Optimum designs for clinical trials in personalized medicine when response variance depends on treatment

We study optimal designs for clinical trials when the value of the response and its variance depend on treatment and covariates are included in the response model. Such designs are generalizations of Neyman allocation, commonly used in personalized medicine when external factors may have differing effects on the response depending on subgroups of patients. We develop theoretical results for D-, A-, E- and D${ }_A$-optimal designs and construct semidefinite programming (SDP) formulations that support their numerical computation. D-, A-, and E-optimal designs are appropriate for efficient estimation of distinct properties of the parameters of the response models. Our formulation allows finding optimal allocation schemes for a general number of treatments and of covariates. Finally, we study frequentist sequential clinical trial allocation within contexts where response parameters and their respective variances remain unknown. We illustrate, with a simulated example and with a redesigned clinical trial on the treatment of neuro-degenerative disease, that both theoretical and SDP results, derived under the assumption of known variances, converge asymptotically to allocations obtained through the sequential scheme. Procedures to use static and sequential allocation are proposed.