Table of Contents
Abstract ii
Declaration and Approval iv
Dedication vii
Acknowledgments x
Chapter 1: Introduction
1.1 Concept of Factorial Designs 1
1.2 Literature review 2
1.3 Statement of The Problem 4
1.4 Objectives of the Study 4
1.5 Methodology 4
1.5.1 Definition of Effects 5
1.5.2 Construction of The Design 9
Chapter 2: Partially Duplicated Fractional Factorial Designs which allow for Estimation up to Two- Factor Interactions
2.1 Introduction 11
2.2 Five Factors Experiment involving 24+8 = 32 Runs 11
2.2.1 Constuction of the Design 11
2.2.2 Method of Analysis 13
2.3 Six Factor Experiment involving 32+8 = 40 Runs 19
2.3.1 Constuction of the Design 19
2.3.2 Method of Analysis 20
2.4 Seven Factor Experiment involving 40+8 = 48 Runs 25
2.4.1 Constuction of the Design 25
2.4.2 Method of Analysis 27
2.5 Eight Factor Experiment involving 48+16 = 64 Runs 30
2.5.1 Construction of The Design 31
2.5.2 Method of Analysis 32
2.6 Nine Factor Experiment involving 64+16 = 80 Runs 35
2.6.1 Constuction of The Design 36
2.6.2 Method of Analysis 37
2.7 Ten Factor Experiment involving 64+16 = 80 Runs 40
2.7.1 Construction of The Design 41
2.7.2 Method of Analysis 42
Chapter 3: Partially Duplicated Fractional Factorial Designs which allow for Estimation up to Three- Factor Interactions
3.1 Six Factor Experiment involving 48+16 = 64 Runs 43
3.1.1 Constuction of the Design 43
3.1.2 Method of Analysis 44
3.2 Seven Factor Experiment involving 64+16 = 80 Runs 47
3.2.1 Constuction of the Design 47
3.2.2 Method of Analysis 49
3.3 Eight Factor Experiment involving 96+32 = 128 Runs 52
3.3.1 Construction of The Design 52
3.3.2 Method of Analysis 55
3.4 Nine Factor Experiment involving 96+32 = 128 Runs 58
3.4.1 Construction of The Design 58
3.4.2 Method of Analysis 60
3.5 Ten Factor Experiment involving 96+32 = 128 Runs 64
3.5.1 Construction of The Design 64
3.5.2 Method of Analysis 67
Chapter 4: Conclusions and Recommendations
4.1 Conclusions 76
4.2 Recommendations 77
Bibliography 78
Chapter 1
Introduction
1.1 Concept of Factorial Designs
Factorial designs, sometimes referred to as Industrial designs were and are still widely used in experiments to study the effect of factors, that is, the significant factors. They also allow one to study the effect of their interactions. For Instance, in clinical trials, one would like to determine if the combination(Interaction) of different type of drugs is efficient in curing a disease.Factorial designs are also used in product development, designing processes and quality improvement of products.
Factorial designs are economical and saves on time as compared to experiments done factor by factor popularly known as one-factor-at-a-time (OFAT) experiments. Such experiments involve alternating the level of one factor at a time while adjusting for the other factor levels. Consequently, one will require additional resources and time. It is important to note that this type of experiment does not allow one to study for interactions. Complete factorials can be considered when the number of factors is sufficiently small. Full factorials contain all possible combination of levels for the factors involved.
In the case of an experiment involving large number of factors and/or limited resources to perform a full factorial, under reasonable assumption the experimenter can adopt to run a fraction of the complete factorial design. Consider a ap Factorial design where a implies the number of levels and p the number of factors. These factors can be at two or more levels. Throughout we shall only discuss designs with factors at two levels. To illustrate the usefulness of a fractional factorial design, consider a 29 factorial design. A complete factorial will involve 512 runs which in real life may not be “practical” considering resources like money and time. However, one can adopt a fraction of the above design, say 1 of the 29 factorial to get 16 runs which is much practical. Such designs are referred to as Fractional Factorial Designs and are much more economical and time-saving. Fractional factorial experiments are widely used in screening designs- designs meant to help identify significant or rather active factors from a number of factors.
Running an unreplicated complete or fractional factorial design with the focus of identifying significant effects only as a way of miminising on cost could lead to biased results while making inference during analysis. This is as a result of not obtaining an estimate for the pure error. Again, assumption made on high-order interactions to be considered as negligible so as to obtain an estimate for the experimental error could be misleading as some of the assumed interactions may not be negligible or absent.
Disregarding some of these interactions could lead to an experimental error variance that is biased. One proposed method of obtaining a better estimate for the error variance is duplicating some of the treatment combinations in the design.
Certainly, complete duplication provides a better estimate for the error variance. However, it is not cost effective due to the additional runs and thus partial duplication is a good alternative. Partial duplication provides for an unbiased estimate of the error variance and estimates of effects which are more specific and reliable.
1.2 Literature review
The concept of factorial design was first presented by Fisher (1935) in his book “The Designs of Experiments”.
Yates (1935) contributed to the concept of factorial design introduced by Fisher (1935) by suggesting an algorithm which could be used to estimate the effects involved in a particular factorial design. This algorithm is now widely used and is known as the Yate’s algorithm.
An experiment can either be symmetrical factorial or asymmetrical factorial. Bose and Kishen (1940) studied in detail symmetrical factorial designs and addressed the problem of confounding in such designs highlighting the usefulness of partial confounding. Only symmetrical factorials will be addressed in this project, that is, experiments whose factors occur at the same number of levels.
Bose (1947) extended the work by Bose and Kishen (1940) on symmetrical factorial experiments giving the mathematical theory behind symmetric factorial designs.
Running a complete factorial experiment with many factors requires many observations(runs). To obtain the error d.f. we shall need to replicate the experiment. Fractional factorial de- signs have become extremely popular and have been explored widely by researchers such as Plackett and Burman, 1946; Dykstra, 1959; Patel, 1961 and Montgomery, 2001 among many others.
Plackett and Burman (1946) obtained orthogonal fractional factorial designs that could be used to estimate main effects only assuming that all other interactions were absent. These designs were used extensively by researchers like Ottieno (1984), Odhiambo (1985) and Manene (1987) among others for computing optimum expected number of runs in Group Screening Designs (GSD) when we have error in observations.
Daniel (1957) during the convocation of the American Society for Quality Control, proposed an error estimation method of partially duplicating a subset of the treatment combinations.
Later on Dykstra (1959) extended the conversation by giving an experimental plan and method of analysis of designs with factors at two levels with partial duplication involved.
Dykstra (1960) extended his work done in 1959 to partial duplication of Response Surface Designs to clear the uncertainty of whether variability remains constant or increases away from a center point resulting to a biased estimate of the error. Dykstra showed that obtaining duplicates over the experimental area could solve that uncertainty or the fear of variance increasing away from the center point.
Patel (1963) extended Dykstra’s work (1959) giving the experimental plan, test procedures and block design for 2p designs that had been duplicated partially. Patel’s designs provided for fewer runs than the corresponding Dykstra’s designs. Both Patel and Dykstra duplicated partially in their designs so as to allow for estimation of the error variance.
Pigeon and McAllister (1989) discussed how it was possible to have partial duplication without interfering with the orthogonality of main effects.
Liau (2008) extended the work by Pigeon and McAllister (1989) by presenting construction techniques on how to get the orthogonal main effect plan with some set of duplicated points.
Liau and Chai (2009) re-examined the 1 fraction of 27 design by Snee (1985) where 2 refers to the number of levels and 7 the number of factors. Snee’s design had four points duplicated. After Liau and Chai re-analysed the design they found out that had the points not been duplicated, the significant effects would have been different. They also concluded that partially duplicated designs are more robust and efficient in screening experiments.
Tsai and Liao (2011) extended the proposed 2p symmetrical factorial by Liau and Chai (2009) to obtain optimality in partially replicated mixed factorial experiments.
Patel (1963) and Dykstra (1959) studied designs that allowed for estimation up to
two-factor interactions. Plackett-Burman designs assumed no interaction in effects. In this project, we are going to include partial duplication in the proposed fractional factorial designs that allow for estimation of effects up to three-factor interaction as some of these interactions assumed to be absent could be actually active.
1.3 Statement of The Problem
It is a commonly accepted practice to obtain an estimate of the error by regarding high order interactions absent and pooling their d.f. and Sum of Squares to be for error.
However, some of these interactions may actually be present leading to a biased estimate of the error. Moreover, we may not understand the extent to which the error term is biased (Dykstra,1959).
Here we are going to show how estimation of effects was done in Patel’s work- estimation of effects up-to two-factor interactions using matrix approach- and extend to designs that estimate up to three factor interactions.
1.4 Objectives of the Study
The main aim of this project is to construct partially duplicated fractional factorial designs with as fewer runs as possible.
The specific objectives of the study are are:
i) To construct partially duplicated fractional factorial designs which allow for estimation up-to two factor interactions.
ii) To construct partially duplicated fractional factorial designs which allow for estimation up-to three factor interactions.
1.5 Methodology
1.5.1 Definition of Effects
Consider a factorial experiment with p factors F1, . . . , Fp each at two levels and factors that appear at x1, . . . , xp levels. Let xi the level of the ith factor be coded as 0, 1 for 1 ≤ i ≤ p. That is, xi = 0, 1 for i = 1, . . . , p
Let the combination of levels of the p factors, that is, the treatment combinations be
denoted by f x1 , . . . , f xp or (x1, . . . , xP) (1.1)
The design described above is a 2p factorial design. Consider a 22 design. The treatment combinations are (1), f1 , f2 , f1 f2. The same treatments can be represented as (0, 0), (1, 0), (0, 1) and (1, 1). In a 23 design the eight treatments are (1), f1 , f2 , f1 f2 , f3 , f1 f3
, f2 f3 and f1 f2 f3 also presented as (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1), (1, 0, 1), (0, 1, 1) and (1, 1, 1).
The parameters in a 22 design are µ, F1 , F2 , F1F2 and their estimates are denoted by µˆ , Fˆ1 , Fˆ2 and F1ˆF2 respectively. They are given by the following equations:-
µˆ = [µ] = 1 [(1) + f1 + f2 + f1 f2]
Fˆ1 = [F1] = 1 [−(1) + f1 − f2 + f1 f2] (1.2)
Fˆ2 = [F2] = 1 [−(1) − f1 + f2 + f1 f2]
F1ˆF2 = [F1F2] = 1 [(1) − f1 − f2 + f1 f2]
where [µ], [F1] , [F2] and [F1F2] are contrasts in the actual observation values of the above treatments.
The parameters in a 3 factor experiment are µ, F1 ,F2 , F3 , F1F2 , F1F3 , F2F3 and F1F2F3. Their estimates are defined by the following eight equations:-
µˆ = 1 [(1) + f1 + f2 + f1 f2 + f3 + f1 f3 +2 f3 + f1 f2 f3]
Fˆ 1
1 = 8 [(−1) + f1 − f2 + f1 f2 − f3 + f1 f3 − f2 f3 + f1 f2 f3]
Fˆ 1
2 = 8 [(−1) − f1 + f2 +1 f2 − f3 − f1 f3 + f2 f3 + f1 f2 f3]
Fˆ 1
3 = 8 [(−1) − f1 − f2 − f1 f2 + f3 + f1 f3 + f2 f3 + f1 f2 f3] (1.3)
F1ˆF2 = 1 [(1) − f1 − f2 + f1 f2 + f3 − f1 f3 − f2 f3 + f1 f2 f3] F1ˆF3 = 1 [(1) − f1 + f2 − f1 f2 − f3 + f1 f3 − f2 f3 + f1 f2 f3] F2ˆF3 = 1 [(1) + f1 − f2 − f1 f2 − f3 − f1 f3 + f2 f3 + f1 f2 f3] F1Fˆ2F3 = 1 [(−1) + f1 + f2 − f1 f2 + f3 − f1 f3 − f2 f3 + f1 f2 f3]
From equations (1.2) we can see that µˆ + Fˆ1 is the average of the observations at high level of factor F1 and µˆ − Fˆ1 is the average of observations corresponding to the low level of F1. The same is true for F2. Using the same logic we can reason for equations (1.3).
µˆ + F1ˆF2 is the average of the observations at the low level of both factors and at the high level of both factors whereas µˆ − F1ˆF2 is the average of observations at high level of F1 and high level of F2. The same argument can be used in equations (1.3).
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