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There are several ways an experienced analyst would find out that the additive model does not represent the data. Among these are the following:

  The first and foremost is the physical consideration that a computer scientist can easily justify that the effects of workload and processors do not add. They multiply, as already argued.
  The range of values covered by y is large. The ratio Ymax/Ymin is 147.90/ 0.0118, or 12,534. Taking an arithmetic mean of 114.17 and 0.013 is inappropriate. This calls for a log transformation, as discussed earlier in Section 15.4.
  A plot of residuals versus predicted response is shown in Figure 18.3. Notice that the vertical and horizontal scales are of the same order of magnitude. The residuals are not small as compared to the response. Also, the spread of the residuals is large at larger value of the response. This is another indicator that calls for a log transformation.
  A normal quantile-quantile plot of the residuals is shown in Figure 18.4. Notice that the larger positive and negative residuals do not follow the same line as those in the middle. As was discussed earlier in Section 12.10, on a quantile-quantile plot, this indicates that the residuals have a distribution that has a longer tail than the normal. Thus, the normality assumption is violated.

Based on any one of these arguments, the analyst should at least try a multiplicative model and compare the results with those obtained using the additive model. The logarithm of responses and the corresponding analysis is shown in Table 18.4. The percentage of variation explained and the confidence intervals of effects of the two models are listed in Table 18.5. Notice that the interaction with a multiplicative model is almost zero. The two main effects explain 49.9% of the variation each. The unexplained variation is only 0.2%, which is 1/50th of the previous model.


FIGURE 18.3  Plot of residuals versus predicted response for the additive model.


FIGURE 18.4  Normal quantile-quantile plot for residuals of the additive model.

TABLE 18.4 Transformed Data for Multiplicative Model Example

I A B AB y Mean

1 –1 –1 1 (1.93, 1.90, 2.17) 2.00
1 1 –1 –1 (–0.05, 0.02, 0.03) 0.00
1 –1 1 –1 (–0.02, –0.03, 0.05) 0.00
1 1 1 1 (–1.83, –1.90, –1.93) –1.89
0.11 –3.89 –3.89 0.11 Total
0.03 –0.97 –0.97 0.03 Total/4

TABLE 18.5. Percentage of Variation Explained by the Two Models
Additive Model Multiplicative Model


Factor Effect Percentage of Variation Confidence
Interval
Effect Percentage of Variation Confidence
Interval

I 26.55 (16.35, 36.74) 0.03 (–0.02, 0.07)a
A –26.04 30.1 (–36.23, –15.84) –0.97 49.9 (–1.02, –0.93)
B –26.04 30.1 (–36.23, –15.84) –0.97 49.9 (–1.02, –0.93)
AB 25.54 29.0 (15.35, 35.74) 0.03 0.0 (–0.02, 0.07)a
e 10.8 0.2


a Not significant.


FIGURE 18.5  Plot of residuals versus predicted response for the multiplicative model.

A plot of residuals versus predicted response is shown in Figure 18.5. Notice that the residuals are an order of magnitude lower than the response and that there is no trend in the spread of the residuals. A normal quantile–quantile plot is shown in Figure 18.6. The distribution is satisfactorily normal. Overall the multiplicative model appears to be considerably better for this data than the additive model.

The results of the multiplicative model are interpreted as follows. The model is

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

or

y = 10q010qAxA10qBxB10qABxAxB10e


FIGURE 18.6  Normal quantile-quantile plot for residuals of the multiplicative model.

Substituting the model parameters, we obtain

or

The time for an average processor on an average benchmark is 1.07. The time on processor A1 is 9 times (0.107–1) that on an average processor. The time on A2 is one-ninth (0.1071) that on an average processor. In other words, time on A1 is 81 times that on A2. Or equivalently, the MIPS rate for processor A2 is 81 times that of A1. Similarly, one can argue that the benchmark B1 executes 81 times more instructions than B2. The interaction is negligible. The results are therefore valid for all benchmark and processor combinations.


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