R

(a) Main effects

(b) Two-factor interactions

RxTxS

(c) Three-factor interactions

Figure 3.2. Geometric representation of contrasts corresponding to main effects and interactions [78].

Similarly,

The interactions of higher number of factors are denoted using the same convention. For example, the three-factor interaction between concentration, temperature, and strain is denoted by R x T x S. Three-factor interactions are computed using similar equations. The interaction between the three factors (factor levels as listed in Table 3.1) and illustrated in Figure 3.2 is computed by using two factor interactions. The interaction between Ri and R2 for one level of Rs(-) is [(2/4 - y3) - (2/2 - 2/i)]/2 and for the other level of i?3(+) [(2/8 — 2/7) — (2/6 ~ 2/s)]/2- Half of their difference (for #3[+] — Rs[—]) is defined as the three factor interaction:

Ri x R2 x R3 = - 2/1 + 2/2 + 2/3 - 2/4 + 2/5 - 2/6 - 2/7 + 2/8)

2/2 + 2/3 + Vh + 2/8 2/1+2/4+2/6+2/7 /„<,>. = -i---'A-' I3"8)

Example. Computation of the two-factor and three-factor interactions for the penicillin fermentation data.

The two-factor interactions are computed using Eqs. 3.6 and 3.7. The three-factor interaction is computed using Eq. 3.8:

The levels of factors such as those displayed in Table 3.2 can be used to generate a table of contrast coefficients that facilitates the computation of the effects (Table 3.4). The signs of the main effects are generated using the signs indicating the factor levels. The signs of the interactions are generated by multiplying the signs of the corresponding experiment levels (main effect signs). For example, the main effect T is calculated by using the signs of the third column:

_ -69.24 - 214.82 + 59.45 + 133.49 - 64.41 - 213.61 + 59.72 + 133.71

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