The residual plot can be easily obtained byThe following code defines example dataset L1 from the FOCUS kinetics report, p. 284:
library("mkin")
## Loading required package: minpack.lm
## Loading required package: rootSolve
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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:53 2015
## Date of summary: Sat Feb 21 14:44:53 2015
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 37 model solutions performed in 0.098 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 89.850000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 92.470 1.36800 89.570 95.370 67.58 4.339e-21
## log_k_parent_sink -2.347 0.04057 -2.433 -2.261 -57.86 5.155e-20
## Pr(>t)
## parent_0 2.170e-21
## log_k_parent_sink 2.577e-20
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.0000 0.6248
## log_k_parent_sink 0.6248 1.0000
##
## Residual standard error: 2.948 on 16 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 92.47000 89.57000 95.3700
## k_parent_sink 0.09561 0.08773 0.1042
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 3.424 2 7
## parent 3.424 2 7
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 7.249 24.08
##
## Data:
## time variable observed predicted residual
## 0 parent 88.3 92.471 -4.1710
## 0 parent 91.4 92.471 -1.0710
## 1 parent 85.6 84.039 1.5610
## 1 parent 84.5 84.039 0.4610
## 2 parent 78.9 76.376 2.5241
## 2 parent 77.6 76.376 1.2241
## 3 parent 72.0 69.412 2.5884
## 3 parent 71.9 69.412 2.4884
## 5 parent 50.3 57.330 -7.0301
## 5 parent 59.4 57.330 2.0699
## 7 parent 47.0 47.352 -0.3515
## 7 parent 45.1 47.352 -2.2515
## 14 parent 27.7 24.247 3.4528
## 14 parent 27.3 24.247 3.0528
## 21 parent 10.0 12.416 -2.4163
## 21 parent 10.4 12.416 -2.0163
## 30 parent 2.9 5.251 -2.3513
## 30 parent 4.0 5.251 -1.2513
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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:55 2015
## Date of summary: Sat Feb 21 14:44:55 2015
##
## Equations:
## d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 611 model solutions performed in 1.509 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 89.850000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 92.47 1.482 89.31 95.63 62.39000 1.546e-19
## log_alpha 11.25 598.200 -1264.00 1286.00 0.01880 9.852e-01
## log_beta 13.60 598.200 -1261.00 1289.00 0.02273 9.822e-01
## Pr(>t)
## parent_0 7.730e-20
## log_alpha 4.926e-01
## log_beta 4.911e-01
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.0000 -0.3016 -0.3016
## log_alpha -0.3016 1.0000 1.0000
## log_beta -0.3016 1.0000 1.0000
##
## Residual standard error: 3.045 on 15 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 92.47 89.31 95.63
## alpha 76830.00 0.00 Inf
## beta 803500.00 0.00 Inf
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 3.619 3 6
## parent 3.619 3 6
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 7.249 24.08 7.249
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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:55 2015
## Date of summary: Sat Feb 21 14:44:55 2015
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 41 model solutions performed in 0.1 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 93.950000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 91.4700 3.8070 82.9800 99.9500 24.030 3.545e-10
## log_k_parent_sink -0.4112 0.1074 -0.6505 -0.1719 -3.828 3.329e-03
## Pr(>t)
## parent_0 1.773e-10
## log_k_parent_sink 1.664e-03
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.0000 0.4295
## log_k_parent_sink 0.4295 1.0000
##
## Residual standard error: 5.51 on 10 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 91.4700 82.9800 99.9500
## k_parent_sink 0.6629 0.5218 0.8421
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 14.38 2 4
## parent 14.38 2 4
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 1.046 3.474
##
## Data:
## time variable observed predicted residual
## 0 parent 96.1 9.147e+01 4.6343
## 0 parent 91.8 9.147e+01 0.3343
## 1 parent 41.4 4.714e+01 -5.7394
## 1 parent 38.7 4.714e+01 -8.4394
## 3 parent 19.3 1.252e+01 6.7790
## 3 parent 22.3 1.252e+01 9.7790
## 7 parent 4.6 8.834e-01 3.7166
## 7 parent 4.6 8.834e-01 3.7166
## 14 parent 2.6 8.532e-03 2.5915
## 14 parent 1.2 8.532e-03 1.1915
## 28 parent 0.3 7.958e-07 0.3000
## 28 parent 0.6 7.958e-07 0.6000
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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:55 2015
## Date of summary: Sat Feb 21 14:44:55 2015
##
## Equations:
## d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 81 model solutions performed in 0.201 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 93.950000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 93.7700 1.8560 89.5700 97.9700 50.5100 2.345e-12 1.173e-12
## log_alpha 0.3180 0.1867 -0.1044 0.7405 1.7030 1.227e-01 6.137e-02
## log_beta 0.2102 0.2943 -0.4555 0.8759 0.7142 4.932e-01 2.466e-01
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.00000 -0.09553 -0.1863
## log_alpha -0.09553 1.00000 0.9757
## log_beta -0.18628 0.97567 1.0000
##
## Residual standard error: 2.628 on 9 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 93.770 89.5700 97.970
## alpha 1.374 0.9009 2.097
## beta 1.234 0.6341 2.401
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 6.205 3 3
## parent 6.205 3 3
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 0.8092 5.356 1.612
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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:57 2015
## Date of summary: Sat Feb 21 14:44:57 2015
##
## Equations:
## 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 Port using 336 model solutions performed in 0.856 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 93.9500000 -Inf Inf
## log_k1 0.0000000 -Inf Inf
## log_k2 -4.6051702 -Inf Inf
## g_ilr 0.9802581 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 93.9500 NA NA NA NA NA NA
## log_k1 3.1210 NA NA NA NA NA NA
## log_k2 -1.0880 NA NA NA NA NA NA
## g_ilr -0.2821 NA NA NA NA NA NA
##
## Parameter correlation:
## Could not estimate covariance matrix; singular system:
##
## Residual standard error: 1.732 on 8 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 93.9500 NA NA
## k1 22.6700 NA NA
## k2 0.3369 NA NA
## g 0.4016 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 0.03058 2.058
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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:57 2015
## Date of summary: Sat Feb 21 14:44:57 2015
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 43 model solutions performed in 0.109 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 97.800000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 74.870 8.4570 54.180 95.57 8.853 1.155e-04
## log_k_parent_sink -3.678 0.3261 -4.476 -2.88 -11.280 2.903e-05
## Pr(>t)
## parent_0 5.776e-05
## log_k_parent_sink 1.451e-05
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.0000 0.5483
## log_k_parent_sink 0.5483 1.0000
##
## Residual standard error: 12.91 on 6 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 74.87000 54.18000 95.57000
## k_parent_sink 0.02527 0.01138 0.05612
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 21.24 2 6
## parent 21.24 2 6
##
## Resulting formation fractions:
## ff
## parent_sink 1
##
## Estimated disappearance times:
## DT50 DT90
## parent 27.43 91.13
##
## Data:
## time variable observed predicted residual
## 0 parent 97.8 74.872 22.9281
## 3 parent 60.0 69.406 -9.4061
## 7 parent 51.0 62.734 -11.7340
## 14 parent 43.0 52.564 -9.5638
## 30 parent 35.0 35.084 -0.0839
## 60 parent 22.0 16.440 5.5602
## 91 parent 15.0 7.511 7.4887
## 120 parent 12.0 3.610 8.3903
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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:58 2015
## Date of summary: Sat Feb 21 14:44:58 2015
##
## Equations:
## d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 83 model solutions performed in 0.203 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 97.800000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 96.9700 4.5500 85.2800 108.7000 21.310 4.216e-06 2.108e-06
## log_alpha -0.8619 0.1704 -1.3000 -0.4238 -5.057 3.911e-03 1.955e-03
## log_beta 0.6193 0.4744 -0.6003 1.8390 1.305 2.486e-01 1.243e-01
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.0000 -0.1512 -0.4271
## log_alpha -0.1512 1.0000 0.9110
## log_beta -0.4271 0.9110 1.0000
##
## Residual standard error: 4.572 on 5 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 96.9700 85.2800 108.7000
## alpha 0.4224 0.2725 0.6546
## beta 1.8580 0.5487 6.2890
##
## 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 7.729 431.2 129.8
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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:58 2015
## Date of summary: Sat Feb 21 14:44:58 2015
##
## Equations:
## 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 Port using 137 model solutions performed in 0.346 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 97.800000 -Inf Inf
## log_k1 -2.302585 -Inf Inf
## log_k2 -4.605170 -Inf Inf
## g_ilr 0.000000 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 97.7500 1.43800 93.7500 101.70000 67.970 2.808e-07 1.404e-07
## log_k1 -0.6612 0.13340 -1.0310 -0.29100 -4.958 7.715e-03 3.858e-03
## log_k2 -4.2860 0.05902 -4.4500 -4.12200 -72.620 2.155e-07 1.077e-07
## g_ilr -0.1229 0.05121 -0.2651 0.01925 -2.401 7.431e-02 3.716e-02
##
## Parameter correlation:
## parent_0 log_k1 log_k2 g_ilr
## parent_0 1.00000 0.1640 0.01315 0.4253
## log_k1 0.16400 1.0000 0.46478 -0.5526
## log_k2 0.01315 0.4648 1.00000 -0.6631
## g_ilr 0.42526 -0.5526 -0.66310 1.0000
##
## Residual standard error: 1.439 on 4 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 97.75000 93.75000 101.70000
## k1 0.51620 0.35650 0.74750
## k2 0.01376 0.01168 0.01621
## g 0.45660 0.40730 0.50680
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 2.225 4 4
## parent 2.225 4 4
##
## Estimated disappearance times:
## DT50 DT90 DT50_k1 DT50_k2
## parent 7.464 123 1.343 50.37
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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:58 2015
## Date of summary: Sat Feb 21 14:44:58 2015
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 46 model solutions performed in 0.109 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 96.600000 -Inf Inf
## log_k_parent_sink -2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|)
## parent_0 96.44 1.94900 91.670 101.200 49.49 4.566e-09
## log_k_parent_sink -5.03 0.07999 -5.225 -4.834 -62.88 1.088e-09
## Pr(>t)
## parent_0 2.283e-09
## log_k_parent_sink 5.438e-10
##
## Parameter correlation:
## parent_0 log_k_parent_sink
## parent_0 1.0000 0.5865
## log_k_parent_sink 0.5865 1.0000
##
## Residual standard error: 3.651 on 6 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 96.440000 91.670000 1.012e+02
## k_parent_sink 0.006541 0.005378 7.955e-03
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 3.287 2 6
## parent 3.287 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: 0.9.35
## R version: 3.1.2
## Date of fit: Sat Feb 21 14:44:58 2015
## Date of summary: Sat Feb 21 14:44:58 2015
##
## Equations:
## d_parent = - (alpha/beta) * ((time/beta) + 1)^-1 * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 66 model solutions performed in 0.161 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 96.600000 -Inf Inf
## log_alpha 0.000000 -Inf Inf
## log_beta 2.302585 -Inf Inf
##
## Fixed parameter values:
## None
##
## Optimised, transformed parameters:
## Estimate Std. Error Lower Upper t value Pr(>|t|) Pr(>t)
## parent_0 99.1400 1.6800 94.820 103.5000 59.0200 2.643e-08 1.322e-08
## log_alpha -0.3506 0.3725 -1.308 0.6068 -0.9414 3.897e-01 1.949e-01
## log_beta 4.1740 0.5635 2.726 5.6230 7.4070 7.059e-04 3.530e-04
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.0000 -0.5365 -0.6083
## log_alpha -0.5365 1.0000 0.9913
## log_beta -0.6083 0.9913 1.0000
##
## Residual standard error: 2.315 on 5 degrees of freedom
##
## Backtransformed parameters:
## Estimate Lower Upper
## parent_0 99.1400 94.8200 103.500
## alpha 0.7042 0.2703 1.835
## beta 64.9800 15.2600 276.600
##
## Chi2 error levels in percent:
## err.min n.optim df
## All data 2.029 3 5
## parent 2.029 3 5
##
## Estimated disappearance times:
## DT50 DT90 DT50back
## parent 108.9 1644 494.9
The error level at which the chi2 test passes is slightly lower for the FOMC model. However, the difference appears negligible.