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diff --git a/vignettes/FOCUS_L.md b/vignettes/FOCUS_L.md new file mode 100644 index 0000000..6c43889 --- /dev/null +++ b/vignettes/FOCUS_L.md @@ -0,0 +1,931 @@ +<!-- +%\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. + |