Optimal experimental designs for nonlinear analysis: Solving the conundrum

To reduce the experimental cost and respondents’ fatigue, optimal experimental designs maximize the information elicited from the respondent, or equivalently minimize the estimator variance. However, in many models the variance depends on the unknown regression parameters ß. Therefore we cannot optimally design the experiment because its efficiency depends on parameters to be estimated from the data. Previous literature dealt with this puzzle by imposing assumptions on the unknown parameters: (1) choosing an arbitrary vector of parameters ß supposedly applying ‘prior knowledge’; or (2) postulating a probability distribution for ß over the parametric space hopefully concentrated around the true value. Therefore, the design is efficient only if these assumptions are correct. Little is known about the robustness of the design, when the true parameters deviate from the assumed values. Moreover, if we knew the value of true parameters, there would be no reason to do the experiment in the first place. We propose a general approach to compute optimal conjoint designs in problems in which the covariance matrix depends on the unknown parameter. We
solve this problem using efficient computational methods for robust optimization, and provide numerical examples for discrete-choice experiments comparing our approach and the classical methods.