On Quantifying the Rotatability of the Hybrid Designs

: A procedure of assessing the rotatability for the Hybrid Designs (HDs) was presented and illustrated. Measuring the rotatability of the three three-variable Hybrid Designs by Roquemore (1976), for example, using the procedure, the rotatability values obtained were quite the same as the values obtained using the procedures and measures presented by other authors. Khuri (1988), Draper and Pukelsheim (1990) and Kshirsagar and Cheng (1996) obtained the rotatability values for the Hybrid Designs, ( ( ) 3,10 HD , ( ) 3,11 A HD and ( ) 3,11 B HD ) to be (94.89%, 99.40% and 98.99%), (99.03%, 99.93% and 99.69%) and (97.16%, 99.82% and 98.46%), respectively. Interestingly, the proposed procedure in this study, and using the measure by Ohaebgulem and Chigbu (2021), yielded 97.16%, 99.82% and 98.46%, respectively, for the same three hybrid designs. The beauty of this procedure is that it has nothing to do with the viewing of contour diagrams, and it is also simple to use.


Introduction
Response Surface Methodology (RSM) is used in the development of an adequate functional relationship between a response of interest, y , and a number of associated input variables, (see, Khuri and Mukhopadhyay, 2010).In the early development of RSM, rotatability (first introduced by Box and Hunter (1957)) is one of the design properties that were considered, and has come to be a significant design criterion since then (see, Khuri, 1988;Park et al., 1993;Verseput, 2000;Del Castillo, 2007;and Montgomery, 2013).For a design that is rotatable, the variance of the predicted values of y is a function of the distance of a point from the design centre, and is not a function of the direction the point lies from the center.In addition, the contours associated with the variance of the predicted values are concentric circles.
According to Draper and Pukelsheim (1990), the quest to know if a given design is rotatable or even if it is not rotatable, trying to know the extent of rotatability attained by the nonrotatable design has always been of important concern to any experimenter.Certainly, this has necessitated the need to determine the extent of rotatability attained by a given design, especially when the design is not "perfectly rotatable".
Experimenters, sometimes in practical situations, encounter "near-rotatable" designs; which are designs that are non-rotatable but

Suggested Citation
Ohaegbulem, E.U. ( 2023).On Quantifying the Rotatability of the Hybrid Designs.European Journal of Theoretical and Applied Sciences, 1(5), 254-266. DOI: 10.59324/ejtas.2023.1(5).17exhibit nearly spherical surfaces of constant prediction variances (see, Khuri (1988) and Nguyen and Borkowski (2008), for example).Khuri (1988) states that "near-rotatability" might possibly happen when a design that is originally perfectly rotatable undergoes deformation as a result of wrong settings of some of the predictor variables or even as a result of certain specified levels of the predictor variables being practically difficult to employ; on the one hand, and when a design that was originally rotatable is subjected to some modifications so as to meet up with the very needs of a given experiment, which may have to involve the addition of new design points, or the shifting of existing design points to acquire more information in a certain region of interest, on the other hand.Khuri (1988) added that if "near-rotatability", or perhaps "non-rotatability" occurs, interest may then be to determine the actual effects the deformation or modifications that were carried out would have on the rotatability property of the design so affected.In the case of attempting to repair a design that is not originally rotatable, the degree of rotatability of the design prior and aprior the 'repair' should be determined, in order to appreciate the amounts of the achieved improvement at each of the stages in the repair process.
The usefulness of measuring rotatability, according to Khuri (1988), are as follows: (i) the quantification of the percent rotatability attained by an originally nonrotatable design, to enable the determination of the "closeness" of the design to being rotatable; (ii) the comparison of designs based on their extents/percents of rotatability; (iii) the assessment of the extent of the departure from rotatability caused by deformation or modification of an originally rotatable design; and (iv) the repairing of a non-rotatable design.
Some measures of design rotatability do exist, and they have also been illustrated on some samples of designs.In this study, however, a procedure of determining the percent rotatability of the Hybrid Designs is introduced and illustrated.

The Second-Order Response Surface Designs
The Second-Order Response Surface Model (SORSM) is mostly used by experimental designers in RSM, other than the First-Order Response Surface Model (FORSM), due to its adequacy.
The model for the second-order design is, and with the X -matrix, ( ) where, iu x is the value of the variable, i x at the th u experimental point, and s u ε ′ are uncorrelated errors with mean, 0 and variance, 2 σ .Also, ( ) 1 D is the design matrix, and it is given by, while, In the fitting of the SORSM, the Central Composite Designs (CCDs), the Box-Behnken Designs (BBDs) and the Hybrid Designs (HDs) are among the designs that are used.Among these designs, the most prominent and generally used design in the estimation of second-order response surfaces is the CCD (see, Myers, 1971;Draper, 1982;Verseput, 2000;Myers and Montgomery, 2002;and Park et al, 2008).

The Central Composite Designs
The design matrix, 1 D , of a CCD consists of a k 2 factorial or a 2 k f − fractional factorial portion/factorial run (usually referred to as a "cube"), with points selected from the k 2 points ( ) ( ) usually of resolution V or higher; a set of k 2 axial points/axial runs (usually called a "star"), at a distance, α , from the origin; and the center point(s), 0 n ; such that, 0 1 n ≥ .Generally, according to Khuri and Cornell (1996), the CCD has a total of N design points; where N is given by, A two-factor CCD with two center points will have the design matrix given in (7); and a three-factor CCD with one center point will have the design matrix as shown in (8).
(3; 1) The Hybrid Designs The construction of a k -factor hybrid design involves the use of a CCD (with one centerpoint) for 1 k − factors which is augmented with one column for Factor k and, perhaps, another of second-order response surface designs that are mostly preferred by experimental designers are the rotatable designs.
Rotatability can simply be characterized in relation to the moments of the design.According to Myers (1971) and Khuri and Cornell (1996), one necessary, as well as sufficient, condition for a response surface model (of order, d , in input variables, k , and having experimental units, N ) to be rotatable, is that the design moments of order, δ ( 0, 1, , 2d δ =


), be expressed in the form, ( ) where, the order of the design moment; with ; and the quantity, δ θ depends on d , δ and N .
Khuri (1988) states that, with a rotatable secondorder model, it would be deduced from ( 9), that the two conditions for rotatability are certainly satisfied given the followings; (i) Every moments having at least one i δ odd are all zeros; and (ii) Pure fourth-order moments are thrice the mixed fourth-order moments.This implies that,

Rotatability in the CCDs
In practice, the mostly used second-order rotatable response surface design is the CCD (see, Myers, 1971).Verseput (2000) states that a rotatable or "near-rotatable" CCD is usually employed when non-allowable operating conditions exist at two or more of the extremes of the design region.In addition, rotatable or "near-rotatable" CCDs traditionally yield reduced error for, and improved estimation of, quadratic (curvature) effects.
Pre-multiplying the X -matrix of the CCD by its transpose gives the information matrix for the CCD; where, ( ) and k I and t I are both identity matrices, with ( ) (with 1 being a column vector with k -components).The p and q in (14) are the pure fourth moments and the mixed fourth moments, respectively, of the matrix of fourth moments, ( ) where, and It could be verified that, for a CCD having the elements in the diagonal of its ( )   3 M all p 's and the elements in the off-diagonal all q 's is a clear indication that the first condition for a secondorder design to be rotatable is automatically satisfied by the CCD.Also, sequel to (10), the second condition for rotatability for a secondorder design will be satisfied if, exactly, According to Oyejola and Nwanya (2015), for example, the axial distance, α , is chosen based on the region of interest.A rotatable CCD is constructed by selecting the axial distance, α , in such a manner that, ( )

Measuring Percentage Rotatability in the CCDs
Some measures of design rotatability exist in the literature.The measures by Khuri (1988), Draper and Pukelsheim (1990), Park et al. (1993), Kshirsagar and Cheng (1996) and Ohaegbulem and Chigbu (2021) are some of them.However, the measure by Ohaegbulem and Chigbu (2021) was specifically centered on the CCDs.

The Measure of Rotatability by Ohaegbulem and Chigbu (2021)
The underlying concept behind the framework being adopted while developing the measure of quantifying the extent of rotatability of desings by Ohaegbulem and Chigbu (2021) hinges on the matrix of fourth moments of the CCD which is being assessed for percent rotatability.
Suppose that ( ) r D x and r X stand for the design matrix and the corresponding X-matrix, respectively, of the CCD being assessed for design rotatability; where, ordinarily, the input variables, ' i x s of the CCDs are coded such that, 20)   and by implication, the design moment, In the circumstance the CCD has undergone modification or deformation, the design moment, i x may still be zero or not.In order to guarantee that the design moments are zeros, the design matrix has to be adjusted by the definition of the new variable, iu z , through the transformation given by, As expected, and, by implication too, the adjusted design matrix will have the design moment given by, Employing the transformation in ( 22 ( ) , , where, (1) z M is exactly equal to exactly equal to 2 k or perhaps less than or greater than it; and (3) z M is expressed as, ( ) The steps followed by Ohaegbulem and Chigbu (2021) in developing their measure of percent rotatability are: Firstly, (3) z M (as given in ( 26)), was fragmented into the possible numbers of ( 2 2 × ) component matrices -with a to reflecting "pairwise interactions" of the k − factors of the CCD.Fragmenting is going to produce the ( 2 2 × ) matrices of fourth moments in (27); (ii) Rationalizing the ( 2 2 × ) Matrix of Fourth Moments: The i B 's are each adjusted with the intent to minimizing the differences between each of the elements of the i B 's and the elements of a corresponding k − factor rotatable CCD.And the motive is to have the i B 's smoothened so as to, as much as, exhibit the structural features of a corresponding k − factor rotatable CCD.To do this, consideration is given to the only offdiagonal element and the two diagonal elements of each of the i B 's, in order to ascertain if and , 3(2 ) Rationalizing and the off-diagonal element will become , ; if 2 and , 3( 2) and 3(2 ) 3  The rationalized ( 2 2 × ) matrices of fourth moments are then adjusted with a view to ensuring that the design moments of each of the design matrices are all zero, and also to satisfy the first condition for the second-order designs to be rotatable.Harmonizing any of the i B ⊕ 's will ensure that the diagonal elements are going to be equal.

Harmonizing the diagonal elements of
going to yield the quantity, p * , which is given by, Hence, the now harmonized i B ⊕ is expressed as, (iv) Normalizing the Harmonized ( 2 2 × ) Matrix of Fourth Moments: Here, the harmonized ( 2 2 × ) matrices of fourth moments are further adjusted to make them satisfy the second condition for the secondorder designs to be rotatable.Standard values are, at this juncture, obtained for the two diagonal elements and the only off-diagonal element of the harmonized ( 2 2 × ) matrices of fourth moments.The major intent is to be able to make the i B * 's to mimic a corresponding ( 2 2 × ) matrices of fourth moments of a corresponding k − factor rotatable CCD.
First, is to determine p + (a quantity herein termed "averaging factor"), where; ( ) Next, is to obtain standard values for the diagonal elements and the off-diagonal element.These are, respectively, given by, ( ) And ; if 2 and , 3 ; if 2 and , 3 or if 2 , 3(2 ) and 3(2 ) Interestingly, it is very obvious, from (37), that the underlying intent of this normalization process is to surely arrive at the expression, 3 p q ⊗ ⊗ = ; that, structurally and technically, mimics (18).

(v) The Measure of Design Rotatability:
Now, the measure of design rotatability by Ohaegbulem and Chigbu (2021) was given as, is an expression of the extent of rotatability in the i-th normalized ( 2 2 × ) matrix of fourth moments, i K .The vector, ( ) R v Ω is in a oneto-one correspondence with the elements of ( ) i u K ; where by, ( ) ( ) See ( 16) and ( 17), respectively, for the definitions of p and q .R Ω is the ( 2 2 × ) matrix of fourth moments a corresponding k − factor rotatable CCD, expressed as, The vector, ( ) i u K , consists of the off-diagonal element and the diagonal elements of i K , and is given by, ( ) ( ) close as possible to R Ω ; and it is defined as, ( ) ( ) The expression, ⋅ , means the length of a vector, or the Euclidean norm.
Equation ( 40) is the (arithmetic) mean of all the possible i G 's obtained for the design matrix of a CCD being assessed for rotatability.As a matter of fact, for a rotatable CCD, , but for a non-rotatable CCD, . Furthermore, a CCD whose extent of rotatability is 95 100 ≤ Φ < is considered to be near-rotatable.

Rotatability in the HDs
Based on the seemingly closeness/resemblance of the HDs to the CCDs in terms of structure, the measure of rotatability by Ohaegbulem and Chigbu (2021) is extendable to the HDs, and will hereby be extended to the designs in this study.The basis of this extension is the scaling up or scaling down, as the case may be, of the matrix of fourth moments of any given HD so that it can mimic the matrix of fourth moments of the corresponding k − factor rotatable CCD.

Scaling Up/Down of the Design Matrix of the HDs
The scaling up or scaling down, as the case may be, of the (3) z M of the given HD is done in order to mimic the matrix of fourth moments of the corresponding k -factor rotatable CCD.This scaling up or scaling down is effected by employing a predetermined multiplier, ϖ ; which is, ; for scaling-up 2 min ' ; for scaling-down An appropriate multiplier ensures that the highest off-diagonal element (if scaled up) or the lowest off-diagonal element (if scaled down) of (3) z M for a given HD is equivalent to 2 k , which is the off-diagonal element of (3) z M for a corresponding k -factor rotatable CCD.The scaling up or scaling down, as the case may be, ensures that at least one of the conditions in each of ( 28), ( 29) and (38) would hold.

Measuring Rotatability in the HDs
With the scaling up or scaling down done, the measure of rotatability described in Ohaegbulem and Chigbu (2021) can now be extended to the HD.

Some Illustrations on Measuring Rotatability for the Hds
The procedure of measuring the rotatability of the HDs was illustrated using the Roquemore (1976)'s three-variable HDs, Now, the five-step procedure of measuring rotatability of the CCDs by Ohaegbulem and Chigbu (2021) were adopted and applied on the scaled (3) z M of each of the three HDs, in order to determine their respective rotatability values.
The step-by-step results obtained in the course of determining the rotatability values for the three scaled (3) z M of each of the three HDs are as summarized in Table 3.The rotatability values obtained for each of the three HDs by using the measure by Ohaegbulem and Chigbu (2021), are presented side-by-side with those obtained by some other authors in Table 4.

Conclusion
In this paper, a new procedure for quantifying the rotatability of the HDs has been presented.The rotatability values so obtained using this procedure seem not to be significantly different from the values earlier obtained from using the procedures by some other authors in the literature.One major advantage this new procedure has is that it does not concern itself with viewing of contour diagrams.Also, this procedure involves lesser number of steps and computations than existing ones.In other words, the new procedure has simplicity to its advantage.
), a new design matrix and the corresponding Z-matrix, ( ) r D z and r Z , respectively, are obtained; which are in semblance to the structural forms of ( ) r D x and r X , apart from the fact that the ' iu z s are used and not the ' iu x s.By premultiplying r Z by its transpose, the following expression is obtained;

Table 1 . Roquemore (1976)'s Three-Variable HDs
HD(as recorded inKhuri (1988: p.  100)) and presented in Table1.The titling (labelling) of these designs are done in such a manner that the two initials tell the design type; the first digit in the subscript shows the number of variables; the next two digits show the number of design points; while designs of the same size are differentiated with letters.