The main effect of temperature is the average of these four differences (—43.93). It is denoted by T (not to be confused by T that is the symbol for the input) and it indicates the average effect of temperature over all conditions of the other factors. The main effects of the other factors can be computed similarly. A more efficient computation can be made by noting that the main effect is the difference between two averages:

where yi+ and yt_ are the average responses for the + and — levels of variable i, respectively. Hence, for T:

rr _ V3 + y4 + 2/7 + V& y\ + 2/2 + t/5 + 2/6 1 ~ 4 4

Similar equations can be developed for other main effects. The main effects of all three factors are T = -43.73, R = 106.38, and S = -6.35. □

All eight observations are used to compute the information on each of the main effects, providing a fourfold replicate of the differences. To secure the same precision in the OVAT approach for estimating the main effect of temperature, eight experiments have to be conducted, four at each level of temperature, while the other two inputs are fixed at one of their respective levels. A total of 24 experiments (a threefold increase) is needed to obtain the estimates of the three main effects. In general, a p-fold (p =number of factors) increase in the number of experiments are needed for OVAT over the full factorial approach. Even if all changes are made with respect to a common experimental condition in OVAT design, (p + l)/2 times more experiments are needed than the full factorial designs [78],

The implicit assumption in the OVAT design is that the main effect observed for one factor will remain the same at different settings of the other factors. In other words, the variables act on the response additively. If this assumption is correct, the results based on the OVAT design will provide complete information about the effects of various factors on the response even though the OVAT design would necessitate more experiments to match the precision of factorial design. If the assumption is not appropriate, data based on factorial design (unlike the OVAT design) can detect and estimate interactions between factors that lead to nonadditivity [78].

Interaction Effects. The effect of a factor may be much greater at one level of another factor than its other level. If the factors do not behave additively, they interact. A geometric representation of contrasts corresponding to main effects and interactions is given in Figure 3.2. The interaction between two factors is called two-factor interaction. Most of the interactions between a larger number of factors are usually smaller. The experimental data collected provide the opportunity to compute and assess the significance of these interactions. A measure of two-factor interaction with iii and i?2 as factors is provided by the difference between the average effect of one factor at one level of the second factor and its average effect at the other level of the second factor. Two factor interactions are denoted as Ri x i?2 or R.\R-2 (when the x can be dropped without causing ambiguity). Thus, the temperature and inoculum strain interaction is denoted by T x 5

Consider the interaction between the first and third factors in Table 3.1. The average effect of the first factor for one level of the third factor (i?3=+) is (j/6 ~ ys)/2 + (2/8 - i/7)/2. and for the other level of the third factor (i?3 = —) is (2/2 — 2/1 )/2 + (y4 — yi)/2. The first and third factor interaction thus is or TS.

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