Robust response surface designs against missing observations
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Even though an experiment is carefully planned, some observations may be lost during the process of collecting data or may be suspicious in some way. Missing observations can be the result from many causes, for example, the loss of experimental units and miscoded data where their correct values are non-trackable. The risk of losing observations usually cannot be ignored in practice. When small response surface designs are used, the effects of missing points may be substantial. The ability to estimate all parameters could be completely lost, or the variances of predicted responses could be incredibly large in a certain part of an experimental region. With respect to design optimality, designs will usually no longer be optimal when missing values exist. The robustness against a missing value of several standard response surface designs has been studied via newly proposed measures. The designs include central composite designs (CCDs), small composite designs (SCDs), hybrid designs, and exact alphabetic optimal designs. In addition, an R package has been developed to visualize the effect of missing data. The behaviors of D-, A-, G-, and IV -efficiencies for CCDs are studied, and the axial distance that makes a spherical CCD more robust to a missing point is found via a numerical search. Results show that the axial distance obtained from Min D criterion is the largest and respectively followed by Min G, Min IV , and Min A. The D-, A-, G-, and IV -optimality criteria, as well as the point-exchange algorithm, are modified for constructing optimal robust exact designs. The resulting robust exact designs are compared to existing optimal exact designs to observe how resulting designs are robust to a missing point. It is observed that D- and IV -optimal robust designs are slightly less optimal but their robustness properties are appreciably improved. The robustness of G-optimal robust designs is usually slightly improved but with considerable loss in design efficiency. The same robust optimality criteria are also applied to construct optimal robust exact mixture designs. Finally, the adaptive designs are introduced. This allows experimenters to change design points once a missing value occurs during experimentation.