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　　4.3.4 Optimal design of the dose-response curves

　　This project attempted to fit all the treatment groups in the experiment using the Four-parameter Logistic Regression model by curves, but the results indicated a large variation between individuals that this model cannot be applied to all groups so that the dose-response curve corresponding to each experimental group must be estimated separately. But this is undoubtedly a very time-consuming process, and in order to solve this problem, we can use the optimal design theory to improve the experimental process. In this study, my design method is to compare different regression models with common parameters, that is, when modeling the dose and response according to the target parameters, some parameters are the same between different experimental groups, while the other parameters are different[13]. Here, I take Emax function in the two groups as an example:

　　(6)

　　Emax is a frequently used model in pharmacokinetics that defines the relationship between drug efficacy and dose levels. Where， represents the placebo effect， represents the maximum effect of dose and represents the value of EC50.
　　Obviously, we can set and ，but in general .
　　Based on this, we establish a generic expression for regression models with common parameters:

　　(7)

　　Where， is an independent random variable that follows a standard normal distribution. There are M groups, and the corresponding dose distribution of each group is ，while for each dose level ，we can get observations.

　　Select a suitable model is the key to the modeling process of dose-response studies, and it is necessary to introduce the Multiple Comparison-modeling Approach (MCP-Mod) here. This approach refers to the introduction of a set of candidate models that can cover the general dose-response relationship. Then, Dose-response is established when at least one of the models is statistically significant, otherwise the experiment is stopped to indicate that there is no significant dose-response relation- ship[13]. After the Dose-response is established, the best model with statistical significance is selected according to AIC and BIC criteria.

　　In addition, the Bayesian Model-Averaging Approach (BMA) can be used to take advantage of a set of simple models, which helps prevent under-fitting and over-fitting problems. Therefore, in this project, I plan to adopt 8 frequently used dose response models (Linear log-dose, Emax, Logistic, Beta model and so on) to conduct the above experimental process.