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authorjranke <jranke@edb9625f-4e0d-4859-8d74-9fd3b1da38cb>2013-11-17 15:52:42 +0000
committerjranke <jranke@edb9625f-4e0d-4859-8d74-9fd3b1da38cb>2013-11-17 15:52:42 +0000
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+<!--
+%\VignetteEngine{knitr::knitr}
+%\VignetteIndexEntry{Example evaluation of FOCUS Laboratory Data L1 to L3}
+-->
+
+# Example evaluation of FOCUS Laboratory Data L1 to L3
+
+## Laboratory Data L1
+
+The following code defines example dataset L1 from the FOCUS kinetics
+report, p. 284
+
+
+```r
+library("mkin")
+```
+
+```
+## Loading required package: FME
+## Loading required package: deSolve
+## Loading required package: rootSolve
+## Loading required package: minpack.lm
+## Loading required package: MASS
+## Loading required package: coda
+## Loading required package: lattice
+```
+
+```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, 71.9, 50.3, 59.4, 47,
+ 45.1, 27.7, 27.3, 10, 10.4, 2.9, 4))
+FOCUS_2006_L1_mkin <- mkin_wide_to_long(FOCUS_2006_L1)
+```
+
+
+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 `parent`.
+
+
+```r
+SFO <- mkinmod(parent = list(type = "SFO"))
+FOMC <- mkinmod(parent = list(type = "FOMC"))
+DFOP <- mkinmod(parent = list(type = "DFOP"))
+```
+
+
+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.
+
+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.
+
+
+```r
+m.L1.SFO <- mkinfit(SFO, FOCUS_2006_L1_mkin, quiet = TRUE)
+summary(m.L1.SFO)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:54 2013
+## Date of summary: Sun Nov 17 15:02:54 2013
+##
+## Equations:
+## [1] d_parent = - k_parent_sink * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for optimised parameters:
+## value type transformed
+## parent_0 100.0 state 100.000
+## k_parent_sink 0.1 deparm -2.303
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 92.50 1.3700 89.60 95.40
+## k_parent_sink -2.35 0.0406 -2.43 -2.26
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 92.5000 89.6000 95.400
+## k_parent_sink 0.0956 0.0877 0.104
+##
+## Residual standard error: 2.95 on 16 degrees of freedom
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 3.42 2 7
+## parent 3.42 2 7
+##
+## Estimated disappearance times:
+## DT50 DT90
+## parent 7.25 24.1
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## parent_0 k_parent_sink
+## parent_0 1.000 0.625
+## k_parent_sink 0.625 1.000
+##
+## 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
+```
+
+
+A plot of the fit is obtained with the plot function for mkinfit objects.
+
+
+```r
+plot(m.L1.SFO)
+```
+
+![plot of chunk unnamed-chunk-4](figure/unnamed-chunk-4.png)
+
+The residual plot can be easily obtained by
+
+
+```r
+mkinresplot(m.L1.SFO, ylab = "Observed", xlab = "Time")
+```
+
+![plot of chunk unnamed-chunk-5](figure/unnamed-chunk-5.png)
+
+
+For comparison, the FOMC model is fitted as well, and the chi^2 error level
+is checked.
+
+
+```r
+m.L1.FOMC <- mkinfit(FOMC, FOCUS_2006_L1_mkin, quiet = TRUE)
+summary(m.L1.FOMC, data = FALSE)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:55 2013
+## Date of summary: Sun Nov 17 15:02:55 2013
+##
+## 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 optimised parameters:
+## value type transformed
+## parent_0 100 state 100.000
+## alpha 1 deparm 0.000
+## beta 10 deparm 2.303
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 92.5 NA NA NA
+## alpha 25.6 NA NA NA
+## beta 28.0 NA NA NA
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 9.25e+01 NA NA
+## alpha 1.35e+11 NA NA
+## beta 1.41e+12 NA NA
+##
+## Residual standard error: 3.05 on 15 degrees of freedom
+##
+## 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
+## parent 7.25 24.1
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## Could not estimate covariance matrix; singular system:
+```
+
+
+Due to the higher number of parameters, and the lower number of degrees of
+freedom of the fit, the chi^2 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.
+
+The chi^2 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^2 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.
+Therefore, the reason for the difference was not investigated further.
+
+## Laboratory Data L2
+
+The following code defines example dataset L2 from the FOCUS kinetics
+report, p. 287
+
+
+```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)
+```
+
+
+Again, the SFO model is fitted and a summary is obtained.
+
+
+```r
+m.L2.SFO <- mkinfit(SFO, FOCUS_2006_L2_mkin, quiet = TRUE)
+summary(m.L2.SFO)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:55 2013
+## Date of summary: Sun Nov 17 15:02:55 2013
+##
+## Equations:
+## [1] d_parent = - k_parent_sink * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for optimised parameters:
+## value type transformed
+## parent_0 100.0 state 100.000
+## k_parent_sink 0.1 deparm -2.303
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 91.500 3.810 83.000 99.900
+## k_parent_sink -0.411 0.107 -0.651 -0.172
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 91.500 83.000 99.900
+## k_parent_sink 0.663 0.522 0.842
+##
+## Residual standard error: 5.51 on 10 degrees of freedom
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 14.4 2 4
+## parent 14.4 2 4
+##
+## Estimated disappearance times:
+## DT50 DT90
+## parent 1.05 3.47
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## parent_0 k_parent_sink
+## parent_0 1.00 0.43
+## k_parent_sink 0.43 1.00
+##
+## 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
+```
+
+
+The chi^2 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.
+
+
+```r
+par(mfrow = c(2, 1))
+plot(m.L2.SFO)
+mkinresplot(m.L2.SFO)
+```
+
+![plot of chunk unnamed-chunk-9](figure/unnamed-chunk-9.png)
+
+
+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.
+
+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 _a
+priori_ 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.
+
+For comparison, the FOMC model is fitted as well, and the chi^2 error level
+is checked.
+
+
+```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)
+```
+
+![plot of chunk unnamed-chunk-10](figure/unnamed-chunk-10.png)
+
+```r
+summary(m.L2.FOMC, data = FALSE)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:56 2013
+## Date of summary: Sun Nov 17 15:02:56 2013
+##
+## 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 optimised parameters:
+## value type transformed
+## parent_0 100 state 100.000
+## alpha 1 deparm 0.000
+## beta 10 deparm 2.303
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 93.800 1.860 89.600 98.000
+## alpha 0.318 0.187 -0.104 0.740
+## beta 0.210 0.294 -0.456 0.876
+##
+## 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
+##
+## Residual standard error: 2.63 on 9 degrees of freedom
+##
+## 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
+## parent 0.809 5.36
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## parent_0 alpha beta
+## parent_0 1.0000 -0.0955 -0.186
+## alpha -0.0955 1.0000 0.976
+## beta -0.1863 0.9757 1.000
+```
+
+
+The error level at which the chi^2 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.
+
+Fitting the four parameter DFOP model further reduces the chi^2 error level.
+
+
+```r
+m.L2.DFOP <- mkinfit(DFOP, FOCUS_2006_L2_mkin, quiet = TRUE)
+plot(m.L2.DFOP)
+```
+
+![plot of chunk unnamed-chunk-11](figure/unnamed-chunk-11.png)
+
+
+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.
+
+
+```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)
+```
+
+![plot of chunk unnamed-chunk-12](figure/unnamed-chunk-12.png)
+
+```r
+summary(m.L2.DFOP, data = FALSE)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:57 2013
+## Date of summary: Sun Nov 17 15:02:57 2013
+##
+## 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 optimised parameters:
+## value type transformed
+## parent_0 1e+02 state 100.0000
+## k1 1e+00 deparm 0.0000
+## k2 1e-02 deparm -4.6052
+## g 8e-01 deparm 0.9803
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 93.900 NA NA NA
+## k1 4.960 NA NA NA
+## k2 -1.090 NA NA NA
+## g -0.282 NA NA NA
+##
+## 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
+##
+## Residual standard error: 1.73 on 8 degrees of freedom
+##
+## 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
+## parent NA NA
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## Could not estimate covariance matrix; singular system:
+```
+
+
+Here, the DFOP model is clearly the best-fit model for dataset L2 based on the
+chi^2 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.
+
+## Laboratory Data L3
+
+The following code defines example dataset L3 from the FOCUS kinetics report,
+p. 290.
+
+
+```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)
+```
+
+
+SFO model, summary and plot:
+
+
+```r
+m.L3.SFO <- mkinfit(SFO, FOCUS_2006_L3_mkin, quiet = TRUE)
+plot(m.L3.SFO)
+```
+
+![plot of chunk unnamed-chunk-14](figure/unnamed-chunk-14.png)
+
+```r
+summary(m.L3.SFO)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:57 2013
+## Date of summary: Sun Nov 17 15:02:57 2013
+##
+## Equations:
+## [1] d_parent = - k_parent_sink * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for optimised parameters:
+## value type transformed
+## parent_0 100.0 state 100.000
+## k_parent_sink 0.1 deparm -2.303
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 74.90 8.460 54.20 95.60
+## k_parent_sink -3.68 0.326 -4.48 -2.88
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 74.9000 54.2000 95.6000
+## k_parent_sink 0.0253 0.0114 0.0561
+##
+## Residual standard error: 12.9 on 6 degrees of freedom
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 21.2 2 6
+## parent 21.2 2 6
+##
+## Estimated disappearance times:
+## DT50 DT90
+## parent 27.4 91.1
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## parent_0 k_parent_sink
+## parent_0 1.000 0.548
+## k_parent_sink 0.548 1.000
+##
+## 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
+```
+
+
+The chi^2 error level of 21% as well as the plot suggest that the model
+does not fit very well.
+
+The FOMC model performs better:
+
+
+```r
+m.L3.FOMC <- mkinfit(FOMC, FOCUS_2006_L3_mkin, quiet = TRUE)
+plot(m.L3.FOMC)
+```
+
+![plot of chunk unnamed-chunk-15](figure/unnamed-chunk-15.png)
+
+```r
+summary(m.L3.FOMC, data = FALSE)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:57 2013
+## Date of summary: Sun Nov 17 15:02:57 2013
+##
+## 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 optimised parameters:
+## value type transformed
+## parent_0 100 state 100.000
+## alpha 1 deparm 0.000
+## beta 10 deparm 2.303
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 97.000 4.550 85.3 109.000
+## alpha -0.862 0.170 -1.3 -0.424
+## beta 0.619 0.474 -0.6 1.840
+##
+## 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
+##
+## Residual standard error: 4.57 on 5 degrees of freedom
+##
+## 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
+## parent 7.73 431
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## parent_0 alpha beta
+## parent_0 1.000 -0.151 -0.427
+## alpha -0.151 1.000 0.911
+## beta -0.427 0.911 1.000
+```
+
+
+The error level at which the chi^2 test passes is 7% in this case.
+
+Fitting the four parameter DFOP model further reduces the chi^2 error level
+considerably:
+
+
+```r
+m.L3.DFOP <- mkinfit(DFOP, FOCUS_2006_L3_mkin, quiet = TRUE)
+plot(m.L3.DFOP)
+```
+
+![plot of chunk unnamed-chunk-16](figure/unnamed-chunk-16.png)
+
+```r
+summary(m.L3.DFOP, data = FALSE)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:58 2013
+## Date of summary: Sun Nov 17 15:02:58 2013
+##
+## 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 optimised parameters:
+## value type transformed
+## parent_0 1e+02 state 100.000
+## k1 1e-01 deparm -2.303
+## k2 1e-02 deparm -4.605
+## g 5e-01 deparm 0.000
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 97.700 1.4400 93.800 102.0000
+## k1 -0.661 0.1330 -1.030 -0.2910
+## k2 -4.290 0.0590 -4.450 -4.1200
+## g -0.123 0.0512 -0.265 0.0193
+##
+## 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
+##
+## Residual standard error: 1.44 on 4 degrees of freedom
+##
+## 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
+## parent 7.46 123
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## parent_0 k1 k2 g
+## parent_0 1.0000 0.164 0.0131 0.425
+## k1 0.1640 1.000 0.4648 -0.553
+## k2 0.0131 0.465 1.0000 -0.663
+## g 0.4253 -0.553 -0.6631 1.000
+```
+
+
+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^2 error
+level criterion for laboratory data L3.
+
+## Laboratory Data L4
+
+The following code defines example dataset L4 from the FOCUS kinetics
+report, p. 293
+
+
+```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))
+FOCUS_2006_L4_mkin <- mkin_wide_to_long(FOCUS_2006_L4)
+```
+
+
+SFO model, summary and plot:
+
+
+```r
+m.L4.SFO <- mkinfit(SFO, FOCUS_2006_L4_mkin, quiet = TRUE)
+plot(m.L4.SFO)
+```
+
+![plot of chunk unnamed-chunk-18](figure/unnamed-chunk-18.png)
+
+```r
+summary(m.L4.SFO, data = FALSE)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:58 2013
+## Date of summary: Sun Nov 17 15:02:58 2013
+##
+## Equations:
+## [1] d_parent = - k_parent_sink * parent
+##
+## Method used for solution of differential equation system:
+## analytical
+##
+## Weighting: none
+##
+## Starting values for optimised parameters:
+## value type transformed
+## parent_0 100.0 state 100.000
+## k_parent_sink 0.1 deparm -2.303
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 96.40 1.95 91.70 101.00
+## k_parent_sink -5.03 0.08 -5.23 -4.83
+##
+## Backtransformed parameters:
+## Estimate Lower Upper
+## parent_0 96.40000 91.70000 1.01e+02
+## k_parent_sink 0.00654 0.00538 7.95e-03
+##
+## Residual standard error: 3.65 on 6 degrees of freedom
+##
+## Chi2 error levels in percent:
+## err.min n.optim df
+## All data 3.29 2 6
+## parent 3.29 2 6
+##
+## Estimated disappearance times:
+## DT50 DT90
+## parent 106 352
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## parent_0 k_parent_sink
+## parent_0 1.000 0.587
+## k_parent_sink 0.587 1.000
+```
+
+
+The chi^2 error level of 3.3% as well as the plot suggest that the model
+fits very well.
+
+The FOMC model for comparison
+
+
+```r
+m.L4.FOMC <- mkinfit(FOMC, FOCUS_2006_L4_mkin, quiet = TRUE)
+plot(m.L4.FOMC)
+```
+
+![plot of chunk unnamed-chunk-19](figure/unnamed-chunk-19.png)
+
+```r
+summary(m.L4.FOMC, data = FALSE)
+```
+
+```
+## mkin version: 0.9.25
+## R version: 3.0.2
+## Date of fit: Sun Nov 17 15:02:59 2013
+## Date of summary: Sun Nov 17 15:02:59 2013
+##
+## 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 optimised parameters:
+## value type transformed
+## parent_0 100 state 100.000
+## alpha 1 deparm 0.000
+## beta 10 deparm 2.303
+##
+## Fixed parameter values:
+## None
+##
+## Optimised, transformed parameters:
+## Estimate Std. Error Lower Upper
+## parent_0 99.100 1.680 94.80 103.000
+## alpha -0.351 0.372 -1.31 0.607
+## beta 4.170 0.564 2.73 5.620
+##
+## 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
+##
+## Residual standard error: 2.31 on 5 degrees of freedom
+##
+## 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
+## parent 109 1644
+##
+## Estimated formation fractions:
+## ff
+## parent_sink 1
+##
+## Parameter correlation:
+## parent_0 alpha beta
+## parent_0 1.000 -0.536 -0.608
+## alpha -0.536 1.000 0.991
+## beta -0.608 0.991 1.000
+```
+
+
+The error level at which the chi^2 test passes is slightly lower for the FOMC
+model. However, the difference appears negligible.
+

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