Journal of Modern Applied Statistical Methods

Volume 2 | Issue 1 Article 26

5-1-2003 Improved Multiple Comparisons With The Best In Response Surface Methodology Laura K. Miller University of North Florida

Ping Sa University of North Florida, [email protected]

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Recommended Citation Miller, Laura K. and Sa, Ping (2003) "Improved Multiple Comparisons With The Best In Response Surface Methodology," Journal of Modern Applied Statistical Methods: Vol. 2 : Iss. 1 , Article 26. DOI: 10.22237/jmasm/1051748760 Available at: http://digitalcommons.wayne.edu/jmasm/vol2/iss1/26

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Early Scholars Improved Multiple Comparisons With The Best In Response Surface Methodology

Laura K. Miller Ping Sa University of North Florida

A method to construct simultaneous confidence intervals about the difference in mean responses at the stationary point and at x for all x within a sphere with radius RI is proposed. Results of an efficiency study to compare the new method and the existing method by Moore and Sa (1999) are provided.

Key words: Comparison with the best, response surface methodology, bounding algorithm.

Introduction where b , b ,b are unknown constants, Response surface methodology uses a polynomial 0 i ij response function to explain and analyze the i,jk=1,2,..., and random error relationship between a response variable y and es~NID(0,)2 . several predictor variables x = (xx1,...,k )'. The mean response is optimized at the stationary point that may be a minimum, Usually the x will be converted to coded i maximum, or a saddle point. After determining variables xi by xi=-(xxiii0)/()sc , where xi0 is the levels of the predictor variables where the a centering constant and (sc)0> is a scaling mean response is optimized, it is possible that this i point is not a reasonable option due to practical constant, ik=1,2,..., . The mean response at considerations, such as expense. In this situation, x = (x1, x2,..., xk )' , Ey(|)x , can be multiple comparisons can be performed with other approximated using the quadratic polynomial points in the region to determine if some other model with k predictor variables points provide responses that are not significantly different from the optimal point. This problem will subsequently be y=b+bxx+b2 ++bexx , 0 åiiååiiiijij referred to as multiple comparisons with the best iiij< (MCB) in response surface methodology (RSM).

The MCB problem was first approached by Hsu (1984) in design of experiments where he Laura K. Miller received her Master degree in considered the problem of comparing the Statistics from the University of North Florida. treatment means under study with the “best” She was also the recipient of the Most Outstanding treatment mean. Moore and Sa (1999) first Statistics Graduate Student and the Most approached the MCB problem in the RSM setting. Outstanding Mathematics Undergraduate Student There has also been other substantial work on for University of North Florida in the year 2000 related problems within the field of response and 1998, respectively. Ping Sa is an Associate surface methodology. Sa and Edwards (1993) and Professor of the Department of Mathematics and Merchant, McCann, and Edwards (1998) Statistics at the University of North Florida. She investigated the multiple comparisons with the received the Ph.D. in Statistics from the University control (MCC) problem. of South Carolina in 1990. She has published 15 Sa and Edwards (1993) first addressed the papers. Her recent schola rly activities have MCC in RSM problem by constructing involved research in multiple comparisons and simultaneous confidence intervals for quality control. Email address for correspondence regarding this article is [email protected]. d C (xx)=-E(|y)Ey(|)0 for all x within a

247 MULTIPLE COMPARISONS WITH RESPONSE SURFAC E METHODOLOGY 248

pre-specified distance RI of the origin, such that where d B ()x represents the difference between k for all x, x'x =£xR22, where R is the the mean response at the stationary point åi=1 iI I x=-1 Bb-1 and the mean response at any other “radius of inference.” They showed that for a 0 2 point x , b = (b1, b2,..., bk)¢, and rotatable design, the bounds of d C (x) Îd$ C (x) ± (rF )1/2s(x) can be improved using a result of a,r,n éùb11bb Casella and Strawderman (1980) where the 1122121L k êú1 1/2 bb222L 2 k Scheffé critical point, ()rFau,,r can be replaced B= êú. êúOM by a smaller value ca depending on a, u, and êú the nature of the predictor constraints as ëûêúsym bkk summarized by two other constraints, an integer m and a distance q2 > 0. This confidence interval is useful in Because the design used in practice is determining whether an alternate point could be often not rotatable, Merchant, McCann, and substituted for the stationary point as an optimizer. Edwards (1998) introduced a new method which Furthermore, it provides how much loss in the mean response can be expected if x is moved away combined the Bonferroni method and the McCann and Edwards (1996) algorithm for two or more from x0 . They investigated both Bonferroni and predictors that gives much sharper intervals than Scheffé type confidence intervals for the MCB the Scheffé and also the Sa and Edwards (1993) problem. They also investigated Scheffé’s F- adaptation of the Casella and Strawderman projection method of constructing conservative method. Merchant, McCann, and Edwards’ confidence intervals. However, the delta method method does not require a rotatable design and is much less conservative than the F-projection method and of course, much easier to use. allows for one-sided bounds for d C ()x . They generated a critical point d with simultaneous It is the purpose of this article to address the MCB problem in RSM, but instead of confidence bounds for considering the entire k-dimensional space, it would be more realistic to restrict the region to d (xx)=-E(|y)Ey(|)0 C provide confidence bounds for d ()x within a B for all x within a specified distance of 0 via a sphere with radius RI for all x such that bounding algorithm that requires only a few 2 x'x £ RI . The method proposed by Merchant, seconds to a few minutes of computer time. McCann, and Edwards (1998) for the MCC Closely related and within the field of problem should be adaptable to the MCB problem RSM, Moore and Sa (1999) addressed the MCB since the requirement for using this method is that problem. They constructed confidence intervals the covariance matrix of the estimators must be about the difference in mean responses at the known. stationary point and alternate points over the entire The delta method will be used to k dimensional hyperplane based on a theory that does not depend on the design of the experiment. approximate the variance of d$ B (x) for the MCB To solve the MCB problem, they utilized the delta problem. The next section explains the theory and method to approximate the variance of the the bounding algorithm used to generate the estimated difference for critical point for the MCB problem. The algorithm is design free, that is, it does not depend on the d (xx)=db(,) design of the experiment and should therefore B B provide consistent results regardless of the design. =-E(y|xx0 )Ey(|) 1 =-x'b--x'BxbBb',-1 4 249 MILLER & SA

Theory Behind the Method where a simultaneous bound over this finite The method proposed by Merchant, collection is calculated. McCann, and Edwards (1998) will be adapted to The critical point d must satisfy solve the MCB problem. The goal is to generate éùddˆ (xx)- () an improved critical point d with simultaneous BB . (2) Pdêúmax11/2 £³-a upper confidence bounds of the form ëûxÎGI s()l'lå

ˆ d B (xx)+ ds() (1) ddˆ (xx)- () For each x, BB~ t , where s()l'lå 1/2 u where tu is the univariate-t distribution with u degrees of freedom. Equation (2) can be rewritten as ˆˆˆˆˆˆ1 ˆˆ-1 d B (xx)=dbB(,)'=-xb--x'BxbBb', 4 PëûéùTj £d,jp=1,2,...,1³-a such that b$ = ( b$1 ,b$ 2 ,...,b$ k )¢ where the b$ i ’s are ˆ the least square estimators for ßi 's and B is the or ˆ matrix such that ßij is substituted into the matrix PéùT>d,jp=£1,2,..., a B. The estimated standard error of dˆ (x) is s(x) ëûj B = s(l'Sl)1/2 derived by Moore and Sa (1999) where T12, TT,..., p have a multivariate t where s2 is the mean square error which satisfies distribution (Dunnett & Sobel 1954) with u us 2 2 2 ~ c (u) for integer u > 0 and is degrees of freedom and underlying correlation s 2 -1 matrix R derived from s LLå ' . The critical independent of all b$ i ’s, å is the (')XX point d is then solved by the following equation, matrix with the first row and the first column deleted and l is the vector of partial derivatives of 1/ d with respect to such that P{E(t)} f()tdt = a for dbB ()x, b ò T 0 p 1112 ' l =(-2m1-x11,...,,--24mxmkk E(t) =>()aUj td , (Brown,1984) 21122 ∪j=1 ---x1,...,44mxkk,m1m2xx12,...,

1 mm- xx)', 4 kk--11kk where f is the probability density function of T, a T random variable such that rT2 is distributed as where m= Bb-1 such that m is the ith i Fr(,)u ; U is a random vector independent of T, component of m and x=(,x12xx,...,)k ¢ is any distributed uniformly on the r-dimensional sphere; point satisfying x¢x £ 2 . ' RI and a j are the rows of the full rank matrix In order to approximate the entire set of A: pr´ such that R= AA' . interest, we adapt the fine grid of inference GI , Finally, the probability P{E(t)}for the suggested by Merchant, McCann, and Edwards MCB problem can be calculated using the same (1998) of individual xj points for j =1, 2,..., p in bounding algorithm proposed by Merchant, the region. This grid is constructed by user McCann, and Edwards (1998) for the MCC defined multiples of these x values within a radius problem. This bounding algorithm is a R radiating from the center 0. This matrix is combination of Bonferroni method and the I McCann and Edwards algorithm (1996) and is for th ' ' defined as L: p´r whose j row is ljj= lx() upper bound only. If a lower bound is required MULTIPLE COMPARISONS WITH RESPONSE SURFAC E METHODOLOGY 250 over this region, that is dˆ (x) - ds(x) , it can be the numerical integration. The Fortran program is B available from the first author. computed by constructing the upper bound for Examples and comparisons -d B (x) = E( y | x) - E( y | x 0 ) because an Box and Draper (1987) give an example from an investigation by Derringer and Suich upper bound for -d B (x) is equivalent to a lower (1980) in which RSM is used to analyze the bound for d (x) . B effects of = hydrated silica level in phr (parts The critical points for the MCB problem x1 were computed using a Fortran program and per hundred) and x2 = silane coupling agent level routines from the IMSL Fortran Numerical in phr on the elongation at break of a tire tread Libraries (1997). These included calling the compound. One of the goals was to maximize y = routines DLINRG to calculate the inverse of a elongation at break. Convert x to the coded matrix, DLFTDS to compute the Cholesky 1 factorization of a matrix, DFDF to evaluate the F variable x1 = (x1 -1.2)/0.5 and x2 to the coded distribution function, and DQDAGS to perform variable x2 = (x2 - 50)/10 . The design points

xi and the responses yi are listed in Table 1.

Table 1. Experimental Results: Elongation at break y of a tire tread compound Versus x1 =(phr silica –

1.2)/0.5 and x2 = (phr silane – 50)/10 (Source: Derringer 1980)

Run x1 x2 y

1 -1 -1 900 2 1 -1 860 3 -1 1 800 4 1 1 2294 5 -1 -1 490 6 1 -1 1289 7 -1 1 1270 8 1 1 1090 9 -1.633 0 770 10 1.633 0 1690 11 0 -1.633 700 12 0 1.633 1540 13 0 0 2184 14 0 0 1784 15 0 0 1300 16 0 0 1300 17 0 0 1145 18 0 0 1090 19 0 0 1260 20 0 0 1344

The estimated polynomial response function is

2 2 y$ = 1412.892 + 268.151x1 + 246.503x2 - 97.794x1 -139.044x2 +69.375x1x2 .

251 MILLER & SA

L-97.794 34.688 O The vector b$ = (268.151, 246.503)¢ , the matrix B$ = , and the matrix NM 34.688 -139.044QP

éù.0750000 êú0.075000 êú å=êú00.075.0050 êú êú00.005.0750 ëûêú0000.125 are calculated. The estimated stationary point for this surface where elongation at break (y) is maximized is xˆ 0 = (1.849, 1.348 ) yielding an estimated response of 1826.91. Figure 1 gives the estimated surface plot.

Y

1823

1251

679 1.65

0.55 107 X2 1.65 -0.55 0.55 -0.55 X1 -1.65 -1.65

Figure 1. Estimated Surface Plot for the Tire Tread Compound Example.

As one can see, the stationary point is out Simultaneous 90% lower confidence of the experimental region and it may not be a bounds are constructed for d B ()x for all x whose reasonable option due to practical considerations values are on the grid defined by multiples of .2 or expense. Therefore, multiple comparisons can be performed with other points in a region to with a radius of RI = 2 radiating from the determine if any other point within the region of center of the region of interest. Figure 2 shows the operability will produce a response that is not contours for the estimated difference and the significantly different from the point that simultaneous lower confidence bounds L(x) = maximizes elongation at break (y). Since the d$ (x) - ds(x) for generated d 2 = 6.871607 by optimal point was a maximum, this suggests that B the method detailed in the previous section. lower bounds for d B (x)=-E(y|xx0)Ey(|) are more important than upper bounds. MULTIPLE COMPARISONS WITH RESPONSE SURFAC E METHODOLOGY 252

Figure 2. Contour Plots

253 MILLER & SA

In Figure 2 (above), the contour plots are éù--1.94.285 for the estimated difference dˆ ()x (top) and the B = êú, and the matrix B ëû--.285.78 simultaneous 90% lower confidence bounds (bottom) for the Tire Tread Compound Example éù.1250000 with generated d 2 = 6.871607 . The points that êú0.125000 lie inside the negative contour lines indicate êú possible alternate points that will produce å=êú00.144.0190 are found. responses that are not significantly different from êú the point that maximizes elongation at break (y). êú00.019.1440 The region inside the negative contour lines ëûêú0000.25 indicates possible region that will produce responses that are not significantly different from The stationary point for this surface is the point that maximizes elongation at break (y). x = (.189, .290) yielding a response of 13.676. 2 0 The squared critical constant d = Assume that this option is not a reasonable option, 6.871607 compares very favorably to that of the multiple comparisons are performed to determine 2 Scheffé method, rFa(r,u ) = (5F.10 (5,14)) = if alternate points can substitute for the stationary 11.534702. Therefore, an experimenter using the point in terms of maximizing peanut yield. Scheffé method would have to increase the Therefore, the critical point d is required to experiment size by a factor of approximately perform these comparisons. 2 2 Three central composite designs were (5F.10 (5,14)) /d = 11.534702 / 6.871607 = chosen. The three designs are a rotatable central 1.6786, in other words, by 67.86% in order to composite design with uniform precision, a achieve a precision of estimation (interval rotatable central composite design without width) equal to what would be obtained using the uniform precision, and a central composite design adapted method’s critical constant. with one centerpoint. These designs will be Next, three different designs will be used referred to as Design 1, 2, and 3 respectively. for an example using the bounding algorithm to Table 2 (following References section) generate improved critical points for the MCB shows the critical points that were generated for problem where the sample -size savings will be one and two replications of the three different compared to the Scheffé and Bonferroni critical designs using multiples of .2, RI = 1, and 2 for points. this example using Merchant, McCann, and Khuri and Cornell (1987) provide an Edwards’ (1998) method in order to compare the example in which they use RSM to investigate the critical values and the approximate sample -size effects of the amounts of two fertilizers, x1 and savings for each design. Because the Bonferroni method is conservative due to the large number of x2 , on the yield of peanuts measured in pounds per acre. For the purpose of the efficiency study, comparisons , only the approximate sample -size the estimated parameters from this example will savings vs the Scheffé method were calculated. be treated as the true parameters of an underlying Considerable improvement (between 34% model. The true quadratic response function is and 47%) over the Scheffé adaptation for all three given by designs is possible using the new method by choosing the radius of inference RI =1. For y=-+13.85.90xx12.56 R = 2 (which is near the limits of the . I 22 experimental region for Designs 1 and 2), the -1.94xx12-.78-+.57xx12 e sample -size savings are 26% to 33% over the Scheffé method. Also, as expected, the increased The vector b=(.90, .56)', the matrix sample sizes produced by replicating the designs resulted in smaller critical values. MULTIPLE COMPARISONS WITH RESPONSE SURFAC E METHODOLOGY 254

Conclusion

In conclusion, this article has addressed the Casella, G., & Strawderman, W. E. problem of multiple comparisons with the best in (1980), Confidence bands for linear regression RSM via simultaneous confidence bounds for with restricted predictor variables, Journal of the American Statistical Association, 75, 862-868. d B (x)=-E(y|xx0)Ey(|) for all x such that 2 Derringer, G. C. & Suich, R. (1980), x'x £ RI . The method proposed by Merchant, Simultaneous optimization of several response McCann, and Edwards (1998) for the variables, Journal of Quality Technology, 12, 214- MCC problem has been adapted to the MCB 219. problem. It has provided confidence bounds for Dunnett, C. W., & Sobel, M. (1954), A an example for two predictors where the critical bivariate generalization of student’s t distribution, values compare favorably to the Bonferroni and with tables for certain special cases, Biometrika, Scheffé methods as shown by Table 2 (following 31, 153-169. page). Hsu, J. C. (1984), Constrained This will also hold true for problems simultaneous confidence intervals for multiple containing more than two predictor variables. For comparisons with the best, Annals of Statistics, 12, the example provided, this method has been shown 1136-1144. to provide approximate sample -size savings of at International Mathematical & Statistical least 25% for three different central composite Libraries, Inc. (1997), Fortran routines for designs. In fact, based on the theory behind the mathematical applications, Visual Numerics, Inc. bounding algorithm, the Merchant, McCann, and Khuri, A. I., & Cornell, J. A. (1987), Edwards' method for the MCB problem will Response surfaces, New York: Marcel Dekker. always outperform the Scheffé and Bonferroni McCann, M., & Edwards, D. (1996), A methods (Merchant, McCann, and Edwards, path-length inequality for the multivariate t 1998). distribution, Journal of the American Statistical Association, 91, 211-216. References Merchant, A., McCann, M., & Edwards, D. (1998), Improved multiple comparisons with a Box, G. E. P. & Draper, N. R. (1987), control in response surface analysis, Empirical model-building and response surfaces, Technometrics, 40, 297-303. New York: John Wiley. Moore, L. J., & Sa, P. (1999), Brown, L. D. (1984), A note on the Comparisons with the best in response surface Tukey-Kramer procedure for pairwise methodology, Statistics and Probability Letters, comparisons of correlated means, in Design of 44, 189-194. experiments: ranking and selection, eds. T. J. Sa, P. & Edwards, D. (1993), Multiple Santner & A. C. Tamhane, New York: Marcel comparisons with a control in response surface Dekker, pp.1-6. methodology, Technometrics, 35, 436-445.

255 MILLER & SA

Table 2. Generated critical points using the improved method (one-sided bounds) and Scheffé and Bonferroni critical points with grid spacing = .2 for each design and approximate Sample -Size Savings of the New Method Versus the Scheffé method.

Improved Scheffé Bonferroni Sample -Size Critical Critical Critical Savings vs Design reps u RI Point d Point Point Scheffé 1 1 1 7 3.233020 3.794733 4.605120 37.78% 2 20 2.803443 3.286335 3.460804 37.42%

2 1 7 3.368990 3.794733 5.207830 26.87% 2 20 2.918108 3.286335 3.756539 26.83% 2 1 1 3 4.423153 5.152669 9.505157 35.71% 2 12 2.980882 3.456877 3.813342 34.49%

2 1 3 4.576950 5.152669 12.008948 26.74% 2 12 3.079915 3.456877 4.195280 25.98% 3 1 1 3 4.255092 5.152669 9.505157 46.64% 2 12 2.875548 3.456877 3.813342 44.52%

2 1 3 4.482340 5.152669 12.008948 32.15% 2 12 3.018781 3.456877 4.195280 31.13%