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----
-title: "Example evaluation of FOCUS Laboratory Data L1 to L3"
-author: "Johannes Ranke"
-date: "`r Sys.Date()`"
-output:
-  html_document:
-    toc: true
-    toc_float:
-      collapsed: false
-    mathjax: null
-    fig_retina: null
-references:
-- id: ranke2014
-  title: <span class="nocase">Prüfung und Validierung von Modellierungssoftware als Alternative zu
-    ModelMaker 4.0</span>
-  author:
-  - family: Ranke
-    given: Johannes
-  type: report
-  issued:
-    year: 2014
-  number: "Umweltbundesamt Projektnummer 27452"
-vignette: >
-  %\VignetteIndexEntry{Example evaluation of FOCUS Laboratory Data L1 to L3}
-  %\VignetteEngine{knitr::rmarkdown}
-  %\VignetteEncoding{UTF-8}
----
-
-```{r, include = FALSE}
-library(knitr)
-opts_chunk$set(tidy = FALSE, cache = FALSE)
-```
-
-# Laboratory Data L1
-
-The following code defines example dataset L1 from the FOCUS kinetics
-report, p. 284:
-
-```{r}
-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.
-
-```{r}
-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.
-
-```{r fig.width = 6, fig.height = 5}
-plot(m.L1.SFO, show_errmin = TRUE, main = "FOCUS L1 - SFO")
-```
-
-The residual plot can be easily obtained by
-
-```{r fig.width = 6, fig.height = 5}
-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.
-
-```{r fig.width = 6, fig.height = 5}
-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].
-
-# Laboratory Data L2
-
-The following code defines example dataset L2 from the FOCUS kinetics
-report, p. 287:
-
-```{r}
-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.
-
-```{r fig.width = 7, fig.height = 6}
-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.
-
-```{r fig.width = 7, fig.height = 6}
-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.
-
-## DFOP fit for L2
-
-Fitting the four parameter DFOP model further reduces the $\chi^2$ error level.
-
-```{r fig.width = 7, fig.height = 6}
-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.
-
-# Laboratory Data L3
-
-The following code defines example dataset L3 from the FOCUS kinetics report,
-p. 290.
-
-```{r}
-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.
-
-```{r fig.height = 8}
-# 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.
-
-```{r fig.height = 5}
-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.
-
-# Laboratory Data L4
-
-The following code defines example dataset L4 from the FOCUS kinetics
-report, p. 293:
-
-```{r}
-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:
-
-```{r fig.height = 6}
-# 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.
-
-
-```{r fig.height = 8}
-summary(mm.L4[["SFO", 1]], data = FALSE)
-summary(mm.L4[["FOMC", 1]], data = FALSE)
-```
-
-
-# References