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CHAPTER 21
TWO-FACTOR FULL FACTORIAL DESIGN WITHOUT REPLICATIONS

No amount of experimentation can ever prove me right; a single experiment can prove me wrong.

—Albert Einstein

A two-factor design is used when there are two parameters that are carefully controlled and varied to study their impact on the performance. For example, one would use a two-factor design to compare several processors using several workloads. In this case, processors are one factor and the workloads are another. In a full factorial design all combinations of processors and workloads will be experimented with. Another example of application of two-factor design is that of determining two configuration parameters, such as cache and memory sizes, number of disk drives and number of processors, and so on. It is assumed that the factors are categorical or they are being treated as such by the analyst. If the factors are quantitative, a regression model may be used instead.

A full factorial design with two factors A and B having a and b levels requires 10 ab experiments. In this chapter, we consider the case where each experiment is conducted only once. The design with replications will be considered in Chapter 22.

Most of the concepts presented in this chapter are extensions of those presented earlier in Chapter 20 on one-factor designs. It is assumed that you have already read that chapter.

21.1 MODEL

The model for a two-factor design without replications is

yij = µ + αj + βi + eij

Here, yij is the observation in the experiment with the first factor A being at level j and the second factor B being at level i, µ is the mean response, αj is the effect of factor A at level j, βi is the effect of factor B at level i, and eij is the error term. The effects αj and βi are computed so that their sums are zero:

Notice that it is assumed that the effects of the two factors add and the errors are additive. An alternative to these assumptions will be discussed later in Section 21.7.

21.2 COMPUTATION OF EFFECTS

The procedure to develop expressions for effects is similar to that used earlier in Section 20.2 for one-factor design. The observations are assumed to be arranged in a two-dimensional matrix of b rows and a columns such that the (i,j)th entry yij corresponds to the response in the experiment in which factor A is at level j and factor B is at level i. In other words, the columns correspond to the levels of A and rows correspond to levels of the factor B.

The values of model parameters µ, αj’s, and βi’s are computed such that the error has a zero mean. This means that the sum of error terms along each column and along each row is zero.

Averaging the jth column produces

Since the last two terms are zero, we have

Similarly, averaging along rows produces

Averaging all observations produces

TABLE 21.1 Measured Processor Times for the Cache Comparison Study

Workload Two Caches One Cache No Cache

ASM 54.0 55.0 106.0
TECO 60.0 60.0 123.0
SIEVE 43.0 43.0 120.0
DHRYSTONE 49.0 52.0 111.0
SORT 49.0 50.0 108.0

Thus, the estimates of the model parameters are

Computation of parameters can be easily carried out using a tabular arrangement of data as illustrated by the following example.

Example 21.1 In a study to compare three different cache choices, the processor times to execute five different workloads was measured on three different configurations of a processor. The three configurations differed only in their cache designs. The three cache options were to use two set associative caches, one set associative cache, or no cache. The measured times in milliseconds are shown in Table 21.1. The analysis to compute the effects of the caches and workloads is shown in Table 21.2. For each row (or column), we compute the mean of observations in that row (or column). Overall sum and means are also computed. The difference between a row (or column) mean and overall mean gives the row (or column) effect.
The results of the analysis are interpreted as follows. An average workload on an average processor requires 72.2 milliseconds of processor time.
TABLE 21.2 Computation of Effects for the Cache Comparison Study

Row Row Row
Workload Two Caches One Cache No Cache Sum Mean Effect

ASM 54.0 55.0 126.0 215.0 71.7 -0.5
TECO 60.0 60.0 123.0 243.0 81.0 8.8
SIEVE 43.0 43.0 120.0 206.0 68.7 -3.5
DHRYSTONE 49.0 52.0 111.0 212.0 70.7 -1.5
SORT 49.0 50.0 128.0 207.0 69.0 -3.2

Column sum 255.0 260.0 568.0 1283.0
Column mean 51.0 52.0 113.6 72.2
Column effect -21.2 -20.2 41.4


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