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----
-title: "Testing covariate modelling in hierarchical parent degradation kinetics with residue data on mesotrione"
-author: Johannes Ranke
-date: Last change on 4 August 2023, last compiled on `r format(Sys.time(), "%e
-  %B %Y")`
-output:
-  pdf_document:
-    extra_dependencies: ["float", "listing"]
-toc: yes
-geometry: margin=2cm
----
-
-```{r setup, echo = FALSE, cache = FALSE}
-options(width = 80) # For summary listings
-knitr::opts_chunk$set(
-  cache = TRUE,
-  comment = "", tidy = FALSE,
-  fig.pos = "H", fig.align = "center"
-)
-```
-
-\clearpage
-
-# 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.
-
-```{r packages, cache = FALSE, message = FALSE}
-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)
-}
-```
-
-\clearpage
-
-## Test data
-
-```{r 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.
-
-```{r show-covar-data, dependson = "data", results = "asis"}
-pH <- attr(meso_ds, "covariates")
-kable(pH, caption = "Covariate data")
-```
-
-\clearpage
-
-```{r show-data, dependson = "data", results = "asis"}
-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))
-}
-```
-
-\clearpage
-
-# 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.
-
-```{r f-sep-const, dependson = "data"}
-deg_mods <- c("SFO", "FOMC", "DFOP", "SFORB", "HS")
-f_sep_const <- mmkin(
-  deg_mods,
-  meso_ds,
-  error_model = "const",
-  cluster = cl,
-  quiet = TRUE)
-```
-
-```{r dependson = "f-sep-const"}
-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.
-
-```{r f-sep-tc, dependson = "f-sep-const"}
-f_sep_tc <- update(f_sep_const, error_model = "tc")
-```
-
-```{r dependson = "f-sep-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.
-
-\clearpage
-
-# 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.
-
-```{r f-saem-1, dependson = c("f-sep-const", "f-sep-tc")}
-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).
-
-```{r dependson = "f-saem-1"}
-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.
-
-```{r dependson = "f-saem-1"}
-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.
-
-```{r f-saem-2, dependson = "f-saem-1"}
-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.
-
-```{r dependson = "f-saem-2"}
-illparms(f_saem_2) |> kable()
-```
-
-No ill-defined errors remain in the fits with constant variance.
-
-\clearpage
-
-# 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
-
-```{r sfo-pH, dependson = "f-sep-const"}
-sfo_pH <- saem(f_sep_const["SFO", ], no_random_effect = "meso_0", covariates = pH,
-  covariate_models = list(log_k_meso ~ pH))
-```
-
-```{r dependson = "sfo-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.
-
-
-```{r dependson = "sfo-pH"}
-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.
-
-\clearpage
-
-```{r dependson = "sfo-pH", plot.height = 5}
-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.
-
-```{r dependson = "sfo-pH", plot.height = 5}
-endpoints(sfo_pH)
-endpoints(sfo_pH, covariate_quantile = 0.9)
-endpoints(sfo_pH, covariates = c(pH = 7.0))
-```
-
-\clearpage
-
-## FOMC
-
-```{r fomc-pH, dependson = "f-sep-const"}
-fomc_pH <- saem(f_sep_const["FOMC", ], no_random_effect = "meso_0", covariates = pH,
-  covariate_models = list(log_alpha ~ pH))
-```
-
-```{r dependson = "fomc-pH"}
-summary(fomc_pH)$confint_trans |> kable(digits = 2)
-```
-
-As in the case of SFO, the confidence interval of the slope parameter
-(here `beta_pH(log_alpha)`) quantifying the influence of soil pH
-does not include zero, and the model comparison clearly indicates
-that the model with covariate influence is preferable.
-However, the random effect for `alpha` is not well-defined any
-more after inclusion of the covariate effect (the confidence
-interval of `SD.log_alpha` includes zero).
-
-```{r dependson = "fomc-pH"}
-illparms(fomc_pH)
-```
-
-Therefore, the model is updated without this random effect, and
-no ill-defined parameters remain.
-
-```{r fomc-pH-2, dependson = "fomc_pH"}
-fomc_pH_2 <- update(fomc_pH, no_random_effect = c("meso_0", "log_alpha"))
-illparms(fomc_pH_2)
-```
-
-```{r dependson = "fomc-pH-2"}
-anova(f_saem_2[["FOMC", "const"]], fomc_pH, fomc_pH_2, test = TRUE)
-```
-
-Model comparison indicates that including pH dependence significantly improves
-the fit, and that the reduced model with covariate influence results in
-the most preferable FOMC fit.
-
-```{r dependson = "fomc-pH"}
-summary(fomc_pH_2)$confint_trans |> kable(digits = 2)
-```
-
-\clearpage
-
-```{r dependson = "fomc-pH", plot.height = 5}
-plot(fomc_pH_2)
-```
-
-```{r dependson = "fomc-pH", plot.height = 5}
-endpoints(fomc_pH_2)
-endpoints(fomc_pH_2, covariates = c(pH = 7))
-```
-
-\clearpage
-
-## DFOP
-
-In the DFOP fits without covariate effects, random effects for two degradation
-parameters (`k2` and `g`) were identifiable.
-
-```{r dependson = "f-saem-2"}
-summary(f_saem_2[["DFOP", "const"]])$confint_trans |> kable(digits = 2)
-```
-
-A fit with pH dependent degradation parameters was obtained by excluding
-the same random effects as in the refined DFOP fit without covariate influence,
-and including covariate models for the two identifiable parameters `k2` and `g`.
-
-```{r dfop-pH, dependson = "f-sep-const"}
-dfop_pH <- saem(f_sep_const["DFOP", ], no_random_effect = c("meso_0", "log_k1"),
-  covariates = pH,
-  covariate_models = list(log_k2 ~ pH, g_qlogis ~ pH))
-```
-
-The corresponding parameters for
-the influence of soil pH are `beta_pH(log_k2)` for the influence of soil pH on
-`k2`, and `beta_pH(g_qlogis)` for its influence on `g`.
-
-```{r dependson = "dfop-pH"}
-summary(dfop_pH)$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.
-
-```{r dfop-pH-2, dependson = "dfop-pH"}
-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.
-
-```{r dfop-pH-3, dependson = "f-sep-const"}
-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.
-
-```{r dfop-pH-4, dependson = "dfop-pH-3"}
-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.
-
-```{r dependson = "dfop-pH-4"}
-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.
-
-\clearpage
-
-```{r dependson = "dfop-pH-2", plot.height = 5}
-plot(dfop_pH_2)
-```
-
-```{r dependson = "dfop-pH-2", plot.height = 5}
-endpoints(dfop_pH_2)
-endpoints(dfop_pH_2, covariates = c(pH = 7))
-```
-
-\clearpage
-
-## SFORB
-
-```{r sforb-pH, dependson = "f-sep-const"}
-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))
-```
-
-```{r dependson = "sforb-pH"}
-summary(sforb_pH)$confint_trans |> kable(digits = 2)
-```
-
-The confidence interval of `beta_pH(log_k_meso_bound_free)` includes zero,
-indicating that the influence of soil pH on `k_meso_bound_free` cannot reliably
-be quantified. Also, the confidence interval for the random effect on this
-parameter (`SD.log_k_meso_bound_free`) includes zero.
-
-Using the `illparms` function, these ill-defined parameters can be found
-more conveniently.
-
-```{r dependson = "sforb-pH"}
-illparms(sforb_pH)
-```
-
-To remove the ill-defined parameters, a second variant of the SFORB model
-with pH influence is fitted. No ill-defined parameters remain.
-
-```{r sforb-pH-2, dependson = "f-sforb-pH"}
-sforb_pH_2 <- update(sforb_pH,
-  no_random_effect = c("meso_free_0", "log_k_meso_free_bound", "log_k_meso_bound_free"),
-  covariate_models = list(log_k_meso_free ~ pH))
-illparms(sforb_pH_2)
-```
-
-The model comparison of the SFORB fits includes the refined model without
-covariate effect, and both versions of the SFORB fit with covariate effect.
-
-```{r dependson = "sforb-pH-2"}
-anova(f_saem_2[["SFORB", "const"]], sforb_pH, sforb_pH_2, test = TRUE)
-```
-The first model including pH influence is preferable based on information criteria
-and the likelihood ratio test. However, as it is not fully identifiable,
-the second model is selected.
-
-```{r dependson = "sforb-pH-2"}
-summary(sforb_pH_2)$confint_trans |> kable(digits = 2)
-```
-\clearpage
-
-```{r dependson = "sforb-pH-2", plot.height = 5}
-plot(sforb_pH_2)
-```
-
-```{r dependson = "sforb-pH-2", plot.height = 5}
-endpoints(sforb_pH_2)
-endpoints(sforb_pH_2, covariates = c(pH = 7))
-```
-
-\clearpage
-
-## HS
-
-```{r hs-pH, dependson = "f-sep-const"}
-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))
-```
-
-```{r dependson = "hs-pH"}
-summary(hs_pH)$confint_trans |> kable(digits = 2)
-illparms(hs_pH)
-```
-
-According to the output of the `illparms` function, the random effect on
-the break time `tb` cannot reliably be quantified, neither can the influence of
-soil pH on `tb`. The fit is repeated without the corresponding covariate
-model, and no ill-defined parameters remain.
-
-```{r hs-pH-2, dependson = "hs-pH"}
-hs_pH_2 <- update(hs_pH, covariate_models = list(log_k1 ~ pH, log_k2 ~ pH))
-illparms(hs_pH_2)
-```
-
-Model comparison confirms that this model is preferable to the fit without
-covariate influence, and also to the first version with covariate influence.
-
-```{r dependson = c("hs-pH-2", "hs-pH-3")}
-anova(f_saem_2[["HS", "const"]], hs_pH, hs_pH_2, test = TRUE)
-```
-
-```{r dependson = "hs-pH-2"}
-summary(hs_pH_2)$confint_trans |> kable(digits = 2)
-```
-
-\clearpage
-
-```{r dependson = "hs-pH-2", plot.height = 5}
-plot(hs_pH_2)
-```
-
-```{r dependson = "hs-pH-2", plot.height = 5}
-endpoints(hs_pH_2)
-endpoints(hs_pH_2, covariates = c(pH = 7))
-```
-
-\clearpage
-
-## Comparison across parent models
-
-After model reduction for all models with pH influence, they are compared with
-each other.
-
-```{r, dependson = c("sfo-pH-2", "fomc-pH-2", "dfop-pH-4", "sforb-pH-1", "hs-pH-3")}
-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.
-
-```{r, dependson = "dfop-pH-2"}
-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.
-
-\clearpage
-
-# Appendix
-
-## Hierarchical fit listings
-
-### Fits without covariate effects
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-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
-
-```{r listing-sfo, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(sfo_pH, "Hierarchichal SFO fit with pH influence")
-```
-
-\clearpage
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(fomc_pH, "Hierarchichal FOMC fit with pH influence")
-```
-
-\clearpage
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(fomc_pH_2, "Refined hierarchichal FOMC fit with pH influence")
-```
-
-\clearpage
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(dfop_pH, "Hierarchichal DFOP fit with pH influence")
-```
-
-\clearpage
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(dfop_pH_2, "Refined hierarchical DFOP fit with pH influence")
-```
-
-\clearpage
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(dfop_pH_4, "Further refined hierarchical DFOP fit with pH influence")
-```
-
-\clearpage
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(sforb_pH, "Hierarchichal SFORB fit with pH influence")
-```
-\clearpage
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(sforb_pH_2, "Refined hierarchichal SFORB fit with pH influence")
-```
-\clearpage
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(hs_pH, "Hierarchichal HS fit with pH influence")
-```
-\clearpage
-
-```{r, cache = FALSE, results = "asis", echo = FALSE}
-tex_listing(hs_pH_2, "Refined hierarchichal HS fit with pH influence")
-```
-
-\clearpage
-
-## Session info
-
-```{r, echo = FALSE, cache = FALSE}
-parallel::stopCluster(cl = cl)
-sessionInfo()
-```
-
-
-## Hardware info
-
-```{r, echo = FALSE}
-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]]))
-}
-```