Scaled Prediction Variances of Equiradial Design under Changing Design Sizes, Axial Distances and Center Runs

The aim of every design choice is to minimize the prediction error, especially at every location of the design space, thus, it is important to measure the error at all locations in the design space ranging from the design center (origin) to the perimeter (distance from the origin). The measure of the errors varies from one design type to another and considerably the distance from the design center. Since this measure is affected by design sizes, it is ideal to scale the variance for the purpose of model comparison. Therefore, we have employed the Scaled Prediction Variance and D – optimality criterion to check the behavior of equiradial designs and compare them under varying axial distances, design sizes and center points. The following similarities were observed: (i) increasing the design radius (axial distance) of an equiradial design changes the maximum determinant of the information matrix by five percent of the new axial distance (5% of 1.414 = 0.07) see Table 3. (ii) increasing the nc center runs pushes the maximum SPV(x) to the furthest distance from the design center (0 0) (iii) changing the design radius changes the location in the design region with maximum SPV(x) by a multiple of the change and (iv) changing the design radius also does not change the maximum SPV(x) at different radial points and center runs . Based on the findings of this research, we therefore recommend consideration of equiradial designs with only two center runs in order to maximize the determinant of the information matrix and minimize the scaled prediction variances.

Optimal Prediction Variance Capabilities of Inscribed Central Composite Designs

This study looks at the effects of replication on prediction variance performances of inscribe central composite design especially those without replication on the factorial and axial portion (ICCD1), inscribe central composite design with replicated axial portion (ICCD2) and inscribe central composite design whose factorial portion is replicated (ICCD3). The G-optimal, I-optimal and FDS plots were used to examine these designs. Inscribe central composite design without replicated factorial and axial portion (ICCD1) has a better maximum scaled prediction variance (SPV) at factors k = 2 to 4 while inscribe central composite design with replicated factorial portion (ICCD3) has a better maximum and average SPV at 5 and 6 factor levels. The fraction of design space (FDS) plots show that the inscribe central composite design is superior to ICCD3 and inscribe central composite design with replicated axial portion (ICCD2) from 0.0 to 0.5 of the design space while inscribe central composite design with replicated factorial portion (ICCD3) is superior to ICCD1 and ICCD2 from 0.6 to 1.0 of the design space for factors k = 2 to 4.