mkinfit.RdThis function maximises the likelihood of the observed data using
the Port algorithm nlminb, and the specified initial or fixed
parameters and starting values. In each step of the optimsation, the kinetic
model is solved using the function mkinpredict. 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.
Per default, parameters in the kinetic models are internally transformed in order to better satisfy the assumption of a normal distribution of their estimators.
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-8, rtol = 1e-10, n.outtimes = 100, error_model = c("const", "obs", "tc"), error_model_algorithm = c("d_3", "direct", "twostep", "threestep", "fourstep", "IRLS"), reweight.tol = 1e-8, reweight.max.iter = 10, trace_parms = FALSE, ...)
| mkinmod | A list of class |
|---|---|
| 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
|
| 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 |
| n.outtimes | The length of the dataseries that is produced by the model prediction
function |
| 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 the error model is "const", the error model algorithm is ignored, because no special algorithm is needed and unweighted (also known as ordinary) least squares fitting can be applied. The default 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" 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. A summary can be obtained by
summary.mkinfit.
Plotting methods plot.mkinfit and mkinparplot.
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.
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, David M. und Lorenzato, Stefan (1995) A two-component model for measurement error in analytical chemistry. Technometrics 37(2), 176-184.
# Use shorthand notation for parent only degradation fit <- mkinfit("FOMC", FOCUS_2006_C, quiet = TRUE) summary(fit)#> mkin version used for fitting: 0.9.49.5 #> R version used for fitting: 3.6.0 #> Date of fit: Thu Jul 4 08:02:21 2019 #> Date of summary: Thu Jul 4 08:02:21 2019 #> #> Equations: #> d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent #> #> Model predictions using solution type analytical #> #> Fitted using 222 model solutions performed in 0.464 s #> #> Error model: Constant variance #> #> Error model algorithm: d_3 #> #> Starting values for parameters to be optimised: #> value type #> parent_0 85.100000 state #> alpha 1.000000 deparm #> beta 10.000000 deparm #> sigma 1.857444 error #> #> 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 #> sigma 1.857444 0 Inf #> #> Fixed parameter values: #> None #> #> 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 verstrichen #> 1.507 0.004 1.511coef(fit)#> NULLendpoints(fit)#> $ff #> parent_sink parent_m1 m1_sink #> 0.485524 0.514476 1.000000 #> #> $SFORB #> logical(0) #> #> $distimes #> DT50 DT90 #> parent 7.022929 23.32967 #> m1 131.760712 437.69961 #># 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: 18915.53 #> Sum of squared residuals at call 2: 18915.53 #> Sum of squared residuals at call 6: 11424.02 #> Sum of squared residuals at call 10: 11424 #> Sum of squared residuals at call 12: 4094.396 #> Sum of squared residuals at call 16: 4094.396 #> Sum of squared residuals at call 19: 1340.595 #> Sum of squared residuals at call 20: 1340.593 #> Sum of squared residuals at call 25: 1072.239 #> Sum of squared residuals at call 28: 1072.236 #> Sum of squared residuals at call 30: 874.2615 #> Sum of squared residuals at call 33: 874.2611 #> Sum of squared residuals at call 35: 616.2375 #> Sum of squared residuals at call 37: 616.237 #> Sum of squared residuals at call 40: 467.4386 #> Sum of squared residuals at call 42: 467.438 #> Sum of squared residuals at call 46: 398.2913 #> Sum of squared residuals at call 48: 398.2913 #> Sum of squared residuals at call 49: 398.2912 #> Sum of squared residuals at call 51: 395.0711 #> Sum of squared residuals at call 54: 395.071 #> Sum of squared residuals at call 56: 378.3298 #> Sum of squared residuals at call 59: 378.3298 #> Sum of squared residuals at call 62: 376.9812 #> Sum of squared residuals at call 64: 376.9811 #> Sum of squared residuals at call 67: 375.2085 #> Sum of squared residuals at call 69: 375.2085 #> Sum of squared residuals at call 70: 375.2085 #> Sum of squared residuals at call 71: 375.2085 #> Sum of squared residuals at call 72: 374.5723 #> Sum of squared residuals at call 74: 374.5723 #> Sum of squared residuals at call 77: 374.0075 #> Sum of squared residuals at call 79: 374.0075 #> Sum of squared residuals at call 80: 374.0075 #> Sum of squared residuals at call 82: 373.1711 #> Sum of squared residuals at call 84: 373.1711 #> Sum of squared residuals at call 87: 372.6445 #> Sum of squared residuals at call 88: 372.1615 #> Sum of squared residuals at call 90: 372.1615 #> Sum of squared residuals at call 91: 372.1615 #> Sum of squared residuals at call 94: 371.6464 #> Sum of squared residuals at call 99: 371.4299 #> Sum of squared residuals at call 101: 371.4299 #> Sum of squared residuals at call 104: 371.4071 #> Sum of squared residuals at call 106: 371.4071 #> Sum of squared residuals at call 107: 371.4071 #> Sum of squared residuals at call 109: 371.2524 #> Sum of squared residuals at call 113: 371.2524 #> Sum of squared residuals at call 114: 371.2136 #> Sum of squared residuals at call 115: 371.2136 #> Sum of squared residuals at call 116: 371.2136 #> Sum of squared residuals at call 119: 371.2134 #> Sum of squared residuals at call 120: 371.2134 #> Sum of squared residuals at call 122: 371.2134 #> Sum of squared residuals at call 123: 371.2134 #> Sum of squared residuals at call 125: 371.2134 #> Sum of squared residuals at call 126: 371.2134 #> Sum of squared residuals at call 135: 371.2134 #> Negative log-likelihood at call 145: 97.22429#>#> User System verstrichen #> 1.078 0.000 1.078coef(fit.deSolve)#> NULLendpoints(fit.deSolve)#> $ff #> parent_sink parent_m1 m1_sink #> 0.485524 0.514476 1.000000 #> #> $SFORB #> logical(0) #> #> $distimes #> DT50 DT90 #> parent 7.022929 23.32967 #> m1 131.760712 437.69961 #># Use stepwise fitting, using optimised parameters from parent only fit, FOMC#># Fit the model to the FOCUS example dataset D using defaults fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_2006_D, quiet = TRUE)#> Warning: Observations with value of zero were removed from the data# Use starting parameters from parent only FOMC fit fit.FOMC = mkinfit("FOMC", FOCUS_2006_D, quiet = TRUE) fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_2006_D, quiet = TRUE, parms.ini = fit.FOMC$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# Weighted fits, including IRLS 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: 0.9.49.5 #> R version used for fitting: 3.6.0 #> Date of fit: Thu Jul 4 08:02:37 2019 #> Date of summary: Thu Jul 4 08:02:37 2019 #> #> 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 deSolve #> #> Fitted using 421 model solutions performed in 1.096 s #> #> Error model: Constant variance #> #> Error model algorithm: d_3 #> #> Starting values for parameters to be optimised: #> value type #> parent_0 100.750000 state #> k_parent 0.100000 deparm #> k_m1 0.100100 deparm #> f_parent_to_m1 0.500000 deparm #> sigma 3.125504 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 3.125504 0 Inf #> #> Fixed parameter values: #> value type #> m1_0 0 state #> #> 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 -2.265e-07 #> log_k_parent 5.174e-01 1.000e+00 -3.263e-01 -5.426e-01 3.785e-07 #> log_k_m1 -1.688e-01 -3.263e-01 1.000e+00 7.478e-01 -1.386e-07 #> f_parent_ilr_1 -5.471e-01 -5.426e-01 7.478e-01 1.000e+00 -3.641e-08 #> sigma -2.265e-07 3.785e-07 -1.386e-07 -3.641e-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 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: 0.9.49.5 #> R version used for fitting: 3.6.0 #> Date of fit: Thu Jul 4 08:02:40 2019 #> Date of summary: Thu Jul 4 08:02:40 2019 #> #> 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 deSolve #> #> Fitted using 979 model solutions performed in 2.574 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_0 100.750000 state #> k_parent 0.100000 deparm #> k_m1 0.100100 deparm #> f_parent_to_m1 0.500000 deparm #> sigma_parent 3.398909 error #> sigma_m1 2.857157 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 3.398909 0 Inf #> sigma_m1 2.857157 0 Inf #> #> Fixed parameter values: #> value type #> m1_0 0 state #> #> 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: 0.9.49.5 #> R version used for fitting: 3.6.0 #> Date of fit: Thu Jul 4 08:02:49 2019 #> Date of summary: Thu Jul 4 08:02:49 2019 #> #> 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 deSolve #> #> Fitted using 2289 model solutions performed in 9.379 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 1.007500e+02 state #> k_parent 1.000000e-01 deparm #> k_m1 1.001000e-01 deparm #> f_parent_to_m1 5.000000e-01 deparm #> sigma_low 5.641148e-03 error #> rsd_high 8.430766e-02 error #> #> Starting values for the transformed parameters actually optimised: #> value lower upper #> parent_0 100.750000000 -Inf Inf #> log_k_parent -2.302585093 -Inf Inf #> log_k_m1 -2.301585593 -Inf Inf #> f_parent_ilr_1 0.000000000 -Inf Inf #> sigma_low 0.005641148 0 Inf #> rsd_high 0.084307660 0 Inf #> #> Fixed parameter values: #> value type #> m1_0 0 state #> #> 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 -1.274339 #> 0 parent 102.04 100.73434 1.305661 #> 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 2.491201 #> 7 parent 55.13 49.82880 5.301201 #> 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