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2018 | OriginalPaper | Chapter

11. Two-Level Fractional-Factorial Designs

Authors : Paul D. Berger, Robert E. Maurer, Giovana B. Celli

Published in: Experimental Design

Publisher: Springer International Publishing

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Abstract

We continue our examination of two-level factorial designs with discussion of a design technique that is very popular because it allows the study of a relatively large number of factors without running all combinations of the levels of the factors, as done in our earlier 2^k designs. In Chap. 10, we introduced confounding schemes, where we ran all 2^k treatment combinations, although in two or more blocks. Here, we introduce the technique of running a fractional design, that is, running only a portion, or fraction, of all the treatment combinations. Of course, whatever fraction of the total number of combinations is going to be run, the specific treatment combinations chosen must be carefully determined. These designs are called fractional-factorial designs and are widely used for many types of practical problems.

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previous chapter Confounding/Blocking in 2 k Designs

next chapter Designs with Factors at Three Levels

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Produce products are agricultural products, especially fresh fruits and vegetables, as distinguished from grains and other staple crops.

Studies are currently being done to determine the effect of mobile phones on this location/impulse buying issue in supermarkets. There seems to be preliminary evidence that the level of impulse buying in supermarkets is decreasing due to customers’ spending the time in line at the cash register using their mobile phones for various activities, instead of pondering impulse purchases – a phenomenon termed by the retail industry as the “mobile blinders.”

Using the logic of earlier chapters, with only four treatment combinations (and, say, no replicates), we have only three degrees of freedom and can thus get estimates of only three effects. The three pairs noted are those three estimates. In a sense, ABC is “wrapped up” in the estimate of the grand mean – the mean of the eight treatments.

If one is interested in seven or fewer estimates to be clean, this is a necessary – but not sufficient – condition to having a design with eight treatments, that is, k − p = 3. Consider, for example, that instead of the main effects, we want A, B, C, D, E, AB, and CD to be clean. There is no 2^5–2 design that works in this case.

Professors are always learning with their students, especially in a class of somewhat advanced material. This easier method for the “good-to-know” rule was proposed by Joe Bush of the U.S. Army Corp of Engineers R&D Center, while a student in the 14.37S course taught by one of the authors at MIT in 2012.

This algorithm was derived and first published by one of the authors. A more detailed discussion can be found in P. D. Berger (1972), “On Yates’ Order in Fractional Factorial Designs.” Technometrics, vol. 14, n. 4, pp. 971–972.

Whether the effects are in columns and the treatment combinations in rows (as in the sign tables in general) or the transpose of that – the effects in rows and the treatment combinations represented by columns (as in the orthogonal matrices of Chap. 5 and the one below the reference to this note) – is simply a matter of the respective traditions.

Note the coincidence in this example: for a 2² design, we divide by 2 when finding the effects and by \( \sqrt{4}=2 \) when forming the orthonormal table. This is not true for other designs. Taking a 2³ design, for example, we would divide by 4 when finding the effects and by \( \sqrt{8} \) when forming the orthonormal table.

We are, of course, aware that we have not used the technically correct wording. To do so, we should say something like, “We reject the hypothesis that the amount of poster deployment has no effect (…),” and for prizes, “We cannot reject the hypothesis that (…).” At times, we prefer to avoid such double negatives.

Title: Two-Level Fractional-Factorial Designs
Authors: Paul D. Berger
Robert E. Maurer
Giovana B. Celli
Publisher: Springer International Publishing
Book: Experimental Design
Print ISBN: 978-3-319-64582-7

Electronic ISBN: 978-3-319-64583-4

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-64583-4_11

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