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R/nlopt.ui.general.R
In UncertainInterval: Uncertain Interval Methods for Three-Way Cut-Point Determination in Test Results
Defines functions
nlopt.ui.general
Documented in
nlopt.ui.general
#' Function for the determination of the population thresholds an uncertain and
#' inconclusive interval for test scores with a known common distribution.
#'
#' @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 distribution Name of the continuous distribution, exact as used in R
#' package stats. Equal to density function minus d. For instance when the
#' density function is 'dnorm', then the distribution is 'norm'.
#' @param parameters.d0 Named vector of population values or estimates of the
#' parameters of the distribution of the test scores of the persons without
#' the targeted condition. For instance \code{c(mean = 0, sd = 1)}. This
#' distribution should have the lower values.
#' @param parameters.d1 Named vector of population values or estimates of the
#' parameters of the distribution of the test scores of the persons with the
#' targeted condition. For instance \code{c(mean = 1, sd = 1)}. The test
#' scores of d1 should have higher values than d0. If not, use -(test scores).
#' This distribution should have the higher values.
#' @param overlap.interval A vector with a raw estimate of the lower and upper
#' relevant of the overlap of the two distributions. If NULL, set to quantile
#' .001 of the distribution of persons with the targeted condition and
#' quantile .999 of the distribution of persons without the condition. Please
#' check whether this is a good estimate of the relevant overlap.
#' @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.
#' @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 The function can be used to determinate the uncertain interval of
#' the two continuous distributions. The Uncertain Interval is defined as an
#' interval below and above the intersection of the two distributions, with a
#' sensitivity and specificity below a desired value (default .55).
#'
#' Important: The test scores of d1 should have higher values than d0. If not,
#' use -(test scores). The distribution with parameters d1 should have the
#' higher test scores.
#'
#' Important: This is a highly complex function, which is less user friendly
#' and more error prone than other functions. It is included to show that the
#' technique also works with continuous distributions other than bi-normal.
#' Use for bi-normal distributions always ui.binormal.
#'
#' Only a single intersection is assumed (or a second intersection where the
#' overlap is negligible).
#'
#' The function uses an optimization algorithm from the nlopt library
#'
#' (https://nlopt.readthedocs.io/en/latest/NLopt_Algorithms/).
#'
#' It uses the sequential quadratic programming (SQP) algorithm for
#' nonlinear constrained gradient-based optimization (supporting both
#' inequality and equality constraints), based on the implementation by Dieter
#' Kraft (1988; 1944).
#'
#' N.B. When a normal distribution is expected, the functions
#' \code{\link{nlopt.ui}} and \code{\link{ui.binormal}} are recommended.
#'
#' @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{$results:
#' }{Vector with the following values:} \itemize{ \item{exp.UI.Sp: }{The
#' population value of the specificity in the Uncertain Interval, given mu0,
#' sd0, mu1 and sd1. This value should be very near the supplied value of Sp.}
#' \item{exp.UI.Se: }{The population value of the sensitivity in the Uncertain
#' Interval, given mu0, sd0, mu1 and sd1. This value should be very near the
#' supplied value of UI.Se.} \item{vector of parameter values of distribution
#' d0, that is, the values that have been supplied in \code{parameters.d0}.}
#' \item{vector of parameter values of distribution d1, that is, the values
#' that have been supplied in \code{parameters.d1}.}} \item{$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
#' @importFrom nloptr nloptr
#' @importFrom MASS fitdistr
#' @importFrom car qqPlot
#' @importFrom stats uniroot
#'
#'
#' @examples
#' # A simple test model:
#' nlopt.ui.general(UI.Se = .55, UI.Sp = .55,
#' distribution = "norm",
#' parameters.d0 = c(mean = 0, sd = 1),
#' parameters.d1 = c(mean = 1, sd = 1),
#' overlap.interval=c(-2,3))
#' # Standard procedure when using a continuous distribution:
#' nlopt.ui.general(parameters.d0 = c(mean = 0, sd = 1),
#' parameters.d1 = c(mean = 1.6, sd = 2))
#'
#' # library(MASS)
#' # library(car)
#' # gamma distributed data
#' set.seed(4)
#' d0 = rgamma(100, shape=2, rate=.5)
#' d1 = rgamma(100, shape=7.5, rate=1)
#' # 1. obtain parameters
#' parameters.d0=MASS::fitdistr(d0, 'gamma')$estimate
#' parameters.d1=MASS::fitdistr(d1, 'gamma')$estimate
#' # 2. test if supposed distributions (gamma) is fitting
#' car::qqPlot(d0, distribution='gamma', shape=parameters.d0['shape'])
#' car::qqPlot(d1, distribution='gamma', shape=parameters.d1['shape'])
#' # 3. draw curves and determine overlap
#' curve(dgamma(x, shape=parameters.d0['shape'], rate=parameters.d0['rate']), from=0, to=16)
#' curve(dgamma(x, shape=parameters.d1['shape'], rate=parameters.d1['rate']), from=0, to=16, add=TRUE)
#' overlap.interval=c(1, 15) # ignore intersection at 0; observe large overlap
#' # 4. get empirical AUC
#' simple_auc(d0, d1)
#' # about .65 --> Poor
#' # .90-1 = excellent (A)
#' # .80-.90 = good (B)
#' # .70-.80 = fair (C)
#' # .60-.70 = poor (D)
#' # .50-.60 = fail (F)
#' # 5. Get uncertain interval
#' (res=nlopt.ui.general (UI.Se = .57,
#' UI.Sp = .57,
#' distribution = 'gamma',
#' parameters.d0 = parameters.d0,
#' parameters.d1 = parameters.d1,
#' overlap.interval,
#' intersection = NULL,
#' start = NULL,
#' print.level = 0))
#' abline(v=c(res$intersection, res$solution))
#' # 6. Assess improvement when diagnosing outside the uncertain interval
#' sel.d0 = d0 < res$solution[1] | d0 > res$solution[2]
#' sel.d1 = d1 < res$solution[1] | d1 > res$solution[2]
#' (percentage.selected.d0 = sum(sel.d0) / length(d0))
#' (percentage.selected.d1 = sum(sel.d1) / length(d1))
#' simple_auc(d0[sel.d0], d1[sel.d1])
#' # AUC for selected scores outside the uncertain interval
#' simple_auc(d0[!sel.d0], d1[!sel.d1])
#' # AUC for deselected scores; worst are deselected
#' # weibull distributed data
#' set.seed(4)
#' d0 = rweibull(100, shape=3, scale=50)
#' d1 = rweibull(100, shape=3, scale=70)
#' # 1. obtain parameters
#' parameters.d0=MASS::fitdistr(d0, 'weibull')$estimate
#' parameters.d1=MASS::fitdistr(d1, 'weibull')$estimate
#' # 2. test if supposed distributions (gamma) is fitting
#' car::qqPlot(d0, distribution='weibull', shape=parameters.d0['shape'])
#' car::qqPlot(d1, distribution='weibull', shape=parameters.d1['shape'])
#' # 3. draw curves and determine overlap
#' curve(dweibull(x, shape=parameters.d0['shape'],
#' scale=parameters.d0['scale']), from=0, to=150)
#' curve(dweibull(x, shape=parameters.d1['shape'],
#' scale=parameters.d1['scale']), from=0, to=150, add=TRUE)
#' overlap.interval=c(1, 100) # ignore intersection at 0; observe overlap
#' # 4. get empirical AUC
#' simple_auc(d0, d1)
#' # about .65 --> Poor
#' # .90-1 = excellent (A)
#' # .80-.90 = good (B)
#' # .70-.80 = fair (C)
#' # .60-.70 = poor (D)
#' # .50-.60 = fail (F)
#' # 5. Get uncertain interval
#' (res=nlopt.ui.general (UI.Se = .55,
#' UI.Sp = .55,
#' distribution = 'weibull',
#' parameters.d0 = parameters.d0,
#' parameters.d1 = parameters.d1,
#' overlap.interval,
#' intersection = NULL,
#' start = NULL,
#' print.level = 0))
#' abline(v=c(res$intersection, res$solution))
#' # 6. Assess improvement when diagnosing outside the uncertain interval
#' sel.d0 = d0 < res$solution[1] | d0 > res$solution[2]
#' sel.d1 = d1 < res$solution[1] | d1 > res$solution[2]
#' (percentage.selected.d0 = sum(sel.d0) / length(d0))
#' (percentage.selected.d1 = sum(sel.d1) / length(d1))
#' simple_auc(d0[sel.d0], d1[sel.d1])
#' # AUC for selected scores outside the uncertain interval
#' simple_auc(d0[!sel.d0], d1[!sel.d1])
#' # AUC for deselected scores; these scores are almost indistinguishable
# UI.Se = .55; UI.Sp = .55; distribution = 'norm'; parameters.d0 = c(mean = 0, sd = 1);
# parameters.d1 = c(mean = 1, sd = 1); overlap.interval = NULL; intersection = NULL;
# start = NULL; print.level = 0
nlopt.ui.general <- function(UI.Se = .55,
UI.Sp = .55,
distribution = 'norm',
parameters.d0 = c(mean = 0, sd = 1),
parameters.d1 = c(mean = 1, sd = 1),
overlap.interval = NULL,
intersection = NULL,
start = NULL,
print.level = 0) {
uniroot.all.copy <- function (f, interval, lower = min(interval), upper = max(interval),
tol = .Machine$double.eps^0.2, maxiter = 1000, n = 100, ...)
{
if (!missing(interval) && length(interval) != 2)
stop("'interval' must be a vector of length 2")
if (!is.numeric(lower) || !is.numeric(upper) || lower >=
upper)
stop("lower < upper is not fulfilled")
xseq <- seq(lower, upper, len = n + 1)
mod <- f(xseq, ...)
Equi <- xseq[which(mod == 0)]
ss <- mod[1:n] * mod[2:(n + 1)]
ii <- which(ss < 0)
for (i in ii) Equi <- c(Equi, uniroot(f, lower = xseq[i],
upper = xseq[i + 1], ...)$root)
return(Equi)
}
if (UI.Se <= .5) stop('Value <= .5 invalid for UI.Se')
if (UI.Sp <= .5) stop('Value <= .5 invalid for UI.Sp')
# if (UI.Se > .6) warning('Value > .6 not recommended for UI.Se')
# if (UI.Sp > .6) warning('Value > .6 not recommended for UI.Sp')
c01 = UI.Sp / (1 - UI.Sp)
c11 = UI.Se / (1 - UI.Se)
d0 = get(paste('d', distribution, sep = ""))
args=formals(d0)
m = match(names(parameters.d0), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d0' specifies names which are not arguments to", deparse(d0)[1]
))
args[m] = parameters.d0
formals(d0) <- args # d0(2) ; dnorm(2, 0, 1)
d1 = get(paste('d', distribution, sep = ""))
m = match(names(parameters.d1), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d1' specifies names which are not arguments to", deparse(d1)[1]
))
args[m] = parameters.d1
formals(d1) <- args # d1(2) ; dnorm(2, 1, 1)
p0 = get(paste('p', distribution, sep = ""))
args=formals(p0)
m = match(names(parameters.d0), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d0' specifies names which are not arguments to", deparse(p0)[1]
))
args[m] = parameters.d0
formals(p0) <- args # p0(2) ; pnorm(2, 0, 1)
p1 = get(paste('p', distribution, sep = ""))
m = match(names(parameters.d1), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d1' specifies names which are not arguments to", deparse(p1)[1]
))
args[m] = parameters.d1
formals(p1) <- args # p1(2) ; pnorm(2, 1, 1)
q0 = get(paste('q', distribution, sep = ""))
args=formals(q0)
m = match(names(parameters.d0), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d0' specifies names which are not arguments to", deparse(q0)[1]
))
args[m] = parameters.d0
formals(q0) <- args # p1(2) ; qnorm(.001, 0, 1)
q1 = get(paste('q', distribution, sep = ""))
m = match(names(parameters.d1), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d1' specifies names which are not arguments to", deparse(q1)[1]
))
args[m] = parameters.d1
formals(q1) <- args # p1(2) ; qnorm(.001, 1, 1); q1(q=.001)
if (print.level > 0) {
cat('function call d0: ', deparse(d0), '\n' )
cat('function call d1: ', deparse(d1), '\n' )
cat('function call p0: ', deparse(p0), '\n' )
cat('function call p1: ', deparse(p1), '\n' )
cat('function call q0: ', deparse(q0), '\n' )
cat('function call q1: ', deparse(q1), '\n' )
}
if (is.null(overlap.interval)) overlap.interval = c(q1(.001), q0(.999))
if (is.null(intersection)) {
f <- function(x) d0(x)-d1(x)
# library(rootSolve) # overlap.interval=c(-2, 2)
I = uniroot.all.copy(f, interval = overlap.interval)
if (length(I) > 1) {
d = d0(I)
I = I[which.max(d)]
warning('More than one point of intersection. Point with highest density used.')
}
} else
I = intersection # abline(v=I)
# objective function -(H - L)^2
eval_f0 <- function(x) {
return(-(x[2] - x[1]) ^ 2)
# return(-(x[2] - x[1]) )
}
eval_grad_f0 <- function(x) {
return(c(2 * (x[2] - x[1]), -2 * (x[2] - x[1])))
# return(c(1,-1))
}
# constraint function
eval_g0 <- function(x) {
a0 = p0(I) - p0(x[1]) # TN within the uncertain interval
b0 = p0(x[2]) - p0(I) # FP
a1 = p1(x[2]) - p1(I) # TP
b1 = p1(I) - p1(x[1]) # FN
return(c(a0 - c01 * b0,
a1 - c11 * b1)) # vector with two constraint values
}
# jacobian of constraints
eval_jac_g0 <- function(x) {
return(rbind(c(
-d0(x[1]) ,-c01 * d0(x[2])
),
c(
c11 * d1(x[1]), d1(x[2])
)))
}
if (is.null(start)){
par0 = c((overlap.interval[1]+I)/2, (overlap.interval[2]+I)/2)
} else {par0 = c(start[1], start[2])}
# eval_g0(par0)
# eval_jac_g0(par0)
# res0 <- nloptr(
# x0 = par0,
# eval_f = eval_f0,
# eval_g_ineq = eval_g0,
# lb = c(overlap.interval[1], I),
# ub = c(I, overlap.interval[2]),
# opts = list(
# "algorithm" = "NLOPT_LN_COBYLA" , # "NLOPT_LN_COBYLA" "NLOPT_LD_SLSQP"
# "xtol_rel" = 1.0e-8,
# "print_level" = print.level #,
# # "check_derivatives" = TRUE,
# # "check_derivatives_print" = "all"
# )
# )
# par0=c(L=I-.5*sd1,H=I+.5*sd0)
# lb = c(I-2*sd1, I)
# ub = c(I, I+2*sd0)
res0 <- nloptr(
x0 = par0,
lb = c(overlap.interval[1], I),
ub = c(I, overlap.interval[2]),
eval_f = eval_f0,
eval_g_ineq = eval_g0,
eval_grad_f = eval_grad_f0,
eval_jac_g_ineq = eval_jac_g0,
opts = list(
"algorithm" = "NLOPT_LD_SLSQP" , # "NLOPT_LN_COBYLA" "NLOPT_LD_MMA" "NLOPT_LD_SLSQP"
"xtol_rel" = 1.0e-8,
"print_level" = print.level #,
# "check_derivatives" = TRUE,
# "check_derivatives_print" = "all"
)
)
TN = p0(I) - p0(res0$solution[1]) # area check UI.Sp: lower area / upper area
FP = p0(res0$solution[2]) - p0(I)
TP = p1(res0$solution[2]) - p1(I) # area check UI.Se: upper area / lower area
FN = p1(I) - p1(res0$solution[1])
res = list()
res$status = res0$status
res$message = res0$message
res$intersection = I
res$results = c(
exp.UI.Sp = ifelse((TN > 1e-4), TN / (FP + TN), NA),
exp.UI.Se = ifelse(TP > 1e-4, TP / (FN + TP), NA),
parameters.d0,
parameters.d1
)
res$solution = c(L = res0$solution[1], U = res0$solution[2])
return(res)
}
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Defines functions nlopt.ui.general
Documented in nlopt.ui.general
#' Function for the determination of the population thresholds an uncertain and
#' inconclusive interval for test scores with a known common distribution.
#'
#' @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 distribution Name of the continuous distribution, exact as used in R
#' package stats. Equal to density function minus d. For instance when the
#' density function is 'dnorm', then the distribution is 'norm'.
#' @param parameters.d0 Named vector of population values or estimates of the
#' parameters of the distribution of the test scores of the persons without
#' the targeted condition. For instance \code{c(mean = 0, sd = 1)}. This
#' distribution should have the lower values.
#' @param parameters.d1 Named vector of population values or estimates of the
#' parameters of the distribution of the test scores of the persons with the
#' targeted condition. For instance \code{c(mean = 1, sd = 1)}. The test
#' scores of d1 should have higher values than d0. If not, use -(test scores).
#' This distribution should have the higher values.
#' @param overlap.interval A vector with a raw estimate of the lower and upper
#' relevant of the overlap of the two distributions. If NULL, set to quantile
#' .001 of the distribution of persons with the targeted condition and
#' quantile .999 of the distribution of persons without the condition. Please
#' check whether this is a good estimate of the relevant overlap.
#' @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.
#' @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 The function can be used to determinate the uncertain interval of
#' the two continuous distributions. The Uncertain Interval is defined as an
#' interval below and above the intersection of the two distributions, with a
#' sensitivity and specificity below a desired value (default .55).
#'
#' Important: The test scores of d1 should have higher values than d0. If not,
#' use -(test scores). The distribution with parameters d1 should have the
#' higher test scores.
#'
#' Important: This is a highly complex function, which is less user friendly
#' and more error prone than other functions. It is included to show that the
#' technique also works with continuous distributions other than bi-normal.
#' Use for bi-normal distributions always ui.binormal.
#'
#' Only a single intersection is assumed (or a second intersection where the
#' overlap is negligible).
#'
#' The function uses an optimization algorithm from the nlopt library
#'
#' (https://nlopt.readthedocs.io/en/latest/NLopt_Algorithms/).
#'
#' It uses the sequential quadratic programming (SQP) algorithm for
#' nonlinear constrained gradient-based optimization (supporting both
#' inequality and equality constraints), based on the implementation by Dieter
#' Kraft (1988; 1944).
#'
#' N.B. When a normal distribution is expected, the functions
#' \code{\link{nlopt.ui}} and \code{\link{ui.binormal}} are recommended.
#'
#' @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{vector of parameter values of distribution
#' d0, that is, the values that have been supplied in \code{parameters.d0}.}
#' \item{vector of parameter values of distribution d1, that is, the values
#' that have been supplied in \code{parameters.d1}.}} \item{$solution:
#' }{Vector with the following values:} \itemize{ \item{L: }{The population
#' value of the lower threshold of the uncertain interval.} \item{U: }{The
#' population value of the upper threshold of the uncertain interval.} } }
#' @references Dieter Kraft, "A software package for sequential quadratic
#' programming", Technical Report DFVLR-FB 88-28, Institut für Dynamik der
#' Flugsysteme, Oberpfaffenhofen, July 1988.
#'
#' Dieter Kraft, "Algorithm 733: TOMP–Fortran modules for optimal control
#' calculations," ACM Transactions on Mathematical Software, vol. 20, no. 3,
#' pp. 262-281 (1994).
#'
#' Landsheer, J. A. (2018). The Clinical Relevance of Methods for Handling
#' Inconclusive Medical Test Results: Quantification of Uncertainty in Medical
#' Decision-Making and Screening. Diagnostics, 8(2), 32.
#' https://doi.org/10.3390/diagnostics8020032
#' @export
#' @importFrom nloptr nloptr
#' @importFrom MASS fitdistr
#' @importFrom car qqPlot
#' @importFrom stats uniroot
#'
#'
#' @examples
#' # A simple test model:
#' nlopt.ui.general(UI.Se = .55, UI.Sp = .55,
#' distribution = "norm",
#' parameters.d0 = c(mean = 0, sd = 1),
#' parameters.d1 = c(mean = 1, sd = 1),
#' overlap.interval=c(-2,3))
#' # Standard procedure when using a continuous distribution:
#' nlopt.ui.general(parameters.d0 = c(mean = 0, sd = 1),
#' parameters.d1 = c(mean = 1.6, sd = 2))
#'
#' # library(MASS)
#' # library(car)
#' # gamma distributed data
#' set.seed(4)
#' d0 = rgamma(100, shape=2, rate=.5)
#' d1 = rgamma(100, shape=7.5, rate=1)
#' # 1. obtain parameters
#' parameters.d0=MASS::fitdistr(d0, 'gamma')$estimate
#' parameters.d1=MASS::fitdistr(d1, 'gamma')$estimate
#' # 2. test if supposed distributions (gamma) is fitting
#' car::qqPlot(d0, distribution='gamma', shape=parameters.d0['shape'])
#' car::qqPlot(d1, distribution='gamma', shape=parameters.d1['shape'])
#' # 3. draw curves and determine overlap
#' curve(dgamma(x, shape=parameters.d0['shape'], rate=parameters.d0['rate']), from=0, to=16)
#' curve(dgamma(x, shape=parameters.d1['shape'], rate=parameters.d1['rate']), from=0, to=16, add=TRUE)
#' overlap.interval=c(1, 15) # ignore intersection at 0; observe large overlap
#' # 4. get empirical AUC
#' simple_auc(d0, d1)
#' # about .65 --> Poor
#' # .90-1 = excellent (A)
#' # .80-.90 = good (B)
#' # .70-.80 = fair (C)
#' # .60-.70 = poor (D)
#' # .50-.60 = fail (F)
#' # 5. Get uncertain interval
#' (res=nlopt.ui.general (UI.Se = .57,
#' UI.Sp = .57,
#' distribution = 'gamma',
#' parameters.d0 = parameters.d0,
#' parameters.d1 = parameters.d1,
#' overlap.interval,
#' intersection = NULL,
#' start = NULL,
#' print.level = 0))
#' abline(v=c(res$intersection, res$solution))
#' # 6. Assess improvement when diagnosing outside the uncertain interval
#' sel.d0 = d0 < res$solution[1] | d0 > res$solution[2]
#' sel.d1 = d1 < res$solution[1] | d1 > res$solution[2]
#' (percentage.selected.d0 = sum(sel.d0) / length(d0))
#' (percentage.selected.d1 = sum(sel.d1) / length(d1))
#' simple_auc(d0[sel.d0], d1[sel.d1])
#' # AUC for selected scores outside the uncertain interval
#' simple_auc(d0[!sel.d0], d1[!sel.d1])
#' # AUC for deselected scores; worst are deselected
#' # weibull distributed data
#' set.seed(4)
#' d0 = rweibull(100, shape=3, scale=50)
#' d1 = rweibull(100, shape=3, scale=70)
#' # 1. obtain parameters
#' parameters.d0=MASS::fitdistr(d0, 'weibull')$estimate
#' parameters.d1=MASS::fitdistr(d1, 'weibull')$estimate
#' # 2. test if supposed distributions (gamma) is fitting
#' car::qqPlot(d0, distribution='weibull', shape=parameters.d0['shape'])
#' car::qqPlot(d1, distribution='weibull', shape=parameters.d1['shape'])
#' # 3. draw curves and determine overlap
#' curve(dweibull(x, shape=parameters.d0['shape'],
#' scale=parameters.d0['scale']), from=0, to=150)
#' curve(dweibull(x, shape=parameters.d1['shape'],
#' scale=parameters.d1['scale']), from=0, to=150, add=TRUE)
#' overlap.interval=c(1, 100) # ignore intersection at 0; observe overlap
#' # 4. get empirical AUC
#' simple_auc(d0, d1)
#' # about .65 --> Poor
#' # .90-1 = excellent (A)
#' # .80-.90 = good (B)
#' # .70-.80 = fair (C)
#' # .60-.70 = poor (D)
#' # .50-.60 = fail (F)
#' # 5. Get uncertain interval
#' (res=nlopt.ui.general (UI.Se = .55,
#' UI.Sp = .55,
#' distribution = 'weibull',
#' parameters.d0 = parameters.d0,
#' parameters.d1 = parameters.d1,
#' overlap.interval,
#' intersection = NULL,
#' start = NULL,
#' print.level = 0))
#' abline(v=c(res$intersection, res$solution))
#' # 6. Assess improvement when diagnosing outside the uncertain interval
#' sel.d0 = d0 < res$solution[1] | d0 > res$solution[2]
#' sel.d1 = d1 < res$solution[1] | d1 > res$solution[2]
#' (percentage.selected.d0 = sum(sel.d0) / length(d0))
#' (percentage.selected.d1 = sum(sel.d1) / length(d1))
#' simple_auc(d0[sel.d0], d1[sel.d1])
#' # AUC for selected scores outside the uncertain interval
#' simple_auc(d0[!sel.d0], d1[!sel.d1])
#' # AUC for deselected scores; these scores are almost indistinguishable
# UI.Se = .55; UI.Sp = .55; distribution = 'norm'; parameters.d0 = c(mean = 0, sd = 1);
# parameters.d1 = c(mean = 1, sd = 1); overlap.interval = NULL; intersection = NULL;
# start = NULL; print.level = 0
nlopt.ui.general <- function(UI.Se = .55,
UI.Sp = .55,
distribution = 'norm',
parameters.d0 = c(mean = 0, sd = 1),
parameters.d1 = c(mean = 1, sd = 1),
overlap.interval = NULL,
intersection = NULL,
start = NULL,
print.level = 0) {
uniroot.all.copy <- function (f, interval, lower = min(interval), upper = max(interval),
tol = .Machine$double.eps^0.2, maxiter = 1000, n = 100, ...)
{
if (!missing(interval) && length(interval) != 2)
stop("'interval' must be a vector of length 2")
if (!is.numeric(lower) || !is.numeric(upper) || lower >=
upper)
stop("lower < upper is not fulfilled")
xseq <- seq(lower, upper, len = n + 1)
mod <- f(xseq, ...)
Equi <- xseq[which(mod == 0)]
ss <- mod[1:n] * mod[2:(n + 1)]
ii <- which(ss < 0)
for (i in ii) Equi <- c(Equi, uniroot(f, lower = xseq[i],
upper = xseq[i + 1], ...)$root)
return(Equi)
}
if (UI.Se <= .5) stop('Value <= .5 invalid for UI.Se')
if (UI.Sp <= .5) stop('Value <= .5 invalid for UI.Sp')
# if (UI.Se > .6) warning('Value > .6 not recommended for UI.Se')
# if (UI.Sp > .6) warning('Value > .6 not recommended for UI.Sp')
c01 = UI.Sp / (1 - UI.Sp)
c11 = UI.Se / (1 - UI.Se)
d0 = get(paste('d', distribution, sep = ""))
args=formals(d0)
m = match(names(parameters.d0), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d0' specifies names which are not arguments to", deparse(d0)[1]
))
args[m] = parameters.d0
formals(d0) <- args # d0(2) ; dnorm(2, 0, 1)
d1 = get(paste('d', distribution, sep = ""))
m = match(names(parameters.d1), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d1' specifies names which are not arguments to", deparse(d1)[1]
))
args[m] = parameters.d1
formals(d1) <- args # d1(2) ; dnorm(2, 1, 1)
p0 = get(paste('p', distribution, sep = ""))
args=formals(p0)
m = match(names(parameters.d0), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d0' specifies names which are not arguments to", deparse(p0)[1]
))
args[m] = parameters.d0
formals(p0) <- args # p0(2) ; pnorm(2, 0, 1)
p1 = get(paste('p', distribution, sep = ""))
m = match(names(parameters.d1), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d1' specifies names which are not arguments to", deparse(p1)[1]
))
args[m] = parameters.d1
formals(p1) <- args # p1(2) ; pnorm(2, 1, 1)
q0 = get(paste('q', distribution, sep = ""))
args=formals(q0)
m = match(names(parameters.d0), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d0' specifies names which are not arguments to", deparse(q0)[1]
))
args[m] = parameters.d0
formals(q0) <- args # p1(2) ; qnorm(.001, 0, 1)
q1 = get(paste('q', distribution, sep = ""))
m = match(names(parameters.d1), names(args))
if (any(is.na(m)))
stop(paste(
"'parameters.d1' specifies names which are not arguments to", deparse(q1)[1]
))
args[m] = parameters.d1
formals(q1) <- args # p1(2) ; qnorm(.001, 1, 1); q1(q=.001)
if (print.level > 0) {
cat('function call d0: ', deparse(d0), '\n' )
cat('function call d1: ', deparse(d1), '\n' )
cat('function call p0: ', deparse(p0), '\n' )
cat('function call p1: ', deparse(p1), '\n' )
cat('function call q0: ', deparse(q0), '\n' )
cat('function call q1: ', deparse(q1), '\n' )
}
if (is.null(overlap.interval)) overlap.interval = c(q1(.001), q0(.999))
if (is.null(intersection)) {
f <- function(x) d0(x)-d1(x)
# library(rootSolve) # overlap.interval=c(-2, 2)
I = uniroot.all.copy(f, interval = overlap.interval)
if (length(I) > 1) {
d = d0(I)
I = I[which.max(d)]
warning('More than one point of intersection. Point with highest density used.')
}
} else
I = intersection # abline(v=I)
# objective function -(H - L)^2
eval_f0 <- function(x) {
return(-(x[2] - x[1]) ^ 2)
# return(-(x[2] - x[1]) )
}
eval_grad_f0 <- function(x) {
return(c(2 * (x[2] - x[1]), -2 * (x[2] - x[1])))
# return(c(1,-1))
}
# constraint function
eval_g0 <- function(x) {
a0 = p0(I) - p0(x[1]) # TN within the uncertain interval
b0 = p0(x[2]) - p0(I) # FP
a1 = p1(x[2]) - p1(I) # TP
b1 = p1(I) - p1(x[1]) # FN
return(c(a0 - c01 * b0,
a1 - c11 * b1)) # vector with two constraint values
}
# jacobian of constraints
eval_jac_g0 <- function(x) {
return(rbind(c(
-d0(x[1]) ,-c01 * d0(x[2])
),
c(
c11 * d1(x[1]), d1(x[2])
)))
}
if (is.null(start)){
par0 = c((overlap.interval[1]+I)/2, (overlap.interval[2]+I)/2)
} else {par0 = c(start[1], start[2])}
# eval_g0(par0)
# eval_jac_g0(par0)
# res0 <- nloptr(
# x0 = par0,
# eval_f = eval_f0,
# eval_g_ineq = eval_g0,
# lb = c(overlap.interval[1], I),
# ub = c(I, overlap.interval[2]),
# opts = list(
# "algorithm" = "NLOPT_LN_COBYLA" , # "NLOPT_LN_COBYLA" "NLOPT_LD_SLSQP"
# "xtol_rel" = 1.0e-8,
# "print_level" = print.level #,
# # "check_derivatives" = TRUE,
# # "check_derivatives_print" = "all"
# )
# )
# par0=c(L=I-.5*sd1,H=I+.5*sd0)
# lb = c(I-2*sd1, I)
# ub = c(I, I+2*sd0)
res0 <- nloptr(
x0 = par0,
lb = c(overlap.interval[1], I),
ub = c(I, overlap.interval[2]),
eval_f = eval_f0,
eval_g_ineq = eval_g0,
eval_grad_f = eval_grad_f0,
eval_jac_g_ineq = eval_jac_g0,
opts = list(
"algorithm" = "NLOPT_LD_SLSQP" , # "NLOPT_LN_COBYLA" "NLOPT_LD_MMA" "NLOPT_LD_SLSQP"
"xtol_rel" = 1.0e-8,
"print_level" = print.level #,
# "check_derivatives" = TRUE,
# "check_derivatives_print" = "all"
)
)
TN = p0(I) - p0(res0$solution[1]) # area check UI.Sp: lower area / upper area
FP = p0(res0$solution[2]) - p0(I)
TP = p1(res0$solution[2]) - p1(I) # area check UI.Se: upper area / lower area
FN = p1(I) - p1(res0$solution[1])
res = list()
res$status = res0$status
res$message = res0$message
res$intersection = I
res$results = c(
exp.UI.Sp = ifelse((TN > 1e-4), TN / (FP + TN), NA),
exp.UI.Se = ifelse(TP > 1e-4, TP / (FN + TP), NA),
parameters.d0,
parameters.d1
)
res$solution = c(L = res0$solution[1], U = res0$solution[2])
return(res)
}
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