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).

+ +

# 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,
+  ...
+)

+ +

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

+ +

Value

+ +

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

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")
#>                     2.5%      97.5%
+#> parent_0      71.8242430 93.1600766
+#> k_parent_sink  0.2109541  0.4440528
+#> sigma          1.9778868  7.3681380

+# \dontrun{
+confint(f, method = "profile")
#> Profiling the likelihood
#>                     2.5%      97.5%
+#> parent_0      73.0641834 92.1392181
+#> k_parent_sink  0.2170293  0.4235348
+#> sigma          3.1307772  8.0628314

+# 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))
#>    user  system elapsed 
+#>   3.707   1.077   3.444 
# 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))
#> Profiling the likelihood
#> Warning: scheduled cores 2, 1, 3 encountered errors in user code, all values of the jobs will be affected
#> Error in dimnames(x) <- dn: length of 'dimnames' [2] not equal to array extent
#> Timing stopped at: 0.011 0.026 0.207
ci_profile
#>                        2.5%        97.5%
+#> parent_0       96.456003640 1.027703e+02
+#> k_parent        0.090911032 1.071578e-01
+#> k_m1            0.003892605 6.702778e-03
+#> f_parent_to_m1  0.471328495 5.611550e-01
+#> sigma           2.535612399 3.985263e+00
ci_quadratic_transformed <- confint(f_d_1, method = "quadratic")
+ci_quadratic_transformed
#>                        2.5%        97.5%
+#> parent_0       96.403839476 1.027931e+02
+#> k_parent        0.090823790 1.072543e-01
+#> k_m1            0.004012216 6.897547e-03
+#> f_parent_to_m1  0.469118713 5.595960e-01
+#> sigma           2.396089689 3.854918e+00
ci_quadratic_untransformed <- confint(f_d_1, method = "quadratic", transformed = FALSE)
+ci_quadratic_untransformed
#>                        2.5%        97.5%
+#> parent_0       96.403839429 1.027931e+02
+#> k_parent        0.090491931 1.069035e-01
+#> k_m1            0.003835483 6.685819e-03
+#> f_parent_to_m1  0.469113364 5.598386e-01
+#> sigma           2.396089689 3.854918e+00
# 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
#>                 2.5% 97.5%
+#> parent_0        TRUE  TRUE
+#> k_parent        TRUE  TRUE
+#> k_m1           FALSE FALSE
+#> f_parent_to_m1  TRUE FALSE
+#> sigma          FALSE  TRUE
signif(rel_diffs_transformed, 3)
#>                    2.5%    97.5%
+#> parent_0       0.000541 0.000222
+#> k_parent       0.000960 0.000900
+#> k_m1           0.030700 0.029100
+#> f_parent_to_m1 0.004690 0.002780
+#> sigma          0.055000 0.032700
signif(rel_diffs_untransformed, 3)
#>                    2.5%    97.5%
+#> parent_0       0.000541 0.000222
+#> k_parent       0.004610 0.002370
+#> k_m1           0.014700 0.002530
+#> f_parent_to_m1 0.004700 0.002350
+#> sigma          0.055000 0.032700

+
+# 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)
#> Profiling the likelihood
ci_profile_ff
#>                        2.5%        97.5%
+#> parent_0       96.456003640 1.027703e+02
+#> k_parent        0.090911032 1.071578e-01
+#> k_m1            0.003892605 6.702778e-03
+#> f_parent_to_m1  0.471328495 5.611550e-01
+#> sigma           2.535612399 3.985263e+00
ci_quadratic_transformed_ff <- confint(f_d_2, method = "quadratic")
+ci_quadratic_transformed_ff
#>                        2.5%        97.5%
+#> parent_0       96.403839476 1.027931e+02
+#> k_parent        0.090823790 1.072543e-01
+#> k_m1            0.004012216 6.897547e-03
+#> f_parent_to_m1  0.469118713 5.595960e-01
+#> sigma           2.396089689 3.854918e+00
ci_quadratic_untransformed_ff <- confint(f_d_2, method = "quadratic", transformed = FALSE)
+ci_quadratic_untransformed_ff
#>                        2.5%        97.5%
+#> parent_0       96.403839429 1.027931e+02
+#> k_parent        0.090491931 1.069035e-01
+#> k_m1            0.003835483 6.685819e-03
+#> f_parent_to_m1  0.469113364 5.598386e-01
+#> sigma           2.396089689 3.854918e+00
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
#>                 2.5% 97.5%
+#> parent_0        TRUE  TRUE
+#> k_parent        TRUE  TRUE
+#> k_m1           FALSE FALSE
+#> f_parent_to_m1  TRUE FALSE
+#> sigma          FALSE  TRUE
rel_diffs_transformed_ff
#>                        2.5%        97.5%
+#> parent_0       0.0005408078 0.0002217796
+#> k_parent       0.0009596417 0.0009003876
+#> k_m1           0.0307277372 0.0290579184
+#> f_parent_to_m1 0.0046884131 0.0027782558
+#> sigma          0.0550252516 0.0327066836
rel_diffs_untransformed_ff
#>                        2.5%       97.5%
+#> parent_0       0.0005408083 0.000221780
+#> k_parent       0.0046100096 0.002373023
+#> k_m1           0.0146746467 0.002530101
+#> f_parent_to_m1 0.0046997600 0.002346022
+#> sigma          0.0550252516 0.032706684

+# 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)
#> Warning: Optimisation did not converge:
+#> iteration limit reached without convergence (10)
confint(f_tc_2, method = "quadratic")
#>                        2.5%        97.5%
+#> parent_0       95.654015524 105.79279749
+#> k_M1            0.037723773   0.04447598
+#> k_M2            0.008586438   0.01078076
+#> f_parent_to_M1  0.230403596   0.61953014
+#> f_parent_to_M2  0.162909765   0.38019017
+#> k1              0.275434628   0.33331386
+#> k2              0.018602188   0.02249211
+#> g               0.675149759   0.73520889
+#> sigma_low       0.251416929   0.84272023
+#> rsd_high        0.040371818   0.07666540
confint(f_tc_2, "parent_0", method = "quadratic")
#>              2.5%    97.5%
+#> parent_0 95.65402 105.7928
# }
+

+ +