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+
+<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;)
+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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:26 2014
+## Date of summary: Mon Jul 14 19:59:26 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 src="data:image/png;base64,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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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:26 2014
+## Date of summary: Mon Jul 14 19:59:26 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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:27 2014
+## Date of summary: Mon Jul 14 19:59:27 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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:28 2014
+## Date of summary: Mon Jul 14 19:59:28 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 src="data:image/png;base64,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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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:28 2014
+## Date of summary: Mon Jul 14 19:59:28 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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:29 2014
+## Date of summary: Mon Jul 14 19:59:29 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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:29 2014
+## Date of summary: Mon Jul 14 19:59:29 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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:30 2014
+## Date of summary: Mon Jul 14 19:59:30 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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:30 2014
+## Date of summary: Mon Jul 14 19:59:30 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: 0.9.32
+## R version: 3.1.1
+## Date of fit: Mon Jul 14 19:59:31 2014
+## Date of summary: Mon Jul 14 19:59:31 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>
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--- a/vignettes/mkin.pdf
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