In mkin: Kinetic Evaluation of Chemical Degradation Data

options(width = 80) # For summary listings
knitr::opts_chunk$set(
  cache = TRUE,
  comment = "", tidy = FALSE,
  fig.pos = "H", fig.align = "center"
)

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Introduction

The purpose of this document is to test demonstrate how nonlinear hierarchical models (NLHM) based on the parent degradation models SFO, FOMC, DFOP and HS can be fitted with the mkin package, also considering the influence of covariates like soil pH on different degradation parameters. Because in some other case studies, the SFORB parameterisation of biexponential decline has shown some advantages over the DFOP parameterisation, SFORB was included in the list of tested models as well.

The mkin package is used in version r packageVersion("mkin"), which is contains the functions that were used for 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)
}

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Test data

data_file <- system.file(
  "testdata", "mesotrione_soil_efsa_2016.xlsx", package = "mkin")
meso_ds <- read_spreadsheet(data_file, parent_only = TRUE)

The following tables show the covariate data and the r length(meso_ds) datasets that were read in from the spreadsheet file.

pH <- attr(meso_ds, "covariates")
kable(pH, caption = "Covariate data")

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for (ds_name in names(meso_ds)) {
  print(
    kable(mkin_long_to_wide(meso_ds[[ds_name]]),
      caption = paste("Dataset", ds_name),
      booktabs = TRUE, row.names = FALSE))
}

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Separate evaluations

In order to obtain suitable starting parameters for the NLHM fits, separate fits of the five 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", "SFORB", "HS")
f_sep_const <- mmkin(
  deg_mods,
  meso_ds,
  error_model = "const",
  cluster = cl,
  quiet = TRUE)

status(f_sep_const[, 1:5]) |> kable()
status(f_sep_const[, 6:18]) |> kable()

In the tables above, OK indicates convergence and C indicates failure to converge. Most separate fits with constant variance converged, with the exception of two FOMC fits, one SFORB fit and one HS fit.

f_sep_tc <- update(f_sep_const, error_model = "tc")

status(f_sep_tc[, 1:5]) |> kable()
status(f_sep_tc[, 6:18]) |> kable()

With the two-component error model, the set of fits that did not converge is larger, with convergence problems appearing for a number of non-SFO fits.

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Hierarchical model fits without covariate effect

The following code fits hierarchical kinetic models for the ten combinations of the five different degradation models with the two different error models in parallel.

f_saem_1 <- mhmkin(list(f_sep_const, f_sep_tc), cluster = cl)
status(f_saem_1) |> kable()

All fits terminate without errors (status OK).

anova(f_saem_1) |> kable(digits = 1)

The model comparisons show that the fits with constant variance are consistently preferable to the corresponding fits with two-component error for these data. This is confirmed by the fact that the parameter b.1 (the relative standard deviation in the fits obtained with the saemix package), is ill-defined in all fits.

illparms(f_saem_1) |> kable()

For obtaining fits with only well-defined random effects, we update the set of fits, excluding random effects that were ill-defined according to the illparms function.

f_saem_2 <- update(f_saem_1, no_random_effect = illparms(f_saem_1))
status(f_saem_2) |> kable()

The updated fits terminate without errors.

illparms(f_saem_2) |> kable()

No ill-defined errors remain in the fits with constant variance.

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Hierarchical model fits with covariate effect

In the following sections, hierarchical fits including a model for the influence of pH on selected degradation parameters are shown for all parent models. Constant variance is selected as the error model based on the fits without covariate effects. Random effects that were ill-defined in the fits without pH influence are excluded. A potential influence of the soil pH is only included for parameters with a well-defined random effect, because experience has shown that only for such parameters a significant pH effect could be found.

SFO

sfo_pH <- saem(f_sep_const["SFO", ], no_random_effect = "meso_0", covariates = pH,
  covariate_models = list(log_k_meso ~ pH))

summary(sfo_pH)$confint_trans |> kable(digits = 2)

The parameter showing the pH influence in the above table is beta_pH(log_k_meso). Its confidence interval does not include zero, indicating that the influence of soil pH on the log of the degradation rate constant is significantly greater than zero.

anova(f_saem_2[["SFO", "const"]], sfo_pH, test = TRUE)

The comparison with the SFO fit without covariate effect confirms that considering the soil pH improves the model, both by comparison of AIC and BIC and by the likelihood ratio test.

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plot(sfo_pH)

Endpoints for a model with covariates are by default calculated for the median of the covariate values. This quantile can be adapted, or a specific covariate value can be given as shown below.

endpoints(sfo_pH)
endpoints(sfo_pH, covariate_quantile = 0.9)
endpoints(sfo_pH, covariates = c(pH = 7.0))

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FOMC

fomc_pH <- saem(f_sep_const["FOMC", ], no_random_effect = "meso_0", covariates = pH,
  covariate_models = list(log_alpha ~ pH))

summary(fomc_pH)$confint_trans |> kable(digits = 2)

As in the case of SFO, the confidence interval of the slope parameter (here beta_pH(log_alpha)) quantifying the influence of soil pH does not include zero, and the model comparison clearly indicates that the model with covariate influence is preferable. However, the random effect for alpha is not well-defined any more after inclusion of the covariate effect (the confidence interval of SD.log_alpha includes zero).

illparms(fomc_pH)

Therefore, the model is updated without this random effect, and no ill-defined parameters remain.

fomc_pH_2 <- update(fomc_pH, no_random_effect = c("meso_0", "log_alpha"))
illparms(fomc_pH_2)

anova(f_saem_2[["FOMC", "const"]], fomc_pH, fomc_pH_2, test = TRUE)

Model comparison indicates that including pH dependence significantly improves the fit, and that the reduced model with covariate influence results in the most preferable FOMC fit.

summary(fomc_pH_2)$confint_trans |> kable(digits = 2)

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plot(fomc_pH_2)

endpoints(fomc_pH_2)
endpoints(fomc_pH_2, covariates = c(pH = 7))

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DFOP

In the DFOP fits without covariate effects, random effects for two degradation parameters (k2 and g) were identifiable.

summary(f_saem_2[["DFOP", "const"]])$confint_trans |> kable(digits = 2)

A fit with pH dependent degradation parameters was obtained by excluding the same random effects as in the refined DFOP fit without covariate influence, and including covariate models for the two identifiable parameters k2 and g.

dfop_pH <- saem(f_sep_const["DFOP", ], no_random_effect = c("meso_0", "log_k1"),
  covariates = pH,
  covariate_models = list(log_k2 ~ pH, g_qlogis ~ pH))

The corresponding parameters for the influence of soil pH are beta_pH(log_k2) for the influence of soil pH on k2, and beta_pH(g_qlogis) for its influence on g.

summary(dfop_pH)$confint_trans |> kable(digits = 2)
illparms(dfop_pH)

Confidence intervals for neither of them include zero, indicating a significant difference from zero. However, the random effect for g is now ill-defined. The fit is updated without this ill-defined random effect.

dfop_pH_2 <- update(dfop_pH,
  no_random_effect = c("meso_0", "log_k1", "g_qlogis"))
illparms(dfop_pH_2)

Now, the slope parameter for the pH effect on g is ill-defined. Therefore, another attempt is made without the corresponding covariate model.

dfop_pH_3 <- saem(f_sep_const["DFOP", ], no_random_effect = c("meso_0", "log_k1"),
  covariates = pH,
  covariate_models = list(log_k2 ~ pH))
illparms(dfop_pH_3)

As the random effect for g is again ill-defined, the fit is repeated without it.

dfop_pH_4 <- update(dfop_pH_3, no_random_effect = c("meso_0", "log_k1", "g_qlogis"))
illparms(dfop_pH_4)

While no ill-defined parameters remain, model comparison suggests that the previous model dfop_pH_2 with two pH dependent parameters is preferable, based on information criteria as well as based on the likelihood ratio test.

anova(f_saem_2[["DFOP", "const"]], dfop_pH, dfop_pH_2, dfop_pH_3, dfop_pH_4)
anova(dfop_pH_2, dfop_pH_4, test = TRUE)

When focussing on parameter identifiability using the test if the confidence interval includes zero, dfop_pH_4 would still be the preferred model. However, it should be kept in mind that parameter confidence intervals are constructed using a simple linearisation of the likelihood. As the confidence interval of the random effect for g only marginally includes zero, it is suggested that this is acceptable, and that dfop_pH_2 can be considered the most preferable model.

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plot(dfop_pH_2)

endpoints(dfop_pH_2)
endpoints(dfop_pH_2, covariates = c(pH = 7))

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SFORB

sforb_pH <- saem(f_sep_const["SFORB", ], no_random_effect = c("meso_free_0", "log_k_meso_free_bound"),
  covariates = pH,
  covariate_models = list(log_k_meso_free ~ pH, log_k_meso_bound_free ~ pH))

summary(sforb_pH)$confint_trans |> kable(digits = 2)

The confidence interval of beta_pH(log_k_meso_bound_free) includes zero, indicating that the influence of soil pH on k_meso_bound_free cannot reliably be quantified. Also, the confidence interval for the random effect on this parameter (SD.log_k_meso_bound_free) includes zero.

Using the illparms function, these ill-defined parameters can be found more conveniently.

illparms(sforb_pH)

To remove the ill-defined parameters, a second variant of the SFORB model with pH influence is fitted. No ill-defined parameters remain.

sforb_pH_2 <- update(sforb_pH,
  no_random_effect = c("meso_free_0", "log_k_meso_free_bound", "log_k_meso_bound_free"),
  covariate_models = list(log_k_meso_free ~ pH))
illparms(sforb_pH_2)

The model comparison of the SFORB fits includes the refined model without covariate effect, and both versions of the SFORB fit with covariate effect.

anova(f_saem_2[["SFORB", "const"]], sforb_pH, sforb_pH_2, test = TRUE)

The first model including pH influence is preferable based on information criteria and the likelihood ratio test. However, as it is not fully identifiable, the second model is selected.

summary(sforb_pH_2)$confint_trans |> kable(digits = 2)

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plot(sforb_pH_2)

endpoints(sforb_pH_2)
endpoints(sforb_pH_2, covariates = c(pH = 7))

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HS

hs_pH <- saem(f_sep_const["HS", ], no_random_effect = c("meso_0"),
  covariates = pH,
  covariate_models = list(log_k1 ~ pH, log_k2 ~ pH, log_tb ~ pH))

summary(hs_pH)$confint_trans |> kable(digits = 2)
illparms(hs_pH)

According to the output of the illparms function, the random effect on the break time tb cannot reliably be quantified, neither can the influence of soil pH on tb. The fit is repeated without the corresponding covariate model, and no ill-defined parameters remain.

hs_pH_2 <- update(hs_pH, covariate_models = list(log_k1 ~ pH, log_k2 ~ pH))
illparms(hs_pH_2)

Model comparison confirms that this model is preferable to the fit without covariate influence, and also to the first version with covariate influence.

anova(f_saem_2[["HS", "const"]], hs_pH, hs_pH_2, test = TRUE)

summary(hs_pH_2)$confint_trans |> kable(digits = 2)

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plot(hs_pH_2)

endpoints(hs_pH_2)
endpoints(hs_pH_2, covariates = c(pH = 7))

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Comparison across parent models

After model reduction for all models with pH influence, they are compared with each other.

anova(sfo_pH, fomc_pH_2, dfop_pH_2, dfop_pH_4, sforb_pH_2, hs_pH_2)

The DFOP model with pH influence on k2 and g and a random effect only on k2 is finally selected as the best fit.

The endpoints resulting from this model are listed below. Please refer to the Appendix for a detailed listing.

endpoints(dfop_pH_2)
endpoints(dfop_pH_2, covariates = c(pH = 7))

Conclusions

These evaluations demonstrate that covariate effects can be included for all types of parent degradation models. These models can then be further refined to make them fully identifiable.

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Appendix

Hierarchical fit listings

Fits without covariate effects

errmods <- c(const = "constant variance", tc = "two-component error")
for (deg_mod in deg_mods) {
  for (err_mod in c("const")) {
    fit <- f_saem_1[[deg_mod, err_mod]]
    if (!inherits(fit$so, "try-error")) {
      caption <- paste("Hierarchical", deg_mod, "fit with", errmods[err_mod])
      tex_listing(fit, caption)
    }
  }
}

Fits with covariate effects

tex_listing(sfo_pH, "Hierarchichal SFO fit with pH influence")

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tex_listing(fomc_pH, "Hierarchichal FOMC fit with pH influence")

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tex_listing(fomc_pH_2, "Refined hierarchichal FOMC fit with pH influence")

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tex_listing(dfop_pH, "Hierarchichal DFOP fit with pH influence")

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tex_listing(dfop_pH_2, "Refined hierarchical DFOP fit with pH influence")

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tex_listing(dfop_pH_4, "Further refined hierarchical DFOP fit with pH influence")

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tex_listing(sforb_pH, "Hierarchichal SFORB fit with pH influence")

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tex_listing(sforb_pH_2, "Refined hierarchichal SFORB fit with pH influence")

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tex_listing(hs_pH, "Hierarchichal HS fit with pH influence")

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tex_listing(hs_pH_2, "Refined hierarchichal HS fit with pH influence")

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Session info

parallel::stopCluster(cl = cl)
sessionInfo()

Hardware info

if(!inherits(try(cpuinfo <- readLines("/proc/cpuinfo")), "try-error")) {
  cat(gsub("model name\t: ", "CPU model: ", cpuinfo[grep("model name", cpuinfo)[1]]))
}
if(!inherits(try(meminfo <- readLines("/proc/meminfo")), "try-error")) {
  cat(gsub("model name\t: ", "System memory: ", meminfo[grep("MemTotal", meminfo)[1]]))
}

mkin documentation built on Nov. 23, 2023, 3:02 p.m.