Factorial designs four-level

Factorial design — four factors at two levels. Running all 16 combinations is the full factorial. Running either the JE s or the O s is called a l/2 replicate. 

A 2 factorial design with two factors requires four runs, or sets of experimental conditions, for which the uncoded levels, coded levels, and responses are shown in Table 14.4. The terms Po> Po> Pfc> and Pafc in equation 14.4 account for, respectively, the mean effect (which is the average response), first-order effects due to factors A and B, and the interaction between the two factors. Estimates for these parameters are given by the following equations... 

As we have noted, eight tests (series A and C) represented a 23 factorial design involving type of interior finish, presence of a fiberglass insulation in the cavity, and load level (Table I). Series B was similar to A and C except for the absence of load and hardboard siding. In most cases, the four tests of series B could also be included in the analysis. 

For the final optimization, a modified factorial design involving three concentration levels of triethylamine and three pH levels was used. From these results, it was clear that the optimum conditions for the analysis of the carboxylic acid were so different from those required for the other compounds studied that it was not sensible to attempt to analyse all four together and indeed that carboxylic acids were better analysed by using conventional reversed-phase HPLC than by using ion-pairing. 

The aim of this work, to develop a RP-HPLC method for the detemtination of four components given above in a new symp preparation by optimizing the experimental conditions using two level fractional factorial design. 

Full factorial designs allow the estimation of all main and interaction effects, which is not really necessary to evaluate robusmess. They can perfectly be applied when the number of examined factors is maximally four, considering the required number of experiments. In references 69 and 70, four and three factors were examined at two levels in 16 and 8 experiments, respectively. When the number of factors exceeds four, the number of experiments increases dramatically, and then the full factorial designs are not feasible anymore. 

When few factors (/ from two to four) are studied, the full factorial design is the most common approach. The full factorial scheme is the basis for all classical experimental designs, which may be used in more complex situations. For a general two-level full factorial design, each factor has to be considered at a low level (coded as —1) and a high level... 

As another example of the reduction in the number of runs, consider an experiment to investigate three design and four environmental variables, all at three levels. A Taguchi crossed array might use a 3 fractional factorial design for the design array and a 3 fractional factorial design... 

As the number of factors increases, the economies of fractional replication become more evident. With six factors, a l/4 replicate of the 64 possible combinations is not unusual and a l/8 replicate of the 256 run eight-factor factorial is fairly common, Cochran and Cox [3] give a very extensive list of fractional factorial designs, including some in which factors are at three and four levels. 

Latin square design — three factors at four levels. This is a 1/4 replicate of a 43 = 64 factorial... 

Factorial design - two factors at two levels. Each box represents one experimental condition. If we can be sure that the effect of concentration of each inhibitor is exactly the same at both concentrations of the other, only three out of four of the conditions needs to be run. 

Four experimental variables were selected sample pH, primary column type, secondary column type, and methanol concentration. By using each of the four variables at two levels, the complete arrangement of experimental runs became a2X2X2X2or24 factorial design requiring 16 runs. Table I represents the design matrix the high and... 

We can use the same approach to expand the design and obtain a data set applicable for polynomials of third order. The respective full factorial design is constructed by combining all of the factors at five levels, giving a total of N = 5", or N = 25 for n = 2, N = 125 for n = 3, N = 625 for n = 4, etc. It is apparent that the number of the experiments grows geometrically with the number of factors, and there are not many applications where the performance of 625 experiments to explore four factors is reasonable. 

The nonregular nature of the fractional factorial design makes it possible to consider interaction effects as well as main effects see also Chapter 7. An interaction between two factors, each with three levels, has four degrees of freedom which can be decomposed into linear x linear, linear x quadratic, quadratic x quadratic, and quadratic x linear effects. The contrast coefficients for these effects are formed by multiplying the coefficients of the corresponding main effect contrasts. 

Two level full factorial designs (also sometimes called saturated factorial designs), as presented in this section, take into account all linear terms, and all possible k way interactions. The number of different types of terms can be predicted by the binomial theorem [given by k /(k — m) m for mth-order interactions and k factors, e.g. there are six two factor (=m) interactions for four factors (= )]. Hence for a four factor, two level design, there will be 16 experiments, the response being described by an equation with a total of 16 terms, of which... 

Table 2.18 Four factor, two level full factorial design.

Three and four level full factorial designs... 

Plackett and Burman published their classical paper in 1946, which has been much cited by chemists. Their work originated from the need for war-time testing of components in equipment manufacture. A large number of factors influenced the quality of these components and efficient procedures were required for screening. They proposed a number of two level factorial designs, where the number of experiments is a multiple of four. Hence designs exist for 4, 8, 12, 16, 20, 24, etc., experiments. The number of experiments exceeds the number of factors, k, by one. 

In calibration it is normal to use several concentration levels to form a model. Indeed, for information on lack-of-fit and so predictive ability, this is essential. Hence two level factorial designs are inadequate and typically four or five concentration levels are required for each compound. However, chemometric techniques are most useful for multicomponent mixtures. Consider an experiment earned out in a mixture of methanol and acetone. What happens if the concentrations of acetone and methanol in a training set are completely correlated If the concentration of acetone increases, so does that of methanol, and similarly with a decrease. Such an experimental arrangement is shown in Figure 2.25. A more satisfactory design is given in Figure 2.26, in which the two... 

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