## 1. Introduction

One of the critical steps in developing a medicinal drug is a proper understanding and characterization of its dose response relationship. Failing to characterize well the dose response relationship may have severe consequences once the drug is available to patients: selecting too high doses may lead to unacceptable safety problems, whereas selecting too low doses may lead to insufficient efficacy. Further applications, where dose response modelling is of particular importance, include the investigation of a new herbicide or fertilizer, a molecular entity, an environmental toxin or an industrial chemical.

Much literature is available on dose response studies including a placebo group (see Ruberg (1995), Ting (2006) and Bretz *et al*. (2008) among many others). However, in some drug development programmes the dose-dependent efficacy relative to a standard treatment is of major interest, especially in preparation for an active controlled confirmatory non-inferiority trial. In addition to regulatory requirements related to drug approval, health technology assessments for national reimbursement decisions may be improved by dose finding studies that evaluate the incremental dose effect as compared with the standard treatment. Furthermore, in many situations the use of placebo could be considered unethical or unfeasible, even in a phase II dose finding study. If no placebo is used, the extrapolation of the dose response from the lowest dose to the zero dose (i.e. placebo) becomes problematic and the use of an active control (AC) could facilitate the assessment of the overall level of efficacy of the dose response curve.

The considerable interest by regulatory agencies in active controlled studies becomes evident from several related guidelines. For example, the tripartite International Conference on Harmonisation E4 guideline on dose finding encourages the inclusion of an active comparator in a dose finding study to improve assay sensitivity of the trial as well as to generate better data on comparative effectiveness and safety (International Conference on Harmonisation, 1994). In addition, several international disease-specific regulatory guidelines recommend the use of an active comparator in pivotal phase III studies ( European Medicines Agency, 2006, 2011). In a more general context, the European Medicines Agency guideline on the choice of a non-inferiority margin states that a placebo controlled trial is usually not sufficient and that the comparison between test and reference will often be of importance in its own right (European Medicines Agency, 2005). It thus becomes evident that, because of the regulatory requirements on active-controlled phase III trials, dose finding studies with an AC contribute significantly to the proper choice of a dose to be used in phase III and lead to a better risk–benefit profile in comparison with a marketed drug.

The research in the present paper is motivated by an active controlled dose finding phase II study to determine the optimal dose of the new compound for the management of acute flare in gout adult patients who are refractory or contraindicated to standard therapies. The primary objective is to determine the target dose of the new compound, which is the dose that leads to the same efficacy as the AC. It will be identified by assessing the dose response relationship of various doses of the new compound with regard to pain intensity in the target joint at 72 h (day 4) post dose measured on a 0–100 mm visual analogue scale. Approximately 200 patients will be included in the study. Patients who meet the entry criteria will be randomized to receive either the AC or one dose of the new compound. An important problem consists in specifying the dose levels for the new compound as well as the allocation ratio of patients across all treatments arms in this study. Once the optimal dose of the new compound has been selected on the basis of this phase II trial, phase III studies will be conducted to evaluate further the efficacy and safety of the new compound in the respective patient population (either acute or chronic gout patients).

It is well known that optimal designs can substantially improve the efficiency of statistical analyses and numerous researchers have worked on the problem of constructing optimal designs for placebo controlled dose response studies (see Miller *et al*. (2007), Dragalin *et al*. (2007), Dette, Bretz, Pepelyshev and Pinheiro (2008) and Dette, Kiss, Bevanda and Bretz (2010) among others). However, to our best knowledge, optimal design problems for active controlled dose finding studies have not been considered in the literature so far. In this paper we propose a strategy to obtain efficient designs for such situations. In Section 'Statistical model for active controlled dose finding studies' we introduce the statistical model that includes an AC together with several dose levels of the compound under investigation. Locally optimal designs for estimating the target dose for the new compound are constructed explicitly. These results are obtained by a non-standard application of the implicit function theorem, which yields a substantial simplification of the asymptotic variance of the target dose estimate. Locally optimal designs require *a priori* information about the unknown model parameters (see Chernoff (1953), Ford *et al*. (1992) and Fang and Hedayat (2008)) and usually serve as benchmarks for commonly used designs (see Section 'Relative efficiencies'). In addition, locally optimal designs serve as a basis for constructing optimal designs with respect to more sophisticated optimality criteria, which are robust against a misspecification of the unknown parameters (see Pronzato and Walter (1985), Chaloner and Verdinelli (1995), Dette (1997) or Imhof (2001) among others). In Section 'Robust optimal active-control-optimal designs' we consider standardized minimax and Bayesian optimal designs, which minimize the maximal efficiency and average efficiency over a given range of the unknown parameters respectively. For several models, including a reparameterization of the widely used EMAX model, it is demonstrated that the robust design problems are related to interpolation optimal design problems for polynomial regression models as considered in Kiefer and Wolfowitz (1964a, b). These results are used to show that the robust optimal designs are saturated (i.e. the number of different experimental conditions coincides with the number of parameters for the underlying model) and optimal designs with respect to these criteria are determined explicitly. As a by-product we also obtain explicit solutions of some of the design problems that were raised in Kiefer and Wolfowitz (1964a, b). Several examples illustrating the results are presented in Section 'Examples', where we also study the efficiency of commonly used designs for the case-study that was described above. Finally, some conclusions and directions for further research are presented in Section 'Discussion', whereas Appendix A contains the proofs of our main results. Motivated by the regulatory International Conference on Harmonisation E4 guidance (International Conference on Harmonisation, 1994) and the cross-industry Pharmaceutical Research and Manufacturers of America working group on ‘Adaptive dose-finding studies’ (Bornkamp *et al*., 2007), we focus on estimating the minimum efficient dose relative to an AC. However, the methodology that is presented in this paper can be extended to other quantities of interest (e.g. estimating the model parameters) and we discuss this briefly in Section 'Optimal designs for estimating model parameters'.