The following code defines example dataset L1 from the FOCUS kinetics report, p. 284
library("mkin")
## Loading required package: minpack.lm
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.
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.
m.L1.SFO <- mkinfit(SFO, FOCUS_2006_L1_mkin, quiet = TRUE)
summary(m.L1.SFO)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:02 2014
## Date of summary: Thu Feb 27 20:16:03 2014
##
## 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.
plot(m.L1.SFO)
The residual plot can be easily obtained by
mkinresplot(m.L1.SFO, ylab = "Observed", xlab = "Time")
For comparison, the FOMC model is fitted as well, and the chi2 error level is checked.
m.L1.FOMC <- mkinfit(FOMC, FOCUS_2006_L1_mkin, quiet = TRUE)
summary(m.L1.FOMC, data = FALSE)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:05 2014
## Date of summary: Thu Feb 27 20:16:05 2014
##
## 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 chi2 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 chi2 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 chi2 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.
The following code defines example dataset L2 from the FOCUS kinetics report, p. 287
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.
m.L2.SFO <- mkinfit(SFO, FOCUS_2006_L2_mkin, quiet = TRUE)
summary(m.L2.SFO)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:06 2014
## Date of summary: Thu Feb 27 20:16:06 2014
##
## 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 chi2 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.
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 chi2 error level is checked.
m.L2.FOMC <- mkinfit(FOMC, FOCUS_2006_L2_mkin, quiet = TRUE)
par(mfrow = c(2, 1))
plot(m.L2.FOMC)
mkinresplot(m.L2.FOMC)
summary(m.L2.FOMC, data = FALSE)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:07 2014
## Date of summary: Thu Feb 27 20:16:07 2014
##
## 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 chi2 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 chi2 error level.
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.
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)
summary(m.L2.DFOP, data = FALSE)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:09 2014
## Date of summary: Thu Feb 27 20:16:09 2014
##
## 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 chi2 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.
The following code defines example dataset L3 from the FOCUS kinetics report, p. 290.
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:
m.L3.SFO <- mkinfit(SFO, FOCUS_2006_L3_mkin, quiet = TRUE)
plot(m.L3.SFO)
summary(m.L3.SFO)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:10 2014
## Date of summary: Thu Feb 27 20:16:10 2014
##
## 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 chi2 error level of 21% as well as the plot suggest that the model does not fit very well.
The FOMC model performs better:
m.L3.FOMC <- mkinfit(FOMC, FOCUS_2006_L3_mkin, quiet = TRUE)
plot(m.L3.FOMC)
summary(m.L3.FOMC, data = FALSE)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:11 2014
## Date of summary: Thu Feb 27 20:16:11 2014
##
## 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 chi2 test passes is 7% in this case.
Fitting the four parameter DFOP model further reduces the chi2 error level considerably:
m.L3.DFOP <- mkinfit(DFOP, FOCUS_2006_L3_mkin, quiet = TRUE)
plot(m.L3.DFOP)
summary(m.L3.DFOP, data = FALSE)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:12 2014
## Date of summary: Thu Feb 27 20:16:12 2014
##
## 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 chi2 error level criterion for laboratory data L3.
The following code defines example dataset L4 from the FOCUS kinetics report, p. 293
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:
m.L4.SFO <- mkinfit(SFO, FOCUS_2006_L4_mkin, quiet = TRUE)
plot(m.L4.SFO)
summary(m.L4.SFO, data = FALSE)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:13 2014
## Date of summary: Thu Feb 27 20:16:13 2014
##
## 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 chi2 error level of 3.3% as well as the plot suggest that the model fits very well.
The FOMC model for comparison
m.L4.FOMC <- mkinfit(FOMC, FOCUS_2006_L4_mkin, quiet = TRUE)
plot(m.L4.FOMC)
summary(m.L4.FOMC, data = FALSE)
## mkin version: 0.9.25
## R version: 3.0.3
## Date of fit: Thu Feb 27 20:16:14 2014
## Date of summary: Thu Feb 27 20:16:14 2014
##
## 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 chi2 test passes is slightly lower for the FOMC model. However, the difference appears negligible.