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Improved Multiple Comparisons with the Best in Response Surface Methodology Laura K

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Improved Multiple Comparisons with the Best in Response Surface Methodology Laura K 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] Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons 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 This Emerging Scholar is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods Copyright Ó 2003 JMASM, Inc. May, 2003, Vol. 2, No. 1, 247-255 1538 – 9472/03/$30.00 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 C (xx)=-E(|y)Ey(|)0 provide confidence bounds for d B ()x within a 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 ˆ éùddBB(xx)- () an improved critical point d with simultaneous . (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 Uj=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 .
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