Z Factorial Designs

The irealmetils or experimental conditions for each run consist of all possible combinations of levels from the different factors. For example, in a complete factorial design (CFD) with two factors A and B, if they have levels c , 02-, 03 and / , b -, respectively, then the treatment combinations are (uj, by), (ui, 112), 1). 2), ( 23, 1) and ( 3, Z)2)- The number of experimental runs 

If a complete factorial design has three levels (low, middle and high) for each of three factors (A = temperature B = pH and C = reaction time), it is said to be a 3 X 3 X 3 or 3 CFD and 27 runs are required, each one corresponding to a particular combination of the factor levels. Furthermore, if we replicate each run k times, then the number of experiments needed is k x 3.  

By means of an analysis of variance (ANOVA), we can detect the influence of each factor (A, B and C), each interaction (AB, AC, BC, ABC) and, if replicates are available, estimate the purely experimental error. If no replicates were performed, according to the ANOVA rationale, we can consider the high-order interaction as an estimation of the pure experimental error. 

Knowing which factor and interaction have an influence on the response (y), we can try to fit an empirical model by the common least-squares criterion  

This model allows us to estimate a response inside the experimental domain defined by the levels of the factors and so we can search for a maximum, a minimum or a zone of interest of the response. There are two main disadvantages of the complete factorial designs. First, when many factors were defined or when each factor has many levels, a large number of experiments is required. Remember the expression number of experiments = replicates x Oevels) (e.g. with 2 replicates, 3 levels for each factor and 3 factors we would need 2 x 3 = 54 experiments). The second disadvantage is the need to use ANOVA and the least-squares method to analyse the responses, two techniques involving no simple calculi. Of course, this is not a problem if proper statistical software is available, but it may be cumbersome otherwise. 

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