On Optimal Partially Replicated Rotatable and Slope Rotatable Central

Composite Designs

by

P.E. Chigbu and N.C. OrisakweDepartment of Statistics

University of Nigeria, Nsukka

DEPARTMENT OF STATISTICSUNIVERSITY OF NIGERIA

NSUKKA

ABSTRACT Experimental designs are often replicated to obtain accurate

estimates of the effects of the input variables on the response variables. Thus when partial replication of experimental units occur, there is need for some optimal replication of the units to avoid bias. Some variations of experimental runs of central composite designs in the presence of partial replication are compared under rotatable and slope rotatable designs restrictions. The optimal choice of the runs replicated are obtained using the A-, D- and E- optimality criteria. The A-, D- and E-values are determined theoretically for two factors. For each variation, the optimal values are calculated and displayed graphically. Comparisons of the variations are given and the results suggest that replicated cubes plus one star variations are better than one cube plus replicated stars ones.

INTRODUCTION• In an experiment we can do partial replication (due to

feasibility constraints) and still obtain an estimate of pure error. However, we are faced with the issues of possible bias and choice of which runs to repeat.

• Second-order designs are used in response surface

methodology as acceptable approximation of true responses: see, for example, Myers (1971).

• Here, we consider the central composite design (ccd): a type of second-order designs.

Why Central Composite Designs (ccd’s)?

• ccd’s are very popular 2nd-order designs because:

it is extremely simple to use, and it allows estimation of all the parameters in a full second-order model: see Huda and Al-Shingiti (2004).

• Among the exact (integer) designs, the ccd’s often have high efficiencies under the commonly used A-, D- and E-optimality criteria.

Components of a ccdA ccd comprises of:

a) A factorial part consisting of 2k-q (q ≥ 0) units of at least resolution V (i.e. the main effects and two-factor interactions are not aliased with any other main effects or two-factor interactions) with each point replicated rf- times, also called the cube. The levels of the factors are coded +1, -1;

b) An axial part consisting of 2k units on the axis of each factor at a distance, α, from the centre of the design (selected based on criteria such as orthogonality, rotatability and slope rotatability); usually called the star. Each point is replicated rα times;

c) N0 replication of the centre points, (0, 0… 0); all of which gives a total of rf 2k-q + rα2k + N0,

where, k is the number of factors, f is the factorial part, r is the number of replications, q is the number of factors subtracted from k (q > 0, implies fractional factorial), α and N0 are as defined in b) and c) above: see, for example, Onukogu and Chigbu (2002).

ccd: an example• Need for partial replication in ccd’s may arise because of natural occurrence of

experimental units, and feasibility of some factor levels. This causes unequal replications of centre, cube and star points, and also leads to the issue of which points to repeat. For instance, in an experiment with two independent variables in fifteen experimental units; assuming curvature of the response in each of the two factors, a ccd is used to estimate the associated full quadratic response surface model.

• Usually, when k = 2, a ccd is made up of nine distinct points: four points made up of the 22 factorial part (±1, ±1), i.e. the cube; another four points at the 2(2) axial points, which are at equal distance, α, from the centre of the design, [(±α, 0), (0, ±α)], i.e. the star and a single centre point.

• There will be unequal replications of the points in order to exhaust all the

available experimental units. Variations of such unequal replications include one cube plus one star plus seven centre points (one cube plus one star variation), two cubes plus one star plus three centre points (replicated cubes plus one star variation) and one cube plus two stars plus three centre points (one cube replicated stars variation).

Variations of a ccd

For any N-point ccd there are three rational variations: the one cube plus one star, the replicated cubes plus one star, and the one cube plus replicated stars: see, for example, Chigbu and Nduka (2006). The replicated cubes plus one star and the one cube plus replicated stars are considered in this work because of the following reasons:

a) To make an unbiased estimate of pure error, the ccd should comprise of three to five centre points;

b) The value of α for a rotatable ccd does not depend on the centre point, α = (F)1/4; F is the cube: see Montgomery (2005).

Definitions

• k N Variations α• • 2 15 Two cubes plus one star 1.6820• Two stars plus one cube 1.4142• • 19 Three cubes plus one star 1.8612• Three stars plus one cube 1.4142• • 23 Four cubes plus one star 2.0000• Four stars plus one cube 1.4142• • 3 25 Two cubes plus one star 2.0000• Two stars plus one cube 1.6820• • 33 Three cubes plus one star 2.2134• Three stars plus one cube 1.6820• • 41 Four cubes plus one star 2.3784• Four stars plus one cube 1.6820• • 4 43 Two cubes plus one star 2.3784• Two stars plus one cube 2.0000• • 59 Three cubes plus one star 2.6322• Three stars plus one cube 2.0000• • 75 Four cubes plus one star 2.8284• Four stars plus one cube 2.0000

• Table 1

Definitions: Optimality Criterion

Theoretical Approach

Theoretical Approach (contd.)

Derivation: D-optimality

Derivation of A-optimality

Derivation of E-optimality

Derivation of E-optimality (contd.)

Discussion of Results The variations considered are: • one cube plus two replicated stars • one cube plus three replicated stars • one cube plus four replicated stars • two replicated cubes plus one star • three replicated cubes plus one star • four replicated cubes plus one star. The results are illustrated in Figures 1 through 6. Each variation is based on α-value which depends on the restriction . The α-values (Table 1) are computed on the basis of given conditions for rotatable and

slope rotatable ccd. Each variation is considered for two to four factors. A-, D- and E-optimal values are obtained and also plotted. From the definition of the ccd, Resolution V will project into a full factorial of four factors. In practice, the experimenter aims at obtaining optimal design with minimum cost. As the number of factors in a 2k factorial design increases, the number of runs required for a complete replicate of the design rapidly exceed budget. By reasonably assuming that certain high-order interactions are negligible, information on the main effects and low-order interactions may be obtained by running only a fraction of the complete factorial experiment: see, for example, Montgomery (2005).

Comparison of replicated cubes plus one star with one cube plus replicated stars in rotatable central composite design

• Computations and Figures show that the D-values for replicated cubes plus one star variation are greater than those of one cube plus replicated stars.

• Applying appropriate criterion also, we find that replicated cubes plus one star ccd based on rotatability restriction are A-optimal.

• Similarly, replicated cubes plus one star ccd based on rotatability restrictions are E-optimal when compared with one cube plus replicated stars ccd of the same basis.

• These results are shown Figures 1, 2, and 3 (RC means Replicated Cubes plus one star while OC means One Cube plus replicated stars in each of the Figures presented).

Figure 1

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

1

2x 10

28

RCOC

alpha

RC

D

-val

ues

D versus alpha plots for rotatable variations

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

5

10x 10

23

Figure 2

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

alpha

A-v

alue

sA versus alpha plots for rotatable restricted variations

RC

OC

Figure 3

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.005

0.01

0.015

0.02

0.025

0.03

RC

OC

alpha

E-v

alue

sE versus alpha plots for rotatable restricted variations

Comparison of replicated cubes plus one star with one cube plus replicated stars in slope rotatable central composite design

• In slope rotatable ccd, the D-values for two cubes plus one star are greater than those of one cube plus two stars, implying D-optimality. The same thing goes for three cubes plus one star and four cubes plus one star variations.

• Comparison of two cubes plus one star, three cubes plus one star and four cubes plus one star variations with one cube plus two stars, one cube plus three stars and one cube plus four stars variations, respectively, shows that replicated cubes plus one star variation are A-optimal.

• However, along the line, it is revealed that when k = 2, the E-optimality fails while for k = 3 or 4 it holds. For k = 2 in slope rotatable central composite design, replicated cubes plus one star variation are not E-optimal. The E-optimality criterion failed in slope rotatability. This is in support of the findings of Park and Kwon (1998) as quoted by Huda and Al-Shiha (1999), which states that the E-optimality criterion in slope rotatability does not have much physical significance: see Figures 4, 5, and 6.

Figure 4

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.80

1

2x 10

30

RC

OC

D-versus alpha plots for slope rotatable restricted variation

alpha

RC

D-v

alue

s

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.80

1

2x 10

26

Figure 5

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.60.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

OC

RC

alpha

A-v

alue

sA versus alpha plots for slope rotatable restricted variation

Figure 6

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.60

0.002

0.004

0.006

0.008

0.01

0.012

0.014

RCOC

alpha

E-v

alue

sE versus alpha plots for slope rotatable restricted variations

ConclusionThe optimal α-values of the rotatable and slope rotatable restricted variations for the selected N-point ccd were obtained and used to find the relevant results. The computational results show that the replicated cube plus one star variation of ccd is better than the one cube plus replicated stars variation under rotatability and slope rotatability. Graphical illustrations of the comparisons as shown in Figures 1-6, also depict the same results.

ReferencesChigbu, P.E. and Nduka, U.C. (2006). On the Optimal choice of Cube and Star replications in restricted

second-order designs. (preprint, IC/2006/113) The Abdul Salam International Centre for Theoretical Physics, December 2006.

Huda, S. and Al-Shiha, A.A. (1999). On D-Optimal designs for estimating slope. The Indian Journal of Statistics, Vol. 61, Series B, 488 – 495.

Huda, S. and Al-Shingiti, A. (2004). Rotatable generalized central composite designs: A – minimax efficiencies for estimating slopes. Pakistani Journal of Statistics. Vol. 20(3), 397-407.

Montgomery, D.C. (2005). Design and analysis of experiments (6th ed.). Massachusetts: John Wiley.

Myers, R.H. (1971). Response surface methodology. Boston, Massachusetts: Allyn and Bacon Inc.

 Onukogu, I.B. and Chigbu, P.E. (eds.) (2002). Super Convergent Line Series in Optimal Design of Experiments and Mathematical Programming. Nsukka: AP Express Publishers.

 Park, S.H., Kim, H.J and Cho, J. (2008). Optimal central composite designs for fitting Second order response surface regression models . Recent advances in Linear Models and Related Areas. Physica – Verlag HD, 323 – 339.

Park, S.H and Kwon, H.T (1998). Slope – rotatable designs with equal maximum directional variance for second-order response surface models. Communications in Statistics –Theory and Methods, 27, 2837–2851.

Victorbabu, B.Re. (2006). Construction of modified second order rotatable and slope rotatable designs using a pair of incomplete block designs. Sri Lankan Journal of Applied Statistics, 7, 39 – 53.