Factorial designs resolution

Some protection against the effect of biases in the estimation of the first-order coefficients can be obtained by running a resolution IV fractional factorial design. With such a design the two-factor interactions are aliased with other two-factor interactions and so would not bias the estimation of the first-order coefficients. In fact the main effects are aliased with three-factor interactions in a resolution IV design and so the first-order effects would be biased if there were third-order coefficients of the form xxx, in... 

An alternative approach to constructing designs for estimating second-order models is to consider building a design from those constructed for the first-order model. In Section 2.2.1, we discussed the use of fractional factorial designs to estimate the coefficients of the first-order model. It was noted that a fractional factorial design of resolution V would yield... 

In general the cube portion might be replicated times and the star portion might be replicated times. Also, it might be possible to use a fractional factorial design of resolution less than V if the experimenter is prepared to assume that certain interactions are negligible. A central composite design in four variables is shown in Table 2.6. In this table, runs 1-16 are the cube portion, runs 17-24 are the star portion, and runs 25-27 are the center points. 

To construct the central composite design to estimate the coefficients of the second-order model (equation (14)), usually a fractional factorial design of at least resolution V is used. In this case, if the model is valid, then all of the estimates of the main effect coefficients, p., and the interaction coefficients, p. are imbiased. An alternative to the central composite designs for estimating the coefficients of the second-order model are the Box-Behnken designs or the designs referenced in Section 2.2.5. 

Table 2.9, shows the minimum number of runs for a single replicate of a fractional factorial design with the desired resolution for p variables, /7=3,...,11. 

Figure 3.3 Normal probability plot of the normalized effects for the resolution between epianhydrotetracycline and tetracycline obtainedfrom the fractional factorial design...

A reflected half-fraction factorial design for three factors (2 ) was performed. The influence of the factors on the responses recovery (%), resolution between peaks R) and Ry-value was calculated. Approximate critical values were obtained using the method given by Youden and Steiner (Ecritical MSE)e)- The standard error was estimated from... 

S.A. Ahmed, C.A. Lau-Cam and S.M. Bolton, Factorial design in the study of the effects of selected liquid chromatographic conditions on resolution and capacity factors. Journal of Liquid Chromatography, 13(3) (1990) 525-541. 

Three-level fractional factorial designs are also very useful, and charting the effects can be very helpful especially where there are more than three factors. The Plackett-Burman designs are often used to confirm (or otherwise ) the robustness of a method from the set value. Figure 17 shows some results from a ruggedness study for an HPLC method for salbutamol where the resolution factor, between it and its main degradation product is critical. 

The resolution of a fractional factorial design is a convenient way to describe the alias relationships ... 

The sparsity of effects principle (see Box and Meyer, 1986) makes resolution III and IV fractional factorial designs particularly effective for factor screening. This principle states that, when many factors are studied in a factorial experiment, the system tends to be dominated by the main effects of some of the factors and a relatively small number of two-factor interactions. Thus resolution IV designs with main effects clear of two-factor interactions are very effective as screening... 

Two level factorial designs are primarily useful for exploratory purposes and calibration designs have special uses in areas such as multivariate calibration where we often expect an independent linear response from each component in a mixture. It is often important, though, to provide a more detailed model of a system. There are two prime reasons. The first is for optimisation - to find the conditions that result in a maximum or minimum as appropriate. An example is when improving die yield of synthetic reaction, or a chromatographic resolution. The second is to produce a detailed quantitative model to predict mathematically how a response relates to die values of various factors. An example may be how the near-infrared spectrum of a manufactured product relates to the nature of the material and processing employed in manufacturing. 

A five-factor (buffer concentration, pH, IPR and organic solvent concentrations, and column temperature) two-level fractional factorial design with four center points was performed. The center points of the design were the midpoints of the range for each factor. The response from the design did not focus on bare retention but on resolution, and particularly on the separation of potential impurities around the main peak. Peak tailing, run time and backpressure were also considered chromatographic responses [76]. 

The most evident design would appear to be a three-level factorial design. An example of a three-level factorial design is shown in Fig. 6.1.. A full three-level factorial design, 3, can be used to obtain quadratic models. However, unless / is small (/ = 2) the design requires a number of experiments (= V) that is not often feasible. An example of the use of a 3" design can be found in Ref. 51]. The two factors, the pH and the percentage acetonitrile were examined to evaluate their influence on the retention and resolution of three isoxazolyl penicillin antibiotics. The centre point experiment was triplicated to evaluate experimental error. 

Degree of resolution, or shortly, the resolution of a fractional factorial design is defined by the length of the shortest "word" in the set of generators.[l] The resolution is commonly specified by roman numeral characters. 

Fractional factorial designs with higher resolution than V are rarely used in sceening experiments. 

The design is a Resolution III design and main effects are confounded with two-variable interaction effects. The design is one quarter of a full, 2 , factorial design. 

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