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diff --git a/vignettes/FOCUS_L.md b/vignettes/FOCUS_L.md deleted file mode 100644 index 6c43889..0000000 --- a/vignettes/FOCUS_L.md +++ /dev/null @@ -1,931 +0,0 @@ -<!-- -%\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) -``` - - - -The residual plot can be easily obtained by - - -```r -mkinresplot(m.L1.SFO, ylab = "Observed", xlab = "Time") -``` - - - - -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) -``` - - - - -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) -``` - - - -```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) -``` - - - - -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) -``` - - - -```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) -``` - - - -```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) -``` - - - -```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) -``` - - - -```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) -``` - - - -```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) -``` - - - -```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. - |