Central composite designs with missing observations: evaluation and comparisons

Abstract

In meticulously designed experiments, it is conceivable that certain observations may vanish, be compromised, or become unattainable due to factors beyond the control of the experimenter. The unavailability of these observations disrupts the orthogonality and balance of the experimental design, thereby influencing the inferences drawn. The objective of this study is to assess the implications of missing data. To evaluate their impact, we employ various criteria tailored to different numbers of factors k.

The relationships between the determinant of the reduced information matrix X′ rXr and the loss incurred due to missing one, two, or three observations in factorial, axial, and center points vary across different combinations. This loss is influenced by factors such as the location of the missing point, the number of components (k), and the distance of the axial point from the experiment’s center (α).

The losses are compared across all possible combinations of missing observations for a range of factors k, thereby completing the sensitivity analysis. These losses exhibit consistent patterns in pairings, resulting in the same outcomes. To ensure robustness against the absence of one, two, or three observations, the efficiency of the designs was tested using the minimax loss criterion. In a previous study by [55], the cases for two and three missing observations were investigated for 2 < k < 6, for specified values α, and our findings align with those results. However, in this study, we extend the analysis to missing two observations for 2 < k < 10 and missing three observations for 2 < k < 7. Additionally, in Akram’s study [55], the D-value was only computed for k = 2, 3 with missing three data, whereas here, we will compute it for missing two observations for 2 < k < 10 and missing three observations for 2 < k < 7.