Planning experiments factorial designs

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."... 

Because earlier experimental results and data analyses (3-10) had led us to anticipate the inadequacy of the simple approach considered above, we also planned and carried out (2) a second order factorial design of experiments and related data analysis. Mathematical analysis (of the results of 11 experiments) based on the second order model showed that all of these results could be represented satisfactorily by an equation of the form... 

A fractional factorial design is often suggested to observe several parameters at the same time. Advantages are, among others, the formal sampling plan, which is easy to evaluate by supervisors and auditors. Moreover, a fractional design only needs a fraction (usually about 50%) of experiments compared to a design that tests the relevant parameters one by one. 

The inlet gas composition was changed in accordance with an extended 2 factorial-designed experiment [10], For each temperature, the experiment was divided into two parts to avoid very long experiments with severe deactivation. Reference conditions were repeated three times during each run. The total experimental plan contained six runs, each with 10 experimental points. The order of the points and the length between the steps (143 - 176 min) were randomized. The experiments conducted at 780 K will be named A1 and A2, while those performed at 800 and 820 K will be called Bl, B2, Cl and C2. 

If each of the 3 factors is fixed at 2 levels, and we do experiments at all possible combinations of those levels, the result is a 2 full factorial design, of 8 experiments. The design, plan and results (turbidity measurements only) are listed in table 3.5a, in the standard order. 

The technical manager thinks that a small-scale experiment should be performed on the stability in storage of the additive in the climatic chamber. In this way, the factorial design plan shown in Table 23.12 arises. The foreman stopped the experiment off after 70 days, because the additive never was longer than 70 days on stock. The commercial director asks after 3 months whether an increase of the shelf-life was achieved, because he just got an unusual inexpensive offer of the contractor when buying still larger quantities of the additive. Try to evaluate the data from Table 23.12. What was wrong at the conception of this experiment ... 

A factorial design with three variables and two levels would lead to a test plan with eight experiments (Table 13-7). In a three-dimensional depiction, these eight tests occupy the comers of a cube (Fig. 13-10). 

The essence of good planning is to design an experiment so that it is able to provide exactly the type of information sought. To do this, one must define the aim of such trials and thus choose the most suitable technique. At first, in a situation of lack of knowledge about the system to be studied, one must make a screening of the variables that exist in this system the most advisable would be to conduct experiments to follow a full factorial design [21]. 

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. 

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. 

Initial screens can be distinguished between methods that are used to determine what factors are most important, and follow-up screens that allow optimization and improvement of crystal quality (Table 14.1). In experimental design, this is known as the Box-Wilson strategy (Box et al., 1978). The first group of screens is generally based on a so-called factorial plan which determines the polynomial coefficients of a function with k variables (factors) fitted to the response surface. It can be shown that the number of necessary experiments n increases with 2 if all interactions are taken into account. Instead of running an unrealistic, large number of initial experiments, the full factorial matrix can... 

This section is organized into two subsections. In the first, we will illustrate the notion of variance component estimation through an example of a nested or hierarchical data collection scheme. In the second, we will discuss some general considerations in the planning of experiments to detail the pattern of influence of factors on responses, consider so-called factorial and fractional factorial experimental designs, illustrate response surface fitting and... 

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