SEVEN-FACTOR CENTRAL COMPOSITE DESIGN ROBUST TO A PAIR OF MISSING OBSERVATIONS

                           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.

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:

1.2.1

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).

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