Example evaluation of FOCUS Laboratory Data L1 to L3 • mkin

- - - - -

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:    1.2.3 
-## R version used for fitting:       4.2.3 
-## Date of fit:     Sun Apr 16 08:35:14 2023 
-## Date of summary: Sun Apr 16 08:35:14 2023 
-## 
-## Equations:
-## d_parent/dt = - k_parent * parent
-## 
-## Model predictions using solution type analytical 
-## 
-## Fitted using 133 model solutions performed in 0.011 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 sqrt(1/diag(V)): NaNs produced

## Warning in cov2cor(ans$covar): diag(.) had 0 or NA entries; non-finite result
-## is doubtful

## mkin version used for fitting:    1.2.3 
-## R version used for fitting:       4.2.3 
-## Date of fit:     Sun Apr 16 08:35:14 2023 
-## Date of summary: Sun Apr 16 08:35:14 2023 
-## 
-## Equations:
-## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
-## 
-## Model predictions using solution type analytical 
-## 
-## Fitted using 369 model solutions performed in 0.025 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
-##   95.88781 99.44929 -43.9439
-## 
-## Optimised, transformed parameters with symmetric confidence intervals:
-##           Estimate Std. Error  Lower  Upper
-## parent_0     92.47     1.2820 89.720 95.220
-## log_alpha    13.78        NaN    NaN    NaN
-## log_beta     16.13        NaN    NaN    NaN
-## sigma         2.78     0.4598  1.794  3.766
-## 
-## Parameter correlation:
-##            parent_0 log_alpha log_beta     sigma
-## parent_0  1.0000000       NaN      NaN 0.0001671
-## log_alpha       NaN         1      NaN       NaN
-## log_beta        NaN       NaN        1       NaN
-## sigma     0.0001671       NaN      NaN 1.0000000
-## 
-## 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    9.658e+05      NA     NA     NA     NA
-## beta     1.010e+07      NA     NA     NA     NA
-## sigma    2.780e+00      NA     NA  1.794  3.766
-## 
-## 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 7.25 24.08     7.25

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:    1.2.3 
-## R version used for fitting:       4.2.3 
-## Date of fit:     Sun Apr 16 08:35:14 2023 
-## Date of summary: Sun Apr 16 08:35:14 2023 
-## 
-## 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 -7.828e-09
-## log_alpha -1.151e-01  1.000e+00  9.741e-01 -1.602e-07
-## log_beta  -2.085e-01  9.741e-01  1.000e+00 -1.372e-07
-## sigma     -7.828e-09 -1.602e-07 -1.372e-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:    1.2.3 
-## R version used for fitting:       4.2.3 
-## Date of fit:     Sun Apr 16 08:35:15 2023 
-## Date of summary: Sun Apr 16 08:35:15 2023 
-## 
-## 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 0.04 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      3.112  1.842e+03 -4353.0000 4359.0000
-## 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  6.783e-07 -3.390e-10  2.665e-01 -2.967e-10
-## log_k1    6.783e-07  1.000e+00  1.116e-04 -2.196e-04 -1.031e-05
-## log_k2   -3.390e-10  1.116e-04  1.000e+00 -7.903e-01  2.917e-09
-## g_qlogis  2.665e-01 -2.196e-04 -7.903e-01  1.000e+00 -4.408e-09
-## sigma    -2.967e-10 -1.031e-05  2.917e-09 -4.408e-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  93.9500 9.397e+01 2.036e-12 91.5900 96.3100
-## k1        22.4800 5.553e-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 0.03084   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:    1.2.3 
-## R version used for fitting:       4.2.3 
-## Date of fit:     Sun Apr 16 08:35:15 2023 
-## Date of summary: Sun Apr 16 08:35:15 2023 
-## 
-## 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 -9.664e-08
-## log_k1    1.732e-01  1.000e+00  4.945e-01 -5.809e-01  7.147e-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 -7.926e-07
-## sigma    -9.664e-08  7.147e-07  1.022e-06 -7.926e-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:    1.2.3 
-## R version used for fitting:       4.2.3 
-## Date of fit:     Sun Apr 16 08:35:15 2023 
-## Date of summary: Sun Apr 16 08:35:16 2023 
-## 
-## Equations:
-## d_parent/dt = - k_parent * parent
-## 
-## Model predictions using solution type analytical 
-## 
-## Fitted using 142 model solutions performed in 0.009 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 3.387e-07
-## log_k_parent 5.938e-01    1.000e+00 5.830e-07
-## sigma        3.387e-07    5.830e-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:    1.2.3 
-## R version used for fitting:       4.2.3 
-## Date of fit:     Sun Apr 16 08:35:15 2023 
-## Date of summary: Sun Apr 16 08:35:16 2023 
-## 
-## Equations:
-## d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
-## 
-## Model predictions using solution type analytical 
-## 
-## Fitted using 224 model solutions performed in 0.014 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 -2.468e-07
-## log_alpha -4.696e-01  1.000e+00  9.889e-01  2.478e-08
-## log_beta  -5.543e-01  9.889e-01  1.000e+00  5.211e-08
-## sigma     -2.468e-07  2.478e-08  5.211e-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. -

- - - -

Example evaluation of FOCUS Laboratory Data L1 -to L3

Johannes -Ranke

Last change 18 May 2022 -(rebuilt 2023-04-16)