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#' Confidence intervals for parameters of mkinfit objects
#'
#' The default method 'profile' is based on the profile likelihood for each
#' parameter. The method uses two nested optimisations. The speed of the method
#' could likely be improved by using the method of Venzon and Moolgavkar (1988).
#'
#' @param object An \code{\link{mkinfit}} object
#' @param parm A vector of names of the parameters which are to be given
#' confidence intervals. If missing, all parameters are considered.
#' @param level The confidence level required
#' @param alpha The allowed error probability, overrides 'level' if specified.
#' @param cutoff Possibility to specify an alternative cutoff for the difference
#' in the log-likelihoods at the confidence boundary. Specifying an explicit
#' cutoff value overrides arguments 'level' and 'alpha'
#' @param method The 'profile' method searches the parameter space for the
#' cutoff of the confidence intervals by means of a likelihood ratio test.
#' The 'quadratic' method approximates the likelihood function at the
#' optimised parameters using the second term of the Taylor expansion, using
#' a second derivative (hessian) contained in the object.
#' @param transformed If the quadratic approximation is used, should it be
#' applied to the likelihood based on the transformed parameters?
#' @param backtransform If we approximate the likelihood in terms of the
#' transformed parameters, should we backtransform the parameters with
#' their confidence intervals?
#' @param distribution For the quadratic approximation, should we use
#' the student t distribution or assume normal distribution for
#' the parameter estimate
#' @param quiet Should we suppress messages?
#' @return A matrix with columns giving lower and upper confidence limits for
#' each parameter.
#' @param \dots Not used
#' @importFrom stats qnorm
#' @references Pawitan Y (2013) In all likelihood - Statistical modelling and
#' inference using likelihood. Clarendon Press, Oxford.
#'
#' Venzon DJ and Moolgavkar SH (1988) A Method for Computing
#' Profile-Likelihood Based Confidence Intervals, Applied Statistics, 37,
#' 87–94.
#' @examples
#' f <- mkinfit("SFO", FOCUS_2006_C, quiet = TRUE)
#' confint(f, method = "quadratic")
#' \dontrun{
#' confint(f, method = "profile")
#' }
#' @export
confint.mkinfit <- function(object, parm,
level = 0.95, alpha = 1 - level, cutoff,
method = c("profile", "quadratic"),
transformed = TRUE, backtransform = TRUE,
distribution = c("student_t", "normal"), quiet = FALSE, ...)
{
tparms <- parms(object, transformed = TRUE)
bparms <- parms(object, transformed = FALSE)
tpnames <- names(tparms)
bpnames <- names(bparms)
return_pnames <- if (missing(parm)) {
if (backtransform) bpnames else tpnames
} else {
parm
}
p <- length(return_pnames)
method <- match.arg(method)
a <- c(alpha / 2, 1 - (alpha / 2))
if (method == "quadratic") {
distribution <- match.arg(distribution)
quantiles <- switch(distribution,
student_t = qt(a, object$df.residual),
normal = qnorm(a))
covar_pnames <- if (missing(parm)) {
if (transformed) tpnames else bpnames
} else {
parm
}
return_parms <- if (backtransform) bparms[return_pnames]
else tparms[return_pnames]
covar_parms <- if (transformed) tparms[covar_pnames]
else bparms[covar_pnames]
if (transformed) {
covar <- try(solve(object$hessian), silent = TRUE)
} else {
covar <- try(solve(object$hessian_notrans), silent = TRUE)
}
# If inverting the covariance matrix failed or produced NA values
if (!is.numeric(covar) | is.na(covar[1])) {
ses <- lci <- uci <- rep(NA, p)
} else {
ses <- sqrt(diag(covar))[covar_pnames]
lci <- covar_parms + quantiles[1] * ses
uci <- covar_parms + quantiles[2] * ses
if (backtransform) {
lci_back <- backtransform_odeparms(lci,
object$mkinmod, object$transform_rates, object$transform_fractions)
lci <- c(lci_back, lci[names(object$errparms)])
uci_back <- backtransform_odeparms(uci,
object$mkinmod, object$transform_rates, object$transform_fractions)
uci <- c(uci_back, uci[names(object$errparms)])
}
}
}
if (method == "profile") {
if (!quiet) message("Profiling the likelihood")
lci <- uci <- rep(NA, p)
names(lci) <- names(uci) <- return_pnames
profile_pnames <- if(missing(parm)) names(parms(object))
else parm
if (missing(cutoff)) {
cutoff <- 0.5 * qchisq(1 - alpha, 1)
}
all_parms <- parms(object)
for (pname in profile_pnames)
{
pnames_free <- setdiff(names(all_parms), pname)
profile_ll <- function(x)
{
pll_cost <- function(P) {
parms_cost <- all_parms
parms_cost[pnames_free] <- P[pnames_free]
parms_cost[pname] <- x
- object$ll(parms_cost)
}
- nlminb(all_parms[pnames_free], pll_cost)$objective
}
cost <- function(x) {
(cutoff - (object$logLik - profile_ll(x)))^2
}
lci[pname] <- optimize(cost, lower = 0, upper = all_parms[pname])$minimum
uci[pname] <- optimize(cost, lower = all_parms[pname], upper = 15 * all_parms[pname])$minimum
}
}
ci <- cbind(lower = lci, upper = uci)
colnames(ci) <- paste0(
format(100 * a, trim = TRUE, scientific = FALSE, digits = 3), "%")
return(ci)
}
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