From 5d5c5b0c102aa9dbd849277c3e3b831c7cdd91fe Mon Sep 17 00:00:00 2001
From: jranke <jranke@edb9625f-4e0d-4859-8d74-9fd3b1da38cb>
Date: Sun, 17 Nov 2013 15:52:42 +0000
Subject: Conflicts: 	README.md 	TODO

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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.
+
-- 
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