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authorJohannes Ranke <jranke@uni-bremen.de>2014-05-08 21:04:21 +0200
committerJohannes Ranke <jranke@uni-bremen.de>2014-05-08 21:04:21 +0200
commit89df28000c4ce8c1e46965ad72b79b406e4d512f (patch)
treed4045738d299d99389352ace5c64b739ef2b85ae /vignettes
parent6b0efae2868f68d1b6ea90b45a2aa4adb287bddc (diff)
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diff --git a/vignettes/FOCUS_L.md b/vignettes/FOCUS_L.md
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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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