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CHAPTER 23
GENERAL FULL FACTORIAL DESIGNS WITH k FACTORS

Work expands to fill the time available for its completion.

—C. Northcote Parkinson

The experimental designs analyzed so far are sufficient to solve problems encountered frequently. They can be easily extended to a larger number of factors. This chapter briefly describes the models used for such designs and illustrates it with a case study involving four factors. Three informal methods are then presented in Section 23.3 for analysis of experimental designs that can be used for larger number of factors as well as for designs discussed in earlier chapters.

23.1 MODEL

The model for a k-factor full factorial design contains 2k - 1 effects. This includes k main effects, (k2) two-factor interactions, (k3) three-factor interactions, and so on. For example, with three factors A, B, and C at a, b, and c levels, respectively and r replications, the model is

where

yijkl =response (observation) in the Ith replication of
experiment with factors A, B, and C at levels i, j, and k, respectively.
μ =mean response
αi =effect of factor A at level i

and

β =effect of factor B at level j
ξk =effect of factor C at level k
γABij =interaction between A and B at levels i and j.
γABCijk =interaction between A, B, C at levels i, j, k.

and so on.

23.2 ANALYSIS OF A GENERAL DESIGN

The analysis of full factorial designs with k factors is similar to that with two factors presented earlier in Chapters 21 and 22. For example, the parameters can be estimated from means taken along various dimensions:

The sums of squares, degrees of freedom, and F-test also extend as expected. The following case study illustrates the analysis of a general k-factor design.

Case Study 23.1 In a study to quantify the effect of various factors on the paging process, a full factorial 3 x 3 x 3 x 3 experimental design was used. The factors were main memory size, problem program, deck arrangement, and replacement algorithm. The effect of four factors each at three levels was used. The factors and their levels are listed in Thble 23.1. The design consisted of a total of 81 experiments for which the number of page swaps (PS) was measured. The measured values are listed in Table 23.2. It is seen from this table that the number of page swaps has a wide range, the ratio of ymax/ymin is 23134/32, or about 723. This suggests that we should try using a log transformation of the data. This is confirmed also by the observation that as a factor is changed from one level to the next, the change in response is approximately proportional to the response. A log transformation was therefore used. The transformed data, which was called log page swaps (LPS), is shown in Table 23.3. The effects of various factors and their interactions can be obtained by averaging along various axes. For example, the effect of algorithm (factor A) at level 1 is obtained as follows:

α = y1... - y... = 2.74-2.90 = -0.16

TABLE 23.1 Fhetors and Levels for Page Swap Study

Symbol Factor Level 1 Level 2 Level 3

A Page replacement algorithm LRUV FIFO RAND
D Deck arrangement GROUP FREQY ALPHA
P Problem program Small Medium Large
M Memory pages 24P 20P 16P


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