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```
#' Function for the determination of the thresholds of an uncertain interval for
#' bi-normal distributed test scores that are considered as inconclusive.
#'
#' @param ref The reference standard. A column in a data frame or a vector
#' indicating the classification by the reference test. The reference standard
#' must be coded either as 0 (absence of the condition) or 1 (presence of the
#' condition).
#' @param test The index test or test under evaluation. A column in a dataset or
#' vector indicating the test results in a continuous scale.
#' @param UI.Se (default = .55). Desired sensitivity of the test scores within the
#' uncertain interval. A value <= .5 is not allowed.
#' @param UI.Sp (default = .55). Desired specificity of the test scores within the
#' uncertain interval. A value <= .5 is not allowed.
#' @param intersection Default NULL. If not null, the supplied value is used as
#' the estimate of the intersection of the two bi-normal distributions.
#' Otherwise, it is calculated using the function
#' \code{\link{get.intersection}}.
#' @param start Default NULL. If not null, the first two values of the supplied
#' vector are used as the starting values for the \code{nloptr} optimization
#' function.
#' @param print.level Default is 0. The option print_level controls how much
#' output is shown during the optimization process. Possible values: 0)
#' (default) no output; 1) show iteration number and value of objective
#' function; 2) 1 + show value of (in)equalities; 3) 2 + show value of controls.
#'
#' @details{ This function can be used for a test with bi-normal distributed
#' scores. The Uncertain Interval is generally defined as an interval below and
#' above the intersection, where the densities of the two distributions of
#' patients with and without the targeted condition are about equal. These test
#' scores are considered as inconclusive for the decision for or against the
#' targeted condition. This function uses for the definition of the uncertain
#' interval a sensitivity and specificity of the uncertain test scores below a
#' desired value (default .55).
#'
#' Only a single intersection is assumed (or a second intersection where the
#' overlap is negligible). If another intersection exists and the overlap around
#' this intersection is considerable, the test with such a non-negligible
#' overlap is problematic and difficult to apply and interpret.
#'
#' In general, when estimating decision thresholds, a sample of sufficient size
#' should be used. It is recommended to use at least a sample of 100 patients
#' with the targeted condition, and a 'healthy' sample (without the targeted
#' condition) of the same size or larger.
#'
#' The function uses an optimization algorithm from the nlopt library
#' (https://nlopt.readthedocs.io/en/latest/NLopt_Algorithms/): the sequential
#' quadratic programming (SQP) algorithm for nonlinearly constrained
#' gradient-based optimization (supporting both inequality and equality
#' constraints), based on the implementation by Dieter Kraft (1988; 1944).
#' }
#'
#' @return List of values: \describe{ \item{$status: }{Integer value with the
#' status of the optimization (0 is success).} \item{$message: }{More
#' informative message with the status of the optimization} \item{$results:
#' }{Vector with the following values:} \itemize{ \item{exp.UI.Sp: }{The
#' population value of the specificity in the Uncertain Interval, given mu0,
#' sd0, mu1 and sd1. This value should be very near the supplied value of Sp.}
#' \item{exp.UI.Se: }{The population value of the sensitivity in the Uncertain
#' Interval, given mu0, sd0, mu1 and sd1. This value should be very near the
#' supplied value of UI.Se.} \item{mu0: }{The value that has been supplied for
#' mu0.} \item{sd0: }{The value that has been supplied for sd0.} \item{mu1:
#' }{The value that has been supplied for mu1.} \item{sd1: }{The value that
#' has been supplied for sd1.} } \item{$solution: }{Vector with the following
#' values:} \itemize{ \item{L: }{The population value of the lower threshold
#' of the Uncertain Interval.} \item{U: }{The population value of the upper
#' threshold of the Uncertain Interval.} } }
#' @references Dieter Kraft, "A software package for sequential quadratic
#' programming", Technical Report DFVLR-FB 88-28, Institut für Dynamik der
#' Flugsysteme, Oberpfaffenhofen, July 1988.
#'
#' Dieter Kraft, "Algorithm 733: TOMP–Fortran modules for optimal control
#' calculations," ACM Transactions on Mathematical Software, vol. 20, no. 3,
#' pp. 262-281 (1994).
#'
#' Landsheer, J. A. (2018). The Clinical Relevance of Methods for Handling
#' Inconclusive Medical Test Results: Quantification of Uncertainty in Medical
#' Decision-Making and Screening. Diagnostics, 8(2), 32.
#' https://doi.org/10.3390/diagnostics8020032
#' @export
#' @examples
#' test=c(rnorm(500,0,1), rnorm(500,1.6,1))
#' ref=c(rep(0,500), rep(1,500))
#' plotMD(ref, test, model='binormal')
#' ui.binormal(ref, test)
#' # test scores controls > patients works correctly from version 0.7 or higher
#' ui.binormal(ref, -test)
#' ref=c(rep(1,500), rep(0,500))
#' plotMD(ref, test, model='binormal')
#' ui.binormal(ref, test)
#'
ui.binormal <- function(ref, test, UI.Se = .55, UI.Sp = .55,
intersection = NULL, start=NULL, print.level=0){
df = check.data(ref, test, model='binormal')
n0 = df$test[ref==0]
n1 = df$test[ref==1]
m0 = mean(n0)
sd0 = sd(n0)
m1 = mean(n1)
sd1 = sd(n1)
# res0=fitdistr(n0,"normal") # MASS
# m0=res0$estimate[1]; sd0=res0$estimate[2]
# res1=fitdistr(n1,"normal") # fitdistrplus
# m1=res1$estimate[1]; sd1=res1$estimate[2]
# UI.Se=.55; UI.Sp=.55;
# mu0 = m0; sd0 = sd0;
# mu1 = m1; sd1 = sd1;
# intersection = NULL; start=NULL; print.level=0
res=nlopt.ui(UI.Se = UI.Se, UI.Sp = UI.Sp,
mu0 = m0, sd0 = sd0,
mu1 = m1, sd1 = sd1,
intersection = intersection,
start=start, print.level=print.level)
return(res)
}
```

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