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).

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,
  ...
)

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  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  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"),
  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))
#>    user  system elapsed 
#>   1.203   0.000   1.203 
# 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))
#> Profiling the likelihood
#>    user  system elapsed 
#>   0.438   0.097   0.301 
ci_profile
#>                       2.5%        97.5%
#> parent_0      96.456003640 1.027703e+02
#> k_parent_sink  0.040762501 5.549764e-02
#> k_parent_m1    0.046786482 5.500879e-02
#> k_m1_sink      0.003892605 6.702778e-03
#> sigma          2.535612399 3.985263e+00
ci_quadratic_transformed <- confint(f_d_1, method = "quadratic")
ci_quadratic_transformed
#>                       2.5%        97.5%
#> parent_0      96.403841640 1.027931e+02
#> k_parent_sink  0.041033378 5.596269e-02
#> k_parent_m1    0.046777902 5.511931e-02
#> k_m1_sink      0.004012217 6.897547e-03
#> 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      96.403841645 102.79312449
#> k_parent_sink  0.040485331   0.05535491
#> k_parent_m1    0.046611582   0.05494364
#> k_m1_sink      0.003835483   0.00668582
#> sigma          2.396089689   3.85491806
# 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      FALSE FALSE
#> k_parent_sink  TRUE FALSE
#> k_parent_m1    TRUE FALSE
#> k_m1_sink     FALSE FALSE
#> sigma         FALSE FALSE
signif(rel_diffs_transformed, 3)
#>                   2.5%    97.5%
#> parent_0      0.000541 0.000222
#> k_parent_sink 0.006650 0.008380
#> k_parent_m1   0.000183 0.002010
#> k_m1_sink     0.030700 0.029100
#> sigma         0.055000 0.032700
signif(rel_diffs_untransformed, 3)
#>                   2.5%    97.5%
#> parent_0      0.000541 0.000222
#> k_parent_sink 0.006800 0.002570
#> k_parent_m1   0.003740 0.001180
#> k_m1_sink     0.014700 0.002530
#> 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            0.003892606 6.702775e-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       96.403833581 102.79311649
#> k_parent        0.090823771   0.10725430
#> k_m1            0.004012219   0.00689755
#> f_parent_to_m1  0.469118824   0.55959615
#> sigma           2.396089689   3.85491806
ci_quadratic_untransformed_ff <- confint(f_d_2, method = "quadratic", transformed = FALSE)
ci_quadratic_untransformed_ff
#>                        2.5%        97.5%
#> parent_0       96.403833586 1.027931e+02
#> k_parent        0.090491913 1.069035e-01
#> k_m1            0.003835485 6.685823e-03
#> f_parent_to_m1  0.469113477 5.598387e-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       FALSE FALSE
#> k_parent        TRUE  TRUE
#> k_m1           FALSE FALSE
#> f_parent_to_m1  TRUE FALSE
#> sigma           TRUE FALSE
rel_diffs_transformed_ff
#>                        2.5%        97.5%
#> parent_0       0.0005408690 0.0002217233
#> k_parent       0.0009598532 0.0009001864
#> k_m1           0.0307283045 0.0290588367
#> f_parent_to_m1 0.0046881768 0.0027780062
#> sigma          0.0550252516 0.0327066836
rel_diffs_untransformed_ff
#>                        2.5%        97.5%
#> parent_0       0.0005408689 0.0002217233
#> k_parent       0.0046102155 0.0023732280
#> k_m1           0.0146740687 0.0025291815
#> f_parent_to_m1 0.0046995210 0.0023457712
#> sigma          0.0550252516 0.0327066836

# 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")
#>                        2.5%        97.5%
#> parent_0       94.596181875 106.19936592
#> k_M1            0.037605432   0.04490757
#> k_M2            0.008568745   0.01087675
#> f_parent_to_M1  0.021464676   0.62023880
#> f_parent_to_M2  0.015167158   0.37975350
#> k1              0.273897535   0.33388072
#> k2              0.018614555   0.02250379
#> g               0.671943738   0.73583261
#> sigma_low       0.251283679   0.83992102
#> rsd_high        0.040411022   0.07662008
confint(f_tc_2, "parent_0", method = "quadratic")
#>              2.5%    97.5%
#> parent_0 94.59618 106.1994
# }