Factorial experiment

An important purpose of a designed experiment is to obtain information about interactions among the primary variables. This is accompbshed by varying factors simultaneously rather than one at a time. Thus in Figure 2, each of the two preparations would be mn at both low and high temperatures using, for example, a full factorial experiment. 

After the preceding considerations have been taken into account, a test plan is developed to best meet the goals of the program. This might involve one of the standard plans developed by statisticians. Such plans are described in various texts (Table 1) and are considered only briefly here. Sometimes, combinations of plans are encountered, such as a factorial experiment conducted in blocks or a central composite design using a fractional factorial base. 

O. L. Davies and co-workers. The Design andAna/ysis of Industria/Experiments, 2nd ed., Hafner, New York, 1956 reprinted by Longman, New York, 1987. This book, which is a sequel to the authors basic text Statistica/Methods in Eesearch and Production, is directed at industrial situations and chemical appHcations. Three chapters are devoted to factorial experiments and one chapter to fractional factorial plans. A lengthy chapter (84 pp.) discusses the deterrnination of optimum conditions and response surface designs, which are associated with the name of G. Box, one of the seven co-authors. Theoretical material is presented in chapter appendices. 

This resulted in a 2 =16 factorial experiments. To these were added 8 outlayers and 3 repeated centerpoint altogether 27 experiments. The levels of variable are shown on the table in Figure 6.4.1. 

Yates, F., 1937. Design and Analysis of Factorial Experiments. Imperial Bureau of Soil Science, London. 

Because all the variables that influence the properties of the final product are known, one can use a statistical design (known as a one-half factorial) to optimize the properties of the GPC/SEC gels. Factorial experiments are described in detail by Hafner (10). For example, four variables at two levels can be examined in eight observations. From these observations the significance of each variable as related to the performance of the gel can be determined. An example of a one-half factorial experiment applied to the production of GPC/SEC gel is set up in Table 5.2. The four variables are the type of DVB, amount of dodecane, type of methocel, and rate of stirring. 

Other variables in the factorial experiment also have an impact on the character of the final product. The amount of nonsolvent is a very important variable to examine as the pore size of the gel depends on the amount of it present in the formulation. The stabilizer acts as a suspending agent and influences the particle size of the GPC/SEC gel. Lower viscosity suspending agents... 

TABLE 5.2 One-Half Factorial Experiment for Optimization of GPOSEC Gels... 

The factorial experiment sketched out above is used in two settings (cf. Ref. 137 for a tutorial) ... 

Carter CW Jr, Carter CW. Protein crystallization using incomplete factorial experiments. J Biol Chem 1979 254 12219-12223. 

Unfortunately there is also some bad news for modelers. Different humic materials bind compounds to dramatically different extents, and the reasons for this are unclear. Figure 6 shows the binding constants of DDT to seven different humic materials. Some of this data is from a factorial experiment which has been published elsewhere.7 Inspection of this data shows that the humic acids and the... 

Because variables in models are often highly correlated, when experimental data are collected, the xrx matrix in Equation 2.9 can be badly conditioned (see Appendix A), and thus the estimates of the values of the coefficients in a model can have considerable associated uncertainty. The method of factorial experimental design forces the data to be orthogonal and avoids this problem. This method allows you to determine the relative importance of each input variable and thus to develop a parsimonious model, one that includes only the most important variables and effects. Factorial experiments also represent efficient experimentation. You systematically plan and conduct experiments in which all of the variables are changed simultaneously rather than one at a time, thus reducing the number of experiments needed. 

A replicated two-level factorial experiment is carried out as follows (the dependent variables are yields) ... 

Fowlie and Bulman [43] have carried out a detailed study of the extraction of anthracene and benzo[tf]pyrene from soil. They carried out a replicated [24] factorial experiment using Soxhlet extraction and Polytron techniques. Soxhlet extraction followed by thin layer chromatography gave higher recoveries of the two polyaromatic hydrocarbons. 

Blrnbaum, A. (1959). On the analysis of factorial experiments without replication. Technometrics 1, 343-357. 

Pilipauskas, D.R., Using Factorial Experiments in the Development of Process Chemistry. In Process Chemistry in the Pharmaceutical Industry (K.G. Gadamsetti, ed.). Chap. 22, Marcel Dekker, Ine., New York, 1999, pp. 411-428. 

Other models can be fit to data from two-level factorial designs. For example, fitting the model expressed by Equation 12.8 to the data used in this section will produce the fitted model given by Equation 12.10. Some models cannot be fit to data from two-level factorial experiments for example, the model... 

Connor, W-S., and Zelen, M. (1959), Fractional Factorial Experiment Designs for Factors at Three Levels, National Bureau of Standards Applied Mathematics Series, 54, 1-37. 

Three-Level, Full-Factorial Experiment Design, Interaction Model Between Pad A and Pad B... 

Confounding a device whereby, in large factorial experiments, the number of runs to be made can be reduced by sacrificing some of the possible comparisons. The comparisons thus sacrificed are said to be confounded. 

Factorial experiment an experiment designed to examine the effect of two or more factors, each applied at least at two levels of operation. The full factorial investigates all possible combinations of these factors at the indicated levels ... 

Fractional replication a factorial experiment in which only a balanced fraction of the possible treatment combinations is run ... 

Fransson JR. Oxidation of human insulin-like growth factor I in formulation studies. 3. Factorial experiments of the effects of ferric ions, EDTA, and visible light on methionine oxidation and covalent aggregation in aqueous solution. J Pharm Sci 1997 86(9) 4046-4050. 

For determining the robustness of a method a number of parameters, such as extraction time, mobile-phase pH, mobile-phase composition, injection volume, source of column lots and/or suppliers, temperature, detection wavelength, and the flow rate, are varied within a realistic range and tlie quantitative influence of the variables is determined. If the influence of a parameter is within a previously specified tolerance, this parameter is said to be witliin the robustness range of the method. These method parameters may be evaluated one factor at a time or simultaneously as part of a factorial experiment. 

In a factorial experiment, a fixed number of levels are selected for each of a number of variables. For a full factorial, experiments that consist of all possible combinations that can be formed from the different factors and their levels are then performed. This approach allows the investigator to study several factors and examine their interactions simultaneously. The object is to obtain a broad picture of the effects of the selected experimental variables and detect major trends that can determine more promising directions for further experimentation. Advantages of a factorial design over single-factor experiments are (1) more than one factor can be varied at a time to allow the examination of interaction effects and (2) the use of all experimental runs in evaluating an effect increases the efficiency of the experiment and provides more complete information. 

In this study, a factorial experiment was set up to evaluate the effects of four variables at two levels on extraction efficiencies by using bonded-phase isolation techniques and a 27-component synthetic test mixture. The compounds studied and the respective mass ions used for quantitation are given in the box. The compounds in the mix vary greatly in water solubility and volatility and, in general, represent a wide class of organic compounds typical of those present in environmental samples. To maximize solute recoveries, the procedure was... 

Use of Half-Normal Plots with Factorial Data. The application of this method to the factorial data is straightforward. If, for any given compound, the data from the factorial experiment occurred simply as the result of random variation about a fixed mean, and the changes in the levels of the variables had no real effect at all on the percent recovery, then the 15 main effects and interactions, representing 15... 

As a result of this anomalous data, part of the main factorial experiments was repeated by using a 2 design that included the same parameters and conventions as the original factorial, except that one factor, the methanol bridge solvent concentration, was excluded from the design. This smaller factorial consisted of eight experiments that were run in duplicate. The design matrix is depicted in Table V. 

Finally, the problem was resolved by irradiating standards and mixtures of standards in a factorial experiment. The experiment design was a full factorial experiment with three variables, mercury, selenium, and ytterbium, at two levels with replication and with a center point added to test higher order effects. The pertinent information on treatments and levels of variables are shown in Table VII. 

Dairy wastes fall into two categories, one of which may be described as an intrinsic waste, and the other as a conditional waste. All dairy factories experience losses that are intrinsically a part of factory operation. For example, a dairy factory that receives 10,000 lb of milk daily may produce each working day about 1250 gal of waste with a milk solids concentration of 0.1 %. Cheese plants, on the other hand, produce whey as a by-product of cheesemaking although whey contains half the nutrients of the milk from which it was derived, it must be treated as a conditional waste—conditional upon the absence of a suitable market for its use. A more detailed discussion on disposal of dairy wastes can be found in a review by Arbuckle (1970). 

Initial Screening of Process Variables. A 2 X 5 factorial experiment was designed for the first test series, using a high and low extreme for each of five factors. Intermediate points of each factor were included to obtain a sense of direction. 

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