Algorithms for optimizing model-based incomplete block designs
TL;DR
This work proposes a model-based approach that optimizes model parameters, and evaluates first- and best-improvement algorithms, simulated annealing (SA), threshold accepting (TA), and two novel algorithms utilizing directional derivatives (dd) to guide exchanges.
Abstract
Because of time limitations or participation burden, the treatments in an experimental design can be too large for a single subject. Instead of addressing this using combinatorial incomplete block designs, we propose a model-based approach that optimizes model parameters. This offers distinct advantages: it incorporates subject-specific covariates to tailor treatment allocation to individual characteristics, allows for varying block sizes, and eliminates the equal-treatment replication requirement. Despite these benefits, model-based approaches are limited by a lack of software and prohibitively large search spaces, making exact optimization computationally intractable. Therefore, we present local search heuristic algorithms and compare them to existing methods. We evaluate first- and best-improvement algorithms, simulated annealing (SA), threshold accepting (TA), and two novel algorithms utilizing directional derivatives (dd) to guide exchanges. Serving as discrete versions of continuous gradient-based methods, these dd algorithms take smaller steps and avoid flat regions by prioritizing large-difference dd exchanges. Our broadly applicable approach uses item calibration in achievement tests as a comparative example to evaluate objective values and computational times. Results demonstrate that for larger problems, the dd algorithms achieve near-optimal solutions significantly faster than SA and TA. Due to computational efficiency, our algorithms offer a highly appealing approach for practical applications.
