Small Response-Surface Designs

Draper, Norman R.; Lin, Dennis K.J.

doi:10.1080/00401706.1990.10484634

Cited by 106 publications

(60 citation statements)

References 19 publications

(35 reference statements)

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“…Design name or definition Defining characteristics One unstructured there is no structure or organization to the treatments Two or more nested structure factors have levels that are not repeated or have the same meaning at all levels of the other factors (Schutz and Cockerham, 1966) Two or more full factorial design each factor has a specific number of levels that are repeated (have the same definition and meaning) over all levels of the other factors; the number of treatment combinations is the product of the number of levels of each factor (Cochran and Cox, 1957) Two or more confounding design a full factorial in which a higher order interaction term is sacrificed as a blocking factor in order to achieve a reduction in block size (Cochran and Cox, 1957;Cox, 1958) Two or more composite design a subset of a factorial designed to severely reduce the number of treatments required to evaluate the main effects and first-order interactions using regression-based modeling (Draper and John, 1988;Draper and Lin, 1990;Lucas, 1976) Two or more fractional factorial a partial factorial arrangement in which only a subset of the factorial treatments is included, usually based on choosing a higher order interaction term as the defining contrast (Cochran and Cox, 1957;Cox, 1958) Two or more repeated measures one or more of the treatment factors is observed over multiple time points without rerandomization of treatments to experimental units (Milliken and Johnson, 2009) programs that are typically located in mountainous regions with little or no level or flat topography. The last three families of designs in Table 4 each contain considerable flexibility and versatility intended to solve particular problems.…”

Section: Number Of Factorsmentioning

confidence: 99%

Fundamentals of Experimental Design: Guidelines for Designing Successful Experiments

Casler

2015

Agronomy Journal

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We often think of experimental designs as analogous to recipes in a cookbook. We look for something that we like, something that satisfies our needs, and frequently return to those that have become our long‐standing favorites. We can easily become complacent, favoring the tried‐and‐true designs (or recipes) over those that contain unknown or untried ingredients or those that are too complex for our tastes and skills. Instead, I prefer to think of experimental designs as a creative series of decisions that are meant to solve one or more problems. These problems may be real or imagined—we may have direct evidence of a past or current problem or we may simply want insurance against future potential problems. The most significant manifestation of a “problem” or a “failed” design is unsatisfactory P values that prevent us from developing inferences about treatment differences. Four basic tenets or pillars of experimental design— replication, randomization, blocking, and size of experimental units— can be used creatively, intelligently, and consciously to solve both real and perceived problems in comparative experiments. Because research is expensive, both in terms of grant funds and the emotional costs invested in grant competition and administration, biological experiments should be designed under the mantra “failure is not an option.” Guidelines and advice provided in this review are designed to reduce the probability of failure for researchers who are willing to question, evaluate, and possibly modify their decision‐making processes.

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Section: Number Of Factorsmentioning

confidence: 99%

Fundamentals of Experimental Design: Guidelines for Designing Successful Experiments

Casler

2015

Agronomy Journal

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“…The resulting design is of course the same in both cases; the second criterion is easier to handle. We use however a criterion proposed by Draper and Lin [3]: critIII = critII/n = {det [(X T X) -1 ]} 1/k /n. When we compare for k = 2 factors the CCD1 (n = 9) with a Doehlert design, where we add two extra centre points (0, 0) to get also n = 9 experimental units, we have for this Doehlert design with n = 9 critI = 1/91.10362 = 0.010977 and critIII = 1.060536.…”

Section: D-optimality Criterion Comparison For Designs With 2 or 3 Famentioning

confidence: 99%

Use of Doehlert Designs for Second-order Polynomial Models

Verdooren¹

2017

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The most popular designs for fitting the second-order polynomial model are the central composite designs of Box and Wilson [2] and the designs of Box and Behnken [1]. For k = 2, 4, 6 and 8, the uniform shell designs of Doehlert [4] require fewer experimental runs than the central composite or Box-Behnken designs. In analytic chemistry the Doehlert designs are widely used. The uniform shell designs are based on a regular simplex, this is the geometric figure formed by k + 1 equally spaced points in a k -dimensional space; an equilateral triangle is a two-dimensional regular simplex. The shell designs are used for fitting a response surface to k independent factors over a spherical region. Doehlert (1930Doehlert ( -1999 proposed in 1970 the design for k = 2 factors starting from an equilateral triangle with sides of length 1, to construct a regular hexagon with a centre point at (0, 0). The n = 7 experimental points are (1, 0), (0.5, 0.866), (0, 0), (-0.5, 0.866), (-1, 0), (-0.5, -0.866) and (0.5, -0.866).The 6 outer points lie on a circle with a radius 1 and centre (0, 0). This Doehlert design has an equally spaced distribution of points over the experimental region, a so-called uniform space filler, where the distances between neighboring experiments are equal. Response surface designs are usually applied by scaling the coded factor ranges to the ranges of the experimental factors. The first factor covers the interval [-1, + 1], the second factor covers the interval [-0.866, + 0.866]. Doehlert design for four factors needs only 21 trials. Doehlert and Klee [5] show how to rotate the uniform shell designs to minimize the number of levels of the factors. Most of the rotated uniform shell designs have no more than five levels of any factor; the central composite design has five levels of every factor. The D-Optimality determinant criterion of the variance matrix of Doehlert designs will be compared with central composite designs and Box-Behnken designs, see Rasch et al. [6].

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“…The design comprised 16 treatments with two replicates (Table 2). The resulting four significant factors (Table 3) were employed to establish a Draper-Lin small central composite design with one replicate (Draper and Lin, 1990) to optimize xylanolytic activity, comprising 9 treatments (Table 4). All results were analyzed using statistical software (Statgraphics v. 5.0).…”

Section: Experimental Designmentioning

confidence: 99%

Xylanolytic enzymes production by Aspergillus niger GS1 from solid-state fermentation on corn stover and their effect on ruminal digestibility

Regalado‐González¹,

Vázquez-Obregón²,

García-Almendárez³

et al. 2011

Electro Journal of Biotech

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Hemicellulosic agricultural by-products such as corn stover (CS) are highly available materials which represent an opportunity to develop value added products. Native Aspergillus niger GS1 was used for solid-state fermentation (SSF) on alkali pre-treated CS (ACS) aimed to optimize xylanolytic enzymes production, and their effect on in vitro ruminal and true digestibility of ACS. Enzyme production was empirically modelled using a fractional factorial design 2 9-5 , and the resulting significant factors were glucose, yeast extract and two mineral salts, which were arranged in a DraperLin optimization design at two levels. Predicted optimum xylanolytic activity of 33.6 U (mg protein) -1 was achieved at 48 hrs of SSF, and was validated by confirmatory experiments. ACS was incubated with a semipurified enzymatic extract (EE) showing a xylanolytic activity of 1600 U kg -1 dry ACS for 12 hrs before exposure to cow's ruminal liquid for 72 hrs, which led to 5% and 10% increase of in vitro ruminal and true digestibility, respectively. CS is a readily available by-product in different regions which after alkaline treatment and partial hydrolysis with the EE, may be advantageously used as supplement for ruminant feed.

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Small Response-Surface Designs

Cited by 106 publications

References 19 publications

Fundamentals of Experimental Design: Guidelines for Designing Successful Experiments

Fundamentals of Experimental Design: Guidelines for Designing Successful Experiments

Use of Doehlert Designs for Second-order Polynomial Models

Xylanolytic enzymes production by Aspergillus niger GS1 from solid-state fermentation on corn stover and their effect on ruminal digestibility

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