Fit a kinetic model to data with one or more state variables

This function maximises the likelihood of the observed data using the Port -algorithm stats::nlminb(), and the specified initial or fixed +algorithm stats::nlminb(), and the specified initial or fixed parameters and starting values. In each step of the optimisation, the -kinetic model is solved using the function mkinpredict(), except +kinetic model is solved using the function mkinpredict(), except if an analytical solution is implemented, in which case the model is solved -using the degradation function in the mkinmod object. The +using the degradation function in the mkinmod object. The parameters of the selected error model are fitted simultaneously with the degradation model parameters, as both of them are arguments of the likelihood function.

mkinfit(
-  mkinmod,
-  observed,
-  parms.ini = "auto",
-  state.ini = "auto",
-  err.ini = "auto",
-  fixed_parms = NULL,
-  fixed_initials = names(mkinmod$diffs)[-1],
-  from_max_mean = FALSE,
-  solution_type = c("auto", "analytical", "eigen", "deSolve"),
-  method.ode = "lsoda",
-  use_compiled = "auto",
-  control = list(eval.max = 300, iter.max = 200),
-  transform_rates = TRUE,
-  transform_fractions = TRUE,
-  quiet = FALSE,
-  atol = 1e-08,
-  rtol = 1e-10,
-  error_model = c("const", "obs", "tc"),
-  error_model_algorithm = c("auto", "d_3", "direct", "twostep", "threestep",
-    "fourstep", "IRLS", "OLS"),
-  reweight.tol = 1e-08,
-  reweight.max.iter = 10,
-  trace_parms = FALSE,
-  test_residuals = FALSE,
-  ...
-)

mkinfit(
+  mkinmod,
+  observed,
+  parms.ini = "auto",
+  state.ini = "auto",
+  err.ini = "auto",
+  fixed_parms = NULL,
+  fixed_initials = names(mkinmod$diffs)[-1],
+  from_max_mean = FALSE,
+  solution_type = c("auto", "analytical", "eigen", "deSolve"),
+  method.ode = "lsoda",
+  use_compiled = "auto",
+  control = list(eval.max = 300, iter.max = 200),
+  transform_rates = TRUE,
+  transform_fractions = TRUE,
+  quiet = FALSE,
+  atol = 1e-08,
+  rtol = 1e-10,
+  error_model = c("const", "obs", "tc"),
+  error_model_algorithm = c("auto", "d_3", "direct", "twostep", "threestep",
+    "fourstep", "IRLS", "OLS"),
+  reweight.tol = 1e-08,
+  reweight.max.iter = 10,
+  trace_parms = FALSE,
+  test_residuals = FALSE,
+  ...
+)

Arguments

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

mkinmod	A list of class mkinmod, containing the kinetic + + Arguments + mkinmod + A list of class mkinmod, containing the kinetic model to be fitted to the data, or one of the shorthand names ("SFO", "FOMC", "DFOP", "HS", "SFORB", "IORE"). If a shorthand name is given, a parent only degradation model is generated for the variable with the -highest value in `observed`.
observed	A dataframe with the observed data. The first column called +highest value in `observed`. + observed + A dataframe with the observed data. The first column called "name" must contain the name of the observed variable for each data point. The second column must contain the times of observation, named "time". The third column must be named "value" and contain the observed values. @@ -212,11 +149,9 @@ Zero values in the "value" column will be removed, with a warning, in order to avoid problems with fitting the two-component error model. This is not expected to be a problem, because in general, values of zero are not observed in degradation data, because there is a lower limit of -detection.
parms.ini	A named vector of initial values for the parameters, +detection. + parms.ini + A named vector of initial values for the parameters, including parameters to be optimised and potentially also fixed parameters as indicated by `fixed_parms`. If set to "auto", initial values for rate constants are set to default values. Using parameter names that are @@ -225,113 +160,83 @@ not in the model gives an error. needs. You can use the parameter lists "bparms.ode" from a previously fitted model, which contains the differential equation parameters from this model. This works nicely if the models are nested. An example is -given below.
state.ini	A named vector of initial values for the state variables of +given below. + state.ini + A named vector of initial values for the state variables of the model. In case the observed variables are represented by more than one model variable, the names will differ from the names of the observed -variables (see `map` component of mkinmod). The default +variables (see `map` component of mkinmod). The default is to set the initial value of the first model variable to the mean of the time zero values for the variable with the maximum observed value, and all others to 0. If this variable has no time zero observations, its initial -value is set to 100.
err.ini	A named vector of initial values for the error model +value is set to 100. + err.ini + A named vector of initial values for the error model parameters to be optimised. If set to "auto", initial values are set to default values. Otherwise, inital values for all error model parameters -must be given.
fixed_parms	The names of parameters that should not be optimised but +must be given. + fixed_parms + The names of parameters that should not be optimised but rather kept at the values specified in `parms.ini`. Alternatively, a named numeric vector of parameters to be fixed, regardless of the values -in parms.ini.
fixed_initials	The names of model variables for which the initial +in parms.ini. + fixed_initials + The names of model variables for which the initial state at time 0 should be excluded from the optimisation. Defaults to all -state variables except for the first one.
from_max_mean	If this is set to TRUE, and the model has only one +state variables except for the first one. + from_max_mean + If this is set to TRUE, and the model has only one observed variable, then data before the time of the maximum observed value (after averaging for each sampling time) are discarded, and this time is subtracted from all remaining time values, so the time of the maximum -observed mean value is the new time zero.
solution_type	If set to "eigen", the solution of the system of +observed mean value is the new time zero. + solution_type + If set to "eigen", the solution of the system of differential equations is based on the spectral decomposition of the coefficient matrix in cases that this is possible. If set to "deSolve", a -numerical ode solver from package deSolve is used. If +numerical ode solver from package deSolve is used. If set to "analytical", an analytical solution of the model is used. This is only implemented for relatively simple degradation models. The default is "auto", which uses "analytical" if possible, otherwise "deSolve" if a compiler is present, and "eigen" if no compiler is present and the model -can be expressed using eigenvalues and eigenvectors.
method.ode	The solution method passed via `mkinpredict()` -to `deSolve::ode()` in case the solution type is "deSolve". The default -"lsoda" is performant, but sometimes fails to converge.
use_compiled	If set to `FALSE`, no compiled version of the -mkinmod model is used in the calls to `mkinpredict()` even if a compiled -version is present.
control	A list of control arguments passed to `stats::nlminb()`.
transform_rates	Boolean specifying if kinetic rate constants should +can be expressed using eigenvalues and eigenvectors. + method.ode + The solution method passed via `mkinpredict()` +to `deSolve::ode()` in case the solution type is "deSolve". The default +"lsoda" is performant, but sometimes fails to converge. + use_compiled + If set to `FALSE`, no compiled version of the +mkinmod model is used in the calls to `mkinpredict()` even if a compiled +version is present. + control + A list of control arguments passed to `stats::nlminb()`. + transform_rates + Boolean specifying if kinetic rate constants should be transformed in the model specification used in the fitting for better compliance with the assumption of normal distribution of the estimator. If TRUE, also alpha and beta parameters of the FOMC model are log-transformed, as well as k1 and k2 rate constants for the DFOP and HS models and the break point tb of the HS model. If FALSE, zero is used as -a lower bound for the rates in the optimisation.
transform_fractions	Boolean specifying if formation fractions +a lower bound for the rates in the optimisation. + transform_fractions + Boolean specifying if formation fractions should be transformed in the model specification used in the fitting for better compliance with the assumption of normal distribution of the estimator. The default (TRUE) is to do transformations. If TRUE, the g parameter of the DFOP model is also transformed. Transformations -are described in transform_odeparms.
quiet	Suppress printing out the current value of the negative -log-likelihood after each improvement?
atol	Absolute error tolerance, passed to `deSolve::ode()`. Default -is 1e-8, which is lower than the default in the `deSolve::lsoda()` -function which is used per default.
rtol	Absolute error tolerance, passed to `deSolve::ode()`. Default -is 1e-10, much lower than in `deSolve::lsoda()`.
error_model	If the error model is "const", a constant standard +are described in transform_odeparms. + quiet + Suppress printing out the current value of the negative +log-likelihood after each improvement? + atol + Absolute error tolerance, passed to `deSolve::ode()`. Default +is 1e-8, which is lower than the default in the `deSolve::lsoda()` +function which is used per default. + rtol + Absolute error tolerance, passed to `deSolve::ode()`. Default +is 1e-10, much lower than in `deSolve::lsoda()`. + error_model + If the error model is "const", a constant standard deviation is assumed. If the error model is "obs", each observed variable is assumed to have its own variance. @@ -340,11 +245,9 @@ error model similar to the one described by Rocke and Lorenzato (1995) is used for setting up the likelihood function. Note that this model deviates from the model by Rocke and Lorenzato, as their model implies that the errors follow a lognormal distribution for large values, not a -normal distribution as assumed by this method.
error_model_algorithm	If "auto", the selected algorithm depends on +normal distribution as assumed by this method. + error_model_algorithm + If "auto", the selected algorithm depends on the error model. If the error model is "const", unweighted nonlinear least squares fitting ("OLS") is selected. If the error model is "obs", or "tc", the "d_3" algorithm is selected. @@ -367,372 +270,375 @@ degradation and error model parameters. unweighted least squares, and then iterates optimization of the error model parameters and subsequent optimization of the degradation model using those error model parameters, until the error model parameters -converge.
reweight.tol	Tolerance for the convergence criterion calculated from -the error model parameters in IRLS fits.
reweight.max.iter	Maximum number of iterations in IRLS fits.
trace_parms	Should a trace of the parameter values be listed?
test_residuals	Should the residuals be tested for normal distribution?
...	Further arguments that will be passed on to -`deSolve::ode()`.

- -

Value

- +converge.

reweight.tol

Tolerance for the convergence criterion calculated from +the error model parameters in IRLS fits.

reweight.max.iter

Maximum number of iterations in IRLS fits.

trace_parms

Should a trace of the parameter values be listed?

test_residuals

Should the residuals be tested for normal distribution?

...

Further arguments that will be passed on to +deSolve::ode().

Value

A list with "mkinfit" in the class attribute.

Details

- +

Details

Per default, parameters in the kinetic models are internally transformed in order to better satisfy the assumption of a normal distribution of their estimators.

Note

- +

Note

When using the "IORE" submodel for metabolites, fitting with "transform_rates = TRUE" (the default) often leads to failures of the numerical ODE solver. In this situation it may help to switch off the internal rate transformation.

References

- +

References

Rocke DM and Lorenzato S (1995) A two-component model for measurement error in analytical chemistry. Technometrics 37(2), 176-184.

Ranke J and Meinecke S (2019) Error Models for the Kinetic Evaluation of Chemical Degradation Data. Environments 6(12) 124 -doi: 10.3390/environments6120124 +doi: 10.3390/environments6120124 .

Author

- +mmkin.

Author

Johannes Ranke

Examples


-# Use shorthand notation for parent only degradation
-fit <- mkinfit("FOMC", FOCUS_2006_C, quiet = TRUE)
-summary(fit)
-
#> mkin version used for fitting:    1.0.4 
-#> R version used for fitting:       4.0.4 
-#> Date of fit:     Wed Mar 31 19:15:41 2021 
-#> Date of summary: Wed Mar 31 19:15:41 2021 
-#> 
-#> Equations:
-#> d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
-#> 
-#> Model predictions using solution type analytical 
-#> 
-#> Fitted using 222 model solutions performed in 0.046 s
-#> 
-#> Error model: Constant variance 
-#> 
-#> Error model algorithm: OLS 
-#> 
-#> Starting values for parameters to be optimised:
-#>          value   type
-#> parent_0  85.1  state
-#> alpha      1.0 deparm
-#> beta      10.0 deparm
-#> 
-#> Starting values for the transformed parameters actually optimised:
-#>               value lower upper
-#> parent_0  85.100000  -Inf   Inf
-#> log_alpha  0.000000  -Inf   Inf
-#> log_beta   2.302585  -Inf   Inf
-#> 
-#> Fixed parameter values:
-#> None
-#> 
-#> Results:
-#> 
-#>        AIC      BIC    logLik
-#>   44.68652 45.47542 -18.34326
-#> 
-#> Optimised, transformed parameters with symmetric confidence intervals:
-#>           Estimate Std. Error    Lower   Upper
-#> parent_0  85.87000     1.8070 81.23000 90.5200
-#> log_alpha  0.05192     0.1353 -0.29580  0.3996
-#> log_beta   0.65100     0.2287  0.06315  1.2390
-#> sigma      1.85700     0.4378  0.73200  2.9830
-#> 
-#> Parameter correlation:
-#>             parent_0  log_alpha   log_beta     sigma
-#> parent_0   1.000e+00 -1.565e-01 -3.142e-01 4.772e-08
-#> log_alpha -1.565e-01  1.000e+00  9.564e-01 1.005e-07
-#> log_beta  -3.142e-01  9.564e-01  1.000e+00 8.541e-08
-#> sigma      4.772e-08  1.005e-07  8.541e-08 1.000e+00
-#> 
-#> Backtransformed parameters:
-#> Confidence intervals for internally transformed parameters are asymmetric.
-#> t-test (unrealistically) based on the assumption of normal distribution
-#> for estimators of untransformed parameters.
-#>          Estimate t value    Pr(>t)   Lower  Upper
-#> parent_0   85.870  47.530 3.893e-08 81.2300 90.520
-#> alpha       1.053   7.393 3.562e-04  0.7439  1.491
-#> beta        1.917   4.373 3.601e-03  1.0650  3.451
-#> sigma       1.857   4.243 4.074e-03  0.7320  2.983
-#> 
-#> FOCUS Chi2 error levels in percent:
-#>          err.min n.optim df
-#> All data   6.657       3  6
-#> parent     6.657       3  6
-#> 
-#> Estimated disappearance times:
-#>         DT50  DT90 DT50back
-#> parent 1.785 15.15     4.56
-#> 
-#> Data:
-#>  time variable observed predicted residual
-#>     0   parent     85.1    85.875  -0.7749
-#>     1   parent     57.9    55.191   2.7091
-#>     3   parent     29.9    31.845  -1.9452
-#>     7   parent     14.6    17.012  -2.4124
-#>    14   parent      9.7     9.241   0.4590
-#>    28   parent      6.6     4.754   1.8460
-#>    63   parent      4.0     2.102   1.8977
-#>    91   parent      3.9     1.441   2.4590
-#>   119   parent      0.6     1.092  -0.4919

-# One parent compound, one metabolite, both single first order.
-# We remove zero values from FOCUS dataset D in order to avoid warnings
-FOCUS_D <- subset(FOCUS_2006_D, value != 0)
-# Use mkinsub for convenience in model formulation. Pathway to sink included per default.
-SFO_SFO <- mkinmod(
-  parent = mkinsub("SFO", "m1"),
-  m1 = mkinsub("SFO"))
-
#> Temporary DLL for differentials generated and loaded

-# Fit the model quietly to the FOCUS example dataset D using defaults
-fit <- mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE)
-plot_sep(fit)
-
# As lower parent values appear to have lower variance, we try an alternative error model
-fit.tc <- mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "tc")
-# This avoids the warning, and the likelihood ratio test confirms it is preferable
-lrtest(fit.tc, fit)
-
#> Likelihood ratio test
-#> 
-#> Model 1: SFO_SFO with error model tc and fixed parameter(s) m1_0
-#> Model 2: SFO_SFO with error model const and fixed parameter(s) m1_0
-#>   #Df  LogLik Df  Chisq Pr(>Chisq)    
-#> 1   6 -64.983                         
-#> 2   5 -97.224 -1 64.483  9.737e-16 ***
-#> ---
-#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# We can also allow for different variances of parent and metabolite as error model
-fit.obs <- mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "obs")
-# The two-component error model has significantly higher likelihood
-lrtest(fit.obs, fit.tc)
-
#> Likelihood ratio test
-#> 
-#> Model 1: SFO_SFO with error model tc and fixed parameter(s) m1_0
-#> Model 2: SFO_SFO with error model obs and fixed parameter(s) m1_0
-#>   #Df  LogLik Df  Chisq Pr(>Chisq)    
-#> 1   6 -64.983                         
-#> 2   6 -96.936  0 63.907  < 2.2e-16 ***
-#> ---
-#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
parms(fit.tc)
-
#>       parent_0       k_parent           k_m1 f_parent_to_m1      sigma_low 
-#>   1.007343e+02   1.005562e-01   5.166712e-03   5.083933e-01   3.049883e-03 
-#>       rsd_high 
-#>   7.928118e-02 
endpoints(fit.tc)
-
#> $ff
-#>   parent_m1 parent_sink 
-#>   0.5083933   0.4916067 
-#> 
-#> $distimes
-#>             DT50      DT90
-#> parent   6.89313  22.89848
-#> m1     134.15634 445.65772
-#> 

-# We can show a quick (only one replication) benchmark for this case, as we
-# have several alternative solution methods for the model. We skip
-# uncompiled deSolve, as it is so slow. More benchmarks are found in the
-# benchmark vignette
-# \dontrun{
-if(require(rbenchmark)) {
-  benchmark(replications = 1, order = "relative", columns = c("test", "relative", "elapsed"),
-    deSolve_compiled = mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "tc",
-      solution_type = "deSolve", use_compiled = TRUE),
-    eigen = mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "tc",
-      solution_type = "eigen"),
-    analytical = mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "tc",
-      solution_type = "analytical"))
-}
-
#> Loading required package: rbenchmark
#>               test relative elapsed
-#> 3       analytical    1.000   0.547
-#> 1 deSolve_compiled    1.717   0.939
-#> 2            eigen    2.644   1.446
# }
-
-# Use stepwise fitting, using optimised parameters from parent only fit, FOMC-SFO
-# \dontrun{
-FOMC_SFO <- mkinmod(
-  parent = mkinsub("FOMC", "m1"),
-  m1 = mkinsub("SFO"))
-
#> Temporary DLL for differentials generated and loaded
fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_D, quiet = TRUE)
-# Again, we get a warning and try a more sophisticated error model
-fit.FOMC_SFO.tc <- mkinfit(FOMC_SFO, FOCUS_D, quiet = TRUE, error_model = "tc")
-# This model has a higher likelihood, but not significantly so
-lrtest(fit.tc, fit.FOMC_SFO.tc)
-
#> Likelihood ratio test
-#> 
-#> Model 1: FOMC_SFO with error model tc and fixed parameter(s) m1_0
-#> Model 2: SFO_SFO with error model tc and fixed parameter(s) m1_0
-#>   #Df  LogLik Df  Chisq Pr(>Chisq)
-#> 1   7 -64.829                     
-#> 2   6 -64.983 -1 0.3075     0.5792
# Also, the missing standard error for log_beta and the t-tests for alpha
-# and beta indicate overparameterisation
-summary(fit.FOMC_SFO.tc, data = FALSE)
-
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: diag(.) had 0 or NA entries; non-finite result is doubtful
#> mkin version used for fitting:    1.0.4 
-#> R version used for fitting:       4.0.4 
-#> Date of fit:     Wed Mar 31 19:15:53 2021 
-#> Date of summary: Wed Mar 31 19:15:53 2021 
-#> 
-#> Equations:
-#> d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
-#> d_m1/dt = + f_parent_to_m1 * (alpha/beta) * 1/((time/beta) + 1) *
-#>            parent - k_m1 * m1
-#> 
-#> Model predictions using solution type deSolve 
-#> 
-#> Fitted using 3729 model solutions performed in 2.798 s
-#> 
-#> Error model: Two-component variance function 
-#> 
-#> Error model algorithm: d_3 
-#> Direct fitting and three-step fitting yield approximately the same likelihood 
-#> 
-#> Starting values for parameters to be optimised:
-#>                 value   type
-#> parent_0       100.75  state
-#> alpha            1.00 deparm
-#> beta            10.00 deparm
-#> k_m1             0.10 deparm
-#> f_parent_to_m1   0.50 deparm
-#> sigma_low        0.10  error
-#> rsd_high         0.10  error
-#> 
-#> Starting values for the transformed parameters actually optimised:
-#>                      value lower upper
-#> parent_0        100.750000  -Inf   Inf
-#> log_k_m1         -2.302585  -Inf   Inf
-#> f_parent_qlogis   0.000000  -Inf   Inf
-#> log_alpha         0.000000  -Inf   Inf
-#> log_beta          2.302585  -Inf   Inf
-#> sigma_low         0.100000     0   Inf
-#> rsd_high          0.100000     0   Inf
-#> 
-#> Fixed parameter values:
-#>      value  type
-#> m1_0     0 state
-#> 
-#> Results:
-#> 
-#>       AIC      BIC    logLik
-#>   143.658 155.1211 -64.82902
-#> 
-#> Optimised, transformed parameters with symmetric confidence intervals:
-#>                   Estimate Std. Error     Lower      Upper
-#> parent_0        101.600000  2.6400000 96.240000 107.000000
-#> log_k_m1         -5.284000  0.0929100 -5.474000  -5.095000
-#> f_parent_qlogis   0.001426  0.0767000 -0.155000   0.157800
-#> log_alpha         5.522000  0.0077320  5.506000   5.538000
-#> log_beta          7.806000        NaN       NaN        NaN
-#> sigma_low         0.002488  0.0002431  0.001992   0.002984
-#> rsd_high          0.079210  0.0093280  0.060180   0.098230
-#> 
-#> Parameter correlation:
-#>                  parent_0  log_k_m1 f_parent_qlogis log_alpha log_beta
-#> parent_0         1.000000 -0.095226        -0.76678   0.70544      NaN
-#> log_k_m1        -0.095226  1.000000         0.51432  -0.14387      NaN
-#> f_parent_qlogis -0.766780  0.514321         1.00000  -0.61396      NaN
-#> log_alpha        0.705444 -0.143872        -0.61396   1.00000      NaN
-#> log_beta              NaN       NaN             NaN       NaN        1
-#> sigma_low        0.016073  0.001586         0.01548   5.87007      NaN
-#> rsd_high         0.006626 -0.011700        -0.05357   0.04849      NaN
-#>                 sigma_low  rsd_high
-#> parent_0         0.016073  0.006626
-#> log_k_m1         0.001586 -0.011700
-#> f_parent_qlogis  0.015476 -0.053566
-#> log_alpha        5.870075  0.048487
-#> log_beta              NaN       NaN
-#> sigma_low        1.000000 -0.652558
-#> rsd_high        -0.652558  1.000000
-#> 
-#> Backtransformed parameters:
-#> Confidence intervals for internally transformed parameters are asymmetric.
-#> t-test (unrealistically) based on the assumption of normal distribution
-#> for estimators of untransformed parameters.
-#>                 Estimate t value    Pr(>t)     Lower     Upper
-#> parent_0       1.016e+02 32.7800 6.311e-26 9.624e+01 1.070e+02
-#> k_m1           5.072e-03 10.1200 1.216e-11 4.196e-03 6.130e-03
-#> f_parent_to_m1 5.004e-01 20.8300 4.317e-20 4.613e-01 5.394e-01
-#> alpha          2.502e+02  0.5624 2.889e-01 2.463e+02 2.542e+02
-#> beta           2.455e+03  0.5549 2.915e-01        NA        NA
-#> sigma_low      2.488e-03  0.4843 3.158e-01 1.992e-03 2.984e-03
-#> rsd_high       7.921e-02  8.4300 8.001e-10 6.018e-02 9.823e-02
-#> 
-#> FOCUS Chi2 error levels in percent:
-#>          err.min n.optim df
-#> All data   6.781       5 14
-#> parent     7.141       3  6
-#> m1         4.640       2  8
-#> 
-#> Resulting formation fractions:
-#>                 ff
-#> parent_m1   0.5004
-#> parent_sink 0.4996
-#> 
-#> Estimated disappearance times:
-#>           DT50  DT90 DT50back
-#> parent   6.812  22.7    6.834
-#> m1     136.661 454.0       NA

-# We can easily use starting parameters from the parent only fit (only for illustration)
-fit.FOMC = mkinfit("FOMC", FOCUS_2006_D, quiet = TRUE, error_model = "tc")
-fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_D, quiet = TRUE,
-  parms.ini = fit.FOMC$bparms.ode, error_model = "tc")
-# }
-

Examples


+# Use shorthand notation for parent only degradation
+fit <- mkinfit("FOMC", FOCUS_2006_C, quiet = TRUE)
+summary(fit)
+#> mkin version used for fitting:    1.1.0 
+#> R version used for fitting:       4.1.2 
+#> Date of fit:     Wed Mar  2 13:30:39 2022 
+#> Date of summary: Wed Mar  2 13:30:39 2022 
+#> 
+#> Equations:
+#> d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
+#> 
+#> Model predictions using solution type analytical 
+#> 
+#> Fitted using 222 model solutions performed in 0.046 s
+#> 
+#> Error model: Constant variance 
+#> 
+#> Error model algorithm: OLS 
+#> 
+#> Starting values for parameters to be optimised:
+#>          value   type
+#> parent_0  85.1  state
+#> alpha      1.0 deparm
+#> beta      10.0 deparm
+#> 
+#> Starting values for the transformed parameters actually optimised:
+#>               value lower upper
+#> parent_0  85.100000  -Inf   Inf
+#> log_alpha  0.000000  -Inf   Inf
+#> log_beta   2.302585  -Inf   Inf
+#> 
+#> Fixed parameter values:
+#> None
+#> 
+#> Results:
+#> 
+#>        AIC      BIC    logLik
+#>   44.68652 45.47542 -18.34326
+#> 
+#> Optimised, transformed parameters with symmetric confidence intervals:
+#>           Estimate Std. Error    Lower   Upper
+#> parent_0  85.87000     1.8070 81.23000 90.5200
+#> log_alpha  0.05192     0.1353 -0.29580  0.3996
+#> log_beta   0.65100     0.2287  0.06315  1.2390
+#> sigma      1.85700     0.4378  0.73200  2.9830
+#> 
+#> Parameter correlation:
+#>             parent_0  log_alpha   log_beta     sigma
+#> parent_0   1.000e+00 -1.565e-01 -3.142e-01 4.772e-08
+#> log_alpha -1.565e-01  1.000e+00  9.564e-01 1.005e-07
+#> log_beta  -3.142e-01  9.564e-01  1.000e+00 8.541e-08
+#> sigma      4.772e-08  1.005e-07  8.541e-08 1.000e+00
+#> 
+#> Backtransformed parameters:
+#> Confidence intervals for internally transformed parameters are asymmetric.
+#> t-test (unrealistically) based on the assumption of normal distribution
+#> for estimators of untransformed parameters.
+#>          Estimate t value    Pr(>t)   Lower  Upper
+#> parent_0   85.870  47.530 3.893e-08 81.2300 90.520
+#> alpha       1.053   7.393 3.562e-04  0.7439  1.491
+#> beta        1.917   4.373 3.601e-03  1.0650  3.451
+#> sigma       1.857   4.243 4.074e-03  0.7320  2.983
+#> 
+#> FOCUS Chi2 error levels in percent:
+#>          err.min n.optim df
+#> All data   6.657       3  6
+#> parent     6.657       3  6
+#> 
+#> Estimated disappearance times:
+#>         DT50  DT90 DT50back
+#> parent 1.785 15.15     4.56
+#> 
+#> Data:
+#>  time variable observed predicted residual
+#>     0   parent     85.1    85.875  -0.7749
+#>     1   parent     57.9    55.191   2.7091
+#>     3   parent     29.9    31.845  -1.9452
+#>     7   parent     14.6    17.012  -2.4124
+#>    14   parent      9.7     9.241   0.4590
+#>    28   parent      6.6     4.754   1.8460
+#>    63   parent      4.0     2.102   1.8977
+#>    91   parent      3.9     1.441   2.4590
+#>   119   parent      0.6     1.092  -0.4919
+
+# One parent compound, one metabolite, both single first order.
+# We remove zero values from FOCUS dataset D in order to avoid warnings
+FOCUS_D <- subset(FOCUS_2006_D, value != 0)
+# Use mkinsub for convenience in model formulation. Pathway to sink included per default.
+SFO_SFO <- mkinmod(
+  parent = mkinsub("SFO", "m1"),
+  m1 = mkinsub("SFO"))
+#> Temporary DLL for differentials generated and loaded
+
+# Fit the model quietly to the FOCUS example dataset D using defaults
+fit <- mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE)
+plot_sep(fit)
+
+# As lower parent values appear to have lower variance, we try an alternative error model
+fit.tc <- mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "tc")
+# This avoids the warning, and the likelihood ratio test confirms it is preferable
+lrtest(fit.tc, fit)
+#> Likelihood ratio test
+#> 
+#> Model 1: SFO_SFO with error model tc and fixed parameter(s) m1_0
+#> Model 2: SFO_SFO with error model const and fixed parameter(s) m1_0
+#>   #Df  LogLik Df  Chisq Pr(>Chisq)    
+#> 1   6 -64.983                         
+#> 2   5 -97.224 -1 64.483  9.737e-16 ***
+#> ---
+#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+# We can also allow for different variances of parent and metabolite as error model
+fit.obs <- mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "obs")
+# The two-component error model has significantly higher likelihood
+lrtest(fit.obs, fit.tc)
+#> Likelihood ratio test
+#> 
+#> Model 1: SFO_SFO with error model tc and fixed parameter(s) m1_0
+#> Model 2: SFO_SFO with error model obs and fixed parameter(s) m1_0
+#>   #Df  LogLik Df  Chisq Pr(>Chisq)    
+#> 1   6 -64.983                         
+#> 2   6 -96.936  0 63.907  < 2.2e-16 ***
+#> ---
+#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+parms(fit.tc)
+#>       parent_0       k_parent           k_m1 f_parent_to_m1      sigma_low 
+#>   1.007343e+02   1.005562e-01   5.166712e-03   5.083933e-01   3.049883e-03 
+#>       rsd_high 
+#>   7.928118e-02 
+endpoints(fit.tc)
+#> $ff
+#>   parent_m1 parent_sink 
+#>   0.5083933   0.4916067 
+#> 
+#> $distimes
+#>             DT50      DT90
+#> parent   6.89313  22.89848
+#> m1     134.15634 445.65772
+#> 
+
+# We can show a quick (only one replication) benchmark for this case, as we
+# have several alternative solution methods for the model. We skip
+# uncompiled deSolve, as it is so slow. More benchmarks are found in the
+# benchmark vignette
+# \dontrun{
+if(require(rbenchmark)) {
+  benchmark(replications = 1, order = "relative", columns = c("test", "relative", "elapsed"),
+    deSolve_compiled = mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "tc",
+      solution_type = "deSolve", use_compiled = TRUE),
+    eigen = mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "tc",
+      solution_type = "eigen"),
+    analytical = mkinfit(SFO_SFO, FOCUS_D, quiet = TRUE, error_model = "tc",
+      solution_type = "analytical"))
+}
+#>               test relative elapsed
+#> 3       analytical    1.000   0.699
+#> 1 deSolve_compiled    1.595   1.115
+#> 2            eigen    2.252   1.574
+# }
+
+# Use stepwise fitting, using optimised parameters from parent only fit, FOMC-SFO
+# \dontrun{
+FOMC_SFO <- mkinmod(
+  parent = mkinsub("FOMC", "m1"),
+  m1 = mkinsub("SFO"))
+#> Temporary DLL for differentials generated and loaded
+fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_D, quiet = TRUE)
+# Again, we get a warning and try a more sophisticated error model
+fit.FOMC_SFO.tc <- mkinfit(FOMC_SFO, FOCUS_D, quiet = TRUE, error_model = "tc")
+# This model has a higher likelihood, but not significantly so
+lrtest(fit.tc, fit.FOMC_SFO.tc)
+#> Likelihood ratio test
+#> 
+#> Model 1: FOMC_SFO with error model tc and fixed parameter(s) m1_0
+#> Model 2: SFO_SFO with error model tc and fixed parameter(s) m1_0
+#>   #Df  LogLik Df  Chisq Pr(>Chisq)
+#> 1   7 -64.829                     
+#> 2   6 -64.983 -1 0.3075     0.5792
+# Also, the missing standard error for log_beta and the t-tests for alpha
+# and beta indicate overparameterisation
+summary(fit.FOMC_SFO.tc, data = FALSE)
+#> mkin version used for fitting:    1.1.0 
+#> R version used for fitting:       4.1.2 
+#> Date of fit:     Wed Mar  2 13:30:53 2022 
+#> Date of summary: Wed Mar  2 13:30:53 2022 
+#> 
+#> Equations:
+#> d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
+#> d_m1/dt = + f_parent_to_m1 * (alpha/beta) * 1/((time/beta) + 1) *
+#>            parent - k_m1 * m1
+#> 
+#> Model predictions using solution type deSolve 
+#> 
+#> Fitted using 3729 model solutions performed in 2.916 s
+#> 
+#> Error model: Two-component variance function 
+#> 
+#> Error model algorithm: d_3 
+#> Direct fitting and three-step fitting yield approximately the same likelihood 
+#> 
+#> Starting values for parameters to be optimised:
+#>                 value   type
+#> parent_0       100.75  state
+#> alpha            1.00 deparm
+#> beta            10.00 deparm
+#> k_m1             0.10 deparm
+#> f_parent_to_m1   0.50 deparm
+#> sigma_low        0.10  error
+#> rsd_high         0.10  error
+#> 
+#> Starting values for the transformed parameters actually optimised:
+#>                      value lower upper
+#> parent_0        100.750000  -Inf   Inf
+#> log_k_m1         -2.302585  -Inf   Inf
+#> f_parent_qlogis   0.000000  -Inf   Inf
+#> log_alpha         0.000000  -Inf   Inf
+#> log_beta          2.302585  -Inf   Inf
+#> sigma_low         0.100000     0   Inf
+#> rsd_high          0.100000     0   Inf
+#> 
+#> Fixed parameter values:
+#>      value  type
+#> m1_0     0 state
+#> 
+#> Results:
+#> 
+#>       AIC      BIC    logLik
+#>   143.658 155.1211 -64.82902
+#> 
+#> Optimised, transformed parameters with symmetric confidence intervals:
+#>                   Estimate Std. Error     Lower      Upper
+#> parent_0        101.600000  2.6400000 96.240000 107.000000
+#> log_k_m1         -5.284000  0.0929100 -5.474000  -5.095000
+#> f_parent_qlogis   0.001426  0.0767000 -0.155000   0.157800
+#> log_alpha         5.522000  0.0077320  5.506000   5.538000
+#> log_beta          7.806000        NaN       NaN        NaN
+#> sigma_low         0.002488  0.0002431  0.001992   0.002984
+#> rsd_high          0.079210  0.0093280  0.060180   0.098230
+#> 
+#> Parameter correlation:
+#>                  parent_0  log_k_m1 f_parent_qlogis log_alpha log_beta
+#> parent_0         1.000000 -0.095226        -0.76678   0.70544      NaN
+#> log_k_m1        -0.095226  1.000000         0.51432  -0.14387      NaN
+#> f_parent_qlogis -0.766780  0.514321         1.00000  -0.61396      NaN
+#> log_alpha        0.705444 -0.143872        -0.61396   1.00000      NaN
+#> log_beta              NaN       NaN             NaN       NaN        1
+#> sigma_low        0.016073  0.001586         0.01548   5.87007      NaN
+#> rsd_high         0.006626 -0.011700        -0.05357   0.04849      NaN
+#>                 sigma_low  rsd_high
+#> parent_0         0.016073  0.006626
+#> log_k_m1         0.001586 -0.011700
+#> f_parent_qlogis  0.015476 -0.053566
+#> log_alpha        5.870075  0.048487
+#> log_beta              NaN       NaN
+#> sigma_low        1.000000 -0.652558
+#> rsd_high        -0.652558  1.000000
+#> 
+#> Backtransformed parameters:
+#> Confidence intervals for internally transformed parameters are asymmetric.
+#> t-test (unrealistically) based on the assumption of normal distribution
+#> for estimators of untransformed parameters.
+#>                 Estimate t value    Pr(>t)     Lower     Upper
+#> parent_0       1.016e+02 32.7800 6.311e-26 9.624e+01 1.070e+02
+#> k_m1           5.072e-03 10.1200 1.216e-11 4.196e-03 6.130e-03
+#> f_parent_to_m1 5.004e-01 20.8300 4.317e-20 4.613e-01 5.394e-01
+#> alpha          2.502e+02  0.5624 2.889e-01 2.463e+02 2.542e+02
+#> beta           2.455e+03  0.5549 2.915e-01        NA        NA
+#> sigma_low      2.488e-03  0.4843 3.158e-01 1.992e-03 2.984e-03
+#> rsd_high       7.921e-02  8.4300 8.001e-10 6.018e-02 9.823e-02
+#> 
+#> FOCUS Chi2 error levels in percent:
+#>          err.min n.optim df
+#> All data   6.781       5 14
+#> parent     7.141       3  6
+#> m1         4.640       2  8
+#> 
+#> Resulting formation fractions:
+#>                 ff
+#> parent_m1   0.5004
+#> parent_sink 0.4996
+#> 
+#> Estimated disappearance times:
+#>           DT50  DT90 DT50back
+#> parent   6.812  22.7    6.834
+#> m1     136.661 454.0       NA
+
+# We can easily use starting parameters from the parent only fit (only for illustration)
+fit.FOMC = mkinfit("FOMC", FOCUS_2006_D, quiet = TRUE, error_model = "tc")
+fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_D, quiet = TRUE,
+  parms.ini = fit.FOMC$bparms.ode, error_model = "tc")
+# }
+

Fit a kinetic model to data with one or more state variables

Arguments

Arguments

Value

Value

Details

Details

Note

Note

References

References

See also

See also

Author

Author

Examples

Examples