- - -

Laboratory Data L1

The following code defines example dataset L1 from the FOCUS kinetics -report, p. 284:

library("mkin", quietly = TRUE)
+
+
+
+
+Example evaluation of FOCUS Laboratory Data L1 to L3
+Johannes Ranke
+2018-07-17
+
+
+
+
+
+Laboratory Data L1
+The following code defines example dataset L1 from the FOCUS kinetics report, p. 284:
+library("mkin", quietly = TRUE)
 FOCUS_2006_L1 = data.frame(
   t = rep(c(0, 1, 2, 3, 5, 7, 14, 21, 30), each = 2),
   parent = c(88.3, 91.4, 85.6, 84.5, 78.9, 77.6,
              72.0, 71.9, 50.3, 59.4, 47.0, 45.1,
              27.7, 27.3, 10.0, 10.4, 2.9, 4.0))
-FOCUS_2006_L1_mkin <- mkin_wide_to_long(FOCUS_2006_L1)
-
-
-Here we use the assumptions of simple first order (SFO), the case of declining
-rate constant over time (FOMC) and the case of two different phases of the
-kinetics (DFOP). For a more detailed discussion of the models, please see the
-FOCUS kinetics report.
-
-Since mkin version 0.9-32 (July 2014), we can use shorthand notation like "SFO"
-for parent only degradation models. The following two lines fit the model and
-produce the summary report of the model fit. This covers the numerical analysis
-given in the FOCUS report.
-
-m.L1.SFO <- mkinfit("SFO", FOCUS_2006_L1_mkin, quiet = TRUE)
-summary(m.L1.SFO)
-
-
-## mkin version:    0.9.46.3 
-## R version:       3.4.3 
-## Date of fit:     Thu Mar  1 14:24:54 2018 
-## Date of summary: Thu Mar  1 14:24:54 2018 
+FOCUS_2006_L1_mkin <- mkin_wide_to_long(FOCUS_2006_L1)
+Here we use the assumptions of simple first order (SFO), the case of declining rate constant over time (FOMC) and the case of two different phases of the kinetics (DFOP). For a more detailed discussion of the models, please see the FOCUS kinetics report.
+Since mkin version 0.9-32 (July 2014), we can use shorthand notation like "SFO" for parent only degradation models. The following two lines fit the model and produce the summary report of the model fit. This covers the numerical analysis given in the FOCUS report.
+m.L1.SFO <- mkinfit("SFO", FOCUS_2006_L1_mkin, quiet = TRUE)
+summary(m.L1.SFO)
+## mkin version used for fitting:    0.9.47.1 
+## R version used for fitting:       3.5.1 
+## Date of fit:     Tue Jul 17 15:54:20 2018 
+## Date of summary: Tue Jul 17 15:54:20 2018 
 ## 
 ## Equations:
 ## d_parent/dt = - k_parent_sink * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted with method Port using 37 model solutions performed in 0.24 s
+## Fitted with method Port using 37 model solutions performed in 0.097 s
 ## 
 ## Weighting: none
 ## 
@@ -311,46 +323,29 @@ summary(m.L1.SFO)
 ##    21   parent     10.0    12.416  -2.4163
 ##    21   parent     10.4    12.416  -2.0163
 ##    30   parent      2.9     5.251  -2.3513
-##    30   parent      4.0     5.251  -1.2513
-
-
+##    30   parent      4.0     5.251  -1.2513

A plot of the fit is obtained with the plot function for mkinfit objects.

- -

plot(m.L1.SFO, show_errmin = TRUE, main = "FOCUS L1 - SFO")
-

- -

plot of chunk unnamed-chunk-4

- +

plot(m.L1.SFO, show_errmin = TRUE, main = "FOCUS L1 - SFO")

The residual plot can be easily obtained by

- -

mkinresplot(m.L1.SFO, ylab = "Observed", xlab = "Time")
-

- -

plot of chunk unnamed-chunk-5

- -

For comparison, the FOMC model is fitted as well, and the \(\chi^2\) error level -is checked.

- -

m.L1.FOMC <- mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet=TRUE)
-plot(m.L1.FOMC, show_errmin = TRUE, main = "FOCUS L1 - FOMC")
-

- -

plot of chunk unnamed-chunk-6

- -

summary(m.L1.FOMC, data = FALSE)
-

- -

## mkin version:    0.9.46.3 
-## R version:       3.4.3 
-## Date of fit:     Thu Mar  1 14:24:56 2018 
-## Date of summary: Thu Mar  1 14:24:57 2018 
+mkinresplot(m.L1.SFO, ylab = "Observed", xlab = "Time")
+
+For comparison, the FOMC model is fitted as well, and the χ² error level is checked.
+m.L1.FOMC <- mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet=TRUE)
+plot(m.L1.FOMC, show_errmin = TRUE, main = "FOCUS L1 - FOMC")
+
+summary(m.L1.FOMC, data = FALSE)
+## mkin version used for fitting:    0.9.47.1 
+## R version used for fitting:       3.5.1 
+## Date of fit:     Tue Jul 17 15:54:22 2018 
+## Date of summary: Tue Jul 17 15:54:22 2018 
 ## 
 ## Equations:
 ## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted with method Port using 611 model solutions performed in 1.376 s
+## Fitted with method Port using 611 model solutions performed in 1.446 s
 ## 
 ## Weighting: none
 ## 
@@ -399,100 +394,50 @@ plot(m.L1.FOMC, show_errmin = TRUE, main = "FOCUS L1 - FOMC")
 ## 
 ## Estimated disappearance times:
 ##         DT50  DT90 DT50back
-## parent 7.249 24.08    7.249
-
-
-We get a warning that the default optimisation algorithm Port did not converge, which
-is an indication that the model is overparameterised, i.e. contains too many
-parameters that are ill-defined as a consequence.
-
-And in fact, due to the higher number of parameters, and the lower number of
-degrees of freedom of the fit, the \(\chi^2\) error level is actually higher for
-the FOMC model (3.6%) than for the SFO model (3.4%). Additionally, the
-parameters log_alpha and log_beta internally fitted in the model have
-excessive confidence intervals, that span more than 25 orders of magnitude (!)
-when backtransformed to the scale of alpha and beta. Also, the t-test
-for significant difference from zero does not indicate such a significant difference,
-with p-values greater than 0.1, and finally, the parameter correlation of log_alpha
-and log_beta is 1.000, clearly indicating that the model is overparameterised.
-
-The \(\chi^2\) error levels reported in Appendix 3 and Appendix 7 to the FOCUS
-kinetics report are rounded to integer percentages and partly deviate by one
-percentage point from the results calculated by mkin. The reason for
-this is not known. However, mkin gives the same \(\chi^2\) error levels
-as the kinfit package and the calculation routines of the kinfit package have
-been extensively compared to the results obtained by the KinGUI
-software, as documented in the kinfit package vignette. KinGUI was the first
-widely used standard package in this field. Also, the calculation of
-\(\chi^2\) error levels was compared with KinGUII, CAKE and DegKin manager in
-a project sponsored by the German Umweltbundesamt [@ranke2014].
-
+## parent 7.249 24.08    7.249

We get a warning that the default optimisation algorithm Port did not converge, which is an indication that the model is overparameterised, i.e. contains too many parameters that are ill-defined as a consequence.

And in fact, due to the higher number of parameters, and the lower number of degrees of freedom of the fit, the χ² error level is actually higher for the FOMC model (3.6%) than for the SFO model (3.4%). Additionally, the parameters log_alpha and log_beta internally fitted in the model have excessive confidence intervals, that span more than 25 orders of magnitude (!) when backtransformed to the scale of alpha and beta. Also, the t-test for significant difference from zero does not indicate such a significant difference, with p-values greater than 0.1, and finally, the parameter correlation of log_alpha and log_beta is 1.000, clearly indicating that the model is overparameterised.

The χ² error levels reported in Appendix 3 and Appendix 7 to the FOCUS kinetics report are rounded to integer percentages and partly deviate by one percentage point from the results calculated by mkin. The reason for this is not known. However, mkin gives the same χ² error levels as the kinfit package and the calculation routines of the kinfit package have been extensively compared to the results obtained by the KinGUI software, as documented in the kinfit package vignette. KinGUI was the first widely used standard package in this field. Also, the calculation of χ² error levels was compared with KinGUII, CAKE and DegKin manager in a project sponsored by the German Umweltbundesamt (Ranke 2014).

Laboratory Data L2

- -

The following code defines example dataset L2 from the FOCUS kinetics -report, p. 287:

- -

FOCUS_2006_L2 = data.frame(
+The following code defines example dataset L2 from the FOCUS kinetics report, p. 287:
+FOCUS_2006_L2 = data.frame(
   t = rep(c(0, 1, 3, 7, 14, 28), each = 2),
   parent = c(96.1, 91.8, 41.4, 38.7,
              19.3, 22.3, 4.6, 4.6,
              2.6, 1.2, 0.3, 0.6))
-FOCUS_2006_L2_mkin <- mkin_wide_to_long(FOCUS_2006_L2)
-
-
+FOCUS_2006_L2_mkin <- mkin_wide_to_long(FOCUS_2006_L2)

SFO fit for L2

- -

Again, the SFO model is fitted and the result is plotted. The residual plot -can be obtained simply by adding the argument show_residuals to the plot -command.

- -

m.L2.SFO <- mkinfit("SFO", FOCUS_2006_L2_mkin, quiet=TRUE)
+Again, the SFO model is fitted and the result is plotted. The residual plot can be obtained simply by adding the argument show_residuals to the plot command.
+m.L2.SFO <- mkinfit("SFO", FOCUS_2006_L2_mkin, quiet=TRUE)
 plot(m.L2.SFO, show_residuals = TRUE, show_errmin = TRUE,
-     main = "FOCUS L2 - SFO")
-
-
-
-
-The \(\chi^2\) error level of 14% suggests that the model does not fit very well.
-This is also obvious from the plots of the fit, in which we have included
-the residual plot.
-
-In the FOCUS kinetics report, it is stated that there is no apparent systematic
-error observed from the residual plot up to the measured DT90 (approximately at
-day 5), and there is an underestimation beyond that point.
-
-We may add that it is difficult to judge the random nature of the residuals just
-from the three samplings at days 0, 1 and 3. Also, it is not clear a
-priori why a consistent underestimation after the approximate DT90 should be
-irrelevant. However, this can be rationalised by the fact that the FOCUS fate
-models generally only implement SFO kinetics.
-
+     main = "FOCUS L2 - SFO")

The χ² error level of 14% suggests that the model does not fit very well. This is also obvious from the plots of the fit, in which we have included the residual plot.

In the FOCUS kinetics report, it is stated that there is no apparent systematic error observed from the residual plot up to the measured DT90 (approximately at day 5), and there is an underestimation beyond that point.

We may add that it is difficult to judge the random nature of the residuals just from the three samplings at days 0, 1 and 3. Also, it is not clear a priori why a consistent underestimation after the approximate DT90 should be irrelevant. However, this can be rationalised by the fact that the FOCUS fate models generally only implement SFO kinetics.

FOMC fit for L2

- -

For comparison, the FOMC model is fitted as well, and the \(\chi^2\) error level -is checked.

- -

m.L2.FOMC <- mkinfit("FOMC", FOCUS_2006_L2_mkin, quiet = TRUE)
+For comparison, the FOMC model is fitted as well, and the χ² error level is checked.
+m.L2.FOMC <- mkinfit("FOMC", FOCUS_2006_L2_mkin, quiet = TRUE)
 plot(m.L2.FOMC, show_residuals = TRUE,
-     main = "FOCUS L2 - FOMC")
-
-
-
-
-summary(m.L2.FOMC, data = FALSE)
-
-
-## mkin version:    0.9.46.3 
-## R version:       3.4.3 
-## Date of fit:     Thu Mar  1 14:24:57 2018 
-## Date of summary: Thu Mar  1 14:24:57 2018 
+     main = "FOCUS L2 - FOMC")
+
+summary(m.L2.FOMC, data = FALSE)
+## mkin version used for fitting:    0.9.47.1 
+## R version used for fitting:       3.5.1 
+## Date of fit:     Tue Jul 17 15:54:23 2018 
+## Date of summary: Tue Jul 17 15:54:23 2018 
 ## 
 ## Equations:
 ## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted with method Port using 81 model solutions performed in 0.169 s
+## Fitted with method Port using 81 model solutions performed in 0.189 s
 ## 
 ## Weighting: none
 ## 
@@ -541,31 +486,21 @@ plot(m.L2.FOMC, show_residuals = TRUE,
 ## 
 ## Estimated disappearance times:
 ##          DT50  DT90 DT50back
-## parent 0.8092 5.356    1.612
-
-
-The error level at which the \(\chi^2\) test passes is much lower in this case.
-Therefore, the FOMC model provides a better description of the data, as less
-experimental error has to be assumed in order to explain the data.
-
+## parent 0.8092 5.356    1.612

The error level at which the χ² test passes is much lower in this case. Therefore, the FOMC model provides a better description of the data, as less experimental error has to be assumed in order to explain the data.

DFOP fit for L2

- -

Fitting the four parameter DFOP model further reduces the \(\chi^2\) error level.

- -

m.L2.DFOP <- mkinfit("DFOP", FOCUS_2006_L2_mkin, quiet = TRUE)
+Fitting the four parameter DFOP model further reduces the χ² error level.
+m.L2.DFOP <- mkinfit("DFOP", FOCUS_2006_L2_mkin, quiet = TRUE)
 plot(m.L2.DFOP, show_residuals = TRUE, show_errmin = TRUE,
-     main = "FOCUS L2 - DFOP")
-
-
-
-
-summary(m.L2.DFOP, data = FALSE)
-
-
-## mkin version:    0.9.46.3 
-## R version:       3.4.3 
-## Date of fit:     Thu Mar  1 14:24:58 2018 
-## Date of summary: Thu Mar  1 14:24:58 2018 
+     main = "FOCUS L2 - DFOP")
+
+summary(m.L2.DFOP, data = FALSE)
+## mkin version used for fitting:    0.9.47.1 
+## R version used for fitting:       3.5.1 
+## Date of fit:     Tue Jul 17 15:54:23 2018 
+## Date of summary: Tue Jul 17 15:54:23 2018 
 ## 
 ## Equations:
 ## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) *
@@ -574,7 +509,7 @@ plot(m.L2.DFOP, show_residuals = TRUE, show_errmin = TRUE,
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted with method Port using 336 model solutions performed in 0.721 s
+## Fitted with method Port using 336 model solutions performed in 0.802 s
 ## 
 ## Weighting: none
 ## 
@@ -602,12 +537,8 @@ plot(m.L2.DFOP, show_residuals = TRUE, show_errmin = TRUE,
 ## log_k2    -1.0880         NA    NA    NA
 ## g_ilr     -0.2821         NA    NA    NA
 ## 
-## Parameter correlation:
-
-
-## Warning in print.summary.mkinfit(x): Could not estimate covariance matrix; singular system:
-
-
+## Parameter correlation:

## Warning in print.summary.mkinfit(x): Could not estimate covariance matrix; singular system:

## Could not estimate covariance matrix; singular system:
 ## 
 ## Residual standard error: 1.732 on 8 degrees of freedom
@@ -629,62 +560,36 @@ plot(m.L2.DFOP, show_residuals = TRUE, show_errmin = TRUE,
 ## 
 ## Estimated disappearance times:
 ##          DT50  DT90 DT50_k1 DT50_k2
-## parent 0.5335 5.311 0.03009   2.058
-

- -

Here, the DFOP model is clearly the best-fit model for dataset L2 based on the -chi² error level criterion. However, the failure to calculate the covariance -matrix indicates that the parameter estimates correlate excessively. Therefore, -the FOMC model may be preferred for this dataset.

- +## parent 0.5335 5.311 0.03009 2.058 +

Here, the DFOP model is clearly the best-fit model for dataset L2 based on the chi^2 error level criterion. However, the failure to calculate the covariance matrix indicates that the parameter estimates correlate excessively. Therefore, the FOMC model may be preferred for this dataset.

Laboratory Data L3

- -

The following code defines example dataset L3 from the FOCUS kinetics report, -p. 290.

- -

FOCUS_2006_L3 = data.frame(
+The following code defines example dataset L3 from the FOCUS kinetics report, p. 290.
+FOCUS_2006_L3 = data.frame(
   t = c(0, 3, 7, 14, 30, 60, 91, 120),
   parent = c(97.8, 60, 51, 43, 35, 22, 15, 12))
-FOCUS_2006_L3_mkin <- mkin_wide_to_long(FOCUS_2006_L3)
-
-
+FOCUS_2006_L3_mkin <- mkin_wide_to_long(FOCUS_2006_L3)

Fit multiple models

- -

As of mkin version 0.9-39 (June 2015), we can fit several models to -one or more datasets in one call to the function mmkin. The datasets -have to be passed in a list, in this case a named list holding only -the L3 dataset prepared above.

- -

# Only use one core here, not to offend the CRAN checks
+As of mkin version 0.9-39 (June 2015), we can fit several models to one or more datasets in one call to the function mmkin. The datasets have to be passed in a list, in this case a named list holding only the L3 dataset prepared above.
+# Only use one core here, not to offend the CRAN checks
 mm.L3 <- mmkin(c("SFO", "FOMC", "DFOP"), cores = 1,
                list("FOCUS L3" = FOCUS_2006_L3_mkin), quiet = TRUE)
-plot(mm.L3)
-
-
-
-
-The \(\chi^2\) error level of 21% as well as the plot suggest that the SFO model
-does not fit very well.  The FOMC model performs better, with an
-error level at which the \(\chi^2\) test passes of 7%.  Fitting the four
-parameter DFOP model further reduces the \(\chi^2\) error level
-considerably.
-
+plot(mm.L3)

The χ² error level of 21% as well as the plot suggest that the SFO model does not fit very well. The FOMC model performs better, with an error level at which the χ² test passes of 7%. Fitting the four parameter DFOP model further reduces the χ² error level considerably.

Accessing mmkin objects

- -

The objects returned by mmkin are arranged like a matrix, with -models as a row index and datasets as a column index.

- -

We can extract the summary and plot for e.g. the DFOP fit, -using square brackets for indexing which will result in the use of -the summary and plot functions working on mkinfit objects.

- -

summary(mm.L3[["DFOP", 1]])
-

- -

## mkin version:    0.9.46.3 
-## R version:       3.4.3 
-## Date of fit:     Thu Mar  1 14:24:59 2018 
-## Date of summary: Thu Mar  1 14:24:59 2018 
+The objects returned by mmkin are arranged like a matrix, with models as a row index and datasets as a column index.
+We can extract the summary and plot for e.g. the DFOP fit, using square brackets for indexing which will result in the use of the summary and plot functions working on mkinfit objects.
+summary(mm.L3[["DFOP", 1]])
+## mkin version used for fitting:    0.9.47.1 
+## R version used for fitting:       3.5.1 
+## Date of fit:     Tue Jul 17 15:54:24 2018 
+## Date of summary: Tue Jul 17 15:54:24 2018 
 ## 
 ## Equations:
 ## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) *
@@ -693,7 +598,7 @@ the summary and plot functions working on mkinfit objects.
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted with method Port using 137 model solutions performed in 0.283 s
+## Fitted with method Port using 137 model solutions performed in 0.318 s
 ## 
 ## Weighting: none
 ## 
@@ -758,64 +663,40 @@ the summary and plot functions working on mkinfit objects.
 ##    30   parent     35.0     35.15 -0.14707
 ##    60   parent     22.0     23.26 -1.25919
 ##    91   parent     15.0     15.18 -0.18181
-##   120   parent     12.0     10.19  1.81395
-
-
-plot(mm.L3[["DFOP", 1]], show_errmin = TRUE)
-
-
-
-
-Here, a look to the model plot, the confidence intervals of the parameters
-and the correlation matrix suggest that the parameter estimates are reliable, and
-the DFOP model can be used as the best-fit model based on the \(\chi^2\) error
-level criterion for laboratory data L3.
-
-This is also an example where the standard t-test for the parameter g_ilr is
-misleading, as it tests for a significant difference from zero. In this case,
-zero appears to be the correct value for this parameter, and the confidence
-interval for the backtransformed parameter g is quite narrow.
-
+##   120   parent     12.0     10.19  1.81395

plot(mm.L3[["DFOP", 1]], show_errmin = TRUE)

Here, a look to the model plot, the confidence intervals of the parameters and the correlation matrix suggest that the parameter estimates are reliable, and the DFOP model can be used as the best-fit model based on the χ² error level criterion for laboratory data L3.

This is also an example where the standard t-test for the parameter g_ilr is misleading, as it tests for a significant difference from zero. In this case, zero appears to be the correct value for this parameter, and the confidence interval for the backtransformed parameter g is quite narrow.

Laboratory Data L4

- -

The following code defines example dataset L4 from the FOCUS kinetics -report, p. 293:

- -

FOCUS_2006_L4 = data.frame(
+The following code defines example dataset L4 from the FOCUS kinetics report, p. 293:
+FOCUS_2006_L4 = data.frame(
   t = c(0, 3, 7, 14, 30, 60, 91, 120),
   parent = c(96.6, 96.3, 94.3, 88.8, 74.9, 59.9, 53.5, 49.0))
-FOCUS_2006_L4_mkin <- mkin_wide_to_long(FOCUS_2006_L4)
-
-
+FOCUS_2006_L4_mkin <- mkin_wide_to_long(FOCUS_2006_L4)

Fits of the SFO and FOMC models, plots and summaries are produced below:

- -

# Only use one core here, not to offend the CRAN checks
+# Only use one core here, not to offend the CRAN checks
 mm.L4 <- mmkin(c("SFO", "FOMC"), cores = 1,
                list("FOCUS L4" = FOCUS_2006_L4_mkin),
                quiet = TRUE)
-plot(mm.L4)
-
-
-
-
-The \(\chi^2\) error level of 3.3% as well as the plot suggest that the SFO model
-fits very well.  The error level at which the \(\chi^2\) test passes is slightly
-lower for the FOMC model. However, the difference appears negligible.
-
-summary(mm.L4[["SFO", 1]], data = FALSE)
-
-
-## mkin version:    0.9.46.3 
-## R version:       3.4.3 
-## Date of fit:     Thu Mar  1 14:24:59 2018 
-## Date of summary: Thu Mar  1 14:24:59 2018 
+plot(mm.L4)
+
+The χ² error level of 3.3% as well as the plot suggest that the SFO model fits very well. The error level at which the χ² test passes is slightly lower for the FOMC model. However, the difference appears negligible.
+summary(mm.L4[["SFO", 1]], data = FALSE)
+## mkin version used for fitting:    0.9.47.1 
+## R version used for fitting:       3.5.1 
+## Date of fit:     Tue Jul 17 15:54:25 2018 
+## Date of summary: Tue Jul 17 15:54:25 2018 
 ## 
 ## Equations:
 ## d_parent/dt = - k_parent_sink * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted with method Port using 46 model solutions performed in 0.098 s
+## Fitted with method Port using 46 model solutions performed in 0.104 s
 ## 
 ## Weighting: none
 ## 
@@ -863,23 +744,19 @@ lower for the FOMC model. However, the difference appears negligible.
 ## 
 ## Estimated disappearance times:
 ##        DT50 DT90
-## parent  106  352
-
-
-summary(mm.L4[["FOMC", 1]], data = FALSE)
-
-
-## mkin version:    0.9.46.3 
-## R version:       3.4.3 
-## Date of fit:     Thu Mar  1 14:24:59 2018 
-## Date of summary: Thu Mar  1 14:24:59 2018 
+## parent  106  352
+summary(mm.L4[["FOMC", 1]], data = FALSE)
+## mkin version used for fitting:    0.9.47.1 
+## R version used for fitting:       3.5.1 
+## Date of fit:     Tue Jul 17 15:54:25 2018 
+## Date of summary: Tue Jul 17 15:54:25 2018 
 ## 
 ## Equations:
 ## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
 ## 
 ## Model predictions using solution type analytical 
 ## 
-## Fitted with method Port using 66 model solutions performed in 0.134 s
+## Fitted with method Port using 66 model solutions performed in 0.154 s
 ## 
 ## Weighting: none
 ## 
@@ -928,11 +805,37 @@ lower for the FOMC model. However, the difference appears negligible.
 ## 
 ## Estimated disappearance times:
 ##         DT50 DT90 DT50back
-## parent 108.9 1644    494.9
-
-
+## parent 108.9 1644    494.9

References

Ranke, Johannes. 2014. “Prüfung und Validierung von Modellierungssoftware als Alternative zu ModelMaker 4.0.” Umweltbundesamt Projektnummer 27452.

+ - +