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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("mkin")
FOCUS_2006_L1 = data.frame(
t = rep(c(0, 1, 2, 3, 5, 7, 14, 21, 30), each = 2),
parent = c(88.3, 91.4, 85.6, 84.5, 78.9, 77.6,
72.0, 71.9, 50.3, 59.4, 47.0, 45.1,
27.7, 27.3, 10.0, 10.4, 2.9, 4.0))
FOCUS_2006_L1_mkin <- mkin_wide_to_long(FOCUS_2006_L1)
</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 <- mkinfit("SFO", FOCUS_2006_L1_mkin, quiet=TRUE)
summary(m.L1.SFO)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:49 2014
## Date of summary: Thu Dec 18 09:52:49 2014
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 37 model solutions performed in 0.214 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 89.850000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 92.470 1.36800 89.570 95.370 67.58 4.339e-21
## log_k_parent_sink -2.347 0.04057 -2.433 -2.261 -57.86 5.155e-20
## Pr(>t)
## parent_0 2.170e-21
## log_k_parent_sink 2.577e-20
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.0000 0.6248
## log_k_parent_sink 0.6248 1.0000
##
## Residual standard error: 2.948 on 16 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 92.47000 89.57000 95.3700
## k_parent_sink 0.09561 0.08773 0.1042
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 3.424 2 7
## parent 3.424 2 7
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 7.249 24.08
##
## Data:
## time variable observed predicted residual
## 0 parent 88.3 92.471 -4.1710
## 0 parent 91.4 92.471 -1.0710
## 1 parent 85.6 84.039 1.5610
## 1 parent 84.5 84.039 0.4610
## 2 parent 78.9 76.376 2.5241
## 2 parent 77.6 76.376 1.2241
## 3 parent 72.0 69.412 2.5884
## 3 parent 71.9 69.412 2.4884
## 5 parent 50.3 57.330 -7.0301
## 5 parent 59.4 57.330 2.0699
## 7 parent 47.0 47.352 -0.3515
## 7 parent 45.1 47.352 -2.2515
## 14 parent 27.7 24.247 3.4528
## 14 parent 27.3 24.247 3.0528
## 21 parent 10.0 12.416 -2.4163
## 21 parent 10.4 12.416 -2.0163
## 30 parent 2.9 5.251 -2.3513
## 30 parent 4.0 5.251 -1.2513
</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"/>
The residual plot can be easily obtained by</p>
<pre><code class="r">mkinresplot(m.L1.SFO, ylab = "Observed", xlab = "Time")
</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 <- mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet=TRUE)
</code></pre>
<pre><code>## Warning in mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet = TRUE): Optimisation by method Port did not converge.
## Convergence code is 1
</code></pre>
<pre><code class="r">summary(m.L1.FOMC, data = FALSE)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:50 2014
## Date of summary: Thu Dec 18 09:52:50 2014
##
##
## Warning: Optimisation by method Port did not converge.
## Convergence code is 1
##
##
## Equations:
## d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 188 model solutions performed in 1.085 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 89.850000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 92.47 1.422 89.44 95.50 65.030 8.317e-20 4.158e-20
## log_alpha 15.43 15.080 -16.71 47.58 1.023 3.224e-01 1.612e-01
## log_beta 17.78 15.090 -14.37 49.93 1.179 2.569e-01 1.284e-01
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.0000 0.1129 0.1112
## log_alpha 0.1129 1.0000 1.0000
## log_beta 0.1112 1.0000 1.0000
##
## Residual standard error: 3.045 on 15 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 9.247e+01 8.944e+01 9.550e+01
## alpha 5.044e+06 5.510e-08 4.618e+20
## beta 5.276e+07 5.732e-07 4.857e+21
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 3.619 3 6
## parent 3.619 3 6
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 7.25 24.08 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 <- 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 <- mkinfit("SFO", FOCUS_2006_L2_mkin, quiet=TRUE)
summary(m.L2.SFO)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:51 2014
## Date of summary: Thu Dec 18 09:52:51 2014
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 41 model solutions performed in 0.238 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 93.950000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 91.4700 3.8070 82.9800 99.9500 24.030 3.545e-10
## log_k_parent_sink -0.4112 0.1074 -0.6505 -0.1719 -3.828 3.329e-03
## Pr(>t)
## parent_0 1.773e-10
## log_k_parent_sink 1.664e-03
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.0000 0.4295
## log_k_parent_sink 0.4295 1.0000
##
## Residual standard error: 5.51 on 10 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 91.4700 82.9800 99.9500
## k_parent_sink 0.6629 0.5218 0.8421
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 14.38 2 4
## parent 14.38 2 4
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 1.046 3.474
##
## Data:
## time variable observed predicted residual
## 0 parent 96.1 9.147e+01 4.6343
## 0 parent 91.8 9.147e+01 0.3343
## 1 parent 41.4 4.714e+01 -5.7394
## 1 parent 38.7 4.714e+01 -8.4394
## 3 parent 19.3 1.252e+01 6.7790
## 3 parent 22.3 1.252e+01 9.7790
## 7 parent 4.6 8.834e-01 3.7166
## 7 parent 4.6 8.834e-01 3.7166
## 14 parent 2.6 8.532e-03 2.5915
## 14 parent 1.2 8.532e-03 1.1915
## 28 parent 0.3 7.958e-07 0.3000
## 28 parent 0.6 7.958e-07 0.6000
</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 <- 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-10"/> </p>
<pre><code class="r">summary(m.L2.FOMC, data = FALSE)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:51 2014
## Date of summary: Thu Dec 18 09:52:51 2014
##
## Equations:
## d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 81 model solutions performed in 0.464 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 93.950000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 93.7700 1.8560 89.5700 97.9700 50.5100 2.345e-12 1.173e-12
## log_alpha 0.3180 0.1867 -0.1044 0.7405 1.7030 1.227e-01 6.137e-02
## log_beta 0.2102 0.2943 -0.4555 0.8759 0.7142 4.932e-01 2.466e-01
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.00000 -0.09553 -0.1863
## log_alpha -0.09553 1.00000 0.9757
## log_beta -0.18628 0.97567 1.0000
##
## Residual standard error: 2.628 on 9 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 93.770 89.5700 97.970
## alpha 1.374 0.9009 2.097
## beta 1.234 0.6341 2.401
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 6.205 3 3
## parent 6.205 3 3
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 0.8092 5.356 1.612
</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 <- 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-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 <- 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-12"/> </p>
<pre><code class="r">summary(m.L2.DFOP, data = FALSE)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:56 2014
## Date of summary: Thu Dec 18 09:52:56 2014
##
## Equations:
## 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 Port using 336 model solutions performed in 1.986 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 93.9500000 -Inf Inf
## log_k1 0.0000000 -Inf Inf
## log_k2 -4.6051702 -Inf Inf
## g_ilr 0.9802581 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 93.9500 NA NA NA NA NA NA
## log_k1 3.1210 NA NA NA NA NA NA
## log_k2 -1.0880 NA NA NA NA NA NA
## g_ilr -0.2821 NA NA NA NA NA NA
##
## Parameter correlation:
## Could not estimate covariance matrix; singular system:
##
## Residual standard error: 1.732 on 8 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 93.9500 NA NA
## k1 22.6700 NA NA
## k2 0.3369 NA NA
## g 0.4016 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 0.03058 2.058
</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 <- mkin_wide_to_long(FOCUS_2006_L3)
</code></pre>
<p>SFO model, summary and plot:</p>
<pre><code class="r">m.L3.SFO <- 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-14"/> </p>
<pre><code class="r">summary(m.L3.SFO)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:56 2014
## Date of summary: Thu Dec 18 09:52:56 2014
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 43 model solutions performed in 0.246 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 97.800000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 74.870 8.4570 54.180 95.57 8.853 1.155e-04
## log_k_parent_sink -3.678 0.3261 -4.476 -2.88 -11.280 2.903e-05
## Pr(>t)
## parent_0 5.776e-05
## log_k_parent_sink 1.451e-05
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.0000 0.5483
## log_k_parent_sink 0.5483 1.0000
##
## Residual standard error: 12.91 on 6 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 74.87000 54.18000 95.57000
## k_parent_sink 0.02527 0.01138 0.05612
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 21.24 2 6
## parent 21.24 2 6
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 27.43 91.13
##
## Data:
## time variable observed predicted residual
## 0 parent 97.8 74.872 22.9281
## 3 parent 60.0 69.406 -9.4061
## 7 parent 51.0 62.734 -11.7340
## 14 parent 43.0 52.564 -9.5638
## 30 parent 35.0 35.084 -0.0839
## 60 parent 22.0 16.440 5.5602
## 91 parent 15.0 7.511 7.4887
## 120 parent 12.0 3.610 8.3903
</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 <- 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-15"/> </p>
<pre><code class="r">summary(m.L3.FOMC, data = FALSE)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:57 2014
## Date of summary: Thu Dec 18 09:52:57 2014
##
## Equations:
## d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 83 model solutions performed in 0.475 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 97.800000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 96.9700 4.5500 85.2800 108.7000 21.310 4.216e-06 2.108e-06
## log_alpha -0.8619 0.1704 -1.3000 -0.4238 -5.057 3.911e-03 1.955e-03
## log_beta 0.6193 0.4744 -0.6003 1.8390 1.305 2.486e-01 1.243e-01
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.0000 -0.1512 -0.4271
## log_alpha -0.1512 1.0000 0.9110
## log_beta -0.4271 0.9110 1.0000
##
## Residual standard error: 4.572 on 5 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 96.9700 85.2800 108.7000
## alpha 0.4224 0.2725 0.6546
## beta 1.8580 0.5487 6.2890
##
## 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 7.729 431.2 129.8
</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 <- 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-16"/> </p>
<pre><code class="r">summary(m.L3.DFOP, data = FALSE)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:58 2014
## Date of summary: Thu Dec 18 09:52:58 2014
##
## Equations:
## 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 Port using 137 model solutions performed in 0.813 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 97.800000 -Inf Inf
## log_k1 -2.302585 -Inf Inf
## log_k2 -4.605170 -Inf Inf
## g_ilr 0.000000 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 97.7500 1.43800 93.7500 101.70000 67.970 2.808e-07 1.404e-07
## log_k1 -0.6612 0.13340 -1.0310 -0.29100 -4.958 7.715e-03 3.858e-03
## log_k2 -4.2860 0.05902 -4.4500 -4.12200 -72.620 2.155e-07 1.077e-07
## g_ilr -0.1229 0.05121 -0.2651 0.01925 -2.401 7.431e-02 3.716e-02
##
## Parameter correlation:
## parent_0 log_k1 log_k2 g_ilr
## parent_0 1.00000 0.1640 0.01315 0.4253
## log_k1 0.16400 1.0000 0.46478 -0.5526
## log_k2 0.01315 0.4648 1.00000 -0.6631
## g_ilr 0.42526 -0.5526 -0.66310 1.0000
##
## Residual standard error: 1.439 on 4 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 97.75000 93.75000 101.70000
## k1 0.51620 0.35650 0.74750
## k2 0.01376 0.01168 0.01621
## g 0.45660 0.40730 0.50680
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 2.225 4 4
## parent 2.225 4 4
##
## Estimated disappearance times:
## DT50 DT90 DT50_k1 DT50_k2
## parent 7.464 123 1.343 50.37
</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 <- mkin_wide_to_long(FOCUS_2006_L4)
</code></pre>
<p>SFO model, summary and plot:</p>
<pre><code class="r">m.L4.SFO <- 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-18"/> </p>
<pre><code class="r">summary(m.L4.SFO, data = FALSE)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:58 2014
## Date of summary: Thu Dec 18 09:52:58 2014
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 46 model solutions performed in 0.264 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 96.600000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 96.44 1.94900 91.670 101.200 49.49 4.566e-09
## log_k_parent_sink -5.03 0.07999 -5.225 -4.834 -62.88 1.088e-09
## Pr(>t)
## parent_0 2.283e-09
## log_k_parent_sink 5.438e-10
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.0000 0.5865
## log_k_parent_sink 0.5865 1.0000
##
## Residual standard error: 3.651 on 6 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 96.440000 91.670000 1.012e+02
## k_parent_sink 0.006541 0.005378 7.955e-03
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 3.287 2 6
## parent 3.287 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 <- 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-19"/> </p>
<pre><code class="r">summary(m.L4.FOMC, data = FALSE)
</code></pre>
<pre><code>## mkin version: 0.9.34
## R version: 3.1.2
## Date of fit: Thu Dec 18 09:52:59 2014
## Date of summary: Thu Dec 18 09:52:59 2014
##
## Equations:
## d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 66 model solutions performed in 0.375 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 96.600000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 99.1400 1.6800 94.820 103.5000 59.0200 2.643e-08 1.322e-08
## log_alpha -0.3506 0.3725 -1.308 0.6068 -0.9414 3.897e-01 1.949e-01
## log_beta 4.1740 0.5635 2.726 5.6230 7.4070 7.059e-04 3.530e-04
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.0000 -0.5365 -0.6083
## log_alpha -0.5365 1.0000 0.9913
## log_beta -0.6083 0.9913 1.0000
##
## Residual standard error: 2.315 on 5 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 99.1400 94.8200 103.500
## alpha 0.7042 0.2703 1.835
## beta 64.9800 15.2600 276.600
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 2.029 3 5
## parent 2.029 3 5
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 108.9 1644 494.9
</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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