x-y. |
Details
The input tauVec=(tau1, tau2), and must have: tau1 < tau2. The relavant definitions are:
tau1 = F^{-1}(1-partial2); tau2 = F^{-1}(1-partial1).
Thus, we must have partial2 > partial1.
This function is testing 3 parameters simultanuously.
It depend on the package emplik2.
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 ~.
Value
A single value that is the "-2LLR" from emplik2::el2.cen.EMm().
Typically should be distributed as chi square df=3, under H_0.
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
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)
Arguments
theta |
The "true" value of the AUC under |
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]]. |
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 ~.
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. |
Author(s)
Mai Zhou <maizhou@gmail.com>.
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
Testing one pAUC(0, p) value by Empirical likelihood.
Description
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.
Usage
el2testPauc(theta, x, y, ind, nuilow, nuiup, partial, epsxy, epsT)
Arguments
theta |
The "true" value of the pAUC(0, p) under |
x |
a vector of observations, length m, for the first sample, test-results with the healthy subjects. |
y |
a vector of observations, length n, for the second sample, test-results with the desease subjects. |
ind |
The (smoothed) indicator function for compare x-y. |
nuilow |
Lower bound for the nuisamce parameter (1-p)-th quantile of X) search. |
nuiup |
Upper bound for nuisance parameter search. |
partial |
The probability p in pAUC(0, p). |
epsxy |
The smoothing parameter when compare x-y. |
epsT |
The smoothing parameter when calculating quantile. |
Details
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 ~.
Value
A list containing
"-2LLR" |
The -2 log empirical likelihood ratio. |
Nupar |
The nuisance parameter value that achieved the minimum. |
Pval |
The p-value, by using chi square distribution with 1 df. |
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
Testing one pAUC(0, p) value and one quantile: F(tau) = 1-p 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. (F is the CDF of X).
The two samples data are in the x-vector and y-vector inputs. It uses EM.
Usage
el2testPaucT(tau, pauc, ind, partial, x, y, epsxy, epsT)
Arguments
tau |
The "true" value of the (1-p)-th quantile of X-distribution F, to be tested. |
pauc |
The |
ind |
A smoothed indicator function, to generate a Matrix of (smoothed) indicator values: I[x[i] < y[j]]. |
partial |
The probability p in pAUC(0, p); also the p in F(tau) = 1-p. |
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. |
epsxy |
The smoothing parameter when compare x-y. |
epsT |
The smoothing parameter when compare x to Tau, for quantile estimation. |
Details
This function is called by el2testPauc( ).
It is listed here stand alone because users may find it useful elsewhere.
It make use of function smooth3( ) and the function el2.cen.EMm( )
from the emplik2 package.
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
It returns one value that is the "-2LLR". It should have chi square df=2 under H_0.
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
Testing one AUC value by Empirical likelihood.
Description
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.
Usage
eltest4aucONE(theta, x, y, ind, tol.u, tol.v, tol.H0)
Arguments
theta |
The "true" value of the AUC under |
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]]. |
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 the constrained NPMLE to satisfy |
Details
This function is similar to el2test4auc, but using our own algorithm (not EM).
It may be slightly different to the above in terms of speed and convergence property.
We listed 3 kind of tol to control convergence.
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. |
Pval |
The p-value. |
iterNum |
The iteration number used in computing. |
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
#### We know the AUC estimator here is 0.75.
#### We may test a hypothesis about the AUC value: H0: AUC= 0.7
eltest4aucONE(theta=0.7, x=x, y=y, ind=smooth3, tol.u=1e-6, tol.v=1e-6, tol.H0=1e-6)
#### Two of the outputs should be '-2LLR'=0.1379561 and Pval=0.7103214
Testing one pAUC value by Empirical likelihood.
Description
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.
Usage
eltest4paucONE(theta,x,y,nuilow,nuiup,ind,partial,epsxy=0.05,epsT=(length(x))^(-0.75))
Arguments
theta |
The "true" value of the pAUC under |
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. |
nuilow |
The lower bound for the nuisance parameter (the (1-p)-th quantile of X CDF F) search. |
nuiup |
The upper bound for the nuisance parameter search. |
ind |
A smoothed indicator function, to generate a Matrix of (smoothed) indicator values: I[x[i] < y[j]]. |
partial |
The probability p in the pAUC(0,p). |
epsxy |
Window width for the smoother, "ind", when compare x-y. |
epsT |
Window width for the smoother, "ind", when define quantile. |
Details
This function calls the function eltest4paucT.
We then use optimize( ) to profile out the nuisance parameter.
Return an empirical likelihood ratio siutable for testing one parameter pAUC(0,p).
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
"-2LLR" |
The -2 log empirical likelihood ratio. |
Nupar |
The nuisance parameter value to achieve the minimum. |
Pval |
The p-value. |
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
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 |
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 |
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 <maizhou@gmail.com>.
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)
Internal emplikAUC functions
Description
These are internal functions called by eltest4aucONE and eltest4paucT. They are not intended
to be called by users.
Usage
lamONE(u0, v0, indicmat, tol.H0)
lam2(u0, v0, indicmat1, indicmat2, tol.H0, start)
Arguments
u0 |
a vector of probabilities, length m, for the first (X) sample. |
v0 |
a vector of propabilities, length n, for the second (Y) sample. |
indicmat |
a Matrix (size m by n) of indicator function values: I[X[i] < Y[j]], But typically smoothed and centered. |
tol.H0 |
The error bound for (checking) if satisfy |
Details
lamONE() find the lambda (tilting parameter) in the two sample Empirical Likelihood,
with an AUC constrain.
This function computes the lambda that satisfy null hypothesis H_0 constraint by
using uniroot.
This function is called by eltest4aucONE.
It is listed here because the user may find it useful elsewhere.
The function lam2, similar to function lamONE,
calculates the tilting parameters (lam1, lam2) for the empirical likelihood of
a joint hypothesis about pAUC(0, p) = theta and F(tau) = p.
The indicmat1, indicmat2 should reflect the joint hypothesis, in the U-stat fashion.
The function lam2( ) is called by eltest4paucT( ), and is
similar to lamONE
except there are 2 lambda's and needs
to call multiroot() to solve the
equation.
And also, with a starting value input to help the
iterations of solving the equation. It (start) is directly passed to multiroot( ) function.
Author: Ding Xue.
Value
Output is a list containing:
u |
the new u vector. |
v |
the new v vector. |
lam |
The solution lambda. |
Finding the Lower bound of a confidence interval for theta by repeatedly testing the hypothesis for the parameter theta.
Description
This function try to find the Lower bound of a confidence interval
by repeatedly testing the
hypothesis for the parameter theta until we reach the
level for the "-2LLR" which return from
the user supplied function fun.
Usage
findLnew(step=0.01, initStep=0, fun, MLE, level=qchisq(0.95,df=1),
tol=.Machine$double.eps^0.5,...)
Arguments
step |
Search step. Must > 0. |
initStep |
The initial step from MLE. May be used if we knew the bound is far away from MLE. This help to speed up things. Should be >=0. |
fun |
The function that should return "-2LLR". |
MLE |
The MLE of the parameter. |
level |
The level of the confidence. Default to 3.84 which is 95 percent confidence. |
tol |
The error bound for achieving the level given. |
... |
Any additional input to be passed to fun. |
Details
This function is similar to findUnew( ).
Value
It returns a list containing
Low |
Lower bound of the confidence interval. |
FstepL |
The error when search for Lower bound. |
Lvalue |
The final likelihood ratio value for Lower bound. Should =~ level. |
Author(s)
Mai Zhou <maizhou@gmail.com>.
References
Zhao, Y., Ding, X. and Zhou (2021). Confidence Intervals of AUC and pAUC by Empirical Likelihood. Tech Report. https://www.ms.uky.edu/~mai/research/eAUC1.pdf
Examples
y <- c(10, 209, 273, 279, 324, 391, 566, 785)
x <- c(21, 38, 39, 51, 77, 185, 240, 289, 524)
Finding the Upper and Lower bound of a confidence interval for theta by repeatedly testing the hypothesis for the parameter theta.
Description
This function try to find the Upper and Lower bound of a confidence interval
by repeatedly testing the
hypothesis for the parameter theta until we reach the
given level for the "-2LLR" which is returned from
the user supplied function fun.
Usage
findULNEW(step=0.01, initStep=0, fun, MLE, level=qchisq(0.95,df=1),
tol=.Machine$double.eps^0.5,...)
Arguments
step |
Search step. Must > 0. |
initStep |
The initial step from MLE. May be used if we knew the bound is far away from MLE. This help to speed up things. Should be >=0. |
fun |
The function that should return "-2LLR". |
MLE |
The MLE of the parameter. |
level |
The level of the confidence. Default to 3.84 which is 95 percent confidence (assume df=1). |
tol |
The error bound for achieving the level given. |
... |
Any additional input to be passed to fun. |
Details
This function just combines the two functions findUnew( ) and findLnew( ).
Value
It returns a list containing
Low |
Lower bound of the confidence interval. |
Up |
Upper bound of the confidence interval. |
FstepL |
The error when search for Lower bound. |
FstepU |
The error when search for Upper bound. |
Lvalue |
The final likelihood ratio value for Lower bound. Should = level. |
Uvalue |
The final likelihood ratio value for Upper bound. Should = level. |
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
Finding the Upper bound of a confidence interval for theta by repeatedly testing the
hypothesis for the parameter theta.
Description
This function try to find the Upper bound of a confidence interval
by repeatedly testing the
hypothesis for the parameter theta until we reach the
level for the "-2LLR" which return from
the user supplied function fun.
Usage
findUnew(step=0.01, initStep=0, fun, MLE, level=qchisq(0.95,df=1),
tol=.Machine$double.eps^0.5,...)
Arguments
step |
Search step. Must > 0. |
initStep |
The initial step from MLE. May be used if we knew the bound is far away from MLE. This help to speed up things. Should be >=0. |
fun |
The user supplied function that should return "-2LLR". |
MLE |
The MLE of the parameter. An approximate value should be OK. |
level |
The level of the confidence. Default to 3.84 which is 95 percent confidence. |
tol |
The error bound for achieving the level given. |
... |
Any additional input to be passed to fun. |
Details
This function is similar to findLnew( ).
The search is separate (for upper and lower)
since Upper and Lower bound may behave differently and require different
nuisance parameters inputs (in ...).
Value
It returns a list containing
Up |
Upper bound of the confidence interval. |
FstepU |
The error when search for Upper bound. |
Uvalue |
The final likelihood ratio value for Upper bound. Should =~ level. |
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
Given the x, y 2-sample data, first estimate the (1-partial)-th quantile of X sample, then estimate the pAUC(0, partial), with the plug-in estimated quantile.
Description
This function computes the estimator using smoothed Indicator function I[x < y] which is a 3rd order polynomial.
eps is a scalar, must > 0. It is the smoothing window width for indicator function when compare x-y.
epsT is a scalar, must > 0. It is the smoothing window width for estimating quantile function when compare x with tau.
Usage
myEstPaucT(x, y, partial, eps=0.05, epsT=(length(x))^(-0.75))
Arguments
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. |
partial |
The probability in pAUC(0, partial). |
eps |
The smoothing window width, for indicator I[x < y]. Must >0. |
epsT |
The smoothing window width for the quantile estimation. Must >0. |
Details
This function gives the estimators that are consistant with our smoothing used in the computation of empirical likelihood. Typically the smoother for quantile should be a bit "smoother" that usual.
Known problem: when input partial is too close to 1 or 0, it will fail.
When partial is equal to 1, then the pAUC reduces to AUC, which can be estimated easily,
as the example below shows.
Why not also give the AUC estimator? May be it is too easy. See example below.
Value
This function returns a list, with
"tau(1-partial)" |
The estimator for the (1-partial)-th quantile of X-distribution, |
and
"Pauc(0, partial)" |
The estimated pAUC. |
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
#### To get an estimator of the AUC (not pAUC), using our smooth3( ), we do
sum(smooth3(x=x, y=y))/(length(x)*length(y))
#### We should get AUC = 0.75.
#### To get the estimator of the pAUC(0, 0.3) AND the 70th quantile of x sample
myEstPaucT(x=x, y=y, partial=0.3)
#### We should get estimator tau(1-0.3)=239.9474 and pAUC(0, 0.3)=0.1416667.
Smoothed quantile estimation from the given x-sample.
Description
This function computes the smoothed quantile estimate,
using the smoothing function smooth3. (or it can be changed easily)
Otherwise it is similar to R function quantile().
Compare to quantile(x, prob, type=9).
myeps is a scalar, must > 0. It is the smoothing window width.
Usage
quantONE(x, prob, myeps=(length(x))^(-0.75))
Arguments
x |
a vector of observations, length m, the sample. |
prob |
a probabilty. |
myeps |
The smoothing window width, must >0. |
Details
This function is called by myEstPaucT.
It is listed here because the user may find it useful elsewhere.
Known problems: when the input prob is too close to 0 or 1 (distance less than 0.03),
the computation will stop.
Also, if the solution to the equation
\tau ~| ~~~ \frac{1}{m} \sum_{i=1}^m smooth3vec(X_i, \tau, myeps) = prob
is not unique, this function only returns one of the solutions.
Value
The function quantONE returns a scalar that is the estimated (prob)-th quantile of X distribution.
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
#### To estimate the 70-th percentile of x distrubution:
quantONE(x=x, prob=0.7)
#### we should get 239.9474.
Non-Smoothed indicator function 0.5I[x=y] + I[x < y].
Description
This function computes the non-smoothed Indicator function 0.5I[x=y] + I[x < y].
Usage
smooth0(x, y)
Arguments
x |
a vector of observations, length m, for the first sample. |
y |
a vector of observations, length n, for the second sample. |
Details
This function is used in the original AUC computation (non-smoothed). You may achieve the same result with a (data dependent) small eps in the function smooth3(x,y,eps). But this is faster and no eps to worry about. It is listed here because users may find it useful.
Value
smooth0( ) returns a matrix of dimension ncol=length(y), nrow=length(x). The entry
of the matrix are values of 0.5I[x[i] = y[j]] + I[x[i] < y[j]].
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
Smoothed indicator function I[x < y], which is the integration of the Epanechnikov kernal.
Description
This function computes the smoothed Indicator function I[x < y] using a 3rd order polynomial.
If |x-y| > eps then the result is the same as the indicator function I[x < y] (either 0 or 1). For |x-y| < eps, it is a 3rd order polynomial.
eps is a scalar, must > 0. It is the smoothing window width.
Usage
smooth3(x, y, eps=0.05)
Arguments
x |
a vector of observations, length m, for the first sample. |
y |
a vector of observations, length n, for the second sample. |
eps |
The smoothing window width, must >0. |
Details
This function is used in many places to replace an indicator function I[x<y].
For example, when estimating the AUC.
It is listed here because users may find it useful elsewhere.
Value
smooth3( ) returns a matrix of dimension ncol=length(y), nrow=length(x). The entry
of the matrix are smoothed values of I[x[i] < y[j]].
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
Smoothed indicator function I[x < const], which is the integration of the Epanechnikov kernal.
Description
This function computes the smoothed Indicator function I[x < const] using a 3rd order polynomial.
If |x - const| > eps then the result is the same as the indicator function I[x < const] (either 0 or 1). For |x - const| < eps, it is a 3rd order polynomial.
eps is a scalar, must > 0. It is the smoothing window width.
Usage
smooth3vec(x, const, eps=0.05)
Arguments
x |
a vector of observations, length m, for the first sample. |
const |
a single number. |
eps |
The smoothing window width, must be >0. Ideally this should be sample size dependent. |
Details
This function is similar to smooth3 but only compare the x vector to a single number
and thus returns a vector instead of matrix.
You may also use the smooth3() with a bit care, for that matter, but this vector
version should be faster and save memory.
It is listed here because the user may find it useful elsewhere.
We used this function to estimate the quantile from the x-sample.
Value
smooth3vec returns a vector of length=length(x). The entry
of the vector are smoothed values of I[x[i] < const].
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
Smoothed indicator function I[x < const], which is the integration of the Quartic kernal.
Description
This function computes the smoothed Indicator function I[x < const] using a 5th order polynomial.
If |x - const| > eps then the result is the same as the indicator function I[x < const] (either 0 or 1). For |x - const| < eps, it is a 5th order polynomial.
eps is a scalar, must > 0. It is the smoothing window width.
Usage
smooth5vec(x, const, eps=0.05)
Arguments
x |
a vector of observations, length m, for the first sample. |
const |
a single number. |
eps |
The smoothing window width, must be >0. Ideally, this should be sample size dependent. |
Details
This function is twice continuously differenciable, smoother than smooth3vec.
It is listed here because the user may need extra smoothness (compare to smooth3vec)
and may find it useful elsewhere.
Value
smooth5vec returns a vector of length=length(x). The entry
of the vector are smoothed values of I[x[i] < const].
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)
Smoothed indicator function I[x < const], linear connecting the 1 and 0 on [-eps, eps].
Description
This function computes the smoothed Indicator function I[x < const] using a liare.
If |x - const| > eps then the result is the same as the indicator function I[x < const] (either 0 or 1). For |x - const| < eps, it is a linear function.
eps is a scalar, must > 0. It is the smoothing window width.
Usage
smoothLvec(x, const, eps=0.05)
Arguments
x |
a vector of observations, length m, for the first sample. |
const |
a single number. |
eps |
The smoothing window width, must be >0. Ideally this should be sample size dependent. |
Details
This function is similar to smooth3vec but only compare the x vector to a single number
and thus returns a vector instead of matrix.
You may also use the smooth3() with a bit care, for that matter, but this vector
version should be faster and save memory.
It is listed here because the user may find it useful elsewhere.
We used this function to estimate the quantile from the x-sample.
Value
smoothLvec returns a vector of length=length(x). The entry
of the vector are smoothed values of I[x[i] < const].
Author(s)
Mai Zhou <maizhou@gmail.com>.
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)