This function maximises the likelihood of the observed data using the Port
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
if an analytical solution is implemented, in which case the model is solved
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, ... )
| 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 | 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. 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,
including parameters to be optimised and potentially also fixed parameters
as indicated by It is possible to only specify a subset of the parameters that the model 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
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 |
| 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
rather kept at the values specified in |
| 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 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 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 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 |
| use_compiled | If set to |
| control | A list of control arguments passed to |
| 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
constants 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 and HS models are also transformed, as they
can also be seen as compositional data. The transformation used for these
transformations is the |
| quiet | Suppress printing out the current value of the negative log-likelihood after each improvement? |
| atol | Absolute error tolerance, passed to |
| rtol | Absolute error tolerance, passed to |
| 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. If the error model is "tc" (two-component error model), a two component 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 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. The algorithm "d_3" will directly minimize the negative log-likelihood and independently also use the three step algorithm described below. The fit with the higher likelihood is returned. The algorithm "direct" will directly minimize the negative log-likelihood. The algorithm "twostep" will minimize the negative log-likelihood after an initial unweighted least squares optimisation step. The algorithm "threestep" starts with unweighted least squares, then optimizes only the error model using the degradation model parameters found, and then minimizes the negative log-likelihood with free degradation and error model parameters. The algorithm "fourstep" starts with unweighted least squares, then optimizes only the error model using the degradation model parameters found, then optimizes the degradation model again with fixed error model parameters, and finally minimizes the negative log-likelihood with free degradation and error model parameters. The algorithm "IRLS" (Iteratively Reweighted Least Squares) starts with 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? |
| ... | Further arguments that will be passed on to
|
A list with "mkinfit" in the class attribute.
Per default, parameters in the kinetic models are internally transformed in order to better satisfy the assumption of a normal distribution of their estimators.
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.
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.
summary.mkinfit, plot.mkinfit, parms and lrtest.
Comparisons of models fitted to the same data can be made using
AIC by virtue of the method logLik.mkinfit.
Fitting of several models to several datasets in a single call to
mmkin.
# Use shorthand notation for parent only degradation fit <- mkinfit("FOMC", FOCUS_2006_C, quiet = TRUE) summary(fit)#> mkin version used for fitting: 0.9.50.3 #> R version used for fitting: 4.0.0 #> Date of fit: Wed May 27 07:43:45 2020 #> Date of summary: Wed May 27 07:43:45 2020 #> #> Equations: #> d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent #> #> Model predictions using solution type analytical #> #> Fitted using 222 model solutions performed in 0.044 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 4.770e-08 #> log_alpha -1.565e-01 1.000e+00 9.564e-01 9.974e-08 #> log_beta -3.142e-01 9.564e-01 1.000e+00 8.468e-08 #> sigma 4.770e-08 9.974e-08 8.468e-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. # Use mkinsub for convenience in model formulation. Pathway to sink included per default. SFO_SFO <- mkinmod( parent = mkinsub("SFO", "m1"), m1 = mkinsub("SFO"))#># Fit the model to the FOCUS example dataset D using defaults print(system.time(fit <- mkinfit(SFO_SFO, FOCUS_2006_D, solution_type = "eigen", quiet = TRUE)))#> Warning: Observations with value of zero were removed from the data#> user system elapsed #> 0.405 0.001 0.407parms(fit)#> parent_0 k_parent k_m1 f_parent_to_m1 sigma #> 99.598481046 0.098697740 0.005260651 0.514475962 3.125503875endpoints(fit)#> $ff #> parent_m1 parent_sink #> 0.514476 0.485524 #> #> $distimes #> DT50 DT90 #> parent 7.022929 23.32966 #> m1 131.760724 437.69965 #># \dontrun{ # deSolve is slower when no C compiler (gcc) was available during model generation print(system.time(fit.deSolve <- mkinfit(SFO_SFO, FOCUS_2006_D, solution_type = "deSolve")))#> Warning: Observations with value of zero were removed from the data#>#> Sum of squared residuals at call 1: 15156.12 #> Sum of squared residuals at call 2: 15156.12 #> Sum of squared residuals at call 6: 8243.645 #> Sum of squared residuals at call 12: 6290.712 #> Sum of squared residuals at call 13: 6290.683 #> Sum of squared residuals at call 15: 6290.452 #> Sum of squared residuals at call 18: 1700.749 #> Sum of squared residuals at call 20: 1700.611 #> Sum of squared residuals at call 24: 1190.923 #> Sum of squared residuals at call 26: 1190.922 #> Sum of squared residuals at call 29: 1017.417 #> Sum of squared residuals at call 31: 1017.417 #> Sum of squared residuals at call 33: 1017.416 #> Sum of squared residuals at call 34: 644.0472 #> Sum of squared residuals at call 36: 644.047 #> Sum of squared residuals at call 38: 644.047 #> Sum of squared residuals at call 39: 590.5025 #> Sum of squared residuals at call 41: 590.5022 #> Sum of squared residuals at call 43: 590.5016 #> Sum of squared residuals at call 44: 543.2196 #> Sum of squared residuals at call 45: 543.2193 #> Sum of squared residuals at call 46: 543.2192 #> Sum of squared residuals at call 50: 391.348 #> Sum of squared residuals at call 51: 391.3479 #> Sum of squared residuals at call 56: 386.479 #> Sum of squared residuals at call 58: 386.479 #> Sum of squared residuals at call 60: 386.4779 #> Sum of squared residuals at call 61: 384.0686 #> Sum of squared residuals at call 63: 384.0686 #> Sum of squared residuals at call 66: 382.7813 #> Sum of squared residuals at call 68: 382.7813 #> Sum of squared residuals at call 70: 382.7813 #> Sum of squared residuals at call 71: 378.9273 #> Sum of squared residuals at call 73: 378.9273 #> Sum of squared residuals at call 75: 378.9272 #> Sum of squared residuals at call 76: 377.4847 #> Sum of squared residuals at call 78: 377.4846 #> Sum of squared residuals at call 81: 375.9738 #> Sum of squared residuals at call 83: 375.9738 #> Sum of squared residuals at call 86: 375.3387 #> Sum of squared residuals at call 88: 375.3387 #> Sum of squared residuals at call 91: 374.5774 #> Sum of squared residuals at call 93: 374.5774 #> Sum of squared residuals at call 95: 374.5774 #> Sum of squared residuals at call 96: 373.5438 #> Sum of squared residuals at call 100: 373.5438 #> Sum of squared residuals at call 102: 373.265 #> Sum of squared residuals at call 104: 373.265 #> Sum of squared residuals at call 107: 372.6825 #> Sum of squared residuals at call 111: 372.6825 #> Sum of squared residuals at call 114: 372.6356 #> Sum of squared residuals at call 116: 372.6356 #> Sum of squared residuals at call 119: 372.6199 #> Sum of squared residuals at call 121: 372.6199 #> Sum of squared residuals at call 123: 372.6199 #> Sum of squared residuals at call 124: 372.5881 #> Sum of squared residuals at call 126: 372.5881 #> Sum of squared residuals at call 129: 372.5418 #> Sum of squared residuals at call 130: 372.4866 #> Sum of squared residuals at call 131: 372.2242 #> Sum of squared residuals at call 132: 371.5237 #> Sum of squared residuals at call 134: 371.5237 #> Sum of squared residuals at call 137: 371.292 #> Sum of squared residuals at call 139: 371.292 #> Sum of squared residuals at call 143: 371.2256 #> Sum of squared residuals at call 144: 371.2256 #> Sum of squared residuals at call 146: 371.2256 #> Sum of squared residuals at call 149: 371.2194 #> Sum of squared residuals at call 150: 371.2147 #> Sum of squared residuals at call 153: 371.2147 #> Sum of squared residuals at call 155: 371.2137 #> Sum of squared residuals at call 156: 371.2137 #> Sum of squared residuals at call 157: 371.2137 #> Sum of squared residuals at call 160: 371.2134 #> Sum of squared residuals at call 164: 371.2134 #> Sum of squared residuals at call 165: 371.2134 #> Sum of squared residuals at call 167: 371.2134 #> Negative log-likelihood at call 177: 97.22429#>#> user system elapsed #> 0.361 0.000 0.361parms(fit.deSolve)#> parent_0 k_parent k_m1 f_parent_to_m1 sigma #> 99.598480300 0.098697739 0.005260651 0.514475968 3.125503874endpoints(fit.deSolve)#> $ff #> parent_m1 parent_sink #> 0.514476 0.485524 #> #> $distimes #> DT50 DT90 #> parent 7.022929 23.32966 #> m1 131.760721 437.69964 #># } # Use stepwise fitting, using optimised parameters from parent only fit, FOMC # \dontrun{ FOMC_SFO <- mkinmod( parent = mkinsub("FOMC", "m1"), m1 = mkinsub("SFO"))#># Fit the model to the FOCUS example dataset D using defaults fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_2006_D, quiet = TRUE)#> Warning: Observations with value of zero were removed from the data# Use starting parameters from parent only FOMC fit fit.FOMC = mkinfit("FOMC", FOCUS_2006_D, quiet = TRUE) fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_2006_D, quiet = TRUE, parms.ini = fit.FOMC$bparms.ode)#> Warning: Observations with value of zero were removed from the data# Use stepwise fitting, using optimised parameters from parent only fit, SFORB SFORB_SFO <- mkinmod( parent = list(type = "SFORB", to = "m1", sink = TRUE), m1 = list(type = "SFO"))#># Fit the model to the FOCUS example dataset D using defaults fit.SFORB_SFO <- mkinfit(SFORB_SFO, FOCUS_2006_D, quiet = TRUE)#> Warning: Observations with value of zero were removed from the datafit.SFORB_SFO.deSolve <- mkinfit(SFORB_SFO, FOCUS_2006_D, solution_type = "deSolve", quiet = TRUE)#> Warning: Observations with value of zero were removed from the data# Use starting parameters from parent only SFORB fit (not really needed in this case) fit.SFORB = mkinfit("SFORB", FOCUS_2006_D, quiet = TRUE) fit.SFORB_SFO <- mkinfit(SFORB_SFO, FOCUS_2006_D, parms.ini = fit.SFORB$bparms.ode, quiet = TRUE)#> Warning: Observations with value of zero were removed from the data#> Warning: Initial parameter(s) k_parent_free_sink not used in the model# } # \dontrun{ # Weighted fits, including IRLS (error_model = "obs") SFO_SFO.ff <- mkinmod(parent = mkinsub("SFO", "m1"), m1 = mkinsub("SFO"), use_of_ff = "max")#>f.noweight <- mkinfit(SFO_SFO.ff, FOCUS_2006_D, quiet = TRUE)#> Warning: Observations with value of zero were removed from the datasummary(f.noweight)#> mkin version used for fitting: 0.9.50.3 #> R version used for fitting: 4.0.0 #> Date of fit: Wed May 27 07:43:50 2020 #> Date of summary: Wed May 27 07:43:50 2020 #> #> Equations: #> d_parent/dt = - k_parent * parent #> d_m1/dt = + f_parent_to_m1 * k_parent * parent - k_m1 * m1 #> #> Model predictions using solution type analytical #> #> Fitted using 421 model solutions performed in 0.125 s #> #> Error model: Constant variance #> #> Error model algorithm: OLS #> #> Starting values for parameters to be optimised: #> value type #> parent_0 100.7500 state #> k_parent 0.1000 deparm #> k_m1 0.1001 deparm #> f_parent_to_m1 0.5000 deparm #> #> Starting values for the transformed parameters actually optimised: #> value lower upper #> parent_0 100.750000 -Inf Inf #> log_k_parent -2.302585 -Inf Inf #> log_k_m1 -2.301586 -Inf Inf #> f_parent_ilr_1 0.000000 -Inf Inf #> #> Fixed parameter values: #> value type #> m1_0 0 state #> #> Results: #> #> AIC BIC logLik #> 204.4486 212.6365 -97.22429 #> #> Optimised, transformed parameters with symmetric confidence intervals: #> Estimate Std. Error Lower Upper #> parent_0 99.60000 1.57000 96.40000 102.8000 #> log_k_parent -2.31600 0.04087 -2.39900 -2.2330 #> log_k_m1 -5.24800 0.13320 -5.51800 -4.9770 #> f_parent_ilr_1 0.04096 0.06312 -0.08746 0.1694 #> sigma 3.12600 0.35850 2.39600 3.8550 #> #> Parameter correlation: #> parent_0 log_k_parent log_k_m1 f_parent_ilr_1 sigma #> parent_0 1.000e+00 5.174e-01 -1.688e-01 -5.471e-01 -3.190e-07 #> log_k_parent 5.174e-01 1.000e+00 -3.263e-01 -5.426e-01 3.168e-07 #> log_k_m1 -1.688e-01 -3.263e-01 1.000e+00 7.478e-01 -1.406e-07 #> f_parent_ilr_1 -5.471e-01 -5.426e-01 7.478e-01 1.000e+00 -1.587e-10 #> sigma -3.190e-07 3.168e-07 -1.406e-07 -1.587e-10 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 99.600000 63.430 2.298e-36 96.400000 1.028e+02 #> k_parent 0.098700 24.470 4.955e-23 0.090820 1.073e-01 #> k_m1 0.005261 7.510 6.165e-09 0.004012 6.898e-03 #> f_parent_to_m1 0.514500 23.070 3.104e-22 0.469100 5.596e-01 #> sigma 3.126000 8.718 2.235e-10 2.396000 3.855e+00 #> #> FOCUS Chi2 error levels in percent: #> err.min n.optim df #> All data 6.398 4 15 #> parent 6.459 2 7 #> m1 4.690 2 8 #> #> Resulting formation fractions: #> ff #> parent_m1 0.5145 #> parent_sink 0.4855 #> #> Estimated disappearance times: #> DT50 DT90 #> parent 7.023 23.33 #> m1 131.761 437.70 #> #> Data: #> time variable observed predicted residual #> 0 parent 99.46 99.59848 -1.385e-01 #> 0 parent 102.04 99.59848 2.442e+00 #> 1 parent 93.50 90.23787 3.262e+00 #> 1 parent 92.50 90.23787 2.262e+00 #> 3 parent 63.23 74.07319 -1.084e+01 #> 3 parent 68.99 74.07319 -5.083e+00 #> 7 parent 52.32 49.91206 2.408e+00 #> 7 parent 55.13 49.91206 5.218e+00 #> 14 parent 27.27 25.01257 2.257e+00 #> 14 parent 26.64 25.01257 1.627e+00 #> 21 parent 11.50 12.53462 -1.035e+00 #> 21 parent 11.64 12.53462 -8.946e-01 #> 35 parent 2.85 3.14787 -2.979e-01 #> 35 parent 2.91 3.14787 -2.379e-01 #> 50 parent 0.69 0.71624 -2.624e-02 #> 50 parent 0.63 0.71624 -8.624e-02 #> 75 parent 0.05 0.06074 -1.074e-02 #> 75 parent 0.06 0.06074 -7.381e-04 #> 1 m1 4.84 4.80296 3.704e-02 #> 1 m1 5.64 4.80296 8.370e-01 #> 3 m1 12.91 13.02400 -1.140e-01 #> 3 m1 12.96 13.02400 -6.400e-02 #> 7 m1 22.97 25.04476 -2.075e+00 #> 7 m1 24.47 25.04476 -5.748e-01 #> 14 m1 41.69 36.69002 5.000e+00 #> 14 m1 33.21 36.69002 -3.480e+00 #> 21 m1 44.37 41.65310 2.717e+00 #> 21 m1 46.44 41.65310 4.787e+00 #> 35 m1 41.22 43.31312 -2.093e+00 #> 35 m1 37.95 43.31312 -5.363e+00 #> 50 m1 41.19 41.21831 -2.831e-02 #> 50 m1 40.01 41.21831 -1.208e+00 #> 75 m1 40.09 36.44703 3.643e+00 #> 75 m1 33.85 36.44703 -2.597e+00 #> 100 m1 31.04 31.98163 -9.416e-01 #> 100 m1 33.13 31.98163 1.148e+00 #> 120 m1 25.15 28.78984 -3.640e+00 #> 120 m1 33.31 28.78984 4.520e+00f.obs <- mkinfit(SFO_SFO.ff, FOCUS_2006_D, error_model = "obs", quiet = TRUE)#> Warning: Observations with value of zero were removed from the datasummary(f.obs)#> mkin version used for fitting: 0.9.50.3 #> R version used for fitting: 4.0.0 #> Date of fit: Wed May 27 07:43:51 2020 #> Date of summary: Wed May 27 07:43:51 2020 #> #> Equations: #> d_parent/dt = - k_parent * parent #> d_m1/dt = + f_parent_to_m1 * k_parent * parent - k_m1 * m1 #> #> Model predictions using solution type analytical #> #> Fitted using 978 model solutions performed in 0.413 s #> #> Error model: Variance unique to each observed variable #> #> 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_0 100.7500 state #> k_parent 0.1000 deparm #> k_m1 0.1001 deparm #> f_parent_to_m1 0.5000 deparm #> sigma_parent 3.0000 error #> sigma_m1 3.0000 error #> #> Starting values for the transformed parameters actually optimised: #> value lower upper #> parent_0 100.750000 -Inf Inf #> log_k_parent -2.302585 -Inf Inf #> log_k_m1 -2.301586 -Inf Inf #> f_parent_ilr_1 0.000000 -Inf Inf #> sigma_parent 3.000000 0 Inf #> sigma_m1 3.000000 0 Inf #> #> Fixed parameter values: #> value type #> m1_0 0 state #> #> Results: #> #> AIC BIC logLik #> 205.8727 215.6982 -96.93634 #> #> Optimised, transformed parameters with symmetric confidence intervals: #> Estimate Std. Error Lower Upper #> parent_0 99.65000 1.70200 96.19000 103.1000 #> log_k_parent -2.31300 0.04376 -2.40200 -2.2240 #> log_k_m1 -5.25000 0.12430 -5.50400 -4.9970 #> f_parent_ilr_1 0.03861 0.06171 -0.08708 0.1643 #> sigma_parent 3.40100 0.56820 2.24400 4.5590 #> sigma_m1 2.85500 0.45240 1.93400 3.7770 #> #> Parameter correlation: #> parent_0 log_k_parent log_k_m1 f_parent_ilr_1 sigma_parent #> parent_0 1.00000 0.51078 -0.19133 -0.59997 0.035670 #> log_k_parent 0.51078 1.00000 -0.37458 -0.59239 0.069833 #> log_k_m1 -0.19133 -0.37458 1.00000 0.74398 -0.026158 #> f_parent_ilr_1 -0.59997 -0.59239 0.74398 1.00000 -0.041369 #> sigma_parent 0.03567 0.06983 -0.02616 -0.04137 1.000000 #> sigma_m1 -0.03385 -0.06627 0.02482 0.03926 -0.004628 #> sigma_m1 #> parent_0 -0.033847 #> log_k_parent -0.066265 #> log_k_m1 0.024823 #> f_parent_ilr_1 0.039256 #> sigma_parent -0.004628 #> sigma_m1 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 99.650000 58.560 2.004e-34 96.190000 1.031e+02 #> k_parent 0.098970 22.850 1.099e-21 0.090530 1.082e-01 #> k_m1 0.005245 8.046 1.732e-09 0.004072 6.756e-03 #> f_parent_to_m1 0.513600 23.560 4.352e-22 0.469300 5.578e-01 #> sigma_parent 3.401000 5.985 5.662e-07 2.244000 4.559e+00 #> sigma_m1 2.855000 6.311 2.215e-07 1.934000 3.777e+00 #> #> FOCUS Chi2 error levels in percent: #> err.min n.optim df #> All data 6.398 4 15 #> parent 6.464 2 7 #> m1 4.682 2 8 #> #> Resulting formation fractions: #> ff #> parent_m1 0.5136 #> parent_sink 0.4864 #> #> Estimated disappearance times: #> DT50 DT90 #> parent 7.003 23.26 #> m1 132.154 439.01 #> #> Data: #> time variable observed predicted residual #> 0 parent 99.46 99.65417 -1.942e-01 #> 0 parent 102.04 99.65417 2.386e+00 #> 1 parent 93.50 90.26332 3.237e+00 #> 1 parent 92.50 90.26332 2.237e+00 #> 3 parent 63.23 74.05306 -1.082e+01 #> 3 parent 68.99 74.05306 -5.063e+00 #> 7 parent 52.32 49.84325 2.477e+00 #> 7 parent 55.13 49.84325 5.287e+00 #> 14 parent 27.27 24.92971 2.340e+00 #> 14 parent 26.64 24.92971 1.710e+00 #> 21 parent 11.50 12.46890 -9.689e-01 #> 21 parent 11.64 12.46890 -8.289e-01 #> 35 parent 2.85 3.11925 -2.692e-01 #> 35 parent 2.91 3.11925 -2.092e-01 #> 50 parent 0.69 0.70679 -1.679e-02 #> 50 parent 0.63 0.70679 -7.679e-02 #> 75 parent 0.05 0.05952 -9.523e-03 #> 75 parent 0.06 0.05952 4.772e-04 #> 1 m1 4.84 4.81075 2.925e-02 #> 1 m1 5.64 4.81075 8.292e-01 #> 3 m1 12.91 13.04196 -1.320e-01 #> 3 m1 12.96 13.04196 -8.196e-02 #> 7 m1 22.97 25.06847 -2.098e+00 #> 7 m1 24.47 25.06847 -5.985e-01 #> 14 m1 41.69 36.70308 4.987e+00 #> 14 m1 33.21 36.70308 -3.493e+00 #> 21 m1 44.37 41.65115 2.719e+00 #> 21 m1 46.44 41.65115 4.789e+00 #> 35 m1 41.22 43.29465 -2.075e+00 #> 35 m1 37.95 43.29465 -5.345e+00 #> 50 m1 41.19 41.19948 -9.479e-03 #> 50 m1 40.01 41.19948 -1.189e+00 #> 75 m1 40.09 36.44035 3.650e+00 #> 75 m1 33.85 36.44035 -2.590e+00 #> 100 m1 31.04 31.98773 -9.477e-01 #> 100 m1 33.13 31.98773 1.142e+00 #> 120 m1 25.15 28.80429 -3.654e+00 #> 120 m1 33.31 28.80429 4.506e+00f.tc <- mkinfit(SFO_SFO.ff, FOCUS_2006_D, error_model = "tc", quiet = TRUE)#> Warning: Observations with value of zero were removed from the datasummary(f.tc)#> mkin version used for fitting: 0.9.50.3 #> R version used for fitting: 4.0.0 #> Date of fit: Wed May 27 07:43:52 2020 #> Date of summary: Wed May 27 07:43:52 2020 #> #> Equations: #> d_parent/dt = - k_parent * parent #> d_m1/dt = + f_parent_to_m1 * k_parent * parent - k_m1 * m1 #> #> Model predictions using solution type analytical #> #> Fitted using 2088 model solutions performed in 0.714 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_0 100.7500 state #> k_parent 0.1000 deparm #> k_m1 0.1001 deparm #> f_parent_to_m1 0.5000 deparm #> sigma_low 0.1000 error #> rsd_high 0.1000 error #> #> Starting values for the transformed parameters actually optimised: #> value lower upper #> parent_0 100.750000 -Inf Inf #> log_k_parent -2.302585 -Inf Inf #> log_k_m1 -2.301586 -Inf Inf #> f_parent_ilr_1 0.000000 -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 #> 141.9656 151.7911 -64.98278 #> #> Optimised, transformed parameters with symmetric confidence intervals: #> Estimate Std. Error Lower Upper #> parent_0 100.70000 2.621000 95.400000 106.10000 #> log_k_parent -2.29700 0.008862 -2.315000 -2.27900 #> log_k_m1 -5.26600 0.091310 -5.452000 -5.08000 #> f_parent_ilr_1 0.02374 0.055300 -0.088900 0.13640 #> sigma_low 0.00305 0.004829 -0.006786 0.01289 #> rsd_high 0.07928 0.009418 0.060100 0.09847 #> #> Parameter correlation: #> parent_0 log_k_parent log_k_m1 f_parent_ilr_1 sigma_low rsd_high #> parent_0 1.00000 0.67644 -0.10215 -0.76822 0.14294 -0.08783 #> log_k_parent 0.67644 1.00000 -0.15102 -0.59491 0.34611 -0.08125 #> log_k_m1 -0.10215 -0.15102 1.00000 0.51808 -0.05236 0.01240 #> f_parent_ilr_1 -0.76822 -0.59491 0.51808 1.00000 -0.13900 0.03248 #> sigma_low 0.14294 0.34611 -0.05236 -0.13900 1.00000 -0.16546 #> rsd_high -0.08783 -0.08125 0.01240 0.03248 -0.16546 1.00000 #> #> 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 1.007e+02 38.4300 1.180e-28 95.400000 1.061e+02 #> k_parent 1.006e-01 112.8000 1.718e-43 0.098760 1.024e-01 #> k_m1 5.167e-03 10.9500 1.171e-12 0.004290 6.223e-03 #> f_parent_to_m1 5.084e-01 26.0100 2.146e-23 0.468600 5.481e-01 #> sigma_low 3.050e-03 0.6314 2.661e-01 -0.006786 1.289e-02 #> rsd_high 7.928e-02 8.4170 6.418e-10 0.060100 9.847e-02 #> #> FOCUS Chi2 error levels in percent: #> err.min n.optim df #> All data 6.475 4 15 #> parent 6.573 2 7 #> m1 4.671 2 8 #> #> Resulting formation fractions: #> ff #> parent_m1 0.5084 #> parent_sink 0.4916 #> #> Estimated disappearance times: #> DT50 DT90 #> parent 6.893 22.9 #> m1 134.156 445.7 #> #> Data: #> time variable observed predicted residual #> 0 parent 99.46 100.73434 -1.274340 #> 0 parent 102.04 100.73434 1.305660 #> 1 parent 93.50 91.09751 2.402486 #> 1 parent 92.50 91.09751 1.402486 #> 3 parent 63.23 74.50141 -11.271410 #> 3 parent 68.99 74.50141 -5.511410 #> 7 parent 52.32 49.82880 2.491200 #> 7 parent 55.13 49.82880 5.301200 #> 14 parent 27.27 24.64809 2.621908 #> 14 parent 26.64 24.64809 1.991908 #> 21 parent 11.50 12.19232 -0.692315 #> 21 parent 11.64 12.19232 -0.552315 #> 35 parent 2.85 2.98327 -0.133266 #> 35 parent 2.91 2.98327 -0.073266 #> 50 parent 0.69 0.66013 0.029874 #> 50 parent 0.63 0.66013 -0.030126 #> 75 parent 0.05 0.05344 -0.003438 #> 75 parent 0.06 0.05344 0.006562 #> 1 m1 4.84 4.88645 -0.046451 #> 1 m1 5.64 4.88645 0.753549 #> 3 m1 12.91 13.22867 -0.318669 #> 3 m1 12.96 13.22867 -0.268669 #> 7 m1 22.97 25.36417 -2.394166 #> 7 m1 24.47 25.36417 -0.894166 #> 14 m1 41.69 37.00974 4.680263 #> 14 m1 33.21 37.00974 -3.799737 #> 21 m1 44.37 41.90133 2.468669 #> 21 m1 46.44 41.90133 4.538669 #> 35 m1 41.22 43.45691 -2.236913 #> 35 m1 37.95 43.45691 -5.506913 #> 50 m1 41.19 41.34199 -0.151985 #> 50 m1 40.01 41.34199 -1.331985 #> 75 m1 40.09 36.61471 3.475295 #> 75 m1 33.85 36.61471 -2.764705 #> 100 m1 31.04 32.20082 -1.160823 #> 100 m1 33.13 32.20082 0.929177 #> 120 m1 25.15 29.04130 -3.891304 #> 120 m1 33.31 29.04130 4.268696# }