I= CENTRAL COMPOSITE DESIGNS ROBUST TO THREE MISSING OBSERVATIONS
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Title of Thesis
CENTRAL COMPOSITE DESIGNS ROBUST TO THREE MISSING OBSERVATIONS

Author(s)
Muhammad Akram
Institute/University/Department Details
Islamia University Bahawalpur/Department of Statistics
Session
1993
Subject
Statistics
Number of Pages
229
Keywords (Extracted from title, table of contents and abstract of thesis)
central composite designs (ccds), cuboidal, spherical, orthogonal, rotatable, minimum variance, box, draper outlier robust designs, missing observations

Abstract
In well-planed experimental work, situation may arise where some observations are lost or destroyed or unavailable due certain reasons that are beyond the control of the experimenter. Unavailability of the observations destroys the orthogonality and the balance of the design and also affects the inference. The purpose of this study is to assess the consequences of missing any combination of m observations (three in our case) of factorial, axial and centre points.

The intensity of the consequences depends upon the size and type of the design. Generally smaller designs are more affected by the missing observations. We emphasized on various types of Central Composite Designs (CCDs) which includes Cuboidal, Spherical, Orthogonal, Rotatable, Minimum Variance, Box and Draper Outlier Robust Designs with an intention to introduce CCDs robust to m missing observations.

It is observed that different relations occur between different combinations of three missing observations of factorial, axial and center pints and the determinant of the reduced information matrix (X’rXr), the main contributor in the definition of the loss of missing observation. This loss also depends the distance of the axial point from the center of the experiment (a), number of factors(k) and the position of the missing point.

A complete sensitivity analysis is conducted by comparing the losses against all possible combinations of missing observations for a variety of a and k values, 1.0≤a≤3.0;2≤k≤6. These losses fall in predetermined groups of combinations producing same losses with a predictable frequency. For each configuration designs robust to one, two and three missing observations are developed under the minimaxloss criterion and are termed as minimaxloss1, minimaxloss2 and minimaxloss3 respectively. The minimaxloss3 design for each k value are compared with other CCD counterparts.

If the loss of missing m observations approaches one, the design breaks down. To avoid this breakdown and as a precautionary measure certain influential points in the design are additionally replicated when there are higher chances of loosing them. The replication of factorial or axial points depends on the values of a and k . It not only refrain the design from breaking down but helps in improving the efficiency of the design by reducing the loss.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents/ Abstract
201.46 KB
2 1 Introduction 1
477.47 KB
  1.1 Response surface methodology 1
  1.2 First order design 8
  1.3 Second order design 10
  1.4 Central composite design 11
  1.5 Orthogonal design 29
  1.6 The Box- Behnken design 31
  1.7 Centre points 33
  1.8 Optimal design 34
  1.9 Robustness of design 37
3 2 Designs with Missing observations 51
304.61 KB
  2.1 Designs with missing observations 51
  2.2 Compound matrices of R matrix 54
  2.3 The losses due to one missing observation 57
  2.4 Two-factor CCD with single replication of factorial and axial parts 58
  2.5 Three-factor CCD with single replication of factorial and axial parts 59
  2.6 Four-factor CCD with single replication of factorial and axial parts 60
  2.7 Five-factor CCD with single replication of factorial and axial parts 61
  2.8 Five-factor CCD with half replication of factorial part and complete replication of axial part 62
  2.9 Six-factor CCD with single replication of factorial and axial parts 63
  2.10 Six-factor CCD with half replication of factorial part and complete Replication of axial part 65
  2.11 Losses due to a pair of missing observations 66
  2.12 Two-factor CCD with single replication of factorial and axial parts 67
  2.13 Three-factor CCD with single replication of factorial and axial parts 69
  2.14 Four-factor CCD with single replication of factorial and axial parts 70
  2.15 Five-factor CCD with single replication of factorial and axial parts 71
  2.16 Five-factor CCD with half replication of factorial and complete replication of axial parts 73
  2.17 Six-factor CCD with single replication of factorial and axial parts 74
  2.18 Six factor CCD with half replication of factorial and complete replication of axial parts. 76
4 3 Designs robust to three missing observations 78
629.4 KB
  3.1 The losses due to three missing design points 78
  3.2 Two-factor central composite design with single replication of factorial and axial points. 80
  3.3 Three-factor design with single replication of factorial and axial points 89
  3.4 Four-factor CCD with single replication of factorial and axial points 99
  3.5 Five-factor CCD with single replication of factorial and axial points 105
  3.6 Five-factor CCD with half replication of factorial and complete replication of axial points 112
  3.7 Six-factor CCD with complete replication of factorial and axial parts 121
  3.8 Six-factor CCD with half replication of factorial and complete replication of axial parts. 129
5 4 Structure of loss-sensitive combinations of three missing observations 138
345.68 KB
  4.1 Formation of the combinations of three missing observations 138
  4.2 The Combinations of three missing observations with high losses 148
  4.3 The Comparison of the losses due to different combinations of the same configuration. 149
  4.4 A combination of three missing factorial points for high losses when k ≥3 150
  4.5 Combinations of three missing observations with two factorial and one axial points 151
  4.6 Combinations of three missing observations with one factorial and two axial points for 3≤k≤6 153
  4.7 Combinations of three missing axial points for k≥3 154
  4.8 Combinations of three missing observations with two coaxial and one centre points 155
  4.9 Design with half replication of factorial parts and complete replication of axial part. 156
  4.10 The combinations of three missing observations with high losses in five factors design with half-replicate of factorial and complete replicate of axial part 158
  4.11 Combinations of three missing observations with high losses in six-factor design with half-replicate of factorial and complete replicate of axial part 160
6 5 Comparison of different central composite designs 165
405.94 KB
  5.1 Structure of R matrix 165
  5.2 Comparison of the minimaxloss design with different central composite designs of the same configuration. 174
  5.3 The Variance of the parameter estimates for the response surface design 188
7 6 The trace criterion for efficiency of the central composite Design 203
170.17 KB
  6.1 Effect of missing observations on trace of information Matrix 203
  6.2 Replication of influential points to the design 209
8 7 Conclusions 220
56.62 KB
  7.1 Losses due to three missing observations 220
  7.2 Central composite designs robust to three missing observations 221
  7.3 The effect of the missing observations on the variances of the estimates 222
  7.4 Breakdown of design and solution there of 222
  7.5 Future extension to the work 223
9 8 References 225
73.01 KB
10 9 Appendix 230
25.31 KB