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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.
+