path: root/vignettes/FOCUS_L.Rmd

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%\VignetteIndexEntry{Example evaluation of FOCUS Laboratory Data L1 to L3}
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```{r, include = FALSE}
library(knitr)
opts_chunk$set(tidy = FALSE, cache = TRUE)
```

# 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.