2 _ VlSl + wj + • • • + I>gS\ Vi + V2 H----+i>

with v = vi + v>2 + • • • + degrees of freedom.

A direct estimate of the variance a2 is not available if there are no replicate runs. An alternate way to estimate a2 may be based on the assumption that the interactions of large number of factors would be negligible and the numerical values computed would measure differences caused by experimental error.

Example. Inspection of the interactions computed for the example problem (Eq. 3.9) shows that RT=-36.69, RS=1M, TS=0.89, and RTS=-0.92. Since there are only three factors and RT is more than an order of (absolute) magnitude greater than the other interactions, one may either use RTS or pool RS, TS, and RTS to compute an estimate of a2. The exclusion of RT may seem arbitrary, but its large magnitude would justify its exclusion. The first approach will yield s = a/(—0.92)2 = 0.92. The second will give s = ^[1.632 + 0.892 + (—0.92)2]/3 = 1.19. □

Once a standard error is estimated, it can be compared with the magnitude of the effects of various factors and interactions to assess their significance.

Example. Determine the dominant effects of the example problem. Recall the main effect values T = -43.93, R = 110.71 and S = -1.39. Either estimate of s indicates that T and R, and their interaction (RT) are more influential than all other factors and interactions. An increase in temperature reduces the product yield while an increase in substrate feed rate increases it. The two strains have no significant influence on the yield. The interaction of temperature and feed rate is such that a joint increase in both variables causes a reduction in the yield. □

Quantile-Quantile plots. A more systematic approach for the assessment of effects is based on comparison of the magnitudes of actual effects to what might be expected from a Normal distribution. This may be done

Table 3.7. Data and computed values for Q-Q plot of the main effects and interactions for the experimental data in Tables 3.1 and 3.2 and standard Normal distribution


Ordered y(i)





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