From 9ae42bd20bc2543a94cf1581ba9820c2f9e3afbd Mon Sep 17 00:00:00 2001 From: Johannes Ranke Date: Thu, 20 Apr 2023 19:53:28 +0200 Subject: Fix and rebuild documentation, see NEWS I had to fix the two pathway vignettes, as they did not work with the released version any more. So they and the multistart vignette which got some small fixes as well were rebuilt. Complete rebuild of the online docs with the released version. The documentation of the 'hierarchial_kinetics' format had to be fixed as well. --- docs/articles/prebuilt/2022_dmta_parent.html | 2223 ++++++++++++++++++++++++++ 1 file changed, 2223 insertions(+) create mode 100644 docs/articles/prebuilt/2022_dmta_parent.html (limited to 'docs/articles/prebuilt/2022_dmta_parent.html') diff --git a/docs/articles/prebuilt/2022_dmta_parent.html b/docs/articles/prebuilt/2022_dmta_parent.html new file mode 100644 index 00000000..378a7e8e --- /dev/null +++ b/docs/articles/prebuilt/2022_dmta_parent.html @@ -0,0 +1,2223 @@ + + + + + + + +Testing hierarchical parent degradation kinetics with residue data on dimethenamid and dimethenamid-P • mkin + + + + + + + + + + + + +
+
+ + + + +
+
+
+

Testing hierarchical parent degradation kinetics +with residue data on dimethenamid and dimethenamid-P

+

Johannes +Ranke

+ +

Last change on 5 January +2023, last compiled on 20 April 2023

+ + Source: vignettes/prebuilt/2022_dmta_parent.rmd + + +
+ + + +
+

Introduction +

+

The purpose of this document is to demonstrate how nonlinear +hierarchical models (NLHM) based on the parent degradation models SFO, +FOMC, DFOP and HS can be fitted with the mkin package.

+

It was assembled in the course of work package 1.1 of Project Number +173340 (Application of nonlinear hierarchical models to the kinetic +evaluation of chemical degradation data) of the German Environment +Agency carried out in 2022 and 2023.

+

The mkin package is used in version 1.2.3. It contains the test data +and the functions used in the evaluations. The saemix +package is used as a backend for fitting the NLHM, but is also loaded to +make the convergence plot function available.

+

This document is processed with the knitr package, which +also provides the kable function that is used to improve +the display of tabular data in R markdown documents. For parallel +processing, the parallel package is used.

+
+library(mkin)
+library(knitr)
+library(saemix)
+library(parallel)
+n_cores <- detectCores()
+if (Sys.info()["sysname"] == "Windows") {
+  cl <- makePSOCKcluster(n_cores)
+} else {
+  cl <- makeForkCluster(n_cores)
+}
+
+
+

Data +

+

The test data are available in the mkin package as an object of class +mkindsg (mkin dataset group) under the identifier +dimethenamid_2018. The following preprocessing steps are +still necessary:

+
    +
  • The data available for the enantiomer dimethenamid-P (DMTAP) are +renamed to have the same substance name as the data for the racemic +mixture dimethenamid (DMTA). The reason for this is that no difference +between their degradation behaviour was identified in the EU risk +assessment.
    • +
  • The data for transformation products and unnecessary columns are +discarded
    • +
  • The observation times of each dataset are multiplied with the +corresponding normalisation factor also available in the dataset, in +order to make it possible to describe all datasets with a single set of +parameters that are independent of temperature
    • +
  • Finally, datasets observed in the same soil (Elliot 1 +and Elliot 2) are combined, resulting in dimethenamid +(DMTA) data from six soils.
    • +
+

The following commented R code performs this preprocessing.

+
+# Apply a function to each of the seven datasets in the mkindsg object to create a list
+dmta_ds <- lapply(1:7, function(i) {
+  ds_i <- dimethenamid_2018$ds[[i]]$data                     # Get a dataset
+  ds_i[ds_i$name == "DMTAP", "name"] <-  "DMTA"              # Rename DMTAP to DMTA
+  ds_i <- subset(ds_i, name == "DMTA", c("name", "time", "value")) # Select data
+  ds_i$time <- ds_i$time * dimethenamid_2018$f_time_norm[i]  # Normalise time
+  ds_i                                                       # Return the dataset
+})
+
+# Use dataset titles as names for the list elements
+names(dmta_ds) <- sapply(dimethenamid_2018$ds, function(ds) ds$title)
+
+# Combine data for Elliot soil to obtain a named list with six elements
+dmta_ds[["Elliot"]] <- rbind(dmta_ds[["Elliot 1"]], dmta_ds[["Elliot 2"]]) #
+dmta_ds[["Elliot 1"]] <- NULL
+dmta_ds[["Elliot 2"]] <- NULL
+

The following tables show the 6 datasets.

+
+for (ds_name in names(dmta_ds)) {
+    print(kable(mkin_long_to_wide(dmta_ds[[ds_name]]),
+      caption = paste("Dataset", ds_name),
+      label = paste0("tab:", ds_name), booktabs = TRUE))
+    cat("\n\\clearpage\n")
+}
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Calke
timeDMTA
095.8
098.7
1460.5
3039.1
5915.2
1204.8
1204.6
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Borstel
timeDMTA
0.000000100.5
0.00000099.6
1.94129591.9
1.94129591.3
6.79453481.8
6.79453482.1
13.58906769.1
13.58906768.0
27.17813551.4
27.17813551.4
56.29756527.6
56.29756526.8
86.38764315.7
86.38764315.3
115.5070737.9
115.5070738.1
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Flaach
timeDMTA
0.000000096.5
0.000000096.8
0.000000097.0
0.623385682.9
0.623385686.7
0.623385687.4
1.870156772.8
1.870156769.9
1.870156771.9
4.363698951.4
4.363698952.9
4.363698948.6
8.727397928.5
8.727397927.3
8.727397927.5
13.091096814.8
13.091096813.4
13.091096814.4
17.45479577.7
17.45479577.3
17.45479578.1
26.18219362.0
26.18219361.5
26.18219361.9
34.90959151.3
34.90959151.0
34.90959151.1
43.63698930.9
43.63698930.7
43.63698930.7
52.36438720.6
52.36438720.4
52.36438720.5
74.80626740.4
74.80626740.3
74.80626740.3
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset BBA 2.2
timeDMTA
0.000000098.09
0.000000098.77
0.767892293.52
0.767892292.03
2.303676588.39
2.303676587.18
5.375245269.38
5.375245271.06
10.750490445.21
10.750490446.81
16.125735530.54
16.125735530.07
21.500980721.60
21.500980720.41
32.25147119.10
32.25147119.70
43.00196146.58
43.00196146.31
53.75245183.47
53.75245183.52
64.50294213.40
64.50294213.67
91.37916801.62
91.37916801.62
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset BBA 2.3
timeDMTA
0.000000099.33
0.000000097.44
0.673393893.73
0.673393893.77
2.020181487.84
2.020181489.82
4.713756571.61
4.713756571.42
9.427513145.60
9.427513145.42
14.141269631.12
14.141269631.68
18.855026223.20
18.855026224.13
28.28253939.43
28.28253939.82
37.71005237.08
37.71005238.64
47.13756544.41
47.13756544.78
56.56507854.92
56.56507855.08
80.13386122.13
80.13386122.23
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Dataset Elliot
timeDMTA
0.00000097.5
0.000000100.7
1.22847886.4
1.22847888.5
3.68543569.8
3.68543577.1
8.59934959.0
8.59934954.2
17.19869731.3
17.19869733.5
25.79804619.6
25.79804620.9
34.39739513.3
34.39739515.8
51.5960926.7
51.5960928.7
68.7947898.8
68.7947898.7
103.1921846.0
103.1921844.4
146.1889283.3
146.1889282.8
223.5830661.4
223.5830661.8
0.00000093.4
0.000000103.2
1.22847889.2
1.22847886.6
3.68543578.2
3.68543578.1
8.59934955.6
8.59934953.0
17.19869733.7
17.19869733.2
25.79804620.9
25.79804619.9
34.39739518.2
34.39739512.7
51.5960927.8
51.5960929.0
68.79478911.4
68.7947899.0
103.1921843.9
103.1921844.4
146.1889282.6
146.1889283.4
223.5830662.0
223.5830661.7
+
+
+

Separate evaluations +

+

In order to obtain suitable starting parameters for the NLHM fits, +separate fits of the four models to the data for each soil are generated +using the mmkin function from the mkin +package. In a first step, constant variance is assumed. Convergence is +checked with the status function.

+
+deg_mods <- c("SFO", "FOMC", "DFOP", "HS")
+f_sep_const <- mmkin(
+  deg_mods,
+  dmta_ds,
+  error_model = "const",
+  quiet = TRUE)
+
+status(f_sep_const) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
CalkeBorstelFlaachBBA 2.2BBA 2.3Elliot
SFOOKOKOKOKOKOK
FOMCOKOKOKOKOKOK
DFOPOKOKOKOKOKOK
HSOKOKOKCOKOK
+

In the table above, OK indicates convergence, and C indicates failure +to converge. All separate fits with constant variance converged, with +the sole exception of the HS fit to the BBA 2.2 data. To prepare for +fitting NLHM using the two-component error model, the separate fits are +updated assuming two-component error.

+
+f_sep_tc <- update(f_sep_const, error_model = "tc")
+status(f_sep_tc) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
CalkeBorstelFlaachBBA 2.2BBA 2.3Elliot
SFOOKOKOKOKOKOK
FOMCOKOKOKOKCOK
DFOPOKOKCOKCOK
HSOKCOKOKOKOK
+

Using the two-component error model, the one fit that did not +converge with constant variance did converge, but other non-SFO fits +failed to converge.

+
+
+

Hierarchichal model fits +

+

The following code fits eight versions of hierarchical models to the +data, using SFO, FOMC, DFOP and HS for the parent compound, and using +either constant variance or two-component error for the error model. The +default parameter distribution model in mkin allows for variation of all +degradation parameters across the assumed population of soils. In other +words, each degradation parameter is associated with a random effect as +a first step. The mhmkin function makes it possible to fit +all eight versions in parallel (given a sufficient number of computing +cores being available) to save execution time.

+

Convergence plots and summaries for these fits are shown in the +appendix.

+
+f_saem <- mhmkin(list(f_sep_const, f_sep_tc), transformations = "saemix")
+

The output of the status function shows that all fits +terminated successfully.

+
+status(f_saem) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
SFOOKOK
FOMCOKOK
DFOPOKOK
HSOKOK
+

The AIC and BIC values show that the biphasic models DFOP and HS give +the best fits.

+
+anova(f_saem) |> kable(digits = 1)
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLik
SFO const5796.3795.3-393.2
SFO tc6798.3797.1-393.2
FOMC const7734.2732.7-360.1
FOMC tc8720.4718.8-352.2
DFOP const9711.8710.0-346.9
HS const9714.0712.1-348.0
DFOP tc10665.5663.4-322.8
HS tc10667.1665.0-323.6
+

The DFOP model is preferred here, as it has a better mechanistic +basis for batch experiments with constant incubation conditions. Also, +it shows the lowest AIC and BIC values in the first set of fits when +combined with the two-component error model. Therefore, the DFOP model +was selected for further refinements of the fits with the aim to make +the model fully identifiable.

+
+

Parameter identifiability based on the Fisher Information +Matrix +

+

Using the illparms function, ill-defined statistical +model parameters such as standard deviations of the degradation +parameters in the population and error model parameters can be +found.

+
+illparms(f_saem) |> kable()
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
consttc
SFOb.1
FOMCsd(DMTA_0)
DFOPsd(k2)sd(k2)
HSsd(tb)
+

According to the illparms function, the fitted standard +deviation of the second kinetic rate constant k2 is +ill-defined in both DFOP fits. This suggests that different values would +be obtained for this standard deviation when using different starting +values.

+

The thus identified overparameterisation is addressed by removing the +random effect for k2 from the parameter model.

+
+f_saem_dfop_tc_no_ranef_k2 <- update(f_saem[["DFOP", "tc"]],
+  no_random_effect = "k2")
+

For the resulting fit, it is checked whether there are still +ill-defined parameters,

+
+illparms(f_saem_dfop_tc_no_ranef_k2)
+

which is not the case. Below, the refined model is compared with the +previous best model. The model without random effect for k2 +is a reduced version of the previous model. Therefore, the models are +nested and can be compared using the likelihood ratio test. This is +achieved with the argument test = TRUE to the +anova function.

+
+anova(f_saem[["DFOP", "tc"]], f_saem_dfop_tc_no_ranef_k2, test = TRUE) |>
+  kable(format.args = list(digits = 4))
+ ++++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
nparAICBICLikChisqDfPr(>Chisq)
f_saem_dfop_tc_no_ranef_k29663.8661.9-322.9NANANA
f_saem[[“DFOP”, “tc”]]10665.5663.4-322.80.280910.5961
+

The AIC and BIC criteria are lower after removal of the ill-defined +random effect for k2. The p value of the likelihood ratio +test is much greater than 0.05, indicating that the model with the +higher likelihood (here the model with random effects for all +degradation parameters f_saem[["DFOP", "tc"]]) does not fit +significantly better than the model with the lower likelihood (the +reduced model f_saem_dfop_tc_no_ranef_k2).

+

Therefore, AIC, BIC and likelihood ratio test suggest the use of the +reduced model.

+

The convergence of the fit is checked visually.

+
+Convergence plot for the NLHM DFOP fit with two-component error and without a random effect on 'k2'

+Convergence plot for the NLHM DFOP fit with two-component error and +without a random effect on ‘k2’ +

+
+

All parameters appear to have converged to a satisfactory degree. The +final fit is plotted using the plot method from the mkin package.

+
+plot(f_saem_dfop_tc_no_ranef_k2)
+
+Plot of the final NLHM DFOP fit

+Plot of the final NLHM DFOP fit +

+
+

Finally, a summary report of the fit is produced.

+
+summary(f_saem_dfop_tc_no_ranef_k2)
+
saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:09 2023 
+Date of summary: Thu Apr 20 14:07:10 2023 
+
+Equations:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 4.175 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+   DMTA_0        k1        k2         g 
+98.759266  0.087034  0.009933  0.930827 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 g
+DMTA_0  98.76  0  0 0
+k1       0.00  1  0 0
+k2       0.00  0  1 0
+g        0.00  0  0 1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  663.8 661.9 -322.9
+
+Optimised parameters:
+               est.     lower     upper
+DMTA_0    98.228939 96.285869 100.17201
+k1         0.064063  0.033477   0.09465
+k2         0.008297  0.005824   0.01077
+g          0.953821  0.914328   0.99331
+a.1        1.068479  0.869538   1.26742
+b.1        0.029424  0.022406   0.03644
+SD.DMTA_0  2.030437  0.404824   3.65605
+SD.k1      0.594692  0.256660   0.93272
+SD.g       1.006754  0.361327   1.65218
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0218                
+k2  0.0556  0.0355        
+g  -0.0516 -0.0284 -0.2800
+
+Random effects:
+            est.  lower  upper
+SD.DMTA_0 2.0304 0.4048 3.6560
+SD.k1     0.5947 0.2567 0.9327
+SD.g      1.0068 0.3613 1.6522
+
+Variance model:
+       est.   lower   upper
+a.1 1.06848 0.86954 1.26742
+b.1 0.02942 0.02241 0.03644
+
+Estimated disappearance times:
+      DT50 DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.45 41.4    12.46   10.82   83.54
+
+
+

Alternative check of parameter identifiability +

+

The parameter check used in the illparms function is +based on a quadratic approximation of the likelihood surface near its +optimum, which is calculated using the Fisher Information Matrix (FIM). +An alternative way to check parameter identifiability (Duchesne et al. 2021) based on a multistart +approach has recently been implemented in mkin.

+

The graph below shows boxplots of the parameters obtained in 50 runs +of the saem algorithm with different parameter combinations, sampled +from the range of the parameters obtained for the individual datasets +fitted separately using nonlinear regression.

+
+f_saem_dfop_tc_multi <- multistart(f_saem[["DFOP", "tc"]], n = 50, cores = 15)
+
+par(mar = c(6.1, 4.1, 2.1, 2.1))
+parplot(f_saem_dfop_tc_multi, lpos = "bottomright", ylim = c(0.3, 10), las = 2)
+
+Scaled parameters from the multistart runs, full model

+Scaled parameters from the multistart runs, full model +

+
+

The graph clearly confirms the lack of identifiability of the +variance of k2 in the full model. The overparameterisation +of the model also indicates a lack of identifiability of the variance of +parameter g.

+

The parameter boxplots of the multistart runs with the reduced model +shown below indicate that all runs give similar results, regardless of +the starting parameters.

+
+f_saem_dfop_tc_no_ranef_k2_multi <- multistart(f_saem_dfop_tc_no_ranef_k2,
+  n = 50, cores = 15)
+
+par(mar = c(6.1, 4.1, 2.1, 2.1))
+parplot(f_saem_dfop_tc_no_ranef_k2_multi, ylim = c(0.5, 2), las = 2,
+  lpos = "bottomright")
+
+Scaled parameters from the multistart runs, reduced model

+Scaled parameters from the multistart runs, reduced model +

+
+

When only the parameters of the top 25% of the fits are shown (based +on a feature introduced in mkin 1.2.2 currently under development), the +scatter is even less as shown below.

+
+par(mar = c(6.1, 4.1, 2.1, 2.1))
+parplot(f_saem_dfop_tc_no_ranef_k2_multi, ylim = c(0.5, 2), las = 2, llquant = 0.25,
+  lpos = "bottomright")
+
+Scaled parameters from the multistart runs, reduced model, fits with the top 25\% likelihood values

+Scaled parameters from the multistart runs, reduced model, fits with the +top 25% likelihood values +

+
+
+
+
+

Conclusions +

+

Fitting the four parent degradation models SFO, FOMC, DFOP and HS as +part of hierarchical model fits with two different error models and +normal distributions of the transformed degradation parameters works +without technical problems. The biphasic models DFOP and HS gave the +best fit to the data, but the default parameter distribution model was +not fully identifiable. Removing the random effect for the second +kinetic rate constant of the DFOP model resulted in a reduced model that +was fully identifiable and showed the lowest values for the model +selection criteria AIC and BIC. The reliability of the identification of +all model parameters was confirmed using multiple starting values.

+
+
+

Acknowledgements +

+

The helpful comments by Janina Wöltjen of the German Environment +Agency are gratefully acknowledged.

+
+
+

References +

+
+
+Duchesne, Ronan, Anissa Guillemin, Olivier Gandrillon, and Fabien +Crauste. 2021. “Practical Identifiability in the Frame of +Nonlinear Mixed Effects Models: The Example of the in Vitro +Erythropoiesis.” BMC Bioinformatics 22 (478). https://doi.org/10.1186/s12859-021-04373-4. +
+
+
+
+

Appendix +

+
+

Hierarchical model fit listings +

+ +Hierarchical mkin fit of the SFO model with error model const + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:02 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - k_DMTA * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 0.982 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+ DMTA_0  k_DMTA 
+97.2953  0.0566 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k_DMTA
+DMTA_0   97.3      0
+k_DMTA    0.0      1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  796.3 795.3 -393.2
+
+Optimised parameters:
+              est.    lower   upper
+DMTA_0    97.28130 95.71113 98.8515
+k_DMTA     0.05665  0.02909  0.0842
+a.1        2.66442  2.35579  2.9731
+SD.DMTA_0  1.54776  0.15447  2.9411
+SD.k_DMTA  0.60690  0.26248  0.9513
+
+Correlation: 
+       DMTA_0
+k_DMTA 0.0168
+
+Random effects:
+            est.  lower  upper
+SD.DMTA_0 1.5478 0.1545 2.9411
+SD.k_DMTA 0.6069 0.2625 0.9513
+
+Variance model:
+     est. lower upper
+a.1 2.664 2.356 2.973
+
+Estimated disappearance times:
+      DT50  DT90
+DMTA 12.24 40.65
+
+
+

+ +Hierarchical mkin fit of the SFO model with error model tc + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:03 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - k_DMTA * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 2.398 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+  DMTA_0   k_DMTA 
+96.99175  0.05603 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k_DMTA
+DMTA_0  96.99      0
+k_DMTA   0.00      1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  798.3 797.1 -393.2
+
+Optimised parameters:
+               est.     lower    upper
+DMTA_0    97.271822 95.703157 98.84049
+k_DMTA     0.056638  0.029110  0.08417
+a.1        2.660081  2.230398  3.08976
+b.1        0.001665 -0.006911  0.01024
+SD.DMTA_0  1.545520  0.145035  2.94601
+SD.k_DMTA  0.606422  0.262274  0.95057
+
+Correlation: 
+       DMTA_0
+k_DMTA 0.0169
+
+Random effects:
+            est.  lower  upper
+SD.DMTA_0 1.5455 0.1450 2.9460
+SD.k_DMTA 0.6064 0.2623 0.9506
+
+Variance model:
+        est.     lower   upper
+a.1 2.660081  2.230398 3.08976
+b.1 0.001665 -0.006911 0.01024
+
+Estimated disappearance times:
+      DT50  DT90
+DMTA 12.24 40.65
+
+
+

+ +Hierarchical mkin fit of the FOMC model with error model const + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:02 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - (alpha/beta) * 1/((time/beta) + 1) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 1.398 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+ DMTA_0   alpha    beta 
+ 98.292   9.909 156.341 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 alpha beta
+DMTA_0  98.29     0    0
+alpha    0.00     1    0
+beta     0.00     0    1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  734.2 732.7 -360.1
+
+Optimised parameters:
+              est.   lower   upper
+DMTA_0     98.3435 96.9033  99.784
+alpha       7.2007  2.5889  11.812
+beta      112.8746 34.8816 190.868
+a.1         2.0459  1.8054   2.286
+SD.DMTA_0   1.4795  0.2717   2.687
+SD.alpha    0.6396  0.1509   1.128
+SD.beta     0.6874  0.1587   1.216
+
+Correlation: 
+      DMTA_0  alpha  
+alpha -0.1125        
+beta  -0.1227  0.3632
+
+Random effects:
+            est.  lower upper
+SD.DMTA_0 1.4795 0.2717 2.687
+SD.alpha  0.6396 0.1509 1.128
+SD.beta   0.6874 0.1587 1.216
+
+Variance model:
+     est. lower upper
+a.1 2.046 1.805 2.286
+
+Estimated disappearance times:
+      DT50  DT90 DT50back
+DMTA 11.41 42.53     12.8
+
+
+

+ +Hierarchical mkin fit of the FOMC model with error model tc + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:04 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - (alpha/beta) * 1/((time/beta) + 1) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 3.044 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+DMTA_0  alpha   beta 
+98.772  4.663 92.597 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 alpha beta
+DMTA_0  98.77     0    0
+alpha    0.00     1    0
+beta     0.00     0    1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  720.4 718.8 -352.2
+
+Optimised parameters:
+              est.    lower     upper
+DMTA_0    98.99136 97.26011 100.72261
+alpha      5.86312  2.57485   9.15138
+beta      88.55571 29.20889 147.90254
+a.1        1.51063  1.24384   1.77741
+b.1        0.02824  0.02040   0.03609
+SD.DMTA_0  1.57436 -0.04867   3.19739
+SD.alpha   0.59871  0.17132   1.02611
+SD.beta    0.72994  0.22849   1.23139
+
+Correlation: 
+      DMTA_0  alpha  
+alpha -0.1363        
+beta  -0.1414  0.2542
+
+Random effects:
+            est.    lower upper
+SD.DMTA_0 1.5744 -0.04867 3.197
+SD.alpha  0.5987  0.17132 1.026
+SD.beta   0.7299  0.22849 1.231
+
+Variance model:
+       est.  lower   upper
+a.1 1.51063 1.2438 1.77741
+b.1 0.02824 0.0204 0.03609
+
+Estimated disappearance times:
+      DT50 DT90 DT50back
+DMTA 11.11 42.6    12.82
+
+
+

+ +Hierarchical mkin fit of the DFOP model with error model const + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:02 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 1.838 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2        g 
+98.64383  0.09211  0.02999  0.76814 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 g
+DMTA_0  98.64  0  0 0
+k1       0.00  1  0 0
+k2       0.00  0  1 0
+g        0.00  0  0 1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC BIC logLik
+  711.8 710 -346.9
+
+Optimised parameters:
+               est.     lower    upper
+DMTA_0    98.092481 96.573898 99.61106
+k1         0.062499  0.030336  0.09466
+k2         0.009065 -0.005133  0.02326
+g          0.948967  0.862079  1.03586
+a.1        1.821671  1.604774  2.03857
+SD.DMTA_0  1.677785  0.472066  2.88350
+SD.k1      0.634962  0.270788  0.99914
+SD.k2      1.033498 -0.205994  2.27299
+SD.g       1.710046  0.428642  2.99145
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0246                
+k2  0.0491  0.0953        
+g  -0.0552 -0.0889 -0.4795
+
+Random effects:
+           est.   lower  upper
+SD.DMTA_0 1.678  0.4721 2.8835
+SD.k1     0.635  0.2708 0.9991
+SD.k2     1.033 -0.2060 2.2730
+SD.g      1.710  0.4286 2.9914
+
+Variance model:
+     est. lower upper
+a.1 1.822 1.605 2.039
+
+Estimated disappearance times:
+      DT50 DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.79 42.8    12.88   11.09   76.46
+
+
+

+ +Hierarchical mkin fit of the DFOP model with error model tc + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:04 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 *
+           time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time)))
+           * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 3.297 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+   DMTA_0        k1        k2         g 
+98.759266  0.087034  0.009933  0.930827 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 g
+DMTA_0  98.76  0  0 0
+k1       0.00  1  0 0
+k2       0.00  0  1 0
+g        0.00  0  0 1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC   BIC logLik
+  665.5 663.4 -322.8
+
+Optimised parameters:
+               est.     lower     upper
+DMTA_0    98.377019 96.447952 100.30609
+k1         0.064843  0.034607   0.09508
+k2         0.008895  0.006368   0.01142
+g          0.949696  0.903815   0.99558
+a.1        1.065241  0.865754   1.26473
+b.1        0.029340  0.022336   0.03634
+SD.DMTA_0  2.007754  0.387982   3.62753
+SD.k1      0.580473  0.250286   0.91066
+SD.k2      0.006105 -4.920337   4.93255
+SD.g       1.097149  0.412779   1.78152
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0235                
+k2  0.0595  0.0424        
+g  -0.0470 -0.0278 -0.2731
+
+Random effects:
+              est.   lower  upper
+SD.DMTA_0 2.007754  0.3880 3.6275
+SD.k1     0.580473  0.2503 0.9107
+SD.k2     0.006105 -4.9203 4.9325
+SD.g      1.097149  0.4128 1.7815
+
+Variance model:
+       est.   lower   upper
+a.1 1.06524 0.86575 1.26473
+b.1 0.02934 0.02234 0.03634
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.36 41.32    12.44   10.69   77.92
+
+
+

+ +Hierarchical mkin fit of the HS model with error model const + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:03 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - ifelse(time <= tb, k1, k2) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 1.972 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Constant variance 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2       tb 
+97.82176  0.06931  0.02997 11.13945 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 tb
+DMTA_0  97.82  0  0  0
+k1       0.00  1  0  0
+k2       0.00  0  1  0
+tb       0.00  0  0  1
+
+Starting values for error model parameters:
+a.1 
+  1 
+
+Results:
+
+Likelihood computed by importance sampling
+  AIC   BIC logLik
+  714 712.1   -348
+
+Optimised parameters:
+              est.    lower    upper
+DMTA_0    98.16102 96.47747 99.84456
+k1         0.07876  0.05261  0.10491
+k2         0.02227  0.01706  0.02747
+tb        13.99089 -7.40049 35.38228
+a.1        1.82305  1.60700  2.03910
+SD.DMTA_0  1.88413  0.56204  3.20622
+SD.k1      0.34292  0.10482  0.58102
+SD.k2      0.19851  0.01718  0.37985
+SD.tb      1.68168  0.58064  2.78272
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0142                
+k2  0.0001 -0.0025        
+tb  0.0165 -0.1256 -0.0301
+
+Random effects:
+            est.   lower  upper
+SD.DMTA_0 1.8841 0.56204 3.2062
+SD.k1     0.3429 0.10482 0.5810
+SD.k2     0.1985 0.01718 0.3798
+SD.tb     1.6817 0.58064 2.7827
+
+Variance model:
+     est. lower upper
+a.1 1.823 1.607 2.039
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 8.801 67.91    20.44   8.801   31.13
+
+
+

+ +Hierarchical mkin fit of the HS model with error model tc + +

+saemix version used for fitting:      3.2 
+mkin version used for pre-fitting:  1.2.3 
+R version used for fitting:         4.2.3 
+Date of fit:     Thu Apr 20 14:07:04 2023 
+Date of summary: Thu Apr 20 14:08:16 2023 
+
+Equations:
+d_DMTA/dt = - ifelse(time <= tb, k1, k2) * DMTA
+
+Data:
+155 observations of 1 variable(s) grouped in 6 datasets
+
+Model predictions using solution type analytical 
+
+Fitted in 3.378 s
+Using 300, 100 iterations and 9 chains
+
+Variance model: Two-component variance function 
+
+Starting values for degradation parameters:
+  DMTA_0       k1       k2       tb 
+98.45190  0.07525  0.02576 19.19375 
+
+Fixed degradation parameter values:
+None
+
+Starting values for random effects (square root of initial entries in omega):
+       DMTA_0 k1 k2 tb
+DMTA_0  98.45  0  0  0
+k1       0.00  1  0  0
+k2       0.00  0  1  0
+tb       0.00  0  0  1
+
+Starting values for error model parameters:
+a.1 b.1 
+  1   1 
+
+Results:
+
+Likelihood computed by importance sampling
+    AIC BIC logLik
+  667.1 665 -323.6
+
+Optimised parameters:
+              est.    lower    upper
+DMTA_0    97.76570 95.81350 99.71791
+k1         0.05855  0.03080  0.08630
+k2         0.02337  0.01664  0.03010
+tb        31.09638 29.38289 32.80987
+a.1        1.08835  0.88590  1.29080
+b.1        0.02964  0.02257  0.03671
+SD.DMTA_0  2.04877  0.42607  3.67147
+SD.k1      0.59166  0.25621  0.92711
+SD.k2      0.30698  0.09561  0.51835
+SD.tb      0.01274 -0.10914  0.13462
+
+Correlation: 
+   DMTA_0  k1      k2     
+k1  0.0160                
+k2 -0.0070 -0.0024        
+tb -0.0668 -0.0103 -0.2013
+
+Random effects:
+             est.    lower  upper
+SD.DMTA_0 2.04877  0.42607 3.6715
+SD.k1     0.59166  0.25621 0.9271
+SD.k2     0.30698  0.09561 0.5183
+SD.tb     0.01274 -0.10914 0.1346
+
+Variance model:
+       est.   lower   upper
+a.1 1.08835 0.88590 1.29080
+b.1 0.02964 0.02257 0.03671
+
+Estimated disappearance times:
+      DT50  DT90 DT50back DT50_k1 DT50_k2
+DMTA 11.84 51.71    15.57   11.84   29.66
+
+
+

+
+
+

Hierarchical model convergence plots +

+
+Convergence plot for the NLHM SFO fit with constant variance

+Convergence plot for the NLHM SFO fit with constant variance +

+
+
+Convergence plot for the NLHM SFO fit with two-component error

+Convergence plot for the NLHM SFO fit with two-component error +

+
+
+Convergence plot for the NLHM FOMC fit with constant variance

+Convergence plot for the NLHM FOMC fit with constant variance +

+
+
+Convergence plot for the NLHM FOMC fit with two-component error

+Convergence plot for the NLHM FOMC fit with two-component error +

+
+
+Convergence plot for the NLHM DFOP fit with constant variance

+Convergence plot for the NLHM DFOP fit with constant variance +

+
+
+Convergence plot for the NLHM DFOP fit with two-component error

+Convergence plot for the NLHM DFOP fit with two-component error +

+
+
+Convergence plot for the NLHM HS fit with constant variance

+Convergence plot for the NLHM HS fit with constant variance +

+
+
+Convergence plot for the NLHM HS fit with two-component error

+Convergence plot for the NLHM HS fit with two-component error +

+
+
+
+

Session info +

+
R version 4.2.3 (2023-03-15)
+Platform: x86_64-pc-linux-gnu (64-bit)
+Running under: Debian GNU/Linux 12 (bookworm)
+
+Matrix products: default
+BLAS:   /usr/lib/x86_64-linux-gnu/openblas-serial/libblas.so.3
+LAPACK: /usr/lib/x86_64-linux-gnu/openblas-serial/libopenblas-r0.3.21.so
+
+locale:
+ [1] LC_CTYPE=de_DE.UTF-8       LC_NUMERIC=C              
+ [3] LC_TIME=de_DE.UTF-8        LC_COLLATE=de_DE.UTF-8    
+ [5] LC_MONETARY=de_DE.UTF-8    LC_MESSAGES=de_DE.UTF-8   
+ [7] LC_PAPER=de_DE.UTF-8       LC_NAME=C                 
+ [9] LC_ADDRESS=C               LC_TELEPHONE=C            
+[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C       
+
+attached base packages:
+[1] parallel  stats     graphics  grDevices utils     datasets  methods  
+[8] base     
+
+other attached packages:
+[1] saemix_3.2 npde_3.3   knitr_1.42 mkin_1.2.3
+
+loaded via a namespace (and not attached):
+ [1] highr_0.10        pillar_1.9.0      bslib_0.4.2       compiler_4.2.3   
+ [5] jquerylib_0.1.4   tools_4.2.3       mclust_6.0.0      digest_0.6.31    
+ [9] tibble_3.2.1      jsonlite_1.8.4    evaluate_0.20     memoise_2.0.1    
+[13] lifecycle_1.0.3   nlme_3.1-162      gtable_0.3.3      lattice_0.21-8   
+[17] pkgconfig_2.0.3   rlang_1.1.0       DBI_1.1.3         cli_3.6.1        
+[21] yaml_2.3.7        pkgdown_2.0.7     xfun_0.38         fastmap_1.1.1    
+[25] gridExtra_2.3     dplyr_1.1.1       stringr_1.5.0     generics_0.1.3   
+[29] desc_1.4.2        fs_1.6.1          vctrs_0.6.1       sass_0.4.5       
+[33] systemfonts_1.0.4 tidyselect_1.2.0  rprojroot_2.0.3   lmtest_0.9-40    
+[37] grid_4.2.3        glue_1.6.2        R6_2.5.1          textshaping_0.3.6
+[41] fansi_1.0.4       rmarkdown_2.21    purrr_1.0.1       ggplot2_3.4.2    
+[45] magrittr_2.0.3    codetools_0.2-19  scales_1.2.1      htmltools_0.5.5  
+[49] colorspace_2.1-0  ragg_1.2.5        utf8_1.2.3        stringi_1.7.12   
+[53] munsell_0.5.0     cachem_1.0.7      zoo_1.8-12
+
+
+

Hardware info +

+
CPU model: AMD Ryzen 9 7950X 16-Core Processor
+
MemTotal:       64936316 kB
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+ + + + + + + + -- cgit v1.2.1