mkin (version 1.2.6)

confint.mkinfit: Confidence intervals for parameters of mkinfit objects

Description

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).

Usage

# S3 method for mkinfit
confint(
  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,
  ...
)

Value

A matrix with columns giving lower and upper confidence limits for each parameter.

Arguments

object

An mkinfit object

parm

A vector of names of the parameters which are to be given confidence intervals. If missing, all parameters are considered.

level

The confidence level required

alpha

The allowed error probability, overrides 'level' if specified.

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'

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.

transformed

If the quadratic approximation is used, should it be applied to the likelihood based on the transformed parameters?

backtransform

If we approximate the likelihood in terms of the transformed parameters, should we backtransform the parameters with their confidence intervals?

cores

The number of cores to be used for multicore processing. On Windows machines, cores > 1 is currently not supported.

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?

quiet

Should we suppress the message "Profiling the likelihood"

...

Not used

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")

if (FALSE) {
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"),
  use_of_ff = "min", 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 if we use at least 4 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")
}