Second-order Designs Research Articles

On D-minimax optimal second-order designs for estimating slopes of second-order response surfaces

Minimization of determinant of the covariance matrix of the estimated axial slopes at a point, maximized over all points in the region of interest, is taken as the design criterion. Optimal designs under this P-minimax criterion are considered for four different second-order models over hypercubic regions. Relative performances of the optimal designs are investigated. Optimal designs from the classes of product designs, central composite designs and Kono designs are also derived. Some efficient exact designs are obtained.

Exact Inference for Response Surface Designs Within a Split-Plot Structure

Many industrial experiments involve factors that are hard to change as well as factors that are easy to change, which leads naturally to split-plot experiments. Unfortunately, the literature for second-order response surface designs traditionally assumes a completely randomized design. By carefully choosing the design for response surface experiments, it is possible to guarantee that the ordinary least-squares estimates of the model are equivalent to the generalized least-squares estimates. This paper uses designs satifying this result to derive conditions for exact tests of almost all of the model parameters. It then sets up the Satterthwaite's procedure for the remaining parameters. An example illustrates the methodology. Comparisons between the exact estimators, estimators based on replicated points, and estimators using restricted maximum likelihood estimators (REML) are done using the example.

Small, balanced, efficient and near rotatable central composite designs

Standard central composite designs for fitting second-order models usually have a large number of design points, especially when the number of factors under consideration is large. In this paper we propose small, balanced and near rotatable central composite designs reducing the design's size and maintaining the ability to estimate all the model parameters. We list the best designs we identified to accommodate 4 ⩽ k ⩽ 9 experimental factors. A detailed comparison between our findings and already known second-order designs has also been made.

Modified second-order slope rotatable designs with equi-spaced levels

In this paper, a new method of construction of modified second-order slope rotatable designs with equi-spaced levels using central composite designs and balanced incomplete block designs is suggested.

Unbalanced and Minimal Point Equivalent Estimation Second-Order Split-Plot Designs

Restricting the randomization of hard-to-change factors in industrial experiments is often required, resulting in a split-plot design structure. From an economic perspective, these designs minimize the experimental cost by reducing the number of resets of the hard-to-change factors. In this paper, unbalanced designs are considered for cases where the subplots are relatively expensive and the experimental apparatus accommodates an unequal number of runs per whole plot. We provide construction methods for unbalanced second-order split-plot designs that possess the equivalent estimation optimality property, providing best linear unbiased estimates of the parameters, independent of the variance components. Unbalanced versions of the central composite and Box–Behnken designs are developed. For cases where the subplot cost approaches the whole-plot cost, minimal point designs are proposed and illustrated with a split-plot Notz design.

Optimal Three-Level Designs for Response Surfaces in Spherical Experimental Regions

Since 1960, Box–Behnken designs have been very popular with experimenters wishing to estimate a second-order model in three or four factors. This popularity is due to these three-level designs' simplicity and high efficiency. However, as the number of factors increases, the run size of Box–Behnken designs increases rapidly, making them less attractive. The purpose of this article is to recommend the use of D-optimal and I-optimal three-level designs for spherical design regions involving three or more factors. Using an optimal design algorithm offers flexibility of run size. In addition, optimal-design criteria permit construction of second-order designs for more complex response surface applications involving mixture or qualitative factors or requiring split-plot randomization. The restriction to three-level designs provides great practical convenience and often little loss in efficiency, provided the design is not nearly saturated.

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 - and 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/ ∼ jobo/bbd/ .

Restricted second-order designs on one on two concentric balls

Designs for quadratic regression on one and two concentric balls are considered in this paper. The balls are defined by The 2 k factorial structure with additional 2 axial points provides the basic framework for the experiment. Restrictions such as orthogonality, rotatability, etc. which induce desirable properties are imposed on the design and their effects on the variance of the quadratic model compared.

Optimizing the medium components in bioemulsifiers production byCandida lipolyticawith response surface method

A response surface methodology was used to study bioemulsifier production by Candida lipolytica. A 2(4) full experimental design was previously carried out to investigate the effects and interactions of the concentrations of corn oil, urea, ammonium sulfate, and potassium dihydrogen orthophosphate on the emulsification activity (EA) of the bioemulsifier produced by C. lipolytica. The best EA value (3.727 units of emulsification activity (UEA)) was obtained with a medium composed of 0.4 g of urea, 1.1 g of ammonium sulfate, 2.04 g of potassium dihydrogen orthophosphate, 5 mL of corn oil, 50 mL of distilled water, and 50 mL of seawater. A curvature check was performed and revealed a lack of fit of the linear approximation. The proximity of the optimum point was evident, as was the need for quadratic model and second-order designs that incorporate the effect of the curvature. Medium constituents were then optimized for the EA using a three-factor central composite design and response surface methodology. The second-order model showed statistical significance and predictive ability. It was found that the maximum EA produced was 4.415 UEA, and the optimum levels of urea, ammonium sulfate, and potassium dihydrogen orthophosphate were, respectively, 0.544% (m/v), 2.131% (m/v), and 2.628% (m/v).

Prediction Variance Properties of Second-Order Designs for Cuboidal Regions

Much information is available about the construction and evaluation of the prediction variance properties of response surface designs for fitting a second-order model in a spherical region of interest. There is less information available about the prediction variance properties of second-order designs for a cuboidal region. In this paper, we construct and evaluate designs that are appropriate for the second-order model in cuboidal regions of interest. Several standard experimental designs are investigated and some new designs created using the G and I optimality criteria. We evaluate the designs using the variance of the predicted response over the design region. Variance dispersion graphs and fraction of design space plots are utilized to characterize the scaled prediction variance properties of the most promising designs.

On Second-Order A-, D- and E-Minimax Designs for Estimating Slopes in Extrapolation and Restricted Interpolation Regions

Summary The concepts of D-, A- and E-minimax optimality criteria of designs for estimating the slopes of a response surface are considered for situations where the region of interest may not be identical to the experimental region. Optimal second-order designs are derived for the situation where the experimental region and the region of interest are both hyperspherical with a common centre. The dependence of the optimal design on the relative sizes of the regions is investigated. Further, the perfomance of designs optimal for one region in estimating slopes in other regions is also examined.

Second-Order Experimental Designs for Simulation Metamodeling

The main purpose of this study is to compare the performance of a group of second-order designs such as Box-Behnken, face-center cube, three-level factorial, central composite, minimum bias, and minimum variance plus bias for estimating a quadratic metamodel. A time-shared computer system is used to demonstrate the ability of the designs in providing good fit of the metamodel to the simulation response. First, for various numbers of center runs, these designs are compared with respect to their efficiency, rotatability, orthogonality, robustness, bias, and prediction variance. Next, second-order metamodels are fit to the data collected using these designs. Metamodel fit is investigated using criteria such as average absolute error, PRESS, and the C p statistic. Results indicate that the minimum variance plus bias design is the most promising design to estimate a metamodel for the case studied.

Three-Level Simplex Designs and Their Use in Sequential Experimentation

In this paper three-level simplex designs of k+1 runs for k factors are presented. Each simplex design is composed of k treatment combinations from a two-level factorial design, plus an additional base run that represents a third level for each factor. These orthogonal, first-order designs are simple to construct. Furthermore, simplex designs can be augmented to construct second-order simplex sum designs. Such designs are particularly attractive when the experimental region of interest is spherical rather than a hypercube. It is also noted that the k treatment combinations in each simplex design coincide with the axial points in the second-order, asymmetric composite designs proposed by Box and Wilson (1951) and later discussed by Lucas (1974). Including the base point permits the inclusion of a block effect in the fitted second-order model.

Experimental designs when there are one or more factor constraints

In response surface methodology, designs of orders one or two are often needed such that some or all the factor levels satisfy one or more linear constraints. A method is discussed for obtaining such designs by projection of a standard design onto the constraint hyperplane. It is shown that a projected design obtained from a rotatable design is also rotatable, and for a rotatable design that is also orthogonal (in particular any orthogonal first-order design) a least squares analysis carried out on the generating design supplies a least squares solution for the constrained design subject to the constraints. Some useful properties of the generating design, such as orthogonal blocking and fractionation are retained in the projected design. Some second-order mixture designs generated by two-level factorials are discussed.

Noncentral Composite Designs

Second-order designs are constructed by adding a second two-level design to an initial 2k or 2k−p experiment. I label such designs as noncentral composite designs since they are composed of two designs with different centers. The location of the second block can be determined following analysis of the initial experiment. Noncentral composite designs provide an attractive alternative to the popular central composite designs when the location of interest shifts following an initial two-level design. An alternative family of asymmetric composite designs is also considered. The methodology is illustrated via a paper-helicopter experiment.

Second-Order Designs in Two Stages

This study examines the comparative information intrinsic to each stage of a two-stage experiment. Aliasing at the first stage, possible confounding of effects between stages, and efficiency comparisons for the two stages are determined directly from the analysis. A practical advantage is that runs to be assigned prospectively to the first and second stages may be evaluated beforehand, independently of empirical data. Motivated by studies in the digestion of seafood wastes, a case study develops a central composite design in k = 3 regressors, first on augmenting a 2 3 design with center mn, then on augmenting a small composite design with center run. Other augmentations are considered as well. The study concludes with some minimal augmentations to a 2 3 design with center run, so as to ensure full estimability in a second-order model in three regressors.

An enhanced response surface methodology (RSM) algorithm using gradient deflection and second-order search strategies

In the optimization of complex systems where at best one can only evaluate the function being optimized, often via a simulation experimental process, Response Surface Methodology (RSM) is frequently the method of choice. However, RSM invariably employs a first-order steepest descent method, presumably because of the prohibitive cost of employing second-order designs, except for the canonical analysis that is conducted at termination. In this paper, we suggest the use of deflected/conjugate gradient approaches that can be gainfully employed to improve the performance of RSM. These methods involve the same effort per iteration and require only first-order designs. However, instead of discarding previously generated information, the method utilizes these computations to advantageously build upon accumulated knowledge regarding the second-order curvature of the function being optimized, hence accelerating convergence by circumventing the characteristic zigzagging of steepest descent methods. The proposed scheme is illustrated using various test examples from the literature for which optimal solutions are known and that serve the purpose of exhibiting the potential of the recommended methodology.This paper attempts to improve the search techniques being currently used in standard Response Surface Methodology (RSM) algorithms. RSM is a collection of mathematical and statistical techniques for experimental optimization. This work presents a novel RSM algorithm that incorporates certain gradient deflection methods, augmented with appropriate restarting criteria, as opposed to using the path of steepest descent as the only search direction. In order to investigate the use of the new RSM algorithm in comparison with the standard existing RSM techniques, a set of standard test functions is used, both with and without random perturbations. Computational results exhibit the improvements achieved under the proposed algorithm.

Principal Predictors and Efficiency in Small Second-Order Designs

Linear parametric functions affected by deleting or augmenting sets of design points are identified explicitly using principal components of the predictive dispersion at those points, offering fresh insight and significant computational savings. Leverages from regression diagnostics are seen to determine efficiencies due to augmenting or deleting single points. Eight small second-order designs are studied in detail with supporting numerical displays. Comparisons are drawn to other approaches from the literature.

Minimax second-order designs over hypercubes for the difference between estimated responses at a point and at the centre

Minimization of the variance of the difference between estimated response at a point and that at the centre maximized over all points in the experimental region is taken as the design criterion. Optimal designs are derived for second-order polynomial models over hypercubes. The performance of best designs among the symmetric product designs and the star point designs is investigated.

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.