#' Confidence intervals for parameters of mkinfit objects
#'
#' The default method 'quadratic' is based on the quadratic approximation of
#' the curvature of the likelihood function at the maximum likelihood parameter
#' estimates.
#' The alternative method 'profile' is based on the profile likelihood for each
#' parameter. The 'profile' method uses two nested optimisations and can take a
#' very long time, even if parallelized by specifying 'cores' on unixoid
#' platforms. 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 '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.
#' The 'profile' method searches the parameter space for the
#' cutoff of the confidence intervals by means of a likelihood ratio test.
#' @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 rel_tol If the method is 'profile', what should be the accuracy
#' of the lower and upper bounds, relative to the estimate obtained from
#' the quadratic method?
#' @param cores The number of cores to be used for multicore processing.
#' On Windows machines, cores > 1 is currently not supported.
#' @param quiet Should we suppress the message "Profiling the likelihood"
#' @return A matrix with columns giving lower and upper confidence limits for
#' each parameter.
#' @param \dots Not used
#' @importFrom stats qnorm
#' @references
#' Bates DM and Watts GW (1988) Nonlinear regression analysis & its applications
#'
#' 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")
#'
#' # Set the number of cores for the profiling method for further examples
#' if (identical(Sys.getenv("NOT_CRAN"), "true")) {
#' n_cores <- parallel::detectCores() - 1
#' } else {
#' n_cores <- 1
#' }
#' if (Sys.getenv("TRAVIS") != "") n_cores = 1
#' if (Sys.info()["sysname"] == "Windows") n_cores = 1
#'
#' SFO_SFO <- mkinmod(parent = mkinsub("SFO", "m1"), m1 = mkinsub("SFO"), quiet = TRUE)
#' SFO_SFO.ff <- mkinmod(parent = mkinsub("SFO", "m1"), m1 = mkinsub("SFO"),
#' use_of_ff = "max", quiet = TRUE)
#' f_d_1 <- mkinfit(SFO_SFO, subset(FOCUS_2006_D, value != 0), quiet = TRUE)
#' system.time(ci_profile <- confint(f_d_1, method = "profile", cores = 1, quiet = TRUE))
#' # Using more cores does not save much time here, as parent_0 takes up most of the time
#' # If we additionally exclude parent_0 (the confidence of which is often of
#' # minor interest), we get a nice performance improvement from about 50
#' # seconds to about 12 seconds if we use at least four cores
#' system.time(ci_profile_no_parent_0 <- confint(f_d_1, method = "profile",
#' c("k_parent_sink", "k_parent_m1", "k_m1_sink", "sigma"), cores = n_cores))
#' ci_profile
#' ci_quadratic_transformed <- confint(f_d_1, method = "quadratic")
#' ci_quadratic_transformed
#' ci_quadratic_untransformed <- confint(f_d_1, method = "quadratic", transformed = FALSE)
#' ci_quadratic_untransformed
#' # Against the expectation based on Bates and Watts (1988), the confidence
#' # intervals based on the internal parameter transformation are less
#' # congruent with the likelihood based intervals. Note the superiority of the
#' # interval based on the untransformed fit for k_m1_sink
#' rel_diffs_transformed <- abs((ci_quadratic_transformed - ci_profile)/ci_profile)
#' rel_diffs_untransformed <- abs((ci_quadratic_untransformed - ci_profile)/ci_profile)
#' rel_diffs_transformed < rel_diffs_untransformed
#' signif(rel_diffs_transformed, 3)
#' signif(rel_diffs_untransformed, 3)
#'
#'
#' # Investigate a case with formation fractions
#' f_d_2 <- mkinfit(SFO_SFO.ff, subset(FOCUS_2006_D, value != 0), quiet = TRUE)
#' ci_profile_ff <- confint(f_d_2, method = "profile", cores = n_cores)
#' ci_profile_ff
#' ci_quadratic_transformed_ff <- confint(f_d_2, method = "quadratic")
#' ci_quadratic_transformed_ff
#' ci_quadratic_untransformed_ff <- confint(f_d_2, method = "quadratic", transformed = FALSE)
#' ci_quadratic_untransformed_ff
#' rel_diffs_transformed_ff <- abs((ci_quadratic_transformed_ff - ci_profile_ff)/ci_profile_ff)
#' rel_diffs_untransformed_ff <- abs((ci_quadratic_untransformed_ff - ci_profile_ff)/ci_profile_ff)
#' # While the confidence interval for the parent rate constant is closer to
#' # the profile based interval when using the internal parameter
#' # transformation, the interval for the metabolite rate constant is 'better
#' # without internal parameter transformation.
#' rel_diffs_transformed_ff < rel_diffs_untransformed_ff
#' rel_diffs_transformed_ff
#' rel_diffs_untransformed_ff
#'
#' # The profiling for the following fit does not finish in a reasonable time,
#' # therefore we use the quadratic approximation
#' m_synth_DFOP_par <- mkinmod(parent = mkinsub("DFOP", c("M1", "M2")),
#' M1 = mkinsub("SFO"),
#' M2 = mkinsub("SFO"),
#' use_of_ff = "max", quiet = TRUE)
#' DFOP_par_c <- synthetic_data_for_UBA_2014[[12]]$data
#' f_tc_2 <- mkinfit(m_synth_DFOP_par, DFOP_par_c, error_model = "tc",
#' error_model_algorithm = "direct", quiet = TRUE)
#' confint(f_tc_2, method = "quadratic")
#' confint(f_tc_2, "parent_0", method = "quadratic")
#' }
#' @export
confint.mkinfit <- function(object, parm,
level = 0.95, alpha = 1 - level, cutoff,
method = c("quadratic", "profile"),
transformed = TRUE, backtransform = TRUE,
cores = parallel::detectCores(), rel_tol = 0.01, quiet = FALSE, ...)
{
tparms <- parms(object, transformed = TRUE)
bparms <- parms(object, transformed = FALSE)
tpnames <- names(tparms)
bpnames <- names(bparms)
return_pnames <- if (missing(parm)) {
if (backtransform) bpnames else tpnames
} else {
parm
}
p <- length(return_pnames)
method <- match.arg(method)
a <- c(alpha / 2, 1 - (alpha / 2))
quantiles <- qt(a, object$df.residual)
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 (transformed & backtransform) {
lci_back <- backtransform_odeparms(lci,
object$mkinmod, object$transform_rates, object$transform_fractions)
uci_back <- backtransform_odeparms(uci,
object$mkinmod, object$transform_rates, object$transform_fractions)
return_errparm_names <- intersect(names(object$errparms), return_pnames)
lci <- c(lci_back, lci[return_errparm_names])
uci <- c(uci_back, uci[return_errparm_names])
}
}
ci <- cbind(lower = lci, upper = uci)
if (method == "profile") {
ci_quadratic <- ci
if (!quiet) message("Profiling the likelihood")
lci <- uci <- rep(NA, p)
names(lci) <- names(uci) <- return_pnames
profile_pnames <- if(missing(parm)) names(parms(object))
else parm
if (missing(cutoff)) {
cutoff <- 0.5 * qchisq(1 - alpha, 1)
}
all_parms <- parms(object)
get_ci <- function(pname) {
pnames_free <- setdiff(names(all_parms), pname)
profile_ll <- function(x)
{
pll_cost <- function(P) {
parms_cost <- all_parms
parms_cost[pnames_free] <- P[pnames_free]
parms_cost[pname] <- x
- object$ll(parms_cost)
}
- nlminb(all_parms[pnames_free], pll_cost)$objective
}
cost <- function(x) {
(cutoff - (object$logLik - profile_ll(x)))^2
}
lower_quadratic <- ci_quadratic["lower"][pname]
upper_quadratic <- ci_quadratic["upper"][pname]
ltol <- if (!is.na(lower_quadratic)) rel_tol * lower_quadratic else .Machine$double.eps^0.25
utol <- if (!is.na(upper_quadratic)) rel_tol * upper_quadratic else .Machine$double.eps^0.25
lci_pname <- optimize(cost, lower = 0, upper = all_parms[pname], tol = ltol)$minimum
uci_pname <- optimize(cost, lower = all_parms[pname],
upper = ifelse(grepl("^f_|^g$", pname), 1, 15 * all_parms[pname]),
tol = utol)$minimum
return(c(lci_pname, uci_pname))
}
ci <- t(parallel::mcmapply(get_ci, profile_pnames, mc.cores = cores))
}
colnames(ci) <- paste0(
format(100 * a, trim = TRUE, scientific = FALSE, digits = 3), "%")
return(ci)
}
