Partial Replication Strategies for Definitive Screening Designs with Second-Order Models

Replicating runs in designed experiments is essential for unbiased error variance estimation, particularly when the fitted model may not account for all active effects. Definitive screening designs (DSDs), while offering orthogonality between main effects and quadratic terms, face critical limitations in error estimation when fitting the full quadratic model. For even numbers of factors (6, 8, 10, 12), the basic DSD provides zero degrees of freedom for error variance estimation, while odd-factor designs (5, 7, 9, 11) retain only two degrees of freedom, both insufficient for reliable statistical inference. This study presents a comprehensive investigation of partial replication strategies for DSDs with even factors (6, 8, 10, 12) and odd factors (5, 7, 9, 11), examining the addition of two, four, six, and eight replicated runs under two distinct configurations: balanced replication (Case 1) and unbalanced replication (Case 2). We systematically evaluate the trade-offs between statistical power, relative standard errors (RSEs), D-efficiency, and A-efficiency as functions of the replication level and number of factors. Our analysis demonstrates that replicating half the total runs achieves a high statistical power of 0.90–0.95 for main effects—retaining at least 90% of full replication’s power—while requiring only 62.5% of the experimental runs. The most significant improvement occurs after replicating four runs, where critical t-values decrease from 12.71 to 2.78, making effect detection practically feasible. The results show that balanced replication (Case 1) yields a lower RSE for quadratic effects, while unbalanced replication (Case 2) provides superior orthogonality with the maximum absolute correlations approaching zero. Comprehensive design matrices and performance metrics are provided in the appendices to ensure reproducibility. Based on empirical and theoretical considerations, we recommend half-partial replication as a highly efficient strategy that balances statistical rigor with resource constraints.