Example evaluation of FOCUS Laboratory Data L1 to L3"

library(knitr)
opts_chunk$set(tidy = FALSE, cache = FALSE)

Laboratory Data L1

The following code defines example dataset L1 from the FOCUS kinetics report, p. 284:

library("mkin", quietly = TRUE)
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)

A plot of the fit is obtained with the plot function for mkinfit objects.

plot(m.L1.SFO, show_errmin = TRUE, main = "FOCUS 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)
plot(m.L1.FOMC, show_errmin = TRUE, main = "FOCUS L1 - FOMC")
summary(m.L1.FOMC, data = FALSE)

We get a warning that the default optimisation algorithm Port did not converge, which is an indication that the model is overparameterised, i.e. contains too many parameters that are ill-defined as a consequence.

And in fact, due to the higher number of parameters, and the lower number of degrees of freedom of the fit, the $\chi^2$ error level is actually higher for the FOMC model (3.6%) than for the SFO model (3.4%). Additionally, the parameters log_alpha and log_beta internally fitted in the model have excessive confidence intervals, that span more than 25 orders of magnitude (!) when backtransformed to the scale of alpha and beta. Also, the t-test for significant difference from zero does not indicate such a significant difference, with p-values greater than 0.1, and finally, the parameter correlation of log_alpha and log_beta is 1.000, clearly 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 and 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 was the first widely used standard package in this field. Also, the calculation of $\chi^2$ error levels was compared with KinGUII, CAKE and DegKin manager in a project sponsored by the German Umweltbundesamt [@ranke2014].

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)

SFO fit for L2

Again, the SFO model is fitted and the result is plotted. The residual plot can be obtained simply by adding the argument show_residuals to the plot command.

m.L2.SFO <- mkinfit("SFO", FOCUS_2006_L2_mkin, quiet=TRUE)
plot(m.L2.SFO, show_residuals = TRUE, show_errmin = TRUE,
     main = "FOCUS L2 - SFO")

The $\chi^2$ error level of 14% suggests that the model does not fit very well. This is also obvious from the plots of the fit, in which we have included the residual plot.

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.

FOMC fit for L2

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)
plot(m.L2.FOMC, show_residuals = TRUE,
     main = "FOCUS L2 - FOMC")
summary(m.L2.FOMC, data = FALSE)

The error level at which the $\chi^2$ test passes is much lower in this case. Therefore, the FOMC model provides a better description of the data, as less experimental error has to be assumed in order to explain the data.

DFOP fit for L2

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, show_residuals = TRUE, show_errmin = TRUE,
     main = "FOCUS L2 - DFOP")
summary(m.L2.DFOP, data = FALSE)

Here, the DFOP model is clearly the best-fit model for dataset L2 based on the chi^2 error level criterion. However, the failure to calculate the covariance matrix indicates that the parameter estimates correlate excessively. Therefore, the FOMC model may be preferred for this dataset.

Laboratory Data L3

The following code defines example dataset L3 from the FOCUS kinetics report, p. 290.

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)

Fit multiple models

As of mkin version 0.9-39 (June 2015), we can fit several models to one or more datasets in one call to the function mmkin. The datasets have to be passed in a list, in this case a named list holding only the L3 dataset prepared above.

# Only use one core here, not to offend the CRAN checks
mm.L3 <- mmkin(c("SFO", "FOMC", "DFOP"), cores = 1,
               list("FOCUS L3" = FOCUS_2006_L3_mkin), quiet = TRUE)
plot(mm.L3)

The $\chi^2$ error level of 21% as well as the plot suggest that the SFO model does not fit very well. The FOMC model performs better, with an error level at which the $\chi^2$ test passes of 7%. Fitting the four parameter DFOP model further reduces the $\chi^2$ error level considerably.

Accessing mmkin objects

The objects returned by mmkin are arranged like a matrix, with models as a row index and datasets as a column index.

We can extract the summary and plot for e.g. the DFOP fit, using square brackets for indexing which will result in the use of the summary and plot functions working on mkinfit objects.

summary(mm.L3[["DFOP", 1]])
plot(mm.L3[["DFOP", 1]], show_errmin = TRUE)

Here, a look to the model plot, the confidence intervals of the parameters and the correlation matrix suggest that the parameter estimates are reliable, and the DFOP model can be used as the best-fit model based on the $\chi^2$ error level criterion for laboratory data L3.

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)

Fits of the SFO and FOMC models, plots and summaries are produced below:

# Only use one core here, not to offend the CRAN checks
mm.L4 <- mmkin(c("SFO", "FOMC"), cores = 1,
               list("FOCUS L4" = FOCUS_2006_L4_mkin),
               quiet = TRUE)
plot(mm.L4)

The $\chi^2$ error level of 3.3% as well as the plot suggest that the SFO model fits very well. The error level at which the $\chi^2$ test passes is slightly lower for the FOMC model. However, the difference appears negligible.

summary(mm.L4[["SFO", 1]], data = FALSE)
summary(mm.L4[["FOMC", 1]], data = FALSE)

References



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mkin documentation built on Sept. 21, 2018, 6:21 p.m.