---
title: "Example evaluation of FOCUS Laboratory Data L1 to L3"
author: "Johannes Ranke"
date: "`r Sys.Date()`"
output:
  html_document:
    css: mkin_vignettes.css
    toc: true
    mathjax: null
    theme: united
vignette: >
  %\VignetteIndexEntry{Example evaluation of FOCUS Laboratory Data L1 to L3}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r, include = FALSE}
library(knitr)
opts_chunk$set(tidy = FALSE, cache = TRUE)
```

## 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)
```

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=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 parameters
`log_alpha` and `log_beta` internally fitted in the model have p-values for the two
sided t-test of 0.18 and 0.125, and their correlation is 1.000, 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.  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. 

## 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.

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)
```

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.