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design selected satisfy the relationship 123 = —45. Consequently, the 123 and 45 interactions are confounded. The individual interactions 123 and 45 are called aliases of each other. A relationship such as 5 = 1234 used to construct the 25-1 design is called the generator of the design. Recall that the numbers 1 to 5 used above or the uppercase letters used in Section 3.3.1 denote a factor and a column of — and + signs indicate its level. The multiplication of the elements of a column by another column having identical elements is represented as 1 x 1 = l2 = I. Similarly 2x2 = 1 and T x T = I. Furthermore, 2x1 = 2. Hence,

The relation 1=12345 is called the defining relation of the design and is the key for determining all confoundings. For example, multiplying both sides of the defining relation with 1 yields 1= 2345, indicating that the main effect 1 is confounded with the four-factor interaction 2345. All confounding patterns for the 25_1 design with the defining relation 1=12345 are given in Table 3.6.

The complementary half-fraction design for 25"1 is made up by all the entries in Table 3.5 without the asterisk in the "half-fraction" column. Its defining relation is I= —12345, the "-" sign indicating that the - level of 1234 interaction is used. Higher fractions such as 1/4 or 1/8 may also be of interest because of limited resources to conduct experiments. Then, additional defining relations must be used to design the experiment plan. The selection of the defining contrasts and confounded effects becomes more challenging as the number of factors and level of the fractions increase, necessitating systematic procedures such as the algorithm proposed by Franklin [164],

Design Resolution. Fractional factorial designs are classified according to their resolutions as well. A design of resolution R has no p-faet.or effect confounded with any other effect containing less than R — p factors. Usually the resolution of the design is indicated with Roman numerals as subscripts such as for the design with the defining relation 1=12345. Referring to Table 3.6, main effects are confounded only with 4-factor interactions (R — 4 = 1, hence R =V) and 3-factor interactions are confounded only with 2-factor interactions (R—3 = 2, hence R =V again). In general, the resolution of a two-level fractional design is equal to the length of the shortest word in the defining relation (1=12345 has R =V) [78]. A design of resolution R =111 does not confound main effects with one another, but confounds main effects with two-factor interactions. A design of resolution R =IV does not confound main factors and two-factor interactions, but confounds two-factor interactions with other two-factor interactions. Consequently, given the number of experiments that will be performed, the design with the highest resolution is sought. The selection of the defining relation plays a critical role in the resolution. In general, to construct a 2P 1 fractional factorial design of highest possible resolution, one writes a full factorial design for the first p — 1 variables and associates the pth variable with the interaction 123 • • • {p - 1) [78].

Exploratory experimentation is an iterative process where results from a small number of experiments are used to obtain some insight about the process and use that information to plan additional experiments for learning more about the process. It is better to conduct sequential experimentation in exploratory studies using fractional factorial designs and use these fractions as the building blocks to design more complete sets of experiments as needed.

### 3.3.3 Analysis of Data from Screening Experiments

Once the numerical values of main and interaction effects are computed, one must decide which effects are significant. Comparison of the estimates of effects and standard errors indicates the dominant effects. Consequently, the standard errors must be computed. If replicate runs are made at each set of experimental conditions, the variation between their outcomes may be used to estimate the standard deviation of a single observation and consequently the standard deviation of the effects [78]. For a specific combination of experimental conditions, n* replicate runs made at the it h set of experimental conditions yield an estimate s2 of the variance a2 having fi = rii — 1 degrees of freedom. In general, the pooled estimate of the run variance for g sets of experimental conditions is