32 factorial design

Factorial design seeks to determine a black-box model of the system that can accurately describe the behaviour of the system in the region studied that also includes interactions or combinations of the different variables (e.g. X1X2). Factorial designs have the advantage that changes in the variables are not made sequentially, but following some type of pattern, so that interactions between the different variables can also be measured. 

The basic factorial design consists of k factors or independent variables and / different levels or points at which the system will be tested. A factorial experiment with I levels and k factors is called an factorial experiment. For the purposes of this discussion, it will be assumed that all factors have the same number of levels. The complete experimental design will be repeated % times, which is referred to as the number of replicates. A treatment refers to a single run of the factorial design with given values for each factor. 

---

Factorial design methods cannot always be applied to QSAR-type studies. For example, i may not be practically possible to make any compounds at all with certain combination of factor values (in contrast to the situation where the factojs are physical properties sucl as temperature or pH, which can be easily varied). Under these circumstances, one woul( like to know which compounds from those that are available should be chosen to give well-balanced set with a wide spread of values in the variable space. D-optimal design i one technique that can be used for such a selection. This technique chooses subsets o... 

The linear regression calculations for a 2 factorial design are straightforward and can be done without the aid of a sophisticated statistical software package. To simplify the computations, factor levels are coded as +1 for the high level, and -1 for the low level. The relationship between a factor s coded level, Xf, and its actual value, Xf, is given as... 

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... 

Example of Uncoded and Coded Factor Levels and Responses for a 2 Factorial Design... 

The computation just outlined is easily extended to any number of factors. For a system with three factors, for example, a 2 factorial design can be used to determine the parameters for the empirical model described by the following equation... 

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

Curved one-factor response surface showing (a) the limitation of a 2 factorial design for modeling second-order effects and (b) the application of a 3 factorial design for modeling second-order effects. 

If the actual response is that represented by the dashed curve, then the empirical model is in error. To fit an empirical model that includes curvature, a minimum of three levels must be included for each factor. The 3 factorial design shown in Figure 14.13b, for example, can be fit to an empirical model that includes second-order effects for the factor. 

In general, an -level factorial design can include single-factor and interaction terms up to the ( - l)th order. 

Four replicate measurements were made at the center of the factorial design, giving responses of 0.334, 0.336, 0.346, and 0.323. Determine if a first-order empirical model is appropriate for this system. Use a 90% confidence interval when accounting for the effect of random error. 

We begin by determining the confidence interval for the response at the center of the factorial design. The mean response is 0.335, with a standard deviation of 0.0094. The 90% confidence interval, therefore, is... 

Because exceeds the confidence interval s upper limit of 0.346, there is reason to believe that a 2 factorial design and a first-order empirical model are inappropriate for this system. A complete empirical model for this system is presented in problem 10 in the end-of-chapter problem set. 

The following set of experiments provides practical examples of the optimization of experimental conditions. Examples include simplex optimization, factorial designs used to develop empirical models of response surfaces, and the fitting of experimental data to theoretical models of the response surface. 

This experiment describes the use of a fractional factorial design to examine the effects of volume of HNO3, molarity of AgN03, volume of AgN03, digestion temperature, and composition of wash water on the gravimetric analysis for chloride. 

This experiment describes a fractional factorial design used to examine the effects of flame height, flame stoichiometry, acetic acid, lamp current, wavelength, and slit width on the flame atomic absorbance obtained using a solution of 2.00-ppm Ag+. 

This experiment examines the effect of reaction time, temperature, and mole ratio of reactants on the synthetic yield of acetylferrocene by a Eriedel-Crafts acylation of ferrocene. A central composite experimental design is used to find the optimum conditions, but the experiment could be modified to use a factorial design. 

A 2 factorial design was used to determine the equation for the response surface in problem lb. The uncoded levels, coded levels, and the responses are shown in the following table. 

Koscielniak and Parczewski investigated the influence of A1 on the determination of Ca by atomic absorption spectrophotometry using the 2 factorial design shown in the following table. ... 

Strange reports the following information for a 2 factorial design used to investigate the yield of a chemical process. ... 

The following data for a 2 factorial design were collected during a study of the effect of temperature, pressure, and residence time on the %yield of a reaction. " ... 

Duarte and colleagues used a factorial design to optimize a flow injection analysis method for determining penicillin potentiometricallyd Three factors were studied—reactor length, carrier flow rate, and sample volume, with the high and low values summarized in the following table. 

Here is a challenge McMinn and co-workers investigated the effect of five factors for optimizing an H2-atmosphere flame ionization detector using a 2 factorial design. The factors and their levels were... 

An additional advantage of biU factorial and fractional factorial designs is that by providing a comprehensive scanning of the experimental region they can often identify, without any further analyses, one or two test conditions that are better than any others. The region around these conditions can then be explored further in subsequent experimentation. 

C. Daniel, App/ications of Statistics to lndustria/Experimentation, ]oE Wiley Sons, Inc., New York, 1976. This book is based on the personal experiences and insights of the author, an eminent practitioner of industrial appHcations of experimental design. It provides extensive discussions and concepts, especially in the areas of factorial and fractional factorial designs. "The book should be of use to experimenters who have some knowledge of elementary statistics and to statisticians who want simple explanations, detailed examples, and a documentation of the variety of outcomes that may be encountered." Some of the unusual features are chapters on "Sequences of fractional repHcates" and "Trend-robust plans," and sections entided, "What is the answer (what is the question )," and "Conclusions and apologies."... 

---