Two-strata rotatability in split-plot central composite designs

2009 ◽  
Vol 26 (4) ◽  
pp. 431-447 ◽  
Author(s):  
Li Wang ◽  
G. Geoffrey Vining ◽  
Scott M. Kowalski
Keyword(s):  
Split Plot ◽  
Central Composite Designs ◽  
Composite Designs ◽  
Central Composite

scholarly journals Missing Observations in Split-Plot Central Composite Designs: The Loss in Relative A-, G-, and V- Efficiency

2021 ◽  
Vol 25 (2) ◽  
pp. 239-247
Author(s):  
Y. Yakubu ◽  
A.U. Chukwu
Keyword(s):  
Model Parameters ◽  
Missing Observations ◽  
Prediction Variance ◽  
Efficiency Losses ◽  
Split Plot ◽  
Negative Effect ◽  
Central Composite Designs ◽  
Plot Structure ◽  
Composite Designs ◽  
Central Composite

The trace (A), maximum average prediction variance (G), and integrated average prediction variance (V) criteria are experimental design evaluation criteria, which are based on precision of estimates of parameters and responses. Central Composite Designs(CCD) conducted within a split-plot structure (split-plot CCDs) consists of factorial (𝑓), whole-plot axial (𝛼), subplot axial (𝛽), and center (𝑐) points, each of which play different role in model estimation. This work studies relative A-, G- and V-efficiency losses due to missing pairs of observations in split-plot CCDs under different ratios (d) of whole-plot and sub-plot error variances. Three candidate designs of different sizes were considered and for each of the criteria, relative efficiency functions were formulated and used to investigate the efficiency of each of the designs when some observations were missing relative to the full one. Maximum A-efficiency losses of 19.1, 10.6, and 15.7% were observed at 𝑑 = 0.5, due to missing pairs 𝑓𝑓, 𝛽𝛽, and 𝑓𝛽, respectively, indicating a negative effect on the precision of estimates of model parameters of these designs. However, missing observations of the pairs- 𝑐𝑐, 𝛼𝛼, 𝛼𝑐, 𝑓𝑐, and 𝑓𝛼 did not exhibit any negative effect on these designs' relative A-efficiency. Maximum G- and Vefficiency losses of 10.1,16.1,0.1% and 0.1, 1.1, 0.2%, were observed, respectively, at 𝑑 = 0.5, when the pairs- 𝑓𝑓, 𝛽𝛽, 𝑐𝑐, were missing, indicating a significant increase in the designs' maximum and average variances of prediction. In all, the efficiency losses become insignificant as d increases. Thus, the study has identified the positive impact of correlated observations on efficiency of experimental designs. Keywords: Missing Observations, Efficiency Loss, Prediction variance

scholarly journals Construction of Modified Central Composite Designs for Non-standard Models

2018 ◽  
Vol 7 (5) ◽  
pp. 95
Author(s):  
Iwundu, M. P.
Keyword(s):  
Loss Function ◽  
Information Matrix ◽  
Missing Observations ◽  
Design Efficiency ◽  
Central Composite Designs ◽  
Standard Models ◽  
Alternative Designs ◽  
Composite Designs ◽  
Hat Matrix ◽  
Central Composite

The use of loss function in studying the reduction in determinant of information matrix due to missing observations has effectively produced designs that are robust to missing observations. Modified central composite designs are constructed for non-standard models using principles of the loss function or equivalently first compound of (I ) matrix associated with hat matrix . Although central composite designs (CCDs) are reasonably robust to model mis-specifications, efficient designs with fewer design points are more economical. By classifying the losses due to missing design points in the CCD portions, where there are multiple losses associated with specified CCD portions, the design points having less influence may be deleted from the full CCD. This leads to a possible increase in design efficiency and offers alternative designs, similar in the structure of CCDs, for non-standard models.

scholarly journals An approach to measuring rotatability in central composite designs

2015 ◽  
Vol 3 (2) ◽  
pp. 126
Author(s):  
Emmanuel Ohaegbulem ◽  
Polycarp Chigbu
Keyword(s):  
New Approach ◽  
Measure Design ◽  
Central Composite Designs ◽  
Composite Designs ◽  
Central Composite

<p>An approach to measure design rotatability and a measure, that quantifies the percentage of rotatability (from 0 to 100) in the central composite designs are introduced. This new approach is quite different from the ones provided by previous authors which assessed design rotatability by the viewing of tediously obtained contour diagrams. This new approach has not practical limitations, and the measure is very easy to compute. Some examples were used to express this approach.</p>

scholarly journals Optimal Prediction Variance Capabilities of Inscribed Central Composite Designs

2020 ◽  
pp. 1-8
Author(s):  
Julius C. Nwanya ◽  
Kelechukwu C. N. Dozie
Keyword(s):  
Central Composite Design ◽  
Design Space ◽  
Optimal Prediction ◽  
Prediction Variance ◽  
Central Composite Designs ◽  
Scaled Prediction Variance ◽  
Composite Designs ◽  
Central Composite

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.

Slope-Rotatable Central Composite Designs

Technometrics ◽  
1978 ◽  
Vol 20 (4) ◽  
pp. 413-417 ◽  
Author(s):  
R. J. Hader ◽  
Sung H. Park
Keyword(s):  
Central Composite Designs ◽  
Composite Designs ◽  
Central Composite
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