Full-factorial design

In order to overcome one of the main disadvantages of SFE, Salafranca et al. [322] have proposed the use of full-factorial design with the objective of attaining optimum extraction conditions. It was considered that... 

Recently, nonliving biomass of S. cucullata has been described as a low-cost absorbent of Cr(VI).106 Optimum conditions for the Cr(VI) adsorption by acid-treated S. cucullata were found out using a full factorial design. The Cr(VI) removal efficiency of the adsorbent was found to increase with the increase in time, temperature, adsorbate concentration, and stirring speed, and to decrease with increase in pH and adsorbent dose. The Fourier transform infrared spectroscopy (FT-IR) analysis revealed that in addition to electrostatic force, the adsorption may be due to... 

ANOVA in these chapters also, back when it was still called Statistics in Spectroscopy [16-19] although, to be sure, our discussions were at a fairly elementary level. The experiment that Philip Brown did is eminently suitable for that type of computation. The experiment was formally a three-factor multilevel full-factorial design. Any nonlinearity in the data will show up in the analysis as what Statisticians call an interaction term, which can even be tested for statistical significance. He then used the wavelengths of maximum linearity to perform calibrations for the various sugars. We will discuss the results below, since they are at the heart of what makes this paper important. 

The absorption spectra of Aspt, Ace-K, Caf and Na-Benz were recorded from 190 to 300 nm. The calibration set was generated by a three-level full factorial design (4).The absorbance valnes were recorded eveiy 5 nm. The calibration samples were measured in random order, so that experimental errors due to drift were not introduced. 

For a partial separation situation after screening, the organic modifier content and temperature are decreased according to a 2 full factorial design. When baseline separation is obtained, the retention factor can be further optimized by changing the... 

A full factorial design contains all possible combinations (L ) between the different factors f and their levels L, with L = 2 for two-level designs. It allows estimating all main and interaction effects between the factors. A FF design will only perform a fraction of the full factorial. A two-level FF design 2 examines factors, each at two levels, in 2 experiments, with 1/2"... 

Two-level full factorial designs were used to determine the CE robusmess of a chiral separation of the local anesthetic ropivacaine in injection solutions and of a separation of the macrolide antibiotic tylosin and its main related substances. Table 13a shows the applied... 

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. 

Number of factor combinations required by full factorial designs as a function of the number of factors. 

The upper left panel of Figure 12.28 shows the nine factor combinations (design points) of a 3 full factorial design in two independent mixture components, Xj and Xj. The third component of the three-component mixture is not independent, and is obtained from the mixture constraint (Equation 12.87) by difference ... 

Figure 12.28 Conversion of a 3 full factorial design to a give a constrained mixture design. See text for details.

A larger three-level full factorial design... 

In this chapter we explore factorial-based experimental designs in more detail. We will show how these designs can be used in their full factorial form how factorial designs can be taken apart into blocks to minimize the effect of (or, if desired, to estimate the effect of) an additional factor and how only a portion of the full factorial design (a fractional replicate) can be used to screen many potentially useful factors in a very small number of experiments. Finally, we will illustrate the use of a Latin square design, a special type of fractionalized design. 

Figure 14.1 A two-factor two-level full factorial design in factors A and B.

Traditional tabular presentation of data for carrying out a 2 full factorial design. 

With simple, symmetrical, orthogonal designs like the full factorial designs, when all of the experiments have been done exactly the same number of times, then the factor effects can be calculated using simple algebra. 

In the classical factorial design literature, a factor effect is defined as the difference in average response between the experiments carried out at the high level of the factor and the experiments carried out at the low level of the factor. Thus, in a 2 full factorial design, the main effect of A would be calculated as ... 

To illustrate this classical approach to the calculation of factor effects, consider the following 2 full factorial design ... 

The Yates algorithm is a formal procedure for estimating the P s for full two-level factorial designs [Yates (1936)]. The Yates algorithm is related to the fast Fourier transform. We describe the Yates algorithm here, and illustrate it s use for the 1 full factorial design discussed in Section 14.2. 

Some comments about full factorial designs... 

Full factorial designs have been especially useful for describing the effects of qualitative factors, factors that are measured on nominal or ordinal scales. This environment of qualitative factors is where factorial designs originated. Because all possible factor combinations are investigated in a full design, the results using qualitative factors are essentially historical and have little, if any, predictive ability. 

Full factorial designs are also useful for describing the effects of quantitative factors, factors that are measured on interval or ratio scales. This is how factorial designs are often used today. Because the factors are quantitative (e.g., 20 lbs or 40 lbs of catalyst, 1 hour or 2 hours reaction time), the results have some predictive ability (e.g., what would the results be with 50 lbs of catalyst and 2.5 hours reaction time ). 

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