Example evaluation of FOCUS Laboratory Data L1 to L3

+ + + +

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 used for fitting:    1.2.10 
+## R version used for fitting:       4.4.2 
+## Date of fit:     Fri Feb 14 08:59:11 2025 
+## Date of summary: Fri Feb 14 08:59:11 2025 
+## 
+## Equations:
+## d_parent/dt = - k_parent * parent
+## 
+## Model predictions using solution type analytical 
+## 
+## Fitted using 133 model solutions performed in 0.01 s
+## 
+## Error model: Constant variance 
+## 
+## Error model algorithm: OLS 
+## 
+## Starting values for parameters to be optimised:
+##          value   type
+## parent_0 89.85  state
+## k_parent  0.10 deparm
+## 
+## Starting values for the transformed parameters actually optimised:
+##                  value lower upper
+## parent_0     89.850000  -Inf   Inf
+## log_k_parent -2.302585  -Inf   Inf
+## 
+## Fixed parameter values:
+## None
+## 
+## Results:
+## 
+##        AIC     BIC    logLik
+##   93.88778 96.5589 -43.94389
+## 
+## Optimised, transformed parameters with symmetric confidence intervals:
+##              Estimate Std. Error  Lower  Upper
+## parent_0       92.470    1.28200 89.740 95.200
+## log_k_parent   -2.347    0.03763 -2.428 -2.267
+## sigma           2.780    0.46330  1.792  3.767
+## 
+## Parameter correlation:
+##                parent_0 log_k_parent      sigma
+## parent_0      1.000e+00    6.186e-01 -1.516e-09
+## log_k_parent  6.186e-01    1.000e+00 -3.124e-09
+## sigma        -1.516e-09   -3.124e-09  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 92.47000   72.13 8.824e-21 89.74000 95.2000
+## k_parent  0.09561   26.57 2.487e-14  0.08824  0.1036
+## sigma     2.78000    6.00 1.216e-05  1.79200  3.7670
+## 
+## FOCUS Chi2 error levels in percent:
+##          err.min n.optim df
+## All data   3.424       2  7
+## parent     3.424       2  7
+## 
+## Estimated disappearance times:
+##         DT50  DT90
+## parent 7.249 24.08
+## 
+## Data:
+##  time variable observed predicted residual
+##     0   parent     88.3    92.471  -4.1710
+##     0   parent     91.4    92.471  -1.0710
+##     1   parent     85.6    84.039   1.5610
+##     1   parent     84.5    84.039   0.4610
+##     2   parent     78.9    76.376   2.5241
+##     2   parent     77.6    76.376   1.2241
+##     3   parent     72.0    69.412   2.5884
+##     3   parent     71.9    69.412   2.4884
+##     5   parent     50.3    57.330  -7.0301
+##     5   parent     59.4    57.330   2.0699
+##     7   parent     47.0    47.352  -0.3515
+##     7   parent     45.1    47.352  -2.2515
+##    14   parent     27.7    24.247   3.4528
+##    14   parent     27.3    24.247   3.0528
+##    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

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")

The residual plot can be easily obtained by

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

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)

## Warning in mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet = TRUE): Optimisation did not converge:
+## false convergence (8)

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

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

## Warning in sqrt(diag(covar)): NaNs produced

## Warning in cov2cor(ans$covar): diag(V) had non-positive or NA entries; the
+## non-finite result may be dubious

## mkin version used for fitting:    1.2.10 
+## R version used for fitting:       4.4.2 
+## Date of fit:     Fri Feb 14 08:59:11 2025 
+## Date of summary: Fri Feb 14 08:59:11 2025 
+## 
+## Equations:
+## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
+## 
+## Model predictions using solution type analytical 
+## 
+## Fitted using 342 model solutions performed in 0.023 s
+## 
+## Error model: Constant variance 
+## 
+## Error model algorithm: OLS 
+## 
+## Starting values for parameters to be optimised:
+##          value   type
+## parent_0 89.85  state
+## alpha     1.00 deparm
+## beta     10.00 deparm
+## 
+## Starting values for the transformed parameters actually optimised:
+##               value lower upper
+## parent_0  89.850000  -Inf   Inf
+## log_alpha  0.000000  -Inf   Inf
+## log_beta   2.302585  -Inf   Inf
+## 
+## Fixed parameter values:
+## None
+## 
+## 
+## Warning(s): 
+## Optimisation did not converge:
+## false convergence (8)
+## 
+## Results:
+## 
+##        AIC      BIC    logLik
+##   95.88782 99.44931 -43.94391
+## 
+## Optimised, transformed parameters with symmetric confidence intervals:
+##           Estimate Std. Error  Lower  Upper
+## parent_0     92.47     1.2820 89.720 95.220
+## log_alpha    13.20        NaN    NaN    NaN
+## log_beta     15.54        NaN    NaN    NaN
+## sigma         2.78     0.4607  1.792  3.768
+## 
+## Parameter correlation:
+##           parent_0 log_alpha log_beta    sigma
+## parent_0  1.000000       NaN      NaN 0.000603
+## log_alpha      NaN         1      NaN      NaN
+## log_beta       NaN       NaN        1      NaN
+## sigma     0.000603       NaN      NaN 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 9.247e+01      NA     NA 89.720 95.220
+## alpha    5.386e+05      NA     NA     NA     NA
+## beta     5.633e+06      NA     NA     NA     NA
+## sigma    2.780e+00      NA     NA  1.792  3.768
+## 
+## FOCUS Chi2 error levels in percent:
+##          err.min n.optim df
+## All data   3.619       3  6
+## parent     3.619       3  6
+## 
+## 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 (Ranke 2014).

Laboratory Data L2 +

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)

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)
+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.

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)
+plot(m.L2.FOMC, show_residuals = TRUE,
+     main = "FOCUS L2 - FOMC")

+summary(m.L2.FOMC, data = FALSE)

## mkin version used for fitting:    1.2.10 
+## R version used for fitting:       4.4.2 
+## Date of fit:     Fri Feb 14 08:59:12 2025 
+## Date of summary: Fri Feb 14 08:59:12 2025 
+## 
+## Equations:
+## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
+## 
+## Model predictions using solution type analytical 
+## 
+## Fitted using 239 model solutions performed in 0.015 s
+## 
+## Error model: Constant variance 
+## 
+## Error model algorithm: OLS 
+## 
+## Starting values for parameters to be optimised:
+##          value   type
+## parent_0 93.95  state
+## alpha     1.00 deparm
+## beta     10.00 deparm
+## 
+## Starting values for the transformed parameters actually optimised:
+##               value lower upper
+## parent_0  93.950000  -Inf   Inf
+## log_alpha  0.000000  -Inf   Inf
+## log_beta   2.302585  -Inf   Inf
+## 
+## Fixed parameter values:
+## None
+## 
+## Results:
+## 
+##        AIC      BIC    logLik
+##   61.78966 63.72928 -26.89483
+## 
+## Optimised, transformed parameters with symmetric confidence intervals:
+##           Estimate Std. Error    Lower   Upper
+## parent_0   93.7700     1.6130 90.05000 97.4900
+## log_alpha   0.3180     0.1559 -0.04149  0.6776
+## log_beta    0.2102     0.2493 -0.36460  0.7850
+## sigma       2.2760     0.4645  1.20500  3.3470
+## 
+## Parameter correlation:
+##             parent_0  log_alpha   log_beta      sigma
+## parent_0   1.000e+00 -1.151e-01 -2.085e-01 -7.436e-09
+## log_alpha -1.151e-01  1.000e+00  9.741e-01 -1.617e-07
+## log_beta  -2.085e-01  9.741e-01  1.000e+00 -1.386e-07
+## sigma     -7.436e-09 -1.617e-07 -1.386e-07  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   93.770  58.120 4.267e-12 90.0500 97.490
+## alpha       1.374   6.414 1.030e-04  0.9594  1.969
+## beta        1.234   4.012 1.942e-03  0.6945  2.192
+## sigma       2.276   4.899 5.977e-04  1.2050  3.347
+## 
+## FOCUS Chi2 error levels in percent:
+##          err.min n.optim df
+## All data   6.205       3  3
+## parent     6.205       3  3
+## 
+## 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.

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)
+plot(m.L2.DFOP, show_residuals = TRUE, show_errmin = TRUE,
+     main = "FOCUS L2 - DFOP")

+summary(m.L2.DFOP, data = FALSE)

## mkin version used for fitting:    1.2.10 
+## R version used for fitting:       4.4.2 
+## Date of fit:     Fri Feb 14 08:59:12 2025 
+## Date of summary: Fri Feb 14 08:59:12 2025 
+## 
+## Equations:
+## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+##            time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+##            * parent
+## 
+## Model predictions using solution type analytical 
+## 
+## Fitted using 581 model solutions performed in 0.042 s
+## 
+## Error model: Constant variance 
+## 
+## Error model algorithm: OLS 
+## 
+## Starting values for parameters to be optimised:
+##          value   type
+## parent_0 93.95  state
+## k1        0.10 deparm
+## k2        0.01 deparm
+## g         0.50 deparm
+## 
+## Starting values for the transformed parameters actually optimised:
+##              value lower upper
+## parent_0 93.950000  -Inf   Inf
+## log_k1   -2.302585  -Inf   Inf
+## log_k2   -4.605170  -Inf   Inf
+## g_qlogis  0.000000  -Inf   Inf
+## 
+## Fixed parameter values:
+## None
+## 
+## Results:
+## 
+##        AIC      BIC    logLik
+##   52.36695 54.79148 -21.18347
+## 
+## Optimised, transformed parameters with symmetric confidence intervals:
+##          Estimate Std. Error      Lower     Upper
+## parent_0   93.950  9.998e-01    91.5900   96.3100
+## log_k1      3.113  1.849e+03 -4369.0000 4375.0000
+## log_k2     -1.088  6.285e-02    -1.2370   -0.9394
+## g_qlogis   -0.399  9.946e-02    -0.6342   -0.1638
+## sigma       1.414  2.886e-01     0.7314    2.0960
+## 
+## Parameter correlation:
+##            parent_0     log_k1     log_k2   g_qlogis      sigma
+## parent_0  1.000e+00  6.763e-07 -8.944e-10  2.665e-01 -1.083e-09
+## log_k1    6.763e-07  1.000e+00  1.112e-04 -2.187e-04 -1.027e-05
+## log_k2   -8.944e-10  1.112e-04  1.000e+00 -7.903e-01  9.464e-09
+## g_qlogis  2.665e-01 -2.187e-04 -7.903e-01  1.000e+00 -1.532e-08
+## sigma    -1.083e-09 -1.027e-05  9.464e-09 -1.532e-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  93.9500 9.397e+01 2.036e-12 91.5900 96.3100
+## k1        22.4900 5.533e-04 4.998e-01  0.0000     Inf
+## k2         0.3369 1.591e+01 4.697e-07  0.2904  0.3909
+## g          0.4016 1.680e+01 3.238e-07  0.3466  0.4591
+## sigma      1.4140 4.899e+00 8.776e-04  0.7314  2.0960
+## 
+## FOCUS Chi2 error levels in percent:
+##          err.min n.optim df
+## All data    2.53       4  2
+## parent      2.53       4  2
+## 
+## Estimated disappearance times:
+##          DT50  DT90 DT50back DT50_k1 DT50_k2
+## parent 0.5335 5.311    1.599 0.03083   2.058

Here, the DFOP model is clearly the best-fit model for dataset L2 +based on the chi^2 error level criterion.

Laboratory Data L3 +

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)

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
+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.

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 used for fitting:    1.2.10 
+## R version used for fitting:       4.4.2 
+## Date of fit:     Fri Feb 14 08:59:12 2025 
+## Date of summary: Fri Feb 14 08:59:12 2025 
+## 
+## Equations:
+## d_parent/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+##            time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+##            * parent
+## 
+## Model predictions using solution type analytical 
+## 
+## Fitted using 376 model solutions performed in 0.024 s
+## 
+## Error model: Constant variance 
+## 
+## Error model algorithm: OLS 
+## 
+## Starting values for parameters to be optimised:
+##          value   type
+## parent_0 97.80  state
+## k1        0.10 deparm
+## k2        0.01 deparm
+## g         0.50 deparm
+## 
+## Starting values for the transformed parameters actually optimised:
+##              value lower upper
+## parent_0 97.800000  -Inf   Inf
+## log_k1   -2.302585  -Inf   Inf
+## log_k2   -4.605170  -Inf   Inf
+## g_qlogis  0.000000  -Inf   Inf
+## 
+## Fixed parameter values:
+## None
+## 
+## Results:
+## 
+##        AIC      BIC    logLik
+##   32.97732 33.37453 -11.48866
+## 
+## Optimised, transformed parameters with symmetric confidence intervals:
+##          Estimate Std. Error   Lower      Upper
+## parent_0  97.7500    1.01900 94.5000 101.000000
+## log_k1    -0.6612    0.10050 -0.9812  -0.341300
+## log_k2    -4.2860    0.04322 -4.4230  -4.148000
+## g_qlogis  -0.1739    0.05270 -0.3416  -0.006142
+## sigma      1.0170    0.25430  0.2079   1.827000
+## 
+## Parameter correlation:
+##            parent_0     log_k1     log_k2   g_qlogis      sigma
+## parent_0  1.000e+00  1.732e-01  2.282e-02  4.009e-01 -9.696e-08
+## log_k1    1.732e-01  1.000e+00  4.945e-01 -5.809e-01  7.148e-07
+## log_k2    2.282e-02  4.945e-01  1.000e+00 -6.812e-01  1.022e-06
+## g_qlogis  4.009e-01 -5.809e-01 -6.812e-01  1.000e+00 -7.930e-07
+## sigma    -9.696e-08  7.148e-07  1.022e-06 -7.930e-07  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 97.75000  95.960 1.248e-06 94.50000 101.00000
+## k1        0.51620   9.947 1.081e-03  0.37490   0.71090
+## k2        0.01376  23.140 8.840e-05  0.01199   0.01579
+## g         0.45660  34.920 2.581e-05  0.41540   0.49850
+## sigma     1.01700   4.000 1.400e-02  0.20790   1.82700
+## 
+## FOCUS Chi2 error levels in percent:
+##          err.min n.optim df
+## All data   2.225       4  4
+## parent     2.225       4  4
+## 
+## Estimated disappearance times:
+##         DT50 DT90 DT50back DT50_k1 DT50_k2
+## parent 7.464  123    37.03   1.343   50.37
+## 
+## Data:
+##  time variable observed predicted residual
+##     0   parent     97.8     97.75  0.05396
+##     3   parent     60.0     60.45 -0.44933
+##     7   parent     51.0     49.44  1.56338
+##    14   parent     43.0     43.84 -0.83632
+##    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.

Laboratory Data L4 +

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)

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

+# 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 used for fitting:    1.2.10 
+## R version used for fitting:       4.4.2 
+## Date of fit:     Fri Feb 14 08:59:13 2025 
+## Date of summary: Fri Feb 14 08:59:13 2025 
+## 
+## Equations:
+## d_parent/dt = - k_parent * parent
+## 
+## Model predictions using solution type analytical 
+## 
+## Fitted using 142 model solutions performed in 0.01 s
+## 
+## Error model: Constant variance 
+## 
+## Error model algorithm: OLS 
+## 
+## Starting values for parameters to be optimised:
+##          value   type
+## parent_0  96.6  state
+## k_parent   0.1 deparm
+## 
+## Starting values for the transformed parameters actually optimised:
+##                  value lower upper
+## parent_0     96.600000  -Inf   Inf
+## log_k_parent -2.302585  -Inf   Inf
+## 
+## Fixed parameter values:
+## None
+## 
+## Results:
+## 
+##        AIC      BIC    logLik
+##   47.12133 47.35966 -20.56067
+## 
+## Optimised, transformed parameters with symmetric confidence intervals:
+##              Estimate Std. Error  Lower   Upper
+## parent_0       96.440    1.69900 92.070 100.800
+## log_k_parent   -5.030    0.07059 -5.211  -4.848
+## sigma           3.162    0.79050  1.130   5.194
+## 
+## Parameter correlation:
+##               parent_0 log_k_parent     sigma
+## parent_0     1.000e+00    5.938e-01 3.430e-07
+## log_k_parent 5.938e-01    1.000e+00 5.885e-07
+## sigma        3.430e-07    5.885e-07 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 96.440000   56.77 1.604e-08 92.070000 1.008e+02
+## k_parent  0.006541   14.17 1.578e-05  0.005455 7.842e-03
+## sigma     3.162000    4.00 5.162e-03  1.130000 5.194e+00
+## 
+## FOCUS Chi2 error levels in percent:
+##          err.min n.optim df
+## All data   3.287       2  6
+## parent     3.287       2  6
+## 
+## Estimated disappearance times:
+##        DT50 DT90
+## parent  106  352

+summary(mm.L4[["FOMC", 1]], data = FALSE)

## mkin version used for fitting:    1.2.10 
+## R version used for fitting:       4.4.2 
+## Date of fit:     Fri Feb 14 08:59:13 2025 
+## Date of summary: Fri Feb 14 08:59:13 2025 
+## 
+## Equations:
+## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
+## 
+## Model predictions using solution type analytical 
+## 
+## Fitted using 224 model solutions performed in 0.013 s
+## 
+## Error model: Constant variance 
+## 
+## Error model algorithm: OLS 
+## 
+## Starting values for parameters to be optimised:
+##          value   type
+## parent_0  96.6  state
+## alpha      1.0 deparm
+## beta      10.0 deparm
+## 
+## Starting values for the transformed parameters actually optimised:
+##               value lower upper
+## parent_0  96.600000  -Inf   Inf
+## log_alpha  0.000000  -Inf   Inf
+## log_beta   2.302585  -Inf   Inf
+## 
+## Fixed parameter values:
+## None
+## 
+## Results:
+## 
+##        AIC      BIC    logLik
+##   40.37255 40.69032 -16.18628
+## 
+## Optimised, transformed parameters with symmetric confidence intervals:
+##           Estimate Std. Error   Lower    Upper
+## parent_0   99.1400     1.2670 95.6300 102.7000
+## log_alpha  -0.3506     0.2616 -1.0770   0.3756
+## log_beta    4.1740     0.3938  3.0810   5.2670
+## sigma       1.8300     0.4575  0.5598   3.1000
+## 
+## Parameter correlation:
+##             parent_0  log_alpha   log_beta      sigma
+## parent_0   1.000e+00 -4.696e-01 -5.543e-01 -2.447e-07
+## log_alpha -4.696e-01  1.000e+00  9.889e-01  2.198e-08
+## log_beta  -5.543e-01  9.889e-01  1.000e+00  4.923e-08
+## sigma     -2.447e-07  2.198e-08  4.923e-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  99.1400  78.250 7.993e-08 95.6300 102.700
+## alpha      0.7042   3.823 9.365e-03  0.3407   1.456
+## beta      64.9800   2.540 3.201e-02 21.7800 193.900
+## sigma      1.8300   4.000 8.065e-03  0.5598   3.100
+## 
+## FOCUS Chi2 error levels in percent:
+##          err.min n.optim df
+## All data   2.029       3  5
+## parent     2.029       3  5
+## 
+## Estimated disappearance times:
+##         DT50 DT90 DT50back
+## parent 108.9 1644    494.9

References +

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

Johannes +Ranke

Last change 18 May 2023 +(rebuilt 2025-02-14)