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authorJohannes Ranke <jranke@uni-bremen.de>2014-07-14 18:25:53 +0200
committerJohannes Ranke <jranke@uni-bremen.de>2014-07-14 18:25:53 +0200
commit759e693e9af8e794bbfa62b001117fabbdbc8bfa (patch)
tree2c4fb2232763595924090c76ae95e0b87dd76f40
parenta9a3b38a2ca5bc7223435f43814cfc6c7a1077bd (diff)
Bugfix release version 0.9-31
Diffstat
-rw-r--r--.gitignore1
-rw-r--r--DESCRIPTION2
-rw-r--r--NEWS.md8
-rw-r--r--R/mkinerrmin.R29
-rw-r--r--R/mkinfit.R36
-rw-r--r--vignettes/FOCUS_L.html1090
-rw-r--r--vignettes/FOCUS_Z.pdfbin213000 -> 212998 bytes
-rw-r--r--vignettes/mkin.pdfbin160326 -> 160326 bytes
8 files changed, 1136 insertions, 30 deletions
diff --git a/.gitignore b/.gitignore
index 9dc50c23..2c416f60 100644
--- a/.gitignore
+++ b/.gitignore
@@ -7,5 +7,4 @@ vignettes/*.log
vignettes/*.out
vignettes/*.toc
vignettes/mkin.tex
-vignettes/FOCUS_L.html
vignettes/FOCUS_Z.tex
diff --git a/DESCRIPTION b/DESCRIPTION
index 245f1643..69889b21 100644
--- a/DESCRIPTION
+++ b/DESCRIPTION
@@ -3,7 +3,7 @@ Type: Package
Title: Routines for fitting kinetic models with one or more state
variables to chemical degradation data
Version: 0.9-31
-Date: 2014-07-11
+Date: 2014-07-14
Authors@R: c(person("Johannes", "Ranke", role = c("aut", "cre", "cph"),
email = "jranke@uni-bremen.de"),
person("Katrin", "Lindenberger", role = "ctb"),
diff --git a/NEWS.md b/NEWS.md
index 6c23f316..94bc566a 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -1,3 +1,11 @@
+# CHANGES in mkin VERSION 0.9-31
+
+## BUG FIXES
+
+- The internal renaming of optimised parameters in Version 0.9-30 led to errors in the determination of the degrees of freedom for the chi2 error level calulations in `mkinerrmin()` used by the summary function.
+
+- Initial values for formation fractions were not set in all cases
+
# CHANGES in mkin VERSION 0.9-30
## NEW FEATURES
diff --git a/R/mkinerrmin.R b/R/mkinerrmin.R
index 671bcaab..9ebac6a4 100644
--- a/R/mkinerrmin.R
+++ b/R/mkinerrmin.R
@@ -48,6 +48,7 @@ mkinerrmin <- function(fit, alpha = 0.05)
n.optim = errmin.overall$n.optim, df = errmin.overall$df)
rownames(errmin) <- "All data"
+ # The degrees of freedom are counted according to FOCUS kinetics (2011, p. 164)
for (obs_var in fit$obs_vars)
{
errdata.var <- subset(errdata, name == obs_var)
@@ -57,21 +58,31 @@ mkinerrmin <- function(fit, alpha = 0.05)
# Rate constants are attributed to the source variable
n.k.optim <- length(grep(paste("^k", obs_var, sep="_"), names(parms.optim)))
-
- # Formation fractions are attributed to the target variable
- n.ff.optim <- length(grep(paste("^f", ".*", obs_var, "$", sep=""), names(parms.optim)))
+ n.k.optim <- n.k.optim + length(grep(paste("^log_k", obs_var, sep="_"),
+ names(parms.optim)))
+
+ n.ff.optim <- 0
+ # Formation fractions are attributed to the target variable, so look
+ # for source compartments with formation fractions
+ for (source_var in fit$obs_vars) {
+ for (target_var in fit$mkinmod$spec[[source_var]]$to) {
+ if (obs_var == target_var) {
+ n.ff.optim <- n.ff.optim +
+ length(grep(paste("^f", source_var, sep = "_"),
+ names(parms.optim)))
+ }
+ }
+ }
n.optim <- n.k.optim + n.initials.optim + n.ff.optim
# FOMC, DFOP and HS parameters are only counted if we are looking at the
# first variable in the model which is always the source variable
if (obs_var == fit$obs_vars[[1]]) {
- if ("alpha" %in% names(parms.optim)) n.optim <- n.optim + 1
- if ("beta" %in% names(parms.optim)) n.optim <- n.optim + 1
- if ("k1" %in% names(parms.optim)) n.optim <- n.optim + 1
- if ("k2" %in% names(parms.optim)) n.optim <- n.optim + 1
- if ("g" %in% names(parms.optim)) n.optim <- n.optim + 1
- if ("tb" %in% names(parms.optim)) n.optim <- n.optim + 1
+ special_parms = c("alpha", "log_alpha", "beta", "log_beta",
+ "k1", "log_k1", "k2", "log_k2",
+ "g", "g_ilr", "tb", "log_tb")
+ n.optim <- n.optim + length(intersect(special_parms, names(parms.optim)))
}
# Calculate and add a line to the results
diff --git a/R/mkinfit.R b/R/mkinfit.R
index b7ca1d74..c6e13b97 100644
--- a/R/mkinfit.R
+++ b/R/mkinfit.R
@@ -105,26 +105,24 @@ mkinfit <- function(mkinmod, observed,
if (parmname == "tb") parms.ini[parmname] = 5
if (parmname == "g") parms.ini[parmname] = 0.5
}
- # Default values for formation fractions in case they are used
- if (mkinmod$use_of_ff == "max") {
- for (box in mod_vars) {
- f_names <- mkinmod$parms[grep(paste0("^f_", box), mkinmod$parms)]
- if (length(f_names) > 0) {
- # We need to differentiate between default and specified fractions
- # and set the unspecified to 1 - sum(specified)/n_unspecified
- f_default_names <- intersect(f_names, defaultpar.names)
- f_specified_names <- setdiff(f_names, defaultpar.names)
- sum_f_specified = sum(parms.ini[f_specified_names])
- if (sum_f_specified > 1) {
- stop("Starting values for the formation fractions originating from ",
- box, " sum up to more than 1.")
- }
- if (mkinmod$spec[[box]]$sink) n_unspecified = length(f_default_names) + 1
- else {
- n_unspecified = length(f_default_names)
- }
- parms.ini[f_default_names] <- (1 - sum_f_specified) / n_unspecified
+ # Default values for formation fractions in case they are present
+ for (box in mod_vars) {
+ f_names <- mkinmod$parms[grep(paste0("^f_", box), mkinmod$parms)]
+ if (length(f_names) > 0) {
+ # We need to differentiate between default and specified fractions
+ # and set the unspecified to 1 - sum(specified)/n_unspecified
+ f_default_names <- intersect(f_names, defaultpar.names)
+ f_specified_names <- setdiff(f_names, defaultpar.names)
+ sum_f_specified = sum(parms.ini[f_specified_names])
+ if (sum_f_specified > 1) {
+ stop("Starting values for the formation fractions originating from ",
+ box, " sum up to more than 1.")
+ }
+ if (mkinmod$spec[[box]]$sink) n_unspecified = length(f_default_names) + 1
+ else {
+ n_unspecified = length(f_default_names)
}
+ parms.ini[f_default_names] <- (1 - sum_f_specified) / n_unspecified
}
}
diff --git a/vignettes/FOCUS_L.html b/vignettes/FOCUS_L.html
new file mode 100644
index 00000000..b44481e9
--- /dev/null
+++ b/vignettes/FOCUS_L.html
@@ -0,0 +1,1090 @@
+<!DOCTYPE html>
+<html>
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
+
+<title>Example evaluation of FOCUS Laboratory Data L1 to L3</title>
+
+<!-- Styles for R syntax highlighter -->
+<style type="text/css">
+ pre .operator,
+ pre .paren {
+ color: rgb(104, 118, 135)
+ }
+
+ pre .literal {
+ color: rgb(88, 72, 246)
+ }
+
+ pre .number {
+ color: rgb(0, 0, 205);
+ }
+
+ pre .comment {
+ color: rgb(76, 136, 107);
+ }
+
+ pre .keyword {
+ color: rgb(0, 0, 255);
+ }
+
+ pre .identifier {
+ color: rgb(0, 0, 0);
+ }
+
+ pre .string {
+ color: rgb(3, 106, 7);
+ }
+</style>
+
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+</head>
+
+<body>
+<!--
+%\VignetteEngine{knitr::knitr}
+%\VignetteIndexEntry{Example evaluation of FOCUS Laboratory Data L1 to L3}
+-->
+
+<h1>Example evaluation of FOCUS Laboratory Data L1 to L3</h1>
+
+<h2>Laboratory Data L1</h2>
+
+<p>The following code defines example dataset L1 from the FOCUS kinetics
+report, p. 284</p>
+
+<pre><code class="r">library(&quot;mkin&quot;)
+</code></pre>
+
+<pre><code>## Loading required package: minpack.lm
+## Loading required package: rootSolve
+</code></pre>
+
+<pre><code class="r">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 &lt;- mkin_wide_to_long(FOCUS_2006_L1)
+</code></pre>
+
+<p>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 <code>parent</code>.</p>
+
+<pre><code class="r">SFO &lt;- mkinmod(parent = list(type = &quot;SFO&quot;))
+FOMC &lt;- mkinmod(parent = list(type = &quot;FOMC&quot;))
+DFOP &lt;- mkinmod(parent = list(type = &quot;DFOP&quot;))
+</code></pre>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<pre><code class="r">m.L1.SFO &lt;- mkinfit(SFO, FOCUS_2006_L1_mkin, quiet=TRUE)
+summary(m.L1.SFO)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:12 2014
+## Date of summary: Mon Jul 14 12:36:12 2014
+##
+## Equations:
+## [1] d_parent = - k_parent_sink * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 100.0 state
+## k_parent_sink 0.1 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.000 -Inf Inf
+## log_k_parent_sink -2.303 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|)
+## parent_0 92.50 1.3700 89.60 95.40 67.6 4.34e-21
+## log_k_parent_sink -2.35 0.0406 -2.43 -2.26 -57.9 5.15e-20
+## Pr(&gt;t)
+## parent_0 2.17e-21
+## log_k_parent_sink 2.58e-20
+##
+## Parameter correlation:
+## parent_0 log_k_parent_sink
+## parent_0 1.000 0.625
+## log_k_parent_sink 0.625 1.000
+##
+## Residual standard error: 2.95 on 16 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 92.5000 89.6000 95.400
+## k_parent_sink 0.0956 0.0877 0.104
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 3.42 2 7
+## parent 3.42 2 7
+##
+## Resulting formation fractions:
+## ff
+## parent_sink 1
+##
+## Estimated disappearance times:
+## DT50 DT90
+## parent 7.25 24.1
+##
+## Data:
+## time variable observed predicted residual
+## 0 parent 88.3 92.47 -4.171
+## 0 parent 91.4 92.47 -1.071
+## 1 parent 85.6 84.04 1.561
+## 1 parent 84.5 84.04 0.461
+## 2 parent 78.9 76.38 2.524
+## 2 parent 77.6 76.38 1.224
+## 3 parent 72.0 69.41 2.588
+## 3 parent 71.9 69.41 2.488
+## 5 parent 50.3 57.33 -7.030
+## 5 parent 59.4 57.33 2.070
+## 7 parent 47.0 47.35 -0.352
+## 7 parent 45.1 47.35 -2.252
+## 14 parent 27.7 24.25 3.453
+## 14 parent 27.3 24.25 3.053
+## 21 parent 10.0 12.42 -2.416
+## 21 parent 10.4 12.42 -2.016
+## 30 parent 2.9 5.25 -2.351
+## 30 parent 4.0 5.25 -1.251
+</code></pre>
+
+<p>A plot of the fit is obtained with the plot function for mkinfit objects.</p>
+
+<pre><code class="r">plot(m.L1.SFO)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-5"/> </p>
+
+<p>The residual plot can be easily obtained by</p>
+
+<pre><code class="r">mkinresplot(m.L1.SFO, ylab = &quot;Observed&quot;, xlab = &quot;Time&quot;)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-6"/> </p>
+
+<p>For comparison, the FOMC model is fitted as well, and the chi<sup>2</sup> error level
+is checked.</p>
+
+<pre><code class="r">m.L1.FOMC &lt;- mkinfit(FOMC, FOCUS_2006_L1_mkin, quiet=TRUE)
+summary(m.L1.FOMC, data = FALSE)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:12 2014
+## Date of summary: Mon Jul 14 12:36:12 2014
+##
+## Equations:
+## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 100 state
+## alpha 1 deparm
+## beta 10 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.000 -Inf Inf
+## log_alpha 0.000 -Inf Inf
+## log_beta 2.303 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|) Pr(&gt;t)
+## parent_0 92.5 NA NA NA NA NA NA
+## log_alpha 25.6 NA NA NA NA NA NA
+## log_beta 28.0 NA NA NA NA NA NA
+##
+## Parameter correlation:
+## Could not estimate covariance matrix; singular system:
+##
+## Residual standard error: 3.05 on 15 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 9.25e+01 NA NA
+## alpha 1.35e+11 NA NA
+## beta 1.41e+12 NA NA
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 3.62 3 6
+## parent 3.62 3 6
+##
+## Estimated disappearance times:
+## DT50 DT90 DT50back
+## parent 7.25 24.1 7.25
+</code></pre>
+
+<p>Due to the higher number of parameters, and the lower number of degrees of
+freedom of the fit, the chi<sup>2</sup> 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.</p>
+
+<p>The chi<sup>2</sup> 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<sup>2</sup> error levels
+as the kinfit package.</p>
+
+<p>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.</p>
+
+<h2>Laboratory Data L2</h2>
+
+<p>The following code defines example dataset L2 from the FOCUS kinetics
+report, p. 287</p>
+
+<pre><code class="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 &lt;- mkin_wide_to_long(FOCUS_2006_L2)
+</code></pre>
+
+<p>Again, the SFO model is fitted and a summary is obtained.</p>
+
+<pre><code class="r">m.L2.SFO &lt;- mkinfit(SFO, FOCUS_2006_L2_mkin, quiet=TRUE)
+summary(m.L2.SFO)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:12 2014
+## Date of summary: Mon Jul 14 12:36:13 2014
+##
+## Equations:
+## [1] d_parent = - k_parent_sink * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 100.0 state
+## k_parent_sink 0.1 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.000 -Inf Inf
+## log_k_parent_sink -2.303 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|)
+## parent_0 91.500 3.810 83.000 99.900 24.00 3.55e-10
+## log_k_parent_sink -0.411 0.107 -0.651 -0.172 -3.83 3.33e-03
+## Pr(&gt;t)
+## parent_0 1.77e-10
+## log_k_parent_sink 1.66e-03
+##
+## Parameter correlation:
+## parent_0 log_k_parent_sink
+## parent_0 1.00 0.43
+## log_k_parent_sink 0.43 1.00
+##
+## Residual standard error: 5.51 on 10 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 91.500 83.000 99.900
+## k_parent_sink 0.663 0.522 0.842
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 14.4 2 4
+## parent 14.4 2 4
+##
+## Resulting formation fractions:
+## ff
+## parent_sink 1
+##
+## Estimated disappearance times:
+## DT50 DT90
+## parent 1.05 3.47
+##
+## Data:
+## time variable observed predicted residual
+## 0 parent 96.1 9.15e+01 4.634
+## 0 parent 91.8 9.15e+01 0.334
+## 1 parent 41.4 4.71e+01 -5.740
+## 1 parent 38.7 4.71e+01 -8.440
+## 3 parent 19.3 1.25e+01 6.779
+## 3 parent 22.3 1.25e+01 9.779
+## 7 parent 4.6 8.83e-01 3.717
+## 7 parent 4.6 8.83e-01 3.717
+## 14 parent 2.6 8.53e-03 2.591
+## 14 parent 1.2 8.53e-03 1.191
+## 28 parent 0.3 7.96e-07 0.300
+## 28 parent 0.6 7.96e-07 0.600
+</code></pre>
+
+<p>The chi<sup>2</sup> 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.</p>
+
+<pre><code class="r">par(mfrow = c(2, 1))
+plot(m.L2.SFO)
+mkinresplot(m.L2.SFO)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-10"/> </p>
+
+<p>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.</p>
+
+<p>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 <em>a
+priori</em> 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.</p>
+
+<p>For comparison, the FOMC model is fitted as well, and the chi<sup>2</sup> error level
+is checked.</p>
+
+<pre><code class="r">m.L2.FOMC &lt;- mkinfit(FOMC, FOCUS_2006_L2_mkin, quiet = TRUE)
+par(mfrow = c(2, 1))
+plot(m.L2.FOMC)
+mkinresplot(m.L2.FOMC)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-11"/> </p>
+
+<pre><code class="r">summary(m.L2.FOMC, data = FALSE)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:13 2014
+## Date of summary: Mon Jul 14 12:36:13 2014
+##
+## Equations:
+## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 100 state
+## alpha 1 deparm
+## beta 10 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.000 -Inf Inf
+## log_alpha 0.000 -Inf Inf
+## log_beta 2.303 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|) Pr(&gt;t)
+## parent_0 93.800 1.860 89.600 98.000 50.500 2.35e-12 1.17e-12
+## log_alpha 0.318 0.187 -0.104 0.740 1.700 1.23e-01 6.14e-02
+## log_beta 0.210 0.294 -0.456 0.876 0.714 4.93e-01 2.47e-01
+##
+## Parameter correlation:
+## parent_0 log_alpha log_beta
+## parent_0 1.0000 -0.0955 -0.186
+## log_alpha -0.0955 1.0000 0.976
+## log_beta -0.1863 0.9757 1.000
+##
+## Residual standard error: 2.63 on 9 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 93.80 89.600 98.0
+## alpha 1.37 0.901 2.1
+## beta 1.23 0.634 2.4
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 6.2 3 3
+## parent 6.2 3 3
+##
+## Estimated disappearance times:
+## DT50 DT90 DT50back
+## parent 0.809 5.36 1.61
+</code></pre>
+
+<p>The error level at which the chi<sup>2</sup> 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.</p>
+
+<p>Fitting the four parameter DFOP model further reduces the chi<sup>2</sup> error level. </p>
+
+<pre><code class="r">m.L2.DFOP &lt;- mkinfit(DFOP, FOCUS_2006_L2_mkin, quiet = TRUE)
+plot(m.L2.DFOP)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-12"/> </p>
+
+<p>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.</p>
+
+<pre><code class="r">m.L2.DFOP &lt;- mkinfit(DFOP, FOCUS_2006_L2_mkin,
+ parms.ini = c(k1 = 1, k2 = 0.01, g = 0.8),
+ quiet=TRUE)
+plot(m.L2.DFOP)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-13"/> </p>
+
+<pre><code class="r">summary(m.L2.DFOP, data = FALSE)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:13 2014
+## Date of summary: Mon Jul 14 12:36:13 2014
+##
+## Equations:
+## [1] d_parent = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time))) * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 1e+02 state
+## k1 1e+00 deparm
+## k2 1e-02 deparm
+## g 8e-01 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.0000 -Inf Inf
+## log_k1 0.0000 -Inf Inf
+## log_k2 -4.6052 -Inf Inf
+## g_ilr 0.9803 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|) Pr(&gt;t)
+## parent_0 93.900 NA NA NA NA NA NA
+## log_k1 4.960 NA NA NA NA NA NA
+## log_k2 -1.090 NA NA NA NA NA NA
+## g_ilr -0.282 NA NA NA NA NA NA
+##
+## Parameter correlation:
+## Could not estimate covariance matrix; singular system:
+##
+## Residual standard error: 1.73 on 8 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 93.900 NA NA
+## k1 142.000 NA NA
+## k2 0.337 NA NA
+## g 0.402 NA NA
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 2.53 4 2
+## parent 2.53 4 2
+##
+## Estimated disappearance times:
+## DT50 DT90 DT50_k1 DT50_k2
+## parent NA NA 0.00487 2.06
+</code></pre>
+
+<p>Here, the DFOP model is clearly the best-fit model for dataset L2 based on the
+chi<sup>2</sup> 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.</p>
+
+<h2>Laboratory Data L3</h2>
+
+<p>The following code defines example dataset L3 from the FOCUS kinetics report,
+p. 290.</p>
+
+<pre><code class="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 &lt;- mkin_wide_to_long(FOCUS_2006_L3)
+</code></pre>
+
+<p>SFO model, summary and plot:</p>
+
+<pre><code class="r">m.L3.SFO &lt;- mkinfit(SFO, FOCUS_2006_L3_mkin, quiet = TRUE)
+plot(m.L3.SFO)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-15"/> </p>
+
+<pre><code class="r">summary(m.L3.SFO)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:14 2014
+## Date of summary: Mon Jul 14 12:36:14 2014
+##
+## Equations:
+## [1] d_parent = - k_parent_sink * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 100.0 state
+## k_parent_sink 0.1 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.000 -Inf Inf
+## log_k_parent_sink -2.303 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|)
+## parent_0 74.90 8.460 54.20 95.60 8.85 0.000116
+## log_k_parent_sink -3.68 0.326 -4.48 -2.88 -11.30 0.000029
+## Pr(&gt;t)
+## parent_0 5.78e-05
+## log_k_parent_sink 1.45e-05
+##
+## Parameter correlation:
+## parent_0 log_k_parent_sink
+## parent_0 1.000 0.548
+## log_k_parent_sink 0.548 1.000
+##
+## Residual standard error: 12.9 on 6 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 74.9000 54.2000 95.6000
+## k_parent_sink 0.0253 0.0114 0.0561
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 21.2 2 6
+## parent 21.2 2 6
+##
+## Resulting formation fractions:
+## ff
+## parent_sink 1
+##
+## Estimated disappearance times:
+## DT50 DT90
+## parent 27.4 91.1
+##
+## Data:
+## time variable observed predicted residual
+## 0 parent 97.8 74.87 22.9273
+## 3 parent 60.0 69.41 -9.4065
+## 7 parent 51.0 62.73 -11.7340
+## 14 parent 43.0 52.56 -9.5634
+## 30 parent 35.0 35.08 -0.0828
+## 60 parent 22.0 16.44 5.5614
+## 91 parent 15.0 7.51 7.4896
+## 120 parent 12.0 3.61 8.3908
+</code></pre>
+
+<p>The chi<sup>2</sup> error level of 21% as well as the plot suggest that the model
+does not fit very well. </p>
+
+<p>The FOMC model performs better:</p>
+
+<pre><code class="r">m.L3.FOMC &lt;- mkinfit(FOMC, FOCUS_2006_L3_mkin, quiet = TRUE)
+plot(m.L3.FOMC)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-16"/> </p>
+
+<pre><code class="r">summary(m.L3.FOMC, data = FALSE)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:14 2014
+## Date of summary: Mon Jul 14 12:36:14 2014
+##
+## Equations:
+## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 100 state
+## alpha 1 deparm
+## beta 10 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.000 -Inf Inf
+## log_alpha 0.000 -Inf Inf
+## log_beta 2.303 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|) Pr(&gt;t)
+## parent_0 97.000 4.550 85.3 109.000 21.30 4.22e-06 2.11e-06
+## log_alpha -0.862 0.170 -1.3 -0.424 -5.06 3.91e-03 1.96e-03
+## log_beta 0.619 0.474 -0.6 1.840 1.31 2.49e-01 1.24e-01
+##
+## Parameter correlation:
+## parent_0 log_alpha log_beta
+## parent_0 1.000 -0.151 -0.427
+## log_alpha -0.151 1.000 0.911
+## log_beta -0.427 0.911 1.000
+##
+## Residual standard error: 4.57 on 5 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 97.000 85.300 109.000
+## alpha 0.422 0.273 0.655
+## beta 1.860 0.549 6.290
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 7.32 3 5
+## parent 7.32 3 5
+##
+## Estimated disappearance times:
+## DT50 DT90 DT50back
+## parent 7.73 431 130
+</code></pre>
+
+<p>The error level at which the chi<sup>2</sup> test passes is 7% in this case.</p>
+
+<p>Fitting the four parameter DFOP model further reduces the chi<sup>2</sup> error level
+considerably:</p>
+
+<pre><code class="r">m.L3.DFOP &lt;- mkinfit(DFOP, FOCUS_2006_L3_mkin, quiet = TRUE)
+plot(m.L3.DFOP)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-17"/> </p>
+
+<pre><code class="r">summary(m.L3.DFOP, data = FALSE)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:14 2014
+## Date of summary: Mon Jul 14 12:36:14 2014
+##
+## Equations:
+## [1] d_parent = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time))) * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 1e+02 state
+## k1 1e-01 deparm
+## k2 1e-02 deparm
+## g 5e-01 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.000 -Inf Inf
+## log_k1 -2.303 -Inf Inf
+## log_k2 -4.605 -Inf Inf
+## g_ilr 0.000 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|) Pr(&gt;t)
+## parent_0 97.700 1.4400 93.800 102.0000 68.00 2.81e-07 1.40e-07
+## log_k1 -0.661 0.1330 -1.030 -0.2910 -4.96 7.72e-03 3.86e-03
+## log_k2 -4.290 0.0590 -4.450 -4.1200 -72.60 2.15e-07 1.08e-07
+## g_ilr -0.123 0.0512 -0.265 0.0193 -2.40 7.43e-02 3.72e-02
+##
+## Parameter correlation:
+## parent_0 log_k1 log_k2 g_ilr
+## parent_0 1.0000 0.164 0.0131 0.425
+## log_k1 0.1640 1.000 0.4648 -0.553
+## log_k2 0.0131 0.465 1.0000 -0.663
+## g_ilr 0.4253 -0.553 -0.6631 1.000
+##
+## Residual standard error: 1.44 on 4 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 97.7000 93.8000 102.0000
+## k1 0.5160 0.3560 0.7480
+## k2 0.0138 0.0117 0.0162
+## g 0.4570 0.4070 0.5070
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 2.23 4 4
+## parent 2.23 4 4
+##
+## Estimated disappearance times:
+## DT50 DT90 DT50_k1 DT50_k2
+## parent 7.46 123 1.34 50.4
+</code></pre>
+
+<p>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<sup>2</sup> error
+level criterion for laboratory data L3.</p>
+
+<h2>Laboratory Data L4</h2>
+
+<p>The following code defines example dataset L4 from the FOCUS kinetics
+report, p. 293</p>
+
+<pre><code class="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 &lt;- mkin_wide_to_long(FOCUS_2006_L4)
+</code></pre>
+
+<p>SFO model, summary and plot:</p>
+
+<pre><code class="r">m.L4.SFO &lt;- mkinfit(SFO, FOCUS_2006_L4_mkin, quiet = TRUE)
+plot(m.L4.SFO)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-19"/> </p>
+
+<pre><code class="r">summary(m.L4.SFO, data = FALSE)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:15 2014
+## Date of summary: Mon Jul 14 12:36:15 2014
+##
+## Equations:
+## [1] d_parent = - k_parent_sink * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 100.0 state
+## k_parent_sink 0.1 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.000 -Inf Inf
+## log_k_parent_sink -2.303 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|)
+## parent_0 96.40 1.95 91.70 101.00 49.5 4.57e-09
+## log_k_parent_sink -5.03 0.08 -5.23 -4.83 -62.9 1.09e-09
+## Pr(&gt;t)
+## parent_0 2.28e-09
+## log_k_parent_sink 5.44e-10
+##
+## Parameter correlation:
+## parent_0 log_k_parent_sink
+## parent_0 1.000 0.587
+## log_k_parent_sink 0.587 1.000
+##
+## Residual standard error: 3.65 on 6 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 96.40000 91.70000 1.01e+02
+## k_parent_sink 0.00654 0.00538 7.95e-03
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 3.29 2 6
+## parent 3.29 2 6
+##
+## Resulting formation fractions:
+## ff
+## parent_sink 1
+##
+## Estimated disappearance times:
+## DT50 DT90
+## parent 106 352
+</code></pre>
+
+<p>The chi<sup>2</sup> error level of 3.3% as well as the plot suggest that the model
+fits very well. </p>
+
+<p>The FOMC model for comparison</p>
+
+<pre><code class="r">m.L4.FOMC &lt;- mkinfit(FOMC, FOCUS_2006_L4_mkin, quiet = TRUE)
+plot(m.L4.FOMC)
+</code></pre>
+
+<p><img 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alt="plot of chunk unnamed-chunk-20"/> </p>
+
+<pre><code class="r">summary(m.L4.FOMC, data = FALSE)
+</code></pre>
+
+<pre><code>## mkin version: 0.9.31
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 12:36:15 2014
+## Date of summary: Mon Jul 14 12:36:15 2014
+##
+## Equations:
+## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for parameters to be optimised:
+## value type
+## parent_0 100 state
+## alpha 1 deparm
+## beta 10 deparm
+##
+## Starting values for the transformed parameters actually optimised:
+## value lower upper
+## parent_0 100.000 -Inf Inf
+## log_alpha 0.000 -Inf Inf
+## log_beta 2.303 -Inf Inf
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper t value Pr(&gt;|t|) Pr(&gt;t)
+## parent_0 99.100 1.680 94.80 103.000 59.000 2.64e-08 1.32e-08
+## log_alpha -0.351 0.372 -1.31 0.607 -0.941 3.90e-01 1.95e-01
+## log_beta 4.170 0.564 2.73 5.620 7.410 7.06e-04 3.53e-04
+##
+## Parameter correlation:
+## parent_0 log_alpha log_beta
+## parent_0 1.000 -0.536 -0.608
+## log_alpha -0.536 1.000 0.991
+## log_beta -0.608 0.991 1.000
+##
+## Residual standard error: 2.31 on 5 degrees of freedom
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 99.100 94.80 103.00
+## alpha 0.704 0.27 1.83
+## beta 65.000 15.30 277.00
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 2.03 3 5
+## parent 2.03 3 5
+##
+## Estimated disappearance times:
+## DT50 DT90 DT50back
+## parent 109 1644 495
+</code></pre>
+
+<p>The error level at which the chi<sup>2</sup> test passes is slightly lower for the FOMC
+model. However, the difference appears negligible.</p>
+
+</body>
+
+</html>
diff --git a/vignettes/FOCUS_Z.pdf b/vignettes/FOCUS_Z.pdf
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--- a/vignettes/mkin.pdf
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