Incorporating Minimum Variances into Weighted Optimality Criteria

Weighted optimality criteria allow an experimenter to express hierarchical interest across estimable functions through a concise weighting system. We show how such criteria can be implicitly influenced by the estimable functions’ minimum variances, leading to nonintuitive variance properties of the optimal designs. To address this, we propose a new optimality and evaluation approach that incorporates these minimum variances. A modified <i>c</i>-optimality criterion is introduced to calculate an estimable function’s minimum variance while requiring estimability of all other functions of interest. These minimum variances are then incorporated into a standardized weighted <i>A</i>-criterion that has an intuitive weighting system. We argue that optimal designs under this criterion tend to satisfy the conditions of a new design property we call weight adherence that sets appropriate expectations for how a given weighting system will influence variance properties. A practical, exploratory approach is then described for weighted optimal design generation and evaluation. Examples of the exploratory approach and weight adherence are provided for two types of factorial experiments.