Effects of one missing observation in central composite designs (CCDs) on estimates of model parameters and their standard errors

Abstract

We may be confronted with a situation in which some observations are lost or unavailable due to some accident or cost constraints and their absence has a very bad impact on the estimates of the regression coefficients. This work investigates the effect of one missing observation of different types of design points on the estimated model for the candidate central composite designs considered. Three different Central Composite Designs (CCD) were studied in this work, which include the 2-factor, 3-factor and the 4-factor CCDs. The regression coefficients and their standard errors were first studied for the full designs and then, similar results were investigated separately for one factorial point missing, one axial point missing and one centre point missing for each of the designs considered. It was observed that missing observations of each of the design points have adverse effect on the regression estimates and standard errors of the model parameters of each of the designs considered. For the 2-factor CCD, the quadratic effect (𝑥1 2 ) for factor one was observed to be the largest but negative on the yield for the full design (𝛽̂ = −0.806), while that of factor two (𝑥2 2 ) was the smallest on the yield (𝛽̂ = 0.069) after the cross-product (𝑥1𝑥2 ) effect (𝛽̂ = 0.150). The standard errors of each of these two quadratic effects are the highest (𝑆𝐸 𝑐𝑜𝑒𝑓 = 0.505) after that of the cross-product effect (𝑆𝐸 𝑐𝑜𝑒𝑓 = 0.639).For the 3-factor CCD, it was observed that the linear effects of 𝑥1and 𝑥3 are the most significant linear effects while the quadratic effects of 𝑥2 and 𝑥3 are the most significant quadratic effects in the case of the full design. When a factorial run is missing, the estimated effect of each of the model parameters remain almost unchanged, indicating that the factorial point looks less influential for the regression estimates for this design. However, the standard error of each of these effects become higher with the linear, quadratic, and cross-product effects behaving similarly for the missing factorial run in this design.