Order Rotatable Designs Research Articles

Use of experimental design in detergency tests with a natural soil

AbstractA balanced incomplete block design is used to obtain detergency data in a test where cloth swatches are soiled by rubbing against the skin. The design provides increased precision in the data by adjusting for differences among soilers. The wash treatments are part of a second order rotatable design in three variables: the ratio of sodium nitrilotriacetate to sodium tripolyphosphate builder, pH of the wash solution, and temperature. The effect of builder ratio was not highly significant. Soil removal increased with higher pH and went through a maximum with increase in wash temperature. Redeposition was also measured by reflectance values obtained for unsoiled areas of the swatches. Redeposition increased with increase in wash temperature.

Four and Six Level Second Order Rotatable Designs

Four and Six Level Second Order Rotatable Designs

On Blocking Second Order Rotatable Designs

On Blocking Second Order Rotatable Designs

Further Second Order Rotatable Designs

This note provides some new second order rotatable designs. The method of construction used is an extension of one introduced by Bose and Draper (1959). Further extensions of the method are briefly suggested.

Group-divisible rotatable designs

By modifying the restrictions imposed on the levels of the factors in a second order rotatable design, an alternative series of response surface designs has been obtained. Thev factors of the design have been split into two groups, and the design is rotatable for each group of factors when the levels of the factors in the other group are held constant. As such, the designs have been called ‘group-divisible rotatable designs’.

Sequential Designs for Spherical Weight Functions

Two papers by Box and Draper (1959, 1963) discussed the selection of first and second order rotatable designs when a spherical region of interest was defined and account had to be taken of possible biases due to the fact that the polynomial model under consideration was inadequate by one order. Here we consider a different but related problem. Suppose the intensity of interest in the factor space is represented, not by a spherical region of interest, but by a symmetric multivariate distribution weight function. (Previously it was assumed that total interest lay uniformly over a spherical region and there was no interest outside it.) We wish to specify the runs of a second order design which can be performed sequentially so that both the first order portion and the full second order design provide protection against biases of one order higher. In particular, we shall consider designs which are either the basic central composite type and consist of a “cube” plus “star” plus center points, or designs which contain an extra star. The extra star is required in some cases to allow certain conditions to be met.

A Method for the Construction of Second Order Rotatable Designs in $k$ Dimensions

A Method for the Construction of Second Order Rotatable Designs in $k$ Dimensions

The Distribution of the Coefficient of Variation; Comment on A criticism made by Koopmans, Owen and Rosenblatt

(1) Biometrika (1963), 50, 335–52. ‘The choice of a second order rotatable design.’ By G. E. P. Box and N. R. DRAPER p. 343. In equation (2.3-1) in the denominator of the third term on the right-hand side read −3kλ2 for −3λ2.

On Construction of Second and Third Order Rotatable Designs Through Pairwise Balanced and Doubly Balanced Designs

On Construction of Second and Third Order Rotatable Designs Through Pairwise Balanced and Doubly Balanced Designs

Two Third Order Rotatable Designs in Four Dimensions

Two Third Order Rotatable Designs in Four Dimensions

The Choice of a Second Order Rotatable Design

The Choice of a Second Order Rotatable Design

On Construction of Second Order Rotatable Designs Through Balanced Incomplete Block Designs with Blocks of Unequal Sizes

On Construction of Second Order Rotatable Designs Through Balanced Incomplete Block Designs with Blocks of Unequal Sizes

The choice of a second order rotatable design

The choice of a second order rotatable design

Construction of Rotatable Designs Through Balanced Incomplete Block Designs

Rotatable designs were introduced by Box and Hunter (1954, 1957) for the exploration of response surfaces. They constructed these designs through geometrical configurations and obtained several second order designs. Afterwards, Gardiner and others (1959) obtained some third order designs through the same technique for two and three factors and a third order design for four factors. Bose and Draper (1959) obtained some second order designs by using a different method. Draper (1960 a) gave a method of construction of an infinite series of second order designs in three and more factors. Recently, Box and Behnken (1960 a) have obtained a class of second order rotatable designs from those of first order. Draper (1960 b) has obtained some third order rotatable designs in three dimensions and a third order rotatable design in four dimensions. Das (1961) has obtained such designs, both second and third orders up to 8 factors as fractional replicates of factorial designs. The method of construction of the designs presented in this paper is essentially based on that presented by Das (1961). After the manuscript of this paper was submitted for publication the authors' attention was drawn to the work of Box and Behnken (1960 b). They have obtained some second order designs by following a procedure which uses balanced incomplete block designs in the same manner as described below. They did not, however, extend the method to include other complementary sets of points which, as will be shown, allow one to obtain rotatable second and third order designs based on any balanced incomplete block design. In the present paper a method has been given by using the properties of balanced incomplete block designs through which second order rotatable designs with any number of factors, with a reasonably small number of points, can be obtained. By extending the method, third order rotatable designs, both sequential and nonsequential, up to 15 factors have been obtained with the help of doubly balanced incomplete block designs and complementary B.I.B. designs.

Third Order Rotatable Designs in Three Factors: Analysis'

Recent work [10] has provided a method of constructing third order rotatable designs in three factors. This paper presents nine particular designs, given previously without explanation [11], together with background material, the formulae needed in order to analyze the results from such designs and a worked example.

Third Order Rotatable Designs in Three Factors: Analysis

Third Order Rotatable Designs in Three Factors: Analysis

Third Order Rotatable Designs in Three Dimensions: Some Specific Designs

A recent paper [3] described a method for constructing infinite classes of third order rotatable designs in three dimensions. This note gives nine specific designs selected from some of the infinite classes that have been shown to exist.

Missing Values in Response Surface Designs

The estimation of missing values for a general design is described and discussed. Formulae are provided for the estimation of missing values for two well-known, three factor, second order rotatable designs, with zero to six center points. A worked example illustrates the use of the formulae in the case of the cube plus octahedron plus one center point design.

Third Order Rotatable Designs in Three Dimensions

Two recent papers by Bose and Draper [1] and Draper [3] showed how it was possible, by combining certain sets of points, to construct infinite classes of second order rotatable designs in three and more dimensions. In this paper, third order rotatable designs in three dimensions are discussed. First, a general theorem is proved that provides the conditions under which a third order rotatable arrangement of points in $k$ dimensions is non-singular. The four previously known third order designs in three dimensions are stated; it is then shown how some of the second order design classes constructed earlier [1] may be combined in pairs to give infinite classes of sequential third order rotatable designs in three dimensions. One example of such a combination is worked out in full and it is shown that two of the four known designs are special cases of this class. A summary of further third order rotatable design classes that have been shown to exist, and that have been tabulated by the author, concludes the paper.

A Third Order Rotatable Design in Four Dimensions

A Third Order Rotatable Design in Four Dimensions