Minimax Designs for the Stability of Slope Estimation on Second-order Response Surfaces

In this paper, designs for the stability of the slope estimation on a second-order response surface are considered. Minimization of the point dispersion measure, which is maximized over all points in the region of interest is taken as the optimality criterion, and the minimax properties in some class of designs are derived in spherical and cubic regions of interest. We study the efficiencies of the minimax designs relative to other optimal designs with various criteria.     

Slope rotatability over all directions designs

Slope rotatability over all directions (SROAD)is a useful concept when the slope of a second-order response is to be studied. SROAD designs ensure that knowledge of the slope is acquired symmetrically, whatever direction later becomes of more interest as the data are analyzed. Some standard second-order designs are SROAD,but here we explore designs for two and three dimensions, which do not have the full symmetries of such standard designs but which still possess the SROAD property.     

Slope rotatability over all directions designs for k = 4

Slope rotatability over all directions (SROAD) is a useful concept when the slope of a second-order response is to be studied. SROAD designs ensure that knowledge jof the slope is acquired symmetrically, whatever direction later becomes of more interest as the data are analyzed. In a prior paper, we explored designs for k=2 and 3 dimensions, which do not have the full symmetries of second-order designs but which still possess the SROAD property.Here, we discuss designs in higher dimensions.The introductory sections 1 and 2 are essentially identical to those of the prior paper.     

A note on second order slope rotatable designs over all directions

Hader and Park (1978) introduced second order slope rotatability in axial directions. Park (1987) introduced second order slope rotatabilty over all directions. It is shown that these designs have the additional properly that the sum of the variances of estimates of slopes in all directions at any point is a function of the distance of the point from the design origin.     

A NEW MEASURE OF ROTATABILITY OF RESPONSE SURFACE DESIGNS

This paper proposes a new measure of rotatability of a response surface design. It is based on a comparison between the design moments of a given design and the design moments of a rotatable design closest to the given design in the least squares sense. This measure is easier to calculate than Khur?s measure. An alternative easier non-geometrical way of deriving Khuri's measure is also presented. Both measures are calculated to illustrate some designs considered by Khuri.     

A measure of robust slope-rotatability for second-order response surface experimental designs

In response surface methodology, rotatability and slope-rotatability are natural and highly desirable properties for second-order regression models. In this paper a measure of robust slope-rotatability for second-order response surface designs with a general correlated error structure is developed and illustrated with different examples for autocorrelated error structure.     

A graphical method for evaluating mixture designs with respect to the slope

Good estimation of the slopes of the mixture response function may be important as well as estimation of mean mixture response. It is possible to evaluate and compare several mixture designs with respect to the slope. A graphical method is proposed that allows us to evaluate a given design's support for the fitted model in terms of slope variance. We can plot variances of slopes along Cox direction or axial direction according to existence of restriction of simplex region or not when comparing several different mixture designs.     

A General Construction Method for Five-Level Second-Order Rotatable Designs

Response surface methodology is widely used for developing, improving, and optimizing processes in various fields. In this article, we present a method for constructing second-order rotatable designs in order to explore and optimize response surfaces based on an infinite class of supplementary difference sets. The produced designs achieve both properties of rotatability and estimation efficiency. Also, they possess good predictive properties.     

Construction of central composite designs for balanced orthogonal blocks

Box & Hunter (1957) recommended a set of orthogonally blocked central composite designs (CCD) when the region of interest is spherical. In order to achieve rotatability along with orthogonal blocking, the block size for those designs becomes unequal and it may not be attractive or practical to use such unequally blocked designs in many practical situations. In this paper, a construction method of orthogonally blocked CCD under the assumption of equal block size is proposed and an index of block orthogonality is introduced.     

Construction of New Three-Level Response Surface Designs

Response surface methodology is widely used for developing, improving, and optimizing processes in various fields. In this article, we present a method for constructing three-level designs in order to explore and optimize response surfaces combining orthogonal arrays and covering arrays in a particular manner. The produced designs achieve the properties of rotatability, predictive performance and efficiency for the estimation of a second-order model.     

Minimum,maximum, and average spherical prediction variances for central composite and box-behnken designs

Single value design optimality criteria are often considered when selecting a response surface design. An alternative to a single value criterion is to evaluate prediction variance properties throughout the experimental region and to graphically display the results in a variance dispersion graph (VDG) (Giovannitti-Jensen and Myers (1989)). Three properties of interest are the spherical average, maximum, and minimum prediction variances. Currently, a computer-intensive optimization algorithm is utilized to evaluate these prediction variance properties. It will be shown that the average, maximum, and minimum spherical prediction variances for central composite designs and Box-Behnken designs can be derived analytically. These three prediction variances can be expressed as functions of the radius and the design parameters. These functions provide exact spherical prediction variance values eliminating the implementation of extensive computing involving algorithms which do not guarantee convergence. This research is concerned with the theoretical development of these analytical forms. Results are presented for hyperspherical and hypercuboidal regions.     

A note on the connection between uniformity and generalized minimum aberration

Discrete discrepancy has been utilized as a uniformity measure for comparing and evaluating factorial designs. In this paper, for asymmetrical factorials, we give some linkages between uniformity measured by the discrete discrepancy and other criteria, such as generalized minimum aberration (Xu and Wu, 2001) and minimum projection variance (Ai and Zhang, 2004). These close linkages show a significant justification for the discrete discrepancy used to measure uniformity of factorial designs, and provide an additional rationale for using uniform designs. This research was partially supported by the NNSF of China (No. 10441001), the Key Project of Chinese Ministry of Education (No. 105119) and the Project-sponsored by SRF for ROCS (SEM) (No. [2004]176).     

Graphical evaluation of robust parameter designs based on extended scaled prediction variance and extended spherical average prediction variance

For any response surface design, there are locations in the design region where responses are estimated well and locations where estimation is relatively poor. Consequently, graphical evaluation—such as variance dispersion graphs and the fraction of design space—is used as an alternative to a single-valued criterion. Such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher. In this article, we propose the extended scaled prediction variance and extended spherical average prediction variance as prediction methods. We also illustrate how graphical methods can be employed to evaluate robust parameter designs.     

Sequential asymmetric third order rotatable designs (SATORDs)

Rotatable designs that are available for process/ product optimization trials are mostly symmetric in nature. In many practical situations, response surface designs (RSDs) with mixed factor (unequal) levels are more suitable as these designs explore more regions in the design space but it is hard to get rotatable designs with a given level of asymmetry. When experimenting with unequal factor levels via asymmetric second order rotatable design (ASORDs), the lack of fit of the model may become significant which ultimately leads to the estimation of parameters based on a higher (or third) order model. Experimenting with a new third order rotatable design (TORD) in such a situation would be expensive as the responses observed from the first stage runs would be kept underutilized. In this paper, we propose a method of constructing asymmetric TORD by sequentially augmenting some additional points to the ASORDs without discarding the runs in the first stage. The proposed designs will be more economical to obtain the optimum response as the design in the first stage can be used to fit the second order model and with some additional runs, third order model can be fitted without discarding the initial design.KEYWORDS: Response surface methodology, rotatability, orthogonal transformation, asymmetric, sequential experimentation, third order designs     

1961 Section Officers,Committees and Association Representatives

The multiple correlation coefficient, R (or r, in the simple case), is frequently used in evaluating regression models. Statistical significance is the usual criterion for judging R. Various views of its practical significance should also be considered. One practical measure is the percent reduction in the standard deviation of the response variable achieved by the model. A graph giving this as a function of R and percent loss in degrees of freedom is presented. This measure is compared with others which are sometimes appropriate.     

New 3-level response surface designs constructed from incomplete block designs

Box and Behnken [1958. Some new three level second-order designs for surface fitting. Statistical Technical Research Group Technical Report No. 26. Princeton University, Princeton, NJ; 1960. Some new three level designs for the study of quantitative variables. Technometrics 2, 455–475.] introduced a class of 3-level second-order designs for fitting the second-order response surface model. These 17 Box–Behnken designs (BB designs) are available for 3–12 and 16 factors. Although BB designs were developed nearly 50 years ago, they and the central-composite designs of Box and Wilson [1951. On the experimental attainment of optimum conditions. J. Royal Statist. Soc., Ser. B 13, 1–45.] are still the most often recommended response surface designs. Of the 17 aforementioned BB designs, 10 were constructed from balanced incomplete block designs (BIBDs) and seven were constructed from partially BIBDs (PBIBDs). In this paper we show that these seven BB designs constructed from PBIBDs can be improved in terms of rotatability as well as average prediction variance, D$D$- and G$G$-efficiency. In addition, we also report new orthogonally blocked solutions for 5, 8, 9, 11 and 13 factors. Note that an 11-factor BB design is available but cannot be orthogonally blocked. All new designs can be found at http://www.math.montana.edu/∼$\sim$jobo/bbd/.     

Does sequential augmenting of simple linear heteroscedastic regression reduce variances of ordinary least-squares estimators?

If uncorrelated random variables have a common expected value and decreasing variances, then the variance of a sample mean is decreasing with the number of observations. Unfortunately, this natural and desirable variance reduction property (VRP) by augmenting data is not automatically inherited by ordinary least-squares (OLS) estimators of parameters. We derive a new decomposition for updating the covariance matrices of the OLS which implies conditions for the OLS to have the VRP. In particular, in the case of a straight-line regression, we show that the OLS estimators of intercept and slope have the VRP if the values of the explanatory variable are increasing. This also holds true for alternating two-point experimental designs.     

A measure of slope-rotatability for second order response surface experimental designs

In the design of experiments for estimating the slope of a response surface, slope-rotatability is a desirable property. In this paper, a measure is introduced that enables us to assess the degree of slope-rotatability for a given response surface design. The measure takes the value zero if and only if the design is slope-rotatable, and becomes larger as the design deviates from a slope-rotatable design. Examples of applying this measure to some response surface designs are also given.     

Some adjusted orthogonal,variance balanced,multidimensional block designs

A multidimensional block design (MBD) is an experimental design with d > 1 blocking criteria geometrically represented as a d-dimensional lattice with treatment varieties assigned to some or all nodes of the lattice. Intrablock analysis of variance tables for some special classes of two- and three-dimensional block designs with some empty nodes are given. Design plans and efficiencies for 31 two-dimensional designs, each universally optimal in defined classes of designs, and 7 three-dimensional designs, each nearly optimal in defined classes of designs, are listed in the appendices. A need for such designs is apparent when the blocking criteria are implemented successively and empty nodes do not represent wasted experimental units.     

Some sufficient conditions for the type 1 optimality of block designs

In this paper, we investigate the problem of determining block designs which are optimal under type 1 optimality criteria within various classes of designs having υ treatments arranged in b blocks of size k. The solutions to two optimization problems are given which are related to a general result obtained by Cheng (1978) and which are useful in this investigation. As one application of the solutions obtained, the definition of a regular graph design given in Mitchell and John (1977) is extended to that of a semi-regular graph design and some sufficient conditions are derived for the existence of a semi-regular graph design which is optimal under a given type 1 criterion. A result is also given which shows how the sufficient conditions derived can be used to establish the optimality under a specific type 1 criterion of some particular types of semi- regular graph designs having both equal and unequal numbers of replicates. Finally,some sufficient conditions are obtained for the dual of an A- or D-optimal design to be A- or D-optimal within an appropriate class of dual designs.