Approximate c-Optimal Experimental Designs with Correlated Observations using Combinatorial Optimisation

We review the use of combinatorial optimisation algorithms to identify approximate c-optimal experimental designs when the assumed data generating process is a generalised linear mixed model and there is correlation both between and within experimental conditions. We show how the optimisation problem can be posed as a supermodular function minimisation problem for which algorithms have theoretical guarantees on their solutions. We compare the performance of four variants of these algorithms for a set of example design problems and also against multiplicative methods in the case where experimental conditions are uncorrelated. We show that a local search starting from either a random design or the output of a greedy algorithm provides good performance with the worst outputs having variance $<10\%$ larger than the best output, and frequently better than $<1\%$. We extend the algorithms to robust optimality and Bayesian c-optimality problems.