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CHAPTER 18
2kr FACTORIAL DESIGNS WITH REPLICATIONS

No experiment is ever a complete failure. It can always serve as a negative example.

—Arthur Bloch

One problem with 2k factorial designs is that it is not possible to estimate experimental errors since no experiment is repeated. Experimental errors can be quantified by repeating the measurements under the same factor-level combinations. If each of the 2k experiments in the 2k design is repeated r times, we will have 2kr observations. Such a design is called 2kr design. Once again, it is helpful to start with a two-factor model, develop all the concepts, and then generalize them to a k-factor model.

18.1 22r FACTORIAL DESIGNS

A 22r factorial design is used when there are two factors each at two levels and the analyst wants to isolate experimental errors. Each of the four experiments is repeated r times. Such a design allows us to add an error term to the model, which now becomes

y = q0 + qAxA + qBxB + qABxAxB + e (18.1)

Here, e is the experimental error and the q’s are the effects as before.

18.2 COMPUTATION OF EFFECTS

The easiest way to analyze a 22r design is to use the sign table as before except that in the y column we put the sample means of r measurements at the given factor levels. The remaining analysis to compute the effects is the same as before. This is illustrated by the following example.

Example 18.1 The memory-cache experiments were repeated three times each. This resulted in the 12 observations shown in column y in Table 18.1. The analysis is also shown in the table. We sum the individual observations and divide by 3 (the number of replications) to get the sample means . The first four columns in the table are sign columns as before. The entries in each of the four columns are multiplied by those in column and the sum is entered under the column. The sums under each column are divided by 4 to give the following effects:

q0 = 41, qA = 21.5, qB = 9.5, qAB = 5

TABLE 18.1 Analysis of a 223 Design

I A B AB y Mean

1 –1 –1 1 (15, 18, 12) 15
1 1 –1 –1 (45, 48, 51) 48
1 –1 1 –1 (25, 28, 19) 24
1 1 1 1 (75, 75, 81) 77
164 86 38 20 Total
41 21.5 9.5 5 Total/4

18.3 ESTIMATION OF EXPERIMENTAL ERRORS

Once the effects have been computed in a 22r design, the model can be used to estimate the response for any given factor values (x-values) as follows:

= q0 + qAxAi + qBxBi + qABxAixBi

Here is the estimated response when the factors A and B are at levels xAi and xBi, respectively.

The difference between the estimate and the measured value yij in the jth replication of the ith experiment represents the experimental errors:

eij = yij = yijq0qAxBiqABxAixBi

We can compute the error in each of the 22r observations. The sum of the errors must be zero. The sum of the squared errors (SSE) can be used to estimate the variance of the errors and also to compute the confidence intervals for the effects:

TABLE 18.2 Computation of Errors in Example 18.2
Effect Estimated Measured

Response, Responses Errors
I A B AB

i 41 21.5 9.5 5 yi1 yi2 yi3 ei1 ei2 ei3

1 1 –1 –1 1 15 15 18 12 0 3 –3
2 1 1 –1 –1 48 45 48 51 –3 0 3
3 1 –1 1 –1 24 25 28 19 1 4 –5
4 1 1 1 1 77 75 75 81 –2 –2 4


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