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-%\VignetteIndexEntry{Example evaluation of FOCUS dataset Z}
-%\VignetteEngine{knitr::knitr}
-\documentclass[12pt,a4paper]{article}
-\usepackage{a4wide}
-\input{header}
-\hypersetup{
-  pdftitle = {Example evaluation of FOCUS dataset Z},
-  pdfsubject = {Manuscript},
-  pdfauthor = {Johannes Ranke},
-  colorlinks = {true},
-  linkcolor = {blue},
-  citecolor = {blue},
-  urlcolor = {red},
-  hyperindex = {true},
-  linktocpage = {true},
-}
-
-\begin{document}
-
-<<include=FALSE>>=
-require(knitr)
-opts_chunk$set(engine='R', tidy = FALSE, cache = TRUE)
-options(width=70)
-@
-
-\title{Example evaluation of FOCUS dataset Z}
-\author{\textbf{Johannes Ranke} \\[0.5cm]
-%EndAName
-Wissenschaftlicher Berater\\
-Kronacher Str. 8, 79639 Grenzach-Wyhlen, Germany\\[0.5cm]
-and\\[0.5cm]
-University of Bremen\\
-}
-\maketitle
-
-\thispagestyle{empty} \setcounter{page}{0}
-
-\clearpage
-
-\tableofcontents
-
-\textbf{Key words}: Kinetics, FOCUS, nonlinear optimisation
-
-\section{The data}
-
-The following code defines the example dataset from Appendix 7 to the FOCUS kinetics
-report \citep{FOCUSkinetics2011}, p.350.
-
-<<FOCUS_2006_Z_data, echo=TRUE, eval=TRUE>>=
-require(mkin)
-LOD = 0.5
-FOCUS_2006_Z = data.frame(
-  t = c(0, 0.04, 0.125, 0.29, 0.54, 1, 2, 3, 4, 7, 10, 14, 21,
-        42, 61, 96, 124),
-  Z0 = c(100, 81.7, 70.4, 51.1, 41.2, 6.6, 4.6, 3.9, 4.6, 4.3, 6.8,
-         2.9, 3.5, 5.3, 4.4, 1.2, 0.7),
-  Z1 = c(0, 18.3, 29.6, 46.3, 55.1, 65.7, 39.1, 36, 15.3, 5.6, 1.1,
-         1.6, 0.6, 0.5 * LOD, NA, NA, NA),
-  Z2 = c(0, NA, 0.5 * LOD, 2.6, 3.8, 15.3, 37.2, 31.7, 35.6, 14.5,
-         0.8, 2.1, 1.9, 0.5 * LOD, NA, NA, NA),
-  Z3 = c(0, NA, NA, NA, NA, 0.5 * LOD, 9.2, 13.1, 22.3, 28.4, 32.5,
-         25.2, 17.2, 4.8, 4.5, 2.8, 4.4))
-
-FOCUS_2006_Z_mkin <- mkin_wide_to_long(FOCUS_2006_Z)
-@
-
-\section{Parent compound and one metabolite}
-
-The next step is to set up the models used for the kinetic analysis. As the
-simultaneous fit of parent and the first metabolite is usually straightforward,
-Step 1 (SFO for parent only) is skipped here. We start with the model 2a,
-with formation and decline of metabolite Z1 and the pathway from parent
-directly to sink included (default in mkin).
-
-<<FOCUS_2006_Z_fits_1, echo=TRUE, fig.height=6>>=
-Z.2a <- mkinmod(Z0 = mkinsub("SFO", "Z1"),
-                Z1 = mkinsub("SFO"))
-m.Z.2a <- mkinfit(Z.2a, FOCUS_2006_Z_mkin, quiet = TRUE)
-plot_sep(m.Z.2a)
-summary(m.Z.2a, data = FALSE)$bpar
-@
-
-As obvious from the parameter summary (the \texttt{bpar} component of 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\_Z0\_sink} is ill-defined and the covariance matrix is not
-returned (not shown, would be visible in the complete summary).
-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.height=6>>=
-Z.2a.ff <- mkinmod(Z0 = mkinsub("SFO", "Z1"),
-                   Z1 = mkinsub("SFO"),
-                   use_of_ff = "max")
-
-m.Z.2a.ff <- mkinfit(Z.2a.ff, FOCUS_2006_Z_mkin, quiet = TRUE)
-plot_sep(m.Z.2a.ff)
-summary(m.Z.2a.ff, data = FALSE)$bpar
-@
-
-Here, the ilr transformed formation fraction fitted in the model takes a very
-large value, and the backtransformed formation fraction from parent Z to Z1 is
-practically unity. Again, the covariance matrix is not returned as the model is
-overparameterised.
-
-The simplified model is obtained by setting the list component \texttt{sink} to
-\texttt{FALSE}.\footnote{If the model formulation without formation fractions
-is used, the same effect can be obtained by fixing the parameter \texttt{k\_Z\_sink}
-to a value of zero.}
-
-In the following, we use the parameterisation with formation fractions in order
-to be able to compare with the results in the FOCUS guidance, and as it
-makes it easier to use parameters obtained in a previous fit when adding a further
-metabolite.
-
-<<FOCUS_2006_Z_fits_3, echo=TRUE, fig.height=6>>=
-Z.3 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-               Z1 = mkinsub("SFO"), use_of_ff = "max")
-m.Z.3 <- mkinfit(Z.3, FOCUS_2006_Z_mkin, quiet = TRUE)
-plot_sep(m.Z.3)
-summary(m.Z.3, data = FALSE)$bpar
-@
-
-As there is only one transformation product for Z0 and no pathway
-to sink, the formation fraction is internally fixed to unity.
-
-\section{Including metabolites Z2 and Z3}
-
-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.
-
-<<FOCUS_2006_Z_fits_5, echo=TRUE, fig.height=7>>=
-Z.5 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-               Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-               Z2 = mkinsub("SFO"), use_of_ff = "max")
-m.Z.5 <- mkinfit(Z.5, FOCUS_2006_Z_mkin, quiet = TRUE)
-plot_sep(m.Z.5)
-@
-
-Finally, metabolite Z3 is added to the model. We use the optimised
-differential equation parameter values from the previous fit in order to
-accelerate the optimization.
-
-<<FOCUS_2006_Z_fits_6, echo=TRUE, fig.height=8>>=
-Z.FOCUS <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-                   Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                   Z2 = mkinsub("SFO", "Z3"),
-                   Z3 = mkinsub("SFO"),
-                   use_of_ff = "max")
-m.Z.FOCUS <- mkinfit(Z.FOCUS, FOCUS_2006_Z_mkin,
-                     parms.ini = m.Z.5$bparms.ode,
-                     quiet = TRUE)
-plot_sep(m.Z.FOCUS)
-summary(m.Z.FOCUS, data = FALSE)$bpar
-endpoints(m.Z.FOCUS)
-@
-
-This fit corresponds to the final result chosen in Appendix 7 of the FOCUS
-report. Confidence intervals returned by mkin are based on internally
-transformed parameters, however.
-
-\section{Using the SFORB model for parent and metabolites}
-
-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.height=8>>=
-Z.mkin.1 <- mkinmod(Z0 = mkinsub("SFO", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFORB"))
-m.Z.mkin.1 <- mkinfit(Z.mkin.1, FOCUS_2006_Z_mkin, quiet = TRUE)
-plot_sep(m.Z.mkin.1)
-summary(m.Z.mkin.1, data = FALSE)$cov.unscaled
-@
-
-Therefore, a further stepwise model building is performed starting from the
-stage of parent and two metabolites, starting from the assumption that the model
-fit for the parent compound can be improved by using the SFORB model.
-
-<<FOCUS_2006_Z_fits_9, echo=TRUE, fig.height=8>>=
-Z.mkin.3 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO"))
-m.Z.mkin.3 <- mkinfit(Z.mkin.3, FOCUS_2006_Z_mkin, quiet = TRUE)
-plot_sep(m.Z.mkin.3)
-@
-
-This results in a much better representation of the behaviour of the parent
-compound Z0.
-
-Finally, Z3 is added as well. These models appear overparameterised (no
-covariance matrix returned) if the sink for Z1 is left in the models.
-
-<<FOCUS_2006_Z_fits_10, echo=TRUE, fig.height=8>>=
-Z.mkin.4 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFO"))
-m.Z.mkin.4 <- mkinfit(Z.mkin.4, FOCUS_2006_Z_mkin,
-                      parms.ini = m.Z.mkin.3$bparms.ode,
-                      quiet = TRUE)
-plot_sep(m.Z.mkin.4)
-@
-
-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.
-
-<<FOCUS_2006_Z_fits_11, echo=TRUE, fig.height=8>>=
-Z.mkin.5 <- mkinmod(Z0 = mkinsub("SFORB", "Z1", sink = FALSE),
-                    Z1 = mkinsub("SFO", "Z2", sink = FALSE),
-                    Z2 = mkinsub("SFO", "Z3"),
-                    Z3 = mkinsub("SFORB"))
-m.Z.mkin.5 <- mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin,
-                      parms.ini = m.Z.mkin.4$bparms.ode[1:4],
-                      quiet = TRUE)
-plot_sep(m.Z.mkin.5)
-@
-
-The summary view of the backtransformed parameters shows that we get no
-confidence intervals due to overparameterisation. As the optimized
-\texttt{k\_Z3\_bound\_free} is excessively small, it seems reasonable to fix it to
-zero.
-
-<<FOCUS_2006_Z_fits_11a, echo=TRUE>>=
-m.Z.mkin.5a <- mkinfit(Z.mkin.5, FOCUS_2006_Z_mkin,
-                       parms.ini = c(m.Z.mkin.5$bparms.ode[1:7],
-                                     k_Z3_bound_free = 0),
-                       fixed_parms = "k_Z3_bound_free",
-                       quiet = TRUE)
-plot_sep(m.Z.mkin.5a)
-@
-
-As expected, the residual plots for Z0 and Z3 are more random than in the case of the
-all SFO model for which they were shown above. In conclusion, the model
-\texttt{Z.mkin.5a} is proposed as the best-fit model for the dataset from
-Appendix 7 of the FOCUS report.
-
-A graphical representation of the confidence intervals can finally be obtained.
-
-<<FOCUS_2006_Z_fits_11b, echo=TRUE>>=
-mkinparplot(m.Z.mkin.5a)
-@
-
-The endpoints obtained with this model are
-
-<<FOCUS_2006_Z_fits_11b_endpoints, echo=TRUE>>=
-endpoints(m.Z.mkin.5a)
-@
-
-It is clear the degradation rate of Z3 towards the end of the experiment
-is very low as DT50\_Z3\_b2 (the second Eigenvalue of the system of two differential
-equations representing the SFORB system for Z3, corresponding to the slower rate
-constant of the DFOP model) is reported to be infinity.  However, this appears
-to be a feature of the data.
-
-\bibliographystyle{plainnat}
-\bibliography{references}
-
-\end{document}
-% vim: set foldmethod=syntax: