Much of classical optimal design theory relies on specifying a model with only a small number of parameters. In many applications, such models will give reasonable approximations. However, they will often be found not to be entirely correct when enough data are at hand. A property of classical optimal design methodology is that the amount of data does not influence the design when a fixed model is used. However, it is reasonable that a low dimensional model is satisfactory only if limited data is available. With more data available, more aspects of the underlying relationship can be assessed. We consider a simple model that is not thought to be fully correct. The model misspecification, that is, the difference between the true mean and the simple model, is explicitly modeled with a stochastic process. This gives a unified approach to handle situations with both limited and rich data. Our objective is to estimate the combined model, which is the sum of the simple model and the assumed misspecification process. In our situation, the low-dimensional model can be viewed as a fixed effect and the misspecification term as a random effect in a mixed-effects model. Our aim is to predict within this model. We describe how we minimize the prediction error using an optimal design. We compute optimal designs for the full model in different cases. The results confirm that the optimal design depends strongly on the sample size. In low-information situations, traditional optimal designs for models with a small number of parameters are sufficient, while the inclusion of the misspecification term lead to very different designs in data-rich cases. 

One of the most complex tasks during the clinical development of a new drug is to find a correct dose. Optimal experimental design has as a goal to find the best ways to perform an experiment considering the available resources and the statistical model. Optimal designs have already been used to determine the design of dose-finding studies. In this thesis, optimal designs are considered for the simultaneous response of efficacy and safety in a bivariate model, for the drug combination trials, and for general regression problems, including but not limited to dose-finding analysis.

The thesis consists of four papers: In Paper I, the dose that maximizes the clinical utility index based on an efficacy-safety Emax model gives us the desirable balance between effects and side effects. In order to make use of a symmetry property, we use a log-transformed dose scale. The geometric characterization of the multivariate Elfving method is used to derive c-optimal points and weights for arbitrary c-vectors. The second paper is an extension of the first one. We still use the log-transformed dose scale bivariate model and consider now also the placebo effect and side-effect. Fedorov’s exchange algorithm is applied in order to derive locally D-optimal designs numerically. 

Optimal experimental design for dose-finding studies often focuses on one drug only. Paper III calculates D-optimal designs for the efficacy Emax model of two drugs that might interact. Three conditions can occur in drug combination trials. When there is a positive interaction, we deal with synergy; when it is negative, we have antagonism; and when the interaction is zero, it is called additivity.

Finally, in Paper IV, we present a low dimensional regression model with a distortion term. The distortion term, which in our case is a stochastic process, contributes to the regression. Thus, we estimate the combined model, which is a mixed effect model. Optimal designs for this model are derived by applying the Fedorov Algorithm.

Dose-finding is an important part of the clinical development of a new drug. The purpose of dose-finding studies is to determine a suitable dose for future development based on both efficacy and safety. Optimal experimental designs have already been used to determine the design of this kind of studies, however, often that design is focused on efficacy only. We consider an efficacy-safety model, which is a simplified version of the bivariate Emax model. We use here the clinical utility index concept, which provides the desirable balance between efficacy and safety. By maximizing the utility of the patients, we get the estimated dose. This desire leads us to locally c-optimal designs. An algebraic solution for c-optimal designs is determined for arbitrary c vectors using a multivariate version of Elfving's method. The solution shows that the expected therapeutic index of the drug is a key quantity determining both the number of doses, the doses itself, and their weights in the optimal design. A sequential design is proposed to solve the complication of parameter dependency, and it is illustrated in a simulation study.

Much of traditional optimal design theory relies on specifying a model with only a small number of parameters. In many applications, such models will give reasonable approximations. However, they will often be found not to be entirely correct when enough data are at hand. We consider a low-dimensional model with a distortion term. Our objective is to estimate the combined model, including the distortion. In our situation, the low-dimensional model can be viewed as a fixed effect and the distortion term as a random effect in a mixed-effects model. Since we are interested in estimating the combination of fixed and random effects, our aim is to predict within the mixed model. We describe how we minimize the prediction error using an optimal design by constructing the Best Linear Unbiased Estimator and Predictor in our model. Many algorithms can be used in order to construct an optimal design. We apply here the Fedorov algorithm, which exchanges observations between the design points. By performing the algorithm built on the distorted model, we present the optimal design in different cases. The results indicate that the optimal design depends strongly on the sample size. In low-information situations, optimal designs are sufficient, while distorted terms produce better designs in data-rich cases.

Optimal experimental designs are an essential part of clinical development of a drug and are used to determine the design of dose-finding studies. However, often, these designs for dose-finding aim on one drug. We consider an efficacy Emax model for the combination of two drugs. The interaction is characterized as synergy if it is positive, antagonism when it is negative and additive when there is no interaction between the two drugs. We calculate D-optimal designs algebraically and numerically. The solutions show that the number of doses, the doses itself, and their weights in the D-optimal design depend on the parameter values in the model, ED₅₀^x, ED₅₀^y and the interaction term γ.

The knowledge of how good or optimal designs look like is essential for dose-finding trials. In many cases, dose-finding trials consider both efficacy and safety. We analyse, therefore, a bivariate model for these two outcomes. In contrast to earlier research, we consider a model also having placebo effects to see the impact on the optimal design. We calculate D-optimal designs algebraically and numerically. We see that one more design point is necessary, but that otherwise, the optimal design has a similar structure compared to the model without placebo effects. We confirm that the drug's therapeutic index has a significant impact on the shape of the optimal design.