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-% $Id: examples.Rnw 66 2010-09-03 08:50:26Z jranke $

-%%\VignetteIndexEntry{Examples for kinetic evaluations using mkin}

-%%VignetteDepends{FME}

-%%\usepackage{Sweave}

-\documentclass[12pt,a4paper]{article}

-\usepackage{a4wide}

-%%\usepackage[lists,heads]{endfloat}

-\input{header}

-\hypersetup{  

-  pdftitle = {Examples for kinetic evaluations using mkin},

-  pdfsubject = {Manuscript},

-  pdfauthor = {Johannes Ranke},

-  colorlinks = {true},

-  linkcolor = {blue},

-  citecolor = {blue},

-  urlcolor = {red},

-  hyperindex = {true},

-  linktocpage = {true},

-}

-\SweaveOpts{engine=R, eps=FALSE, keep.source = TRUE}

-<<setup, echo = FALSE, results = hide>>=

-options(prompt = "R> ")

-options(width = 70)

-options(SweaveHooks = list(

-  cex = function() par(cex.lab = 1.3, cex.axis = 1.3)))

-@

-\begin{document}

-\title{Examples for kinetic evaluations using mkin}

-\author{\textbf{Johannes Ranke} \\[0.5cm]

-%EndAName

-Eurofins Regulatory AG\\

-Weidenweg 15, CH--4310 Rheinfelden, Switzerland\\[0.5cm]

-and\\[0.5cm]

-University of Bremen\\

-}

-\maketitle

-

-%\begin{abstract}

-%\end{abstract}

-

-

-\thispagestyle{empty} \setcounter{page}{0}

-

-\clearpage

-

-\tableofcontents

-

-

-\textbf{Key words}: Kinetics, FOCUS, nonlinear optimisation

-

-\section{Kinetic evaluations for parent compounds}

-

-These examples are also evaluated in a parallel vignette of the

-\Rpackage{kinfit} package \citep{pkg:kinfit}. The datasets are from Appendix 3,

-of the FOCUS kinetics report \citep{FOCUS2006, FOCUSkinetics2011}.

-

-\subsection{Laboratory Data L1}

-

-The following code defines example dataset L1 from the FOCUS kinetics

-report, p. 284

-

-<<FOCUS_2006_L1_data, echo=TRUE, eval=TRUE>>=

-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 \texttt{parent}.

-

-<<Simple_models, echo=TRUE>>=

-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.

-

-<<L1_SFO, echo=TRUE>>=

-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.

-

-<<L1_SFO_plot, fig=TRUE, echo=TRUE, height=4>>=

-plot(m.L1.SFO)

-@

-

-The residual plot can be easily obtained by

-

-<<L1_SFO_residuals, fig=TRUE, echo=TRUE, height=4>>=

-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.

-

-<<L1_FOMC, echo=TRUE>>=

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

-summary(m.L1.FOMC)

-@

-

-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.

-

-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 \texttt{mkin}. The reason for this is not known. However,

-\texttt{mkin} gives the same $\chi^2$ error levels as the \Rpackage{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.

-

-\subsection{Laboratory Data L2}

-

-The following code defines example dataset L2 from the FOCUS kinetics

-report, p. 287

-

-<<FOCUS_2006_L2_data, echo=TRUE, eval=TRUE>>=

-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.

-

-<<L2_SFO, echo=TRUE>>=

-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.

-

-<<L2_SFO_plot, fig=TRUE, echo=TRUE, 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 \textit{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.

-

-<<L2_FOMC, echo=TRUE, fig=TRUE, 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. 

-

-<<L2_DFOP, echo=TRUE, fig=TRUE, height=4>>=

-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.

-

-<<L2_DFOP_2, echo=TRUE, fig=TRUE, height=4>>=

-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.

-

-\subsection{Laboratory Data L3}

-

-The following code defines example dataset L3 from the FOCUS kinetics report,

-p. 290.

-

-<<FOCUS_2006_L3_data, echo=TRUE, eval=TRUE>>=

-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:

-

-<<L3_SFO, echo=TRUE, fig=TRUE, height=4>>=

-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 22\% as well as the plot suggest that the model

-does not fit very well. 

-

-The FOMC model performs better:

-

-<<L3_FOMC, echo=TRUE, fig=TRUE, height=4>>=

-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:

-

-<<L3_DFOP, echo=TRUE, fig=TRUE, height=4>>=

-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.

-

-\subsection{Laboratory Data L4}

-

-The following code defines example dataset L4 from the FOCUS kinetics

-report, p. 293

-

-<<FOCUS_2006_L4_data, echo=TRUE, eval=TRUE>>=

-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:

-

-<<L4_SFO, echo=TRUE, fig=TRUE, height=4>>=

-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

-

-<<L4_FOMC, echo=TRUE, fig=TRUE, height=4>>=

-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.

-

-\section{Kinetic evaluations for parent and metabolites}

-

-\subsection{Laboratory Data for example compound Z}

-

-The following code defines the example dataset from Appendix 7 to the FOCUS kinetics

-report, p.350 

-

-<<FOCUS_2006_Z_data, echo=TRUE, eval=TRUE>>=

-LOD = 0.5

-FOCUS_2006_Z = data.frame(

-  t = c(0, 0.04, 0.125, 0.29, 0.54, 1, 2, 3, 4, 7, 10, 14, 21, 

-        42, 61, 96, 124),

-  Z0 = c(100, 81.7, 70.4, 51.1, 41.2, 6.6, 4.6, 3.9, 4.6, 4.3, 6.8, 

-         2.9, 3.5, 5.3, 4.4, 1.2, 0.7),

-  Z1 = c(0, 18.3, 29.6, 46.3, 55.1, 65.7, 39.1, 36, 15.3, 5.6, 1.1, 

-         1.6, 0.6, 0.5 * LOD, NA, NA, NA),

-  Z2 = c(0, NA, 0.5 * LOD, 2.6, 3.8, 15.3, 37.2, 31.7, 35.6, 14.5, 

-         0.8, 2.1, 1.9, 0.5 * LOD, NA, NA, NA),

-  Z3 = c(0, NA, NA, NA, NA, 0.5 * LOD, 9.2, 13.1, 22.3, 28.4, 32.5, 

-         25.2, 17.2, 4.8, 4.5, 2.8, 4.4))

-

-FOCUS_2006_Z_mkin <- mkin_wide_to_long(FOCUS_2006_Z)

-@

-

-The next step is to set up the models used for the kinetic analysis. As the 

-simultaneous fit of parent and the first metabolite is usually straightforward,

-Step 1 (SFO for parent only) is skipped here. We start with the model 2a, 

-with formation and decline of metabolite Z1 and the pathway from parent

-directly to sink included (default in mkin).

-

-<<FOCUS_2006_Z_fits_1, echo=TRUE, fig=TRUE, height=4>>=

-Z.2a <- mkinmod(Z0 = list(type = "SFO", to = "Z1"),

-                Z1 = list(type = "SFO"))

-m.Z.2a <- mkinfit(Z.2a, FOCUS_2006_Z_mkin, quiet = TRUE)

-plot(m.Z.2a)

-summary(m.Z.2a, data = FALSE)

-@

-

-As obvious from the summary, the kinetic rate constant from parent compound Z to sink

-is negligible. Accordingly, the exact magnitude of the fitted parameter 

-\texttt{log k\_Z\_sink} is ill-defined and the covariance matrix is not returned.

-This suggests, in agreement with the analysis in the FOCUS kinetics report, to simplify 

-the model by removing the pathway to sink.

-

-A similar result can be obtained when formation fractions are used in the model formulation:

-

-<<FOCUS_2006_Z_fits_2, echo=TRUE, fig=TRUE, height=4>>=

-Z.2a.ff <- mkinmod(Z0 = list(type = "SFO", to = "Z1"),

-                Z1 = list(type = "SFO"), use_of_ff = "max")

-

-m.Z.2a.ff <- mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE)

-plot(m.Z.2a.ff)

-summary(m.Z.2a.ff, data = FALSE)

-@

-

-Here, the ilr transformed formation fraction fitted in the model takes a very large value, 

-and the backtransformed formation fraction from parent Z to Z1 is practically unity. Again,

-the covariance matrix is not returned as the model is overparameterised. 

-

-The simplified model is obtained by setting the list component \texttt{sink} to

-\texttt{FALSE}. This model definition is not supported when formation fractions 

-are used.

-

-<<FOCUS_2006_Z_fits_3, echo=TRUE, fig=TRUE, height=4>>=

-Z.3 <- mkinmod(Z0 = list(type = "SFO", to = "Z1", sink = FALSE),

-               Z1 = list(type = "SFO"))

-m.Z.3 <- mkinfit(Z.3, FOCUS_2006_Z_mkin, parms.ini = c(k_Z0_Z1 = 0.5), 

-                quiet = TRUE)

-#m.Z.3 <- mkinfit(Z.3, FOCUS_2006_Z_mkin, solution_type = "deSolve") 

-plot(m.Z.3)

-summary(m.Z.3, data = FALSE)

-@

-

-The first attempt to fit the model failed, as the default solution type chosen

-by mkinfit is based on eigenvalues, and the system defined by the starting

-parameters is identified as being singular to the solver. This is caused by the

-fact that the rate constants for both state variables are the same using the

-default starting paramters. Setting a different starting value for one of the

-parameters overcomes this. Alternatively, the \Rpackage{deSolve} based model

-solution can be chosen, at the cost of a bit more computing time.

-

-<<FOCUS_2006_Z_fits_4, echo=TRUE, fig=TRUE, height=4>>=

-Z.4a <- mkinmod(Z0 = list(type = "SFO", to = "Z1", sink = FALSE),

-                Z1 = list(type = "SFO", to = "Z2"),

-                Z2 = list(type = "SFO"))

-m.Z.4a <- mkinfit(Z.4a, FOCUS_2006_Z_mkin, parms.ini = c(k_Z0_Z1 = 0.5),

-                  quiet = TRUE)

-plot(m.Z.4a)

-summary(m.Z.4a, data = FALSE)

-@

-

-As suggested in the FOCUS report, the pathway to sink was removed for metabolite Z1 as

-well in the next step. While this step appears questionable on the basis of the above results, it 

-is followed here for the purpose of comparison. Also, in the FOCUS report, it is 

-assumed that there is additional empirical evidence that Z1 quickly and exclusively

-hydrolyses to Z2. Again, in order to avoid a singular system when using default starting

-parameters, the starting parameter for the pathway without sink term has to be adapted.

-

-<<FOCUS_2006_Z_fits_5, echo=TRUE, fig=TRUE, height=4>>=

-Z.5 <- mkinmod(Z0 = list(type = "SFO", to = "Z1", sink = FALSE),

-                Z1 = list(type = "SFO", to = "Z2", sink = FALSE),

-                Z2 = list(type = "SFO"))

-m.Z.5 <- mkinfit(Z.5, FOCUS_2006_Z_mkin, 

-                  parms.ini = c(k_Z0_Z1 = 0.5, k_Z1_Z2 = 0.2), quiet = TRUE)

-plot(m.Z.5)

-summary(m.Z.5, data = FALSE)

-@

-

-Finally, metabolite Z3 is added to the model.

-

-<<FOCUS_2006_Z_fits_6, echo=TRUE, fig=TRUE, height=4>>=

-Z.FOCUS <- mkinmod(Z0 = list(type = "SFO", to = "Z1", sink = FALSE),

-                Z1 = list(type = "SFO", to = "Z2", sink = FALSE),

-                Z2 = list(type = "SFO", to = "Z3"),

-                Z3 = list(type = "SFO"))

-m.Z.FOCUS <- mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin, 

-                  parms.ini = c(k_Z0_Z1 = 0.5, k_Z1_Z2 = 0.2, k_Z2_Z3 = 0.3),

-                  quiet = TRUE)

-plot(m.Z.FOCUS)

-summary(m.Z.FOCUS, data = FALSE)

-@

-

-This is the fit corresponding to the final result chosen in Appendix 7 of the 

-FOCUS report. The residual plots can be obtained by

-

-<<FOCUS_2006_Z_residuals_6, echo=TRUE, fig=TRUE>>=

-par(mfrow = c(2, 2))

-mkinresplot(m.Z.FOCUS, "Z0", lpos = "bottomright")

-mkinresplot(m.Z.FOCUS, "Z1", lpos = "bottomright")

-mkinresplot(m.Z.FOCUS, "Z2", lpos = "bottomright")

-mkinresplot(m.Z.FOCUS, "Z3", lpos = "bottomright")

-@

-

-As the FOCUS report states, there is a certain tailing of the time course of metabolite 

-Z3. Also, the time course of the parent compound is not fitted very well using the 

-SFO model, as residues at a certain low level remain.

-

-Therefore, an additional model is offered here, using the single first-order 

-reversible binding (SFORB) model for metabolite Z3. As expected, the $\chi^2$

-error level is lower for metabolite Z3 using this model and the graphical 

-fit for Z3 is improved. However, the covariance matrix is not returned.

-

-<<FOCUS_2006_Z_fits_7, echo=TRUE, fig=TRUE, height=4>>=

-Z.mkin.1 <- mkinmod(Z0 = list(type = "SFO", to = "Z1", sink = FALSE),

-                Z1 = list(type = "SFO", to = "Z2", sink = FALSE),

-                Z2 = list(type = "SFO", to = "Z3"),

-                Z3 = list(type = "SFORB"))

-m.Z.mkin.1 <- mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, 

-                  parms.ini = c(k_Z0_Z1 = 0.5, k_Z1_Z2 = 0.3),

-                  quiet = TRUE)

-plot(m.Z.mkin.1)

-summary(m.Z.mkin.1, data = FALSE)

-@

-

-Therefore, a further stepwise model building is performed starting from the

-stage of parent and one metabolite, starting from the assumption that the model

-fit for the parent compound can be improved by using the SFORB model.

-

-<<FOCUS_2006_Z_fits_8, echo=TRUE, fig=TRUE, height=4>>=

-Z.mkin.2 <- mkinmod(Z0 = list(type = "SFORB", to = "Z1", sink = FALSE),

-                Z1 = list(type = "SFO"))

-m.Z.mkin.2 <- mkinfit(Z.mkin.2, FOCUS_2006_Z_mkin, quiet = TRUE)

-plot(m.Z.mkin.2)

-summary(m.Z.mkin.2, data = FALSE)

-@

-

-When metabolite Z2 is added, the additional sink for Z1 is turned off again,

-for the same reasons as in the original analysis.

-

-<<FOCUS_2006_Z_fits_9, echo=TRUE, fig=TRUE, height=4>>=

-Z.mkin.3 <- mkinmod(Z0 = list(type = "SFORB", to = "Z1", sink = FALSE),

-                Z1 = list(type = "SFO", to = "Z2"),

-                Z2 = list(type = "SFO"))

-m.Z.mkin.3 <- mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE)

-plot(m.Z.mkin.3)

-summary(m.Z.mkin.3, data = FALSE)

-@

-

-This results in a much better representation of the behaviour of the parent 

-compound Z0.

-

-Finally, Z3 is added as well. This model appears overparameterised (no

-covariance matrix returned) if the sink for Z1 is left in the model.

-

-<<FOCUS_2006_Z_fits_10, echo=TRUE, fig=TRUE, height=4>>=

-Z.mkin.4 <- mkinmod(Z0 = list(type = "SFORB", to = "Z1", sink = FALSE),

-                Z1 = list(type = "SFO", to = "Z2", sink = FALSE),

-                Z2 = list(type = "SFO", to = "Z3"),

-                Z3 = list(type = "SFO"))

-m.Z.mkin.4 <- mkinfit(Z.mkin.4, FOCUS_2006_Z_mkin, 

-  parms.ini = c(k_Z1_Z2 = 0.05), quiet = TRUE)

-plot(m.Z.mkin.4)

-summary(m.Z.mkin.4, data = FALSE)

-@

-

-The error level of the fit, but especially of metabolite Z3, can be improved if

-the SFORB model is chosen for this metabolite, as this model is capable of

-representing the tailing of the metabolite decline phase.

-

-Using the SFORB additionally for Z1 or Z2 did not further improve the result.

-

-<<FOCUS_2006_Z_fits_11, echo=TRUE, fig=TRUE, height=4>>=

-Z.mkin.5 <- mkinmod(Z0 = list(type = "SFORB", to = "Z1", sink = FALSE),

-                Z1 = list(type = "SFO", to = "Z2", sink = FALSE),

-                Z2 = list(type = "SFO", to = "Z3"),

-                Z3 = list(type = "SFORB"))

-m.Z.mkin.5 <- mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin, 

-  parms.ini = c(k_Z1_Z2 = 0.2), quiet = TRUE)

-plot(m.Z.mkin.5)

-summary(m.Z.mkin.5, data = FALSE)

-@

-

-Looking at the confidence intervals of the SFORB model parameters of Z3, it is

-clear that nothing can be said about the degradation rate of Z3 towards the end 

-of the experiment. However, this appears to be a feature of the data.

-

-<<FOCUS_2006_Z_residuals_11, fig=TRUE>>=

-par(mfrow = c(2, 2))

-mkinresplot(m.Z.mkin.5, "Z0", lpos = "bottomright")

-mkinresplot(m.Z.mkin.5, "Z1", lpos = "bottomright")

-mkinresplot(m.Z.mkin.5, "Z2", lpos = "bottomright")

-mkinresplot(m.Z.mkin.5, "Z3", lpos = "bottomright")

-@

-

-As expected, the residual plots are much more random than in the case of the 

-all SFO model for which they were shown above. In conclusion, the model

-\texttt{Z.mkin.5} is proposed as the best-fit model for the dataset from

-Appendix 7 of the FOCUS report.

-

-\bibliographystyle{plainnat}

-\bibliography{references}

-

-

-\end{document}

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