Second-order Designs Research Articles

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.

Efficiencies of some small second-order designs

Hybrid designs H310 and H311B are compared with Box—Behnken (BBD) and central composite (CCD) designs for estimation and testing in second-order models having k = 3 regressor variables. For linear coefficients these designs are ordered in efficiency as CCD ≻ BBD ≻ H311B ≻ H310, for interaction coefficients as BBD ≻ CCD ≻ H310 ≻ H311B, and for the intercept as H310 ≻ {H311B, BBD, CCD}. Conclusions are less direct regarding the pure quadratic coefficients, which are examined in detail.

Application of experimental design in ball burnishing

The ball burnishing process is used in place of other traditional methods to finish ASSAB XW-5 tool steel on a vertical machining centre. From an initial roughness of about R tm 4 μm, the specimen could be finished to a roughness of 0.27 μm. The response surface methodology (RSM) was used to establish mathematical models correlating three process parameters: depth of penetration; feed; and burnishing speed. The experimental design and method of analysis for fitting polynomials of the first and second degrees (first-and second-order designs) are discussed in this paper.

Second-order space focusing in two-field time-of-flight mass spectrometers

Design equations are given for second-order space focusing in two-field time-of-flight mass spectrometers. Simulations and experimental tests confirm that, for diffuse sources and high-energy fragments, second-order designs give superior resolution and linearity of flight time deviations to initial ion velocities.

Efficiency comparisons in linear inference

A new approach to comparative design efficiency is given in linear estimation and in tests for linear hypotheses. In contrast to others, this approach gives a complete assessment of two designs for any model on identifying precisely the subspaces of parameters for which one design is more efficient than another, or two designs are equally efficient. Local and global bounds on relative design efficiencies are given with reference to these subspaces, and connections to information functionals are noted. The relative influences of design points with regard to deletion and augmentation are examined and related to measures of influence from the literature. Our approach provides numerical diagnostics for use in design evaluation before an experiment has been run. The concepts are illustrated with reference to selected second-order designs.

Robustness of some designs against missing data

This article studies the robustness of several types of designs against missing data. The robustness of orthogonal resolution III fractional factorial designs and second-order rotatable designs is studied when a single observation is missing. We also study the robustness of balanced incomplete block designs when a block is missing and of Youden square designs when a column is missing.

On Performance of D- and A-Optimal Second-Order Designs in Spherical Regions under Model Variations

For spherical experimental regions designs are considered which are D- and A-optimal among the rotatable designs for all the parameters and also for two subsets of the parameters in a second order model. Performance of these optimal designs under model variation is examined. The best central composite designs fot the above two criteria and for the three models are also presented.

Response Surfaces: Designs and Analyses

Introduction to response surface methodology matrix algebra, least squares, the analysis of variance, and principles of experimental design first-order models and designs second-order models and designs determining optimum conditions methods of estimating response surfaces that rival least squares based on the integrated mean squared error criterion analysis of multiresponse experiments response surface models with block effects mixture designs and analyses nonlinear response surface models Taguchi's robust parameter design additional topics and some directions for future research. Appendix: solutions to selected exercises.

An Index of Rotatability

An index of rotatability is suggested that will enable the experimenter to obtain an immediate appreciation of the overall shape of specified variance contours for symmetrical second-order designs. Values of the index are tabulated for the central composite designs for two to eight factors. Three designs of Roquemore (1976) are assessed via the index. Comparisons are made with an entirely different index suggested by Khuri (1988). It is concluded that both indexes are useful and sensibly consistent.

An Index of Rotatability

An index of rotatability is suggested that will enable the experimenter to obtain an immediate appreciation of the overall shape of specified variance contours for symmetrical second-order designs. Values of the index are tabulated for the central composite designs for two to eight factors. Three designs of Roquemore (1976) are assessed via the index. Comparisons are made with an entirely different index suggested by Khuri (1988). It is concluded that both indexes are useful and sensibly consistent.

Minimax second-order designs for difference between estimated responses in extrapolation region

Minimization of the variance of the difference between estimated response at two response at two points maximized over all pairs of points in the extrapolation region is taken as the criterion for selecting designs. Optimal designs under the criterion are derived for second-order models.

Measures of Lack of Fit for Response Surface Designs and Predictor Variable Transformations

Some first-order (2 k–p two-level factorials and fractional factorials plus center points) and second-order (cube plus star plus center-point composite) response surface designs are discussed from the point of view of their ability to detect certain likely kinds of lack of fit of degree one higher than has been fitted. This leads to consideration of conditions for representationa adequacy of first- and second-order models in transformed predictor variables. It is shown how to use the estimated regression coefficients from the higher degree model to check if power transformations of the predictor variables could eliminate the lack of fit, and also actually to estimate the transformations.

A Method for Constructing Second-Order Rotatable Designs

A method for constructing second-order rotatable desians in k dimensions from second-order rotatable designs in k - l dimensions (1 ⩽ l ⩽ k -1) is proposed. The method makes use of balanced incomplete block designs. The method always works and, further, the experiments performed according to the ( k - l)-dimensional design need not be discarded.

An Investigation of First-Order and Second-Order Designs for Extrapolation outside a Hypersphere

An Investigation of First-Order and Second-Order Designs for Extrapolation outside a Hypersphere

Corrections and Amendments: D-Optimum and Quasi-D-Optimum Second-Order Designs on a Cube

Corrections and Amendments: D-Optimum and Quasi-D-Optimum Second-Order Designs on a Cube

‘D-optimum and quasi-D-optimum second-order designs on a cube’

I should like to point out the error in construction of minimal second-order symmetric set I44–2 and absolutely minimal design ξ 4 in my paper. The proper set I44–1 may be constructed as a combination of four half-replicates of the 2 4 factorials with the defining relations, x2x3x4 = I , x1x3x4 = I , x1x2x4 = I and x1x2x3 = I correspondingly for the subset B4 and of four duplicated half-replicates of the 2 3 factorials with the same but negative defining relations for the subsets B4–1 ( i ); the number of different points in B4 is 15 and in I4 , 4–l is 31, although the number of trials is 64, against corresponding numbers 36 and 40 in Table 1 of the paper. For the construction of absolutely minimal design ξ 4 we need to take some modification of set I44–1 with 34 points and we obtain in the same way as in the paper a D -optimum design ξ 4 with the number of points 37 instead of 45 presented in Table 2.

D-optimum and quasi-D-optimum second-order designs on a cube

SUMMARY The D-optimum designs and their sets of support are studied for quadratic regression on a cube. The structure of such designs is found and the designs presented have either the least possible, or considerably fewer, points of support than Kono's D-optimum designs and the designs of Farrell, Kiefer & Walbran (1967). The sets of support of the D-optimum designs are used for the construction of integer designs, the characteristics of which are tabulated.

D-Optimum and Quasi-D-Optimum Second-Order Designs on a Cube

D-Optimum and Quasi-D-Optimum Second-Order Designs on a Cube

Economical Second-Order Designs Based on Irregular Fractions of the 3nFactorial

There are presented some new economical, nonorthogonal, second-order designs based on irregular fractions of partially balanced type of the 3 n factorial for any number of factors n ≥ 3. The designs generalize those of Rechtschaffner [10]. The complete estimation, variance-covariance, trace, and determinant structures have been derived as explicit functions of n and these quantities compared for the best saturated versus the best augmented design. The efficiencies of the parameter estimates relative to those in other designs of comparable size have been computed.

Economical Second-Order Designs Based on Irregular Fractions of the 3 n Factorial

Economical Second-Order Designs Based on Irregular Fractions of the 3 n Factorial