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"Double central composite" designs

The present paper deals with the type of designs known as "Double Central Composite". Basically they are a symmetric type composed of two factorials (or fractional factorials) at levels ±1 and ±W, two stars at levels ±a and ±ga for each factor and central points, presenting five, seven or nine levels for each factor. Due to the higher flexibility of this type of design in comparison with the central composite, properties like orthogonality of the coefficients, orthogonal blocking and rotatability are easily put together in the same design. As these designs can be arranged with a fractionary part of a 2k factorial, they can explore more levels in the range considered with just a few more points than the correspondent central composite design. In agriculture, opposite to technological experiments, and principally in fertilizer research, due to the soil and climatic variability, the experiments should be performed with all treatments allocated at the same time. The responses obtained may be higher and enlarged in the good years and smaller or with a "plateau response" in the bad years. The double central composite design may fit better this situation, because by its structure they cover better a broader range of dosages. If an adequate calibration was originally adopted and the year was a good one, the larger range will fit better the response. If the year was bad, the response is smaller and may present some type of "plateau response" to the higher dosages; the analysis of the complete design may cause a "bias" in the coefficients of the model and in the determination of the economical dosages. In this case the design makes possible to contour the problem by just analysing the central part of the design, as a central composite, using the levels ±1 for the factorial and ± ato the axial, getting an "unbiased" estimate of the coefficients; for the same range of dosages originally utilized, this could not be obtained through an original central composite or with a 3x3x3 factorial. Basically this paper presents three types of design. In Table 1, designs are presented that, besides being orthogonal, are orthogonally blocked in 2, 3 or 5 blocks. The levels were chosen in such a manner as to become the W levels of all designs, smaller than two (levels higher than two are seldom used in practice). The designs of the second group are orthogonal, orthogonally blocked with the external points all of them on the surface of two hyperspheres of rays a= Ök and ga» WÖk, containing two or four central points; these designs may be considered as composed of two central composites, each one having, according with Lucas, quasi optimum D-efficiency. In the third group, presented in Table 3, the designs are orthogonal, orthogonally blocked and with full rotatability or quasi rotatability (the last one presented with an asterisk), in Box and Hunter sense, with enough central points to furnish good estimates of the experimental error, allowing a more precise test for the adequacy of the model. The double central composite design should be of value for the research workers or for the statisticians in those cases in which blocking is very important as in field agriculture experiments and particularly in fertilizer experiments when we are looking for the economical response to NPK. They may be utilized in other areas of research with similar problems, in cases in which we want to evaluate the response for several levels of each factor.


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