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<title>Example evaluation of FOCUS Laboratory Data L1 to L3</title>

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<!--
%\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>Here we use the assumptions of simple first order (SFO), the case of declining
rate constant over time (FOMC) and the case of two different phases of the
kinetics (DFOP). For a more detailed discussion of the models, please see the
FOCUS kinetics report.</p>

<p>Since mkin version 0.9-32 (July 2014), we can use shorthand notation like <code>SFO</code>
for parent only degradation models. The following two lines fit the model and
produce the summary report of the model fit. This covers the numerical analysis
given in the FOCUS report. </p>

<pre><code class="r">m.L1.SFO &lt;- mkinfit(&quot;SFO&quot;, 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 21 09:14:29 2014 
## Date of summary: Mon Jul 21 09:14:29 2014 
## 
## Equations:
## [1] d_parent = - k_parent_sink * parent
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 14 model solutions performed in 0.081 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##               value   type
## parent_0      89.85  state
## k_parent_sink  0.10 deparm
## 
## Starting values for the transformed parameters actually optimised:
##                    value lower upper
## parent_0          89.850  -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 5.16e-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-4"/> </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-5"/> </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(&quot;FOMC&quot;, 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 21 09:14:30 2014 
## Date of summary: Mon Jul 21 09:14:30 2014 
## 
## Equations:
## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 53 model solutions performed in 0.32 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##          value   type
## parent_0 89.85  state
## alpha     1.00 deparm
## beta     10.00 deparm
## 
## Starting values for the transformed parameters actually optimised:
##            value lower upper
## parent_0  89.850  -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       1.45 89.40  95.6   63.60 1.17e-19 5.85e-20
## log_alpha     14.9      10.60 -7.75  37.5    1.40 1.82e-01 9.08e-02
## log_beta      17.2      10.60 -5.38  39.8    1.62 1.25e-01 6.26e-02
## 
## Parameter correlation:
##           parent_0 log_alpha log_beta
## parent_0     1.000      0.24    0.238
## log_alpha    0.240      1.00    1.000
## log_beta     0.238      1.00    1.000
## 
## Residual standard error: 3.05 on 15 degrees of freedom
## 
## Backtransformed parameters:
##          Estimate    Lower    Upper
## parent_0 9.25e+01 8.94e+01 9.56e+01
## alpha    2.85e+06 4.32e-04 1.88e+16
## beta     2.98e+07 4.59e-03 1.93e+17
## 
## 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 parameters
<code>log_alpha</code> and <code>log_beta</code> internally fitted in the model have p-values for the two
sided t-test of 0.18 and 0.125, and their correlation is 1.000, indicating that
the model is overparameterised. </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.  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. </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(&quot;SFO&quot;, 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 21 09:14:30 2014 
## Date of summary: Mon Jul 21 09:14:30 2014 
## 
## Equations:
## [1] d_parent = - k_parent_sink * parent
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 29 model solutions performed in 0.155 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##               value   type
## parent_0      93.95  state
## k_parent_sink  0.10 deparm
## 
## Starting values for the transformed parameters actually optimised:
##                    value lower upper
## parent_0          93.950  -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    4.635
##     0   parent     91.8  9.15e+01    0.335
##     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-9"/> </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(&quot;FOMC&quot;, 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-10"/> </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 21 09:14:31 2014 
## Date of summary: Mon Jul 21 09:14:31 2014 
## 
## Equations:
## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 35 model solutions performed in 0.199 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##          value   type
## parent_0 93.95  state
## alpha     1.00 deparm
## beta     10.00 deparm
## 
## Starting values for the transformed parameters actually optimised:
##            value lower upper
## parent_0  93.950  -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(&quot;DFOP&quot;, FOCUS_2006_L2_mkin, quiet = TRUE)
plot(m.L2.DFOP)
</code></pre>

<p><img 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alt="plot of chunk unnamed-chunk-11"/> </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(&quot;DFOP&quot;, FOCUS_2006_L2_mkin, 
  parms.ini = c(k1 = 1, k2 = 0.01, g = 0.8),
  quiet=TRUE)
plot(m.L2.DFOP)
</code></pre>

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alt="plot of chunk unnamed-chunk-12"/> </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 21 09:14:31 2014 
## Date of summary: Mon Jul 21 09:14:31 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
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 43 model solutions performed in 0.241 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##          value   type
## parent_0 93.95  state
## k1        1.00 deparm
## k2        0.01 deparm
## g         0.80 deparm
## 
## Starting values for the transformed parameters actually optimised:
##            value lower upper
## parent_0 93.9500  -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   94.000         NA    NA    NA      NA       NA     NA
## log_k1      6.160         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   94.000    NA    NA
## k1        476.000    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 0.00146    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(&quot;SFO&quot;, FOCUS_2006_L3_mkin, quiet = TRUE)
plot(m.L3.SFO)
</code></pre>

<p><img 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alt="plot of chunk unnamed-chunk-14"/> </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 21 09:14:32 2014 
## Date of summary: Mon Jul 21 09:14:32 2014 
## 
## Equations:
## [1] d_parent = - k_parent_sink * parent
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 44 model solutions performed in 0.242 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##               value   type
## parent_0       97.8  state
## k_parent_sink   0.1 deparm
## 
## Starting values for the transformed parameters actually optimised:
##                    value lower upper
## parent_0          97.800  -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  22.9274
##     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(&quot;FOMC&quot;, FOCUS_2006_L3_mkin, quiet = TRUE)
plot(m.L3.FOMC)
</code></pre>

<p><img 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alt="plot of chunk unnamed-chunk-15"/> </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 21 09:14:32 2014 
## Date of summary: Mon Jul 21 09:14:32 2014 
## 
## Equations:
## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 26 model solutions performed in 0.143 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##          value   type
## parent_0  97.8  state
## alpha      1.0 deparm
## beta      10.0 deparm
## 
## Starting values for the transformed parameters actually optimised:
##            value lower upper
## parent_0  97.800  -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(&quot;DFOP&quot;, FOCUS_2006_L3_mkin, quiet = TRUE)
plot(m.L3.DFOP)
</code></pre>

<p><img 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alt="plot of chunk unnamed-chunk-16"/> </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 21 09:14:32 2014 
## Date of summary: Mon Jul 21 09:14:32 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
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 37 model solutions performed in 0.21 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##          value   type
## parent_0 97.80  state
## k1        0.10 deparm
## k2        0.01 deparm
## g         0.50 deparm
## 
## Starting values for the transformed parameters actually optimised:
##           value lower upper
## parent_0 97.800  -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>

<p>This is also an example where the standard t-test for the parameter <code>g_ilr</code> is
misleading, as it tests for a significant difference from zero. In this case, 
zero appears to be the correct value for this parameter, and the confidence 
interval for the backtransformed parameter <code>g</code> is quite narrow.</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(&quot;SFO&quot;, FOCUS_2006_L4_mkin, quiet = TRUE)
plot(m.L4.SFO)
</code></pre>

<p><img 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alt="plot of chunk unnamed-chunk-18"/> </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 21 09:14:33 2014 
## Date of summary: Mon Jul 21 09:14:33 2014 
## 
## Equations:
## [1] d_parent = - k_parent_sink * parent
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 20 model solutions performed in 0.109 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##               value   type
## parent_0       96.6  state
## k_parent_sink   0.1 deparm
## 
## Starting values for the transformed parameters actually optimised:
##                    value lower upper
## parent_0          96.600  -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(&quot;FOMC&quot;, FOCUS_2006_L4_mkin, quiet = TRUE)
plot(m.L4.FOMC)
</code></pre>

<p><img 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alt="plot of chunk unnamed-chunk-19"/> </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 21 09:14:33 2014 
## Date of summary: Mon Jul 21 09:14:33 2014 
## 
## Equations:
## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
## 
## Model predictions using solution type analytical 
## 
## Fitted with method Marq using 48 model solutions performed in 0.26 s
## 
## Weighting: none
## 
## Starting values for parameters to be optimised:
##          value   type
## parent_0  96.6  state
## alpha      1.0 deparm
## beta      10.0 deparm
## 
## Starting values for the transformed parameters actually optimised:
##            value lower upper
## parent_0  96.600  -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>

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