ABSTRACT
In this work, the seven-factor central composite design is studied in respect of a pair of missing values using the minimax loss criterion. It was observed empirically that seven-factor central composite design with,and is robust at and variance robust at. We also observed that the loss effect of missing a pair of factorial points is a decreasing function of increasing, while the loss effect of a pair of axial points is a decreasing and increasing function of increasing. The loss effect of missing a factorial and axial points has no specific direction of increase or decrease on increasing values.
TABLE OF CONTENTS
Title Page – – – – – – – – – i
Certification – – – – – – – – – ii
Dedication – – – – – – – – – iii
Acknowledgement – – – – – – – – – iv
Abstract – – – – – – – – – v
Table of Contents – – – – – – – – – vi
List of Tables – – – – – – – – – vii
List of Figures – – – – – – – – – viii
CHAPTER ONE: INTRODUCTION – – – – – 1
1.1 Background of the Study- – – – – – – 1
1.2 Response Surface – – – – – – – 3
1.3 Central Composite Design – – – – – 4
1.4 Statement of Problem – – – – – – 7
1.5 Objective of the Study – – – – – – – 7
1.6 Scope of the Study- – – – – – – – 8
1.7 Significance of the Study- – – – – 8
CHAPTER TWO: LITERATURE REVIEW – 9
2.1 Rotatable Central Composite Design – – – – – 10
2.2 Orthogonality- – – – – – – – – 12
2.3 The Spherical – – – – – – – – 14
2.4 Centre Runs in the – – – – – – – – 14
2.5 Cuboidal Region of Interest- – – – – – – 14
2.6 Optimal Design- – – – – – – – – 15
2.7 Robustness of Design- – – – – – – – 18
2.8 Designs Robust to Outliers- – – – – – – 19
CHAPTER THREE: METHODOLOGY – – – – 21
3.1 Introduction – – – – – – – – – 21
3.2 Minimax loss Criterion – – – – – – – 23
3.3 Measure of Optimality based on the Eignvalues of
Information Matrix – – – – – – – 26
3.4 The Trace Criterion of Efficiency of the – – – 27
CHAPTER FOUR: RESULTS OF ANALYSIS – – – 30
4.1 Introduction- – – – – – – – – 30
4.2 Losses Due to One Missing Observation- – – – – 31
4.3 Losses Due to a Pair of Missing Observations- – – – 32
4.4 A Pair of Missing Values in Seven Factor with Single Replication of Factorial and Axial Parts- – – – – 37
4.5 Comparison of the Minimax loss 2 Design with Designs of the Same Configuration- – – – – – 37
4.6 Different Groups of Pairs with Similar Losses – – – 40
4.7 Formation of the Combinations of Two Missing Observations – 44
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS 46
5.1 Summary of Findings- – – – – – – – 46
5.2 Conclusion – – – – – – – – – 47
5.3 Recommendations – – – – – – – – 47
REFERENCES – – – – – – – – 48
APPENDIX ‘ A’ – – – – – – – – 51
CHAPTER ONE
INTRODUCTION
1.1 Background of the Study
Response Surface Methodology (RSM) is defined by Montgomery (2005, Chapter 11) as a collection of Mathematical and Statistical techniques useful for the modeling and analysis of problems in which a response of interest is influenced by several variables and the objective is to optimize this response. Onukogu (1997, Chapter.1) and Carley et al. (2004) posit that RSM is extensively applied in situations where several input variables potentially influence some performance measure or quality characteristic of the process. The input variables are sometimes called independent or predictor variables and are subject to the control of the experimenter, while the performance measure or quality characteristic is called response. Bradley (2007) stated the objectives of studying RSM to include:
- understanding the topography of the response surface (local maximum, local minimum, ridge lines); and
(ii) finding the region where the optimal response occurs.
The goal is to move rapidly and efficiently along a path to get a maximum or a minimum response so that the response is optimized: see also Montgomery (2005, Chapter 11) and Lenth (2009).
The major goal of any experimental design is to adjust the experimental conditions so that maximal information is gained from experiment. In accordance with the preceding assertion, Lenth (2009) explained that RSM comprises a body of methods for exploring for optimum operating conditions through experimental methods. Adding that, it involves doing several experiments and using the results of one experiment to provide direction for what to do next.
The development of RSM was originated by Box and Wilson (1951) and has since become an efficient tool of modern statistics, which is used to study the relationship between one or more responses and a number of quantitative treatment factors. Wang et al. (2009) has demonstrated the application of RSM in the production of caffeic acid from tobacco waste. Response surface methodology has found its applications in the area of chemical and food industries, biological, biomedical and biopharmaceutical fields and Agricultural Science, see Mead and Pike (1975), Ahmad and Gilmour (2010). Many other recent applications of RSM in the field of scientific experimentation may be found in: Balkin and Lin (2000), Montgomery (2005, Chapter 11) and Bradley (2007).
In many applications of Response Surface Methodology, good estimation of the derivatives of the response function may be as important or perhaps more important than estimation of mean response. Certainly, the computation of a stationary point in a second-order analysis or the use of gradient techniques for example, steepest ascent or ridge analysis depends heavily on the partial derivation of the estimated response function with respect to the design variables. Since designs that attain certain properties in (estimated response) do not enjoy the same properties for the estimated derivatives (slope), it is important for the user to consider experimental designs that are constructed with the derivatives in mind: see Victorbabu (2009).
1.2 Response Surface
Given a response of interest, and a vector of independent factors, that influence, the relationship between and can be written as follows:
where represent random error which is assumed to be normally distributed with mean zero and variance. Since the true response surface function is usually unknown, a response surface of is created to approximate. Predicted values are then obtained using The most widely used response surface approximating functions are simple low-order polynomials. If little curvature appears to exist, the first-order polynomial given in equation (1.2.2) can be employed. If significant curvature exists, the second-order polynomial in equation (1.2.3) including all the two-factor interactions can be used.
The parameters of the polynomials in Equations (1.2.2) and (1.2.3) are usually determined using a least squares regression analysis to fit these response surface approximations to existing data. These approximations are normally used for prediction within Response Surface Methodology: see Simpson et al. (1997).
1.3 Central Composite Design
A design which consists of two-level factorial or fractional factorial chosen as to allow the estimation of all first-order and two-factor interaction terms augmented with further points which allow pure quadratic effects to be estimated is called central composite design . It is one of the most important and commonly used classes of experimental designs for second-order models. These designs are very useful in industry due to its versatility. Box and Wilson (1951) were the pioneers who introduced this class of designs. The practical application of usually arises through sequential experimentation. It consists of factorial or fraction of factorial design augmented with axial and centre points. In it is assumed that the response relationship with the design variables is adequately estimated by the second order polynomial model:
If the design used consists of experimental runs, then the said model in the matrix form may be written as where is a vector of response values, is an matrix of rank with row ( ) of the form is vector of unknown parameters and is a vector of random error having normal distribution with mean zero vector and variance-covariance matrix. In summary, consists of three parts:
(i) A complete or fractional replication of factorial design points with k factors consisting of coded design points of the form usually called “cube” as it is best displayed as cube for and hypercube for. For complete factorial design v
(ii) a set of axial points with coordinates usually called a “star”, where is the distance of the axial points from the design centre;
(iii) a set of centre points at
If the above three parts are replicated only once, then the total number of design points is: For more information, see, for example, Onukogu et al. (2002, Chapter 4) and Draper and Lin (1996).
The choice of parameters, the distance of the axial points from the design centre and , number of centre points, are important for the adequate performance of . The value of specifies the location of the axial point, which can be chosen by the experimenter to satisfy various conditions. A can be made spherical , rotatable , orthogonal , cuboidal and minimum variance , by choosing appropriate values of Akram (2002), has it that, all these designs are scaled so that the design perimeter is at radius . A graphical layout of two-variable with is given as illustration in Figure (1.3.1) below, where there are five levels, (that is: ) for each of the two variables. The four factorial points are placed on a square with its centre at and four axial points of which two are on each of and axes passing through the centre, at distance from the centre. That is, all factorial and axial design points lie at equal distance on the circle of radius and the centre points lie at the design centre.
