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+---
+title: Short demo of the multistart method
+author: Johannes Ranke
+date: Last change 26 September 2022 (rebuilt `r Sys.Date()`)
+output:
+  html_document
+vignette: >
+  %\VignetteEngine{knitr::rmarkdown}
+  %\VignetteIndexEntry{Short demo of the multistart method}
+  %\VignetteEncoding{UTF-8}
+---
+
+The dimethenamid data from 2018 from seven soils is used as example data in this vignette.
+
+```{r}
+library(mkin)
+dmta_ds <- lapply(1:7, function(i) {
+  ds_i <- dimethenamid_2018$ds[[i]]$data
+  ds_i[ds_i$name == "DMTAP", "name"] <-  "DMTA"
+  ds_i$time <- ds_i$time * dimethenamid_2018$f_time_norm[i]
+  ds_i
+})
+names(dmta_ds) <- sapply(dimethenamid_2018$ds, function(ds) ds$title)
+dmta_ds[["Elliot"]] <- rbind(dmta_ds[["Elliot 1"]], dmta_ds[["Elliot 2"]])
+dmta_ds[["Elliot 1"]] <- dmta_ds[["Elliot 2"]] <- NULL
+```
+
+First, we check the DFOP model with the two-component error model and
+random effects for all degradation parameters.
+
+```{r}
+f_mmkin <- mmkin("DFOP", dmta_ds, error_model = "tc", cores = 7, quiet = TRUE)
+f_saem_full <- saem(f_mmkin)
+illparms(f_saem_full)
+```
+We see that not all variability parameters are identifiable. The `illparms`
+function tells us that the confidence interval for the standard deviation
+of 'log_k2' includes zero. We check this assessment using multiple runs
+with different starting values.
+
+```{r}
+f_saem_full_multi <- multistart(f_saem_full, n = 16, cores = 16)
+parplot(f_saem_full_multi)
+```
+
+This confirms that the variance of k2 is the most problematic parameter, so we
+reduce the parameter distribution model by removing the intersoil variability
+for k2.
+
+```{r}
+f_saem_reduced <- update(f_saem_full, no_random_effect = "log_k2")
+illparms(f_saem_reduced)
+f_saem_reduced_multi <- multistart(f_saem_reduced, n = 16, cores = 16)
+parplot(f_saem_reduced_multi, lpos = "topright")
+```
+
+The results confirm that all remaining parameters can be determined with sufficient
+certainty.
+
+We can also analyse the log-likelihoods obtained in the multiple runs:
+
+```{r}
+llhist(f_saem_reduced_multi)
+```
+
+The parameter histograms can be further improved by excluding the result with
+the low likelihood.
+
+```{r}
+parplot(f_saem_reduced_multi, lpos = "topright", llmin = -326, ylim = c(0.5, 2))
+```
+
+We can use the `anova` method to compare the models, including a likelihood ratio
+test if the models are nested.
+
+```{r}
+anova(f_saem_full, best(f_saem_reduced_multi), test = TRUE)
+```
+
+While AIC and BIC are lower for the reduced model, the likelihood ratio test
+does not indicate a significant difference between the fits.
+