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Unformatted text preview: Chapter 8. Supplemental Text Material S8-1. Yatess Method for the Analysis of Fractional Factorials Computer programs are almost always used for the analysis of fractional factorial. However, we may use Yates's algorithm for the analysis of a 2 k-1 fractional factorial design by initially considering the data as having been obtained from a full factorial in k - 1 variables. The treatment combinations for this full factorial are listed in standard order, and then an additional letter (or letters) is added in parentheses to these treatment combinations to produce the actual treatment combinations run. Yates's algorithm then proceeds as usual. The actual effects estimated are identified by multiplying the effects associated with the treatment combinations in the full 2 k-1 design by the defining relation of the 2 k-1 fractional factorial. The procedure is demonstrated in Table 1 below using the data from Example 8-1. This is a 2 4-1 fractional. The data are arranged as a full 2 3 design in the factors A , B, and C . Then the letter d is added in parentheses to yield the actual treatment combinations that were performed. The effect estimated by, say, the second row in this table is A + BCD since A and BCD are aliases. Table 1. Yates's Algorithm for the 2 Fractional Factorial in Example 8-1 4 1 IV Treatment Combination Response (1) (2) (3) Effect Effect Estimate 2 3 ( ) / N (1) 45 145 255 566 - - a ( d) 100 110 311 76 A+BCD 19.00 b ( d) 45 135 75 6 B+ACD 1.5 ab 65 176 1 -4 AB+CD -1.00 c ( d ) 75 55 -35 56 C+ABD 14.00 ac 60 20 41 -74 AC+BD -18.50 bc 80 -15 -15 76 BC+AD 19.00 abc ( d ) 96 16 16 66 ABC+D 16.50 S8-2 Alias Structures in Fractional Factorials and Other Designs In this chapter we show how to find the alias relationships in a 2 k-p fractional factorial design by use of the complete defining relation. This method works well in simple designs, such as the regular fractions we use most frequently, but it does not work as well in more complex settings, such as some of the irregular fractions and partial fold-over designs. Furthermore, there are some fractional factorials that do not have defining relations, such as Plackett-Burman designs, so the defining relation method will not work for these types of designs at all. Fortunately, there is a general method available that works satisfactorily in many situations. The method uses the polynomial or regression model representation of the model, say y X = + 1 1 where y is an n 1 vector of the responses, X 1 is an n p 1 matrix containing the design matrix expanded to the form of the model that the experimenter is fitting, 1 is an p 1 1 vector of the model parameters, and is an n 1 vector of errors. 1 vector of errors....
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This note was uploaded on 03/20/2011 for the course STATISTIC 101 taught by Professor Fandia during the Spring '10 term at UCLA.
- Spring '10