library(knitr) opts_chunk$set(tidy = FALSE, cache = FALSE)

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)

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) plot(m.L1.FOMC, show_errmin = TRUE, main = "FOCUS L1 - FOMC") summary(m.L1.FOMC, data = FALSE)

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].

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)

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.

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)

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.

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)

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.

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)

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.

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]]) 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.

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

**Any scripts or data that you put into this service are public.**

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.