The following code defines example dataset L1 from the FOCUS kinetics report, p. 284:
library("mkin")
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,
72.0, 71.9, 50.3, 59.4, 47.0, 45.1,
27.7, 27.3, 10.0, 10.4, 2.9, 4.0))
FOCUS_2006_L1_mkin <- mkin_wide_to_long(FOCUS_2006_L1)
Here we use the assumptions 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.
Since mkin version 0.9-32 (July 2014), we can use shorthand notation like SFO
for parent only degradation models. 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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:09 2014
## Date of summary: Thu Jul 24 10:32:09 2014
##
## Equations:
## [1] d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 14 model solutions performed in 0.081 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 89.85 state
## k_parent_sink 0.10 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 89.850 -Inf Inf
## log_k_parent_sink -2.303 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 92.50 1.3700 89.60 95.40 67.6 4.34e-21
## log_k_parent_sink -2.35 0.0406 -2.43 -2.26 -57.9 5.16e-20
## Pr(>t)
## parent_0 2.17e-21
## log_k_parent_sink 2.58e-20
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.000 0.625
## log_k_parent_sink 0.625 1.000
##
## Residual standard error: 2.95 on 16 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 92.5000 89.6000 95.400
## k_parent_sink 0.0956 0.0877 0.104
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 3.42 2 7
## parent 3.42 2 7
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 7.25 24.1
##
## 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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:10 2014
## Date of summary: Thu Jul 24 10:32:11 2014
##
## Equations:
## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 53 model solutions performed in 0.321 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 89.85 state
## alpha 1.00 deparm
## beta 10.00 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 89.850 -Inf Inf
## log_alpha 0.000 -Inf Inf
## log_beta 2.303 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 92.5 1.45 89.40 95.6 63.60 1.17e-19 5.85e-20
## log_alpha 14.9 10.60 -7.75 37.5 1.40 1.82e-01 9.08e-02
## log_beta 17.2 10.60 -5.38 39.8 1.62 1.25e-01 6.26e-02
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.000 0.24 0.238
## log_alpha 0.240 1.00 1.000
## log_beta 0.238 1.00 1.000
##
## Residual standard error: 3.05 on 15 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 9.25e+01 8.94e+01 9.56e+01
## alpha 2.85e+06 4.32e-04 1.88e+16
## beta 2.98e+07 4.59e-03 1.93e+17
##
## 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 DT50back
## parent 7.25 24.1 7.25
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 parameters
log_alpha and log_beta internally fitted in the model have p-values for the two
sided t-test of 0.18 and 0.125, and their correlation is 1.000, indicating that
the model is overparameterised.
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.
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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:11 2014
## Date of summary: Thu Jul 24 10:32:11 2014
##
## Equations:
## [1] d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 29 model solutions performed in 0.196 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 93.95 state
## k_parent_sink 0.10 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 93.950 -Inf Inf
## log_k_parent_sink -2.303 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 91.500 3.810 83.000 99.900 24.00 3.55e-10
## log_k_parent_sink -0.411 0.107 -0.651 -0.172 -3.83 3.33e-03
## Pr(>t)
## parent_0 1.77e-10
## log_k_parent_sink 1.66e-03
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.00 0.43
## log_k_parent_sink 0.43 1.00
##
## Residual standard error: 5.51 on 10 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 91.500 83.000 99.900
## k_parent_sink 0.663 0.522 0.842
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 14.4 2 4
## parent 14.4 2 4
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 1.05 3.47
##
## Data:
## time variable observed predicted residual
## 0 parent 96.1 9.15e+01 4.635
## 0 parent 91.8 9.15e+01 0.335
## 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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:11 2014
## Date of summary: Thu Jul 24 10:32:11 2014
##
## Equations:
## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 35 model solutions performed in 0.223 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 93.95 state
## alpha 1.00 deparm
## beta 10.00 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 93.950 -Inf Inf
## log_alpha 0.000 -Inf Inf
## log_beta 2.303 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 93.800 1.860 89.600 98.000 50.500 2.35e-12 1.17e-12
## log_alpha 0.318 0.187 -0.104 0.740 1.700 1.23e-01 6.14e-02
## log_beta 0.210 0.294 -0.456 0.876 0.714 4.93e-01 2.47e-01
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.0000 -0.0955 -0.186
## log_alpha -0.0955 1.0000 0.976
## log_beta -0.1863 0.9757 1.000
##
## Residual standard error: 2.63 on 9 degrees of freedom
##
## 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
##
## 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 DT50back
## parent 0.809 5.36 1.61
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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:12 2014
## Date of summary: Thu Jul 24 10:32:12 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
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 43 model solutions performed in 0.271 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 93.95 state
## k1 1.00 deparm
## k2 0.01 deparm
## g 0.80 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 93.9500 -Inf Inf
## log_k1 0.0000 -Inf Inf
## log_k2 -4.6052 -Inf Inf
## g_ilr 0.9803 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 94.000 NA NA NA NA NA NA
## log_k1 6.160 NA NA NA NA NA NA
## log_k2 -1.090 NA NA NA NA NA NA
## g_ilr -0.282 NA NA NA NA NA NA
##
## Parameter correlation:
## Could not estimate covariance matrix; singular system:
##
## Residual standard error: 1.73 on 8 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 94.000 NA NA
## k1 476.000 NA NA
## k2 0.337 NA NA
## g 0.402 NA NA
##
## 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 DT50_k1 DT50_k2
## parent NA NA 0.00146 2.06
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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:14 2014
## Date of summary: Thu Jul 24 10:32:14 2014
##
## Equations:
## [1] d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 44 model solutions performed in 0.251 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 97.8 state
## k_parent_sink 0.1 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 97.800 -Inf Inf
## log_k_parent_sink -2.303 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 74.90 8.460 54.20 95.60 8.85 0.000116
## log_k_parent_sink -3.68 0.326 -4.48 -2.88 -11.30 0.000029
## Pr(>t)
## parent_0 5.78e-05
## log_k_parent_sink 1.45e-05
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.000 0.548
## log_k_parent_sink 0.548 1.000
##
## Residual standard error: 12.9 on 6 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 74.9000 54.2000 95.6000
## k_parent_sink 0.0253 0.0114 0.0561
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 21.2 2 6
## parent 21.2 2 6
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 27.4 91.1
##
## Data:
## time variable observed predicted residual
## 0 parent 97.8 74.87 22.9274
## 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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:14 2014
## Date of summary: Thu Jul 24 10:32:14 2014
##
## Equations:
## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 26 model solutions performed in 0.154 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 97.8 state
## alpha 1.0 deparm
## beta 10.0 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 97.800 -Inf Inf
## log_alpha 0.000 -Inf Inf
## log_beta 2.303 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 97.000 4.550 85.3 109.000 21.30 4.22e-06 2.11e-06
## log_alpha -0.862 0.170 -1.3 -0.424 -5.06 3.91e-03 1.96e-03
## log_beta 0.619 0.474 -0.6 1.840 1.31 2.49e-01 1.24e-01
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.000 -0.151 -0.427
## log_alpha -0.151 1.000 0.911
## log_beta -0.427 0.911 1.000
##
## Residual standard error: 4.57 on 5 degrees of freedom
##
## 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
##
## 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 DT50back
## parent 7.73 431 130
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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:14 2014
## Date of summary: Thu Jul 24 10:32:14 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
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 37 model solutions performed in 0.228 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 97.80 state
## k1 0.10 deparm
## k2 0.01 deparm
## g 0.50 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 97.800 -Inf Inf
## log_k1 -2.303 -Inf Inf
## log_k2 -4.605 -Inf Inf
## g_ilr 0.000 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 97.700 1.4400 93.800 102.0000 68.00 2.81e-07 1.40e-07
## log_k1 -0.661 0.1330 -1.030 -0.2910 -4.96 7.72e-03 3.86e-03
## log_k2 -4.290 0.0590 -4.450 -4.1200 -72.60 2.15e-07 1.08e-07
## g_ilr -0.123 0.0512 -0.265 0.0193 -2.40 7.43e-02 3.72e-02
##
## Parameter correlation:
## parent_0 log_k1 log_k2 g_ilr
## parent_0 1.0000 0.164 0.0131 0.425
## log_k1 0.1640 1.000 0.4648 -0.553
## log_k2 0.0131 0.465 1.0000 -0.663
## g_ilr 0.4253 -0.553 -0.6631 1.000
##
## Residual standard error: 1.44 on 4 degrees of freedom
##
## 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
##
## 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 DT50_k1 DT50_k2
## parent 7.46 123 1.34 50.4
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.
This is also an example where the standard t-test for the parameter g_ilr is
misleading, as it tests for a significant difference from zero. In this case,
zero appears to be the correct value for this parameter, and the confidence
interval for the backtransformed parameter g is quite narrow.
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, 49.0))
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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:15 2014
## Date of summary: Thu Jul 24 10:32:15 2014
##
## Equations:
## [1] d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 20 model solutions performed in 0.141 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 96.6 state
## k_parent_sink 0.1 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 96.600 -Inf Inf
## log_k_parent_sink -2.303 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 96.40 1.95 91.70 101.00 49.5 4.57e-09
## log_k_parent_sink -5.03 0.08 -5.23 -4.83 -62.9 1.09e-09
## Pr(>t)
## parent_0 2.28e-09
## log_k_parent_sink 5.44e-10
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.000 0.587
## log_k_parent_sink 0.587 1.000
##
## Residual standard error: 3.65 on 6 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 96.40000 91.70000 1.01e+02
## k_parent_sink 0.00654 0.00538 7.95e-03
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 3.29 2 6
## parent 3.29 2 6
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 106 352
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: 0.9.32
## R version: 3.1.1
## Date of fit: Thu Jul 24 10:32:15 2014
## Date of summary: Thu Jul 24 10:32:15 2014
##
## Equations:
## [1] d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Marq using 48 model solutions performed in 0.296 s
##
## Weighting: none
##
## Starting values for parameters to be optimised:
## value type
## parent_0 96.6 state
## alpha 1.0 deparm
## beta 10.0 deparm
##
## Starting values for the transformed parameters actually optimised:
## value lower upper
## parent_0 96.600 -Inf Inf
## log_alpha 0.000 -Inf Inf
## log_beta 2.303 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 99.100 1.680 94.80 103.000 59.000 2.64e-08 1.32e-08
## log_alpha -0.351 0.372 -1.31 0.607 -0.941 3.90e-01 1.95e-01
## log_beta 4.170 0.564 2.73 5.620 7.410 7.06e-04 3.53e-04
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.000 -0.536 -0.608
## log_alpha -0.536 1.000 0.991
## log_beta -0.608 0.991 1.000
##
## Residual standard error: 2.31 on 5 degrees of freedom
##
## 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
##
## 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 DT50back
## parent 109 1644 495
The error level at which the chi2 test passes is slightly lower for the FOMC model. However, the difference appears negligible.