# Approaches to Estimate the Benchmark Dose for Dose-Response Data with Hierarchical Structure

**Specialeforsvar:** Jannat Ahmad

**Titel**: Approaches to Estimate the Benchmark Dose for Dose-Response Data with Hierarchical Structure

**Abstract:** In this thesis the benchmark dose (BMD) is introduced, and is defined as the dose corresponding to the response, which is a percentage decrease or increase from the control response. To estimate the BMD, log-logistic models can be used, but the BMD is not a direct parameter in these models, so it can be estimated in an after-fitting step. The lower limit of the confidence interval for the BMD estimate, denoted BMDL, is used to derive reference values for assessing the safety of exposure to a toxic substance. In practice more than one experiment is usually conducted, so it is required that we can get one combined estimate based on all of the subexperiments, so in this context four approaches are presented. The non-linear mixed-effects modelling approach is used to obtain an overall average BMD estimate over all existing experiments, while taking the within and in-between experiment variation into account. Fitting the non-linear mixed-effects model can be complex, and convergence issues are occurred. Another approach which also takes the within experiment variation, and in-between experiment variation into account, but does not have this issue is the meta-analytic random-effects modelling approach. This approach estimates

the BMD estimates obtained from fitting the log-logistic model individually to each of the subexperiments, and then these estimates is combined into a weighted average of the BMD estimates, where the within and in-between experiment variation is incorporated in the weights. Another simpler approach, which is also based on meta-analysis, but does not take the within and in-between experiment variation into account consists of estimating the BMD parameter individually for each subexperiment, and then taking the simple average of these estimates, which will serve as the combined estimate for the subexperiments. This

approach is denoted as the simple average approach. Lastly, we have the pooled experiment approach, which does not harmonize with the data used in this thesis. This approach consider all of the subexperiments as one big experiment to which the log-logisitc model is fitted, and the BMD parameter is estimated. The aim of this thesis is to compare these four approaches using a simulation study and a real plant dataset. We conclude that the metaanalytic random-effects modelling approach would be the most appropriate approach to use to estimate BMD and the corresponding BMDL data from an experiment consisting

of multiple subexperiments.

**Vejledere**: Helle Sørensen

Signe Marie Jensen, PLEN**Censor:** Sören Möller, SDU