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+<!--

+%\VignetteEngine{knitr::knitr}

+%\VignetteIndexEntry{Example evaluation of FOCUS Laboratory Data L1 to L3}

+-->

+

+# 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

+

+```{r}

+library("mkin")

+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)

+```

+

+The next step is to set up the models used for the kinetic analysis. Note that

+the model definitions contain the names of the observed variables in the data.

+In this case, there is only one variable called `parent`.

+

+```{r}

+SFO <- mkinmod(parent = list(type = "SFO"))

+FOMC <- mkinmod(parent = list(type = "FOMC"))

+DFOP <- mkinmod(parent = list(type = "DFOP"))

+```

+

+The three models cover the first assumption 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.

+

+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=7, fig.height = 5}

+plot(m.L1.SFO)

+```

+The residual plot can be easily obtained by

+

+```{r fig.width=7, 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}

+m.L1.FOMC <- mkinfit(FOMC, FOCUS_2006_L1_mkin, quiet=TRUE)

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

+```

+

+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 covariance

+matrix can not be obtained, indicating overparameterisation of the model.

+As a consequence, no standard errors for transformed parameters nor

+confidence intervals for backtransformed parameters are available.

+

+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.

+

+Furthermore, 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 is a widely used standard package in this field.

+Therefore, the reason for the difference was not investigated further.

+

+## 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)

+```

+

+Again, the SFO model is fitted and a summary is obtained.

+

+```{r}

+m.L2.SFO <- mkinfit(SFO, FOCUS_2006_L2_mkin, quiet=TRUE)

+summary(m.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 and the residuals.

+

+```{r fig.height = 8}

+par(mfrow = c(2, 1))

+plot(m.L2.SFO)

+mkinresplot(m.L2.SFO)

+```

+

+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.

+

+```{r fig.height = 8}

+m.L2.FOMC <- mkinfit(FOMC, FOCUS_2006_L2_mkin, quiet = TRUE)

+par(mfrow = c(2, 1))

+plot(m.L2.FOMC)

+mkinresplot(m.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. 

+

+```{r fig.height = 5}

+m.L2.DFOP <- mkinfit(DFOP, FOCUS_2006_L2_mkin, quiet = TRUE)

+plot(m.L2.DFOP)

+```

+

+Here, the default starting parameters for the DFOP model obviously do not lead

+to a reasonable solution. Therefore the fit is repeated with different starting

+parameters.

+

+```{r fig.height = 5}

+m.L2.DFOP <- mkinfit(DFOP, FOCUS_2006_L2_mkin, 

+  parms.ini = c(k1 = 1, k2 = 0.01, g = 0.8),

+  quiet=TRUE)

+plot(m.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)

+```

+

+SFO model, summary and plot:

+

+```{r fig.height = 5}

+m.L3.SFO <- mkinfit(SFO, FOCUS_2006_L3_mkin, quiet = TRUE)

+plot(m.L3.SFO)

+summary(m.L3.SFO)

+```

+

+The chi^2 error level of 21% as well as the plot suggest that the model

+does not fit very well. 

+

+The FOMC model performs better:

+

+```{r fig.height = 5}

+m.L3.FOMC <- mkinfit(FOMC, FOCUS_2006_L3_mkin, quiet = TRUE)

+plot(m.L3.FOMC)

+summary(m.L3.FOMC, data = FALSE)

+```

+

+The error level at which the chi^2 test passes is 7% in this case.

+

+Fitting the four parameter DFOP model further reduces the chi^2 error level

+considerably:

+

+```{r fig.height = 5}

+m.L3.DFOP <- mkinfit(DFOP, FOCUS_2006_L3_mkin, quiet = TRUE)

+plot(m.L3.DFOP)

+summary(m.L3.DFOP, data = FALSE)

+```

+

+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.

+

+## 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)

+```

+

+SFO model, summary and plot:

+

+```{r fig.height = 5}

+m.L4.SFO <- mkinfit(SFO, FOCUS_2006_L4_mkin, quiet = TRUE)

+plot(m.L4.SFO)

+summary(m.L4.SFO, data = FALSE)

+```

+

+The chi^2 error level of 3.3% as well as the plot suggest that the model

+fits very well. 

+

+The FOMC model for comparison

+

+```{r fig.height = 5}

+m.L4.FOMC <- mkinfit(FOMC, FOCUS_2006_L4_mkin, quiet = TRUE)

+plot(m.L4.FOMC)

+summary(m.L4.FOMC, data = FALSE)

+```

+

+The error level at which the chi^2 test passes is slightly lower for the FOMC 

+model. However, the difference appears negligible.

+