Example evaluation of FOCUS Laboratory Data L1 to L3
Johannes Ranke
2016-03-23
Laboratory Data L1
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: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:06 2015
## Date of summary: Mon Jul 20 14:51:07 2015
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 37 model solutions performed in 0.087 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 92.470 1.36800 89.570 95.370
## log_k_parent_sink -2.347 0.04057 -2.433 -2.261
##
## 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:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 92.47000 67.58 2.170e-21 89.57000 95.3700
## k_parent_sink 0.09561 24.65 1.867e-14 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 -1.2513A 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 chi^2 error level is checked.
m.L1.FOMC <- mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet=TRUE)## Warning in mkinfit("FOMC", FOCUS_2006_L1_mkin, quiet = TRUE): Optimisation by method Port did not converge.
## Convergence code is 1summary(m.L1.FOMC, data = FALSE)## mkin version: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:07 2015
## Date of summary: Mon Jul 20 14:51:07 2015
##
##
## Warning: Optimisation by method Port did not converge.
## Convergence code is 1
##
##
## Equations:
## d_parent = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 188 model solutions performed in 0.43 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 92.47 1.422 89.44 95.50
## log_alpha 15.43 15.080 -16.71 47.58
## log_beta 17.78 15.090 -14.37 49.93
##
## Parameter correlation:
## parent_0 log_alpha log_beta
## parent_0 1.0000 0.1129 0.1112
## log_alpha 0.1129 1.0000 1.0000
## log_beta 0.1112 1.0000 1.0000
##
## Residual standard error: 3.045 on 15 degrees of freedom
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 9.247e+01 65.150 4.044e-20 8.944e+01 9.550e+01
## alpha 5.044e+06 1.271 1.115e-01 5.510e-08 4.618e+20
## beta 5.276e+07 1.259 1.137e-01 5.732e-07 4.857e+21
##
## 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 7.25 24.08 7.25Due 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 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 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.
Laboratory Data L2
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: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:07 2015
## Date of summary: Mon Jul 20 14:51:07 2015
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 41 model solutions performed in 0.094 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 91.4700 3.8070 82.9800 99.9500
## log_k_parent_sink -0.4112 0.1074 -0.6505 -0.1719
##
## 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:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 91.4700 24.03 1.773e-10 82.9800 99.9500
## k_parent_sink 0.6629 9.31 1.525e-06 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 0.6000The 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.
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.
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: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:08 2015
## Date of summary: Mon Jul 20 14:51:08 2015
##
## Equations:
## d_parent = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 81 model solutions performed in 0.18 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 93.7700 1.8560 89.5700 97.9700
## log_alpha 0.3180 0.1867 -0.1044 0.7405
## log_beta 0.2102 0.2943 -0.4555 0.8759
##
## 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:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 93.770 50.510 1.173e-12 89.5700 97.970
## alpha 1.374 5.355 2.296e-04 0.9009 2.097
## beta 1.234 3.398 3.949e-03 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 1.612The 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.
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: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:10 2015
## Date of summary: Mon Jul 20 14:51:10 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 0.793 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 93.9500 NA NA NA
## log_k1 3.1210 NA NA NA
## log_k2 -1.0880 NA NA NA
## g_ilr -0.2821 NA NA NA
##
## Parameter correlation:
## Could not estimate covariance matrix; singular system:
##
## Residual standard error: 1.732 on 8 degrees of freedom
##
## Backtransformed parameters:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 93.9500 NA NA NA NA
## k1 22.6700 NA NA NA NA
## k2 0.3369 NA NA NA NA
## g 0.4016 NA NA 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 0.5335 5.311 0.03058 2.058Here, 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.
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: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:10 2015
## Date of summary: Mon Jul 20 14:51:10 2015
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 43 model solutions performed in 0.103 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 74.870 8.4570 54.180 95.57
## log_k_parent_sink -3.678 0.3261 -4.476 -2.88
##
## 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:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 74.87000 8.853 5.776e-05 54.18000 95.57000
## k_parent_sink 0.02527 3.067 1.102e-02 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 8.3903The 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:
m.L3.FOMC <- mkinfit("FOMC", FOCUS_2006_L3_mkin, quiet = TRUE)
plot(m.L3.FOMC)
summary(m.L3.FOMC, data = FALSE)## mkin version: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:11 2015
## Date of summary: Mon Jul 20 14:51:11 2015
##
## Equations:
## d_parent = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 83 model solutions performed in 0.187 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 96.9700 4.5500 85.2800 108.7000
## log_alpha -0.8619 0.1704 -1.3000 -0.4238
## log_beta 0.6193 0.4744 -0.6003 1.8390
##
## 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:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 96.9700 21.310 2.108e-06 85.2800 108.7000
## alpha 0.4224 5.867 1.020e-03 0.2725 0.6546
## beta 1.8580 2.108 4.444e-02 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 7.729 431.2 129.8The 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:
m.L3.DFOP <- mkinfit("DFOP", FOCUS_2006_L3_mkin, quiet = TRUE)
plot(m.L3.DFOP)
summary(m.L3.DFOP, data = FALSE)## mkin version: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:11 2015
## Date of summary: Mon Jul 20 14:51:11 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 0.322 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 97.7500 1.43800 93.7500 101.70000
## log_k1 -0.6612 0.13340 -1.0310 -0.29100
## log_k2 -4.2860 0.05902 -4.4500 -4.12200
## g_ilr -0.1229 0.05121 -0.2651 0.01925
##
## 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:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 97.75000 67.970 1.404e-07 93.75000 101.70000
## k1 0.51620 7.499 8.460e-04 0.35650 0.74750
## k2 0.01376 16.940 3.557e-05 0.01168 0.01621
## g 0.45660 25.410 7.121e-06 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 50.37Here, 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.
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.
Laboratory Data L4
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: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:12 2015
## Date of summary: Mon Jul 20 14:51:12 2015
##
## Equations:
## d_parent = - k_parent_sink * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 46 model solutions performed in 0.101 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 96.44 1.94900 91.670 101.200
## log_k_parent_sink -5.03 0.07999 -5.225 -4.834
##
## 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:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 96.440000 49.49 2.283e-09 91.670000 1.012e+02
## k_parent_sink 0.006541 12.50 8.008e-06 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 352The chi^2 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: 0.9.40
## R version: 3.2.1
## Date of fit: Mon Jul 20 14:51:12 2015
## Date of summary: Mon Jul 20 14:51:12 2015
##
## Equations:
## d_parent = - (alpha/beta) * 1/((time/beta) + 1) * parent
##
## Model predictions using solution type analytical
##
## Fitted with method Port using 66 model solutions performed in 0.146 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 with symmetric confidence intervals:
## Estimate Std. Error Lower Upper
## parent_0 99.1400 1.6800 94.820 103.5000
## log_alpha -0.3506 0.3725 -1.308 0.6068
## log_beta 4.1740 0.5635 2.726 5.6230
##
## 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:
## Confidence intervals for internally transformed parameters are asymmetric.
## t-test (unrealistically) based on the assumption of normal distribution
## for estimators of untransformed parameters.
## Estimate t value Pr(>t) Lower Upper
## parent_0 99.1400 59.020 1.322e-08 94.8200 103.500
## alpha 0.7042 2.685 2.178e-02 0.2703 1.835
## beta 64.9800 1.775 6.807e-02 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 494.9The error level at which the chi^2 test passes is slightly lower for the FOMC model. However, the difference appears negligible.