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+hljs.initHighlightingOnLoad(); +</script> + +<!-- MathJax scripts --> +<script type="text/javascript" src="https://c328740.ssl.cf1.rackcdn.com/mathjax/2.0-latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> +</script> + + +</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("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>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 <- mkinmod(parent = list(type = "SFO")) +FOMC <- mkinmod(parent = list(type = "FOMC")) +DFOP <- mkinmod(parent = list(type = "DFOP")) +</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 <- 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(>|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(>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 = "Observed", xlab = "Time") +</code></pre> + +<p><img src="data:image/png;base64,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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 <- 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(>|t|) Pr(>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 <- 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: 0.9.32 +## 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(>|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(>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 <- 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(>|t|) Pr(>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 <- 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 <- 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(>|t|) Pr(>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 <- 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-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(>|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(>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 <- 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(>|t|) Pr(>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 <- 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(>|t|) Pr(>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 <- 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-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(>|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(>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 <- 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(>|t|) Pr(>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 Binary files differindex a002f831..a3f0ce1e 100644 --- a/vignettes/FOCUS_Z.pdf +++ b/vignettes/FOCUS_Z.pdf diff --git a/vignettes/mkin.pdf b/vignettes/mkin.pdf Binary files differindex dca871a9..991e54d3 100644 --- a/vignettes/mkin.pdf +++ b/vignettes/mkin.pdf |