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CHAPTER 22
TWO-FACTOR FULL FACTORIAL DESIGN WITH REPLICATIONS

To estimate the time it takes to do a task: estimate the time you think it should take, multiply by 2, and change the unit of measure to the next highest unit. Thus, we allocate 2 days for a one-hour task.

—Westheimer’s Rule

The two-factor full factorial design discussed in Chapter 21 helps estimate the effect of each of the two factors being varied. However, the interactions between the factors were assumed negligible and, hence, ignored as errors. Replicating a full factorial design allows separating out the interactions from experimental errors. Thus, if it is known that there is a significant interaction between the factors, the design with replications discussed in this chapter should be used.

22.1 MODEL

Consider a design with r replications of each of the ab experiments corresponding to the a levels of factor A and b levels of factor B. The model in this case is

Here,

yijk = response (observation) in the kth replication of experiment with factor A at level j and factor B at level i
µ = mean response
αj = effect of factor A at level j
βi = effect of factor B at level i
γij = effect of interaction between factor A at level j and factor B at level i
eijk = experimental error

The effects are computed so that their sum is zero:

The interactions are computed so that their row as well as column sums are zero:

The errors in each experiment add to zero:

22.2 COMPUTATION OF EFFECTS

The expressions for effects can be obtained in a manner similar to that used in Section 21.2 for two-factor designs without replications. The observations are assumed to be arranged in ab cells arranged as a matrix of b rows and a columns. Each cell contains r observations belonging to the replications of one experiment. Averaging the observations in each cell produces

Similarly, averaging across columns and rows and over all observations produces

From these equations, we obtain the following expressions for effects:

Computation of parameters can be easily carried out using a tabular arrangement similar to that used in Section 21.2 except that ab cell means are used in each cell to compute row and column effects.

Example 22.1 Consider the data of Table 22.1, which represents code sizes of five different workloads on four different computers. Three different programmers with similar backgrounds independently wrote the programs and so the sizes represent three replications of each experiment.
The physical considerations as well as the response range dictates the need for a multiplicative model for this problem. A log transformation of the data is shown in Table 22.2. The computation of row and column effects in this case is similar to that in the analysis of two-factor design without replication. The means of all observations in each cell are used in the analysis, as shown in Table 22.3.
TABLE 22.1 Data for Code Size Study with Replications

Workload W X Y Z

I 7,006 12,042 29,061 9,903
6,593 11,794 27,045 9,206
7,302 13,074 30,057 10,035
J 3,207 5,123 8,960 4,153
2,883 5,632 8,064 4,257
3,523 4,608 9,677 4,065
K 4,707 9,407 19,740 7,089
4,935 8,933 19,345 6,982
4,465 9,964 21,122 6,678
L 5,107 5,613 22,340 5,356
5,508 5,947 23,102 5,734
4,743 5,161 21,446 4,965
W 6,807 12,243 28,560 9,803
6,392 11,995 26,846 9,306
7,208 12,974 30,559 10,233


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