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Design of Experiments and Its Deployment in SAS and R
Published in Tanya Kolosova, Samuel Berestizhevsky, Supervised Machine Learning, 2020
Tanya Kolosova, Samuel Berestizhevsky
In statistics, a full-factorial experiment is defined as an experiment with a design consisting of two or more factors, each with two or more discrete levels, so that all possible combinations of the levels across all the factors are applied to the experimental units (Box et al., 1978). All the possible combinations of the levels across all the factors in a full-factorial design can be presented on a hypercube in the k-dimensional design space defined by the minimum and maximum levels of each of the k factors. The experimental points are sometimes called factorial points. The total number of experimental runs is calculated as a product of the numbers of levels of the factors. Let’s say we conduct a full-factorial experiment with three factors A, B, and C, where A, B and C are measured on three levels. Then, the experiment has 33 = 27 runs, which are shown as a hypercube in Figure 5.1.
Principles of Experimental Design
Published in William M. Mendenhall, Terry L. Sincich, Statistics for Engineering and the Sciences, 2016
William M. Mendenhall, Terry L. Sincich
One potential disadvantage of a complete factorial experiment is that it may require a large number of treatments. For example, an experiment involving 10 factors each at two levels would require 210 = 1,024 treatments! This might occur in an exploratory study where we are attempting to determine which of a large set of factors affect the response y. Several volume-increasing designs are available that employ only a fraction of the total number of treatments in a complete factorial experiment. For this reason, they are called fractional factorial experiments. Fractional factorials permit the estimation of the β parameters of lower-order terms (e.g., main effects and two-way interactions); however, β estimates of certain higher-order terms (e.g., three-way and four-way interactions) will be the same as some lower-order terms, thus confounding the results of the experiment. Consequently, a great deal of expertise is required to run and interpret fractional factorial experiments. Consult the references for details on fractional factorials and other more complex, volume-increasing designs.
Problem Solving and Corrective Action
Published in Fred W. Kear, Statistical Process Control in Manufacturing Practice, 2020
In some cases, there is more than one cause to the problem identified by the SPC program. When this is the case, design of experiments can be used to test more than one variable in the process. Depending on the number of variables tested and the number of levels for each variable, the experiment may be simple or extremely complex. Analysis of variance (ANOVA) is used to quantify the data resulting from multi-factorial experiment in terms of which variables have the greatest effect on the problem and how they may need to be adjusted to make improvements.
Application of statistical analysis for optimizing of column flotation with pine oil for oil shale cleaning
Published in International Journal of Coal Preparation and Utilization, 2022
Ahmed Sobhy, Ahmed Yehia, F.I. El Hosiny, S.S. Ibrahim, Rasha Amin
Screening designs are utilized when many factors would affect a specific operation(Collins, Dziak, and Li 2009), and a complete factorial design is not always a choice since levels combinations of the parameters might create unsuitable experimental conditions. Implementation of a complete factorial experiment might be impossible due to resource restrictions even when all combinations of variables are reasonable. Thus, designs that need fewer experiments are required to minimize implementation efforts and unreasonable expenses. Furthermore, reduced designs are often required to make a simultaneous investigation of many independent factors viable. Besides, removing some experimental conditions in the reduced design combines some impacts, so that their interactions only, not the individual effects, can be evaluated (Collins, Dziak, and Li 2009). Therefore, optimization of oil shale column flotation with pine oil as a collector was studied with a reduced three-factor interaction (3FI) experimental design conducted in a previous work by the authors (Sobhy et al. 2019). The design was used to investigate the effects of seven independent factors shown in Table 1. The parameters’ lower and upper levels were identified based on the exploratory experiments. In the meantime, analysis of variance (ANOVA) has been applied to investigate the experimental data to optimize the process parameter and their interactions. Afterward, two confirmation tests were performed to validate the results from the reduced three-factor interactions experimental design.
Recovery of unburned carbonaceous matter (UCM) from sugar mill bottom ash
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2020
Diler Katircioglu-Bayel, Hüseyin Serencam, Metin Uçurum
The statistical design of experiments has several advantages over the classical method of treating one variable at a time. The full factorial experiment is the method of design of experiments in which a statistical analysis is performed to evaluate the significance of the main and interaction effects as evaluated from the experimental results. In particular, they are used when several factors have to be studied in order to determine their main effects and interaction. The experiments can be conducted in an organized manner and can be analyzed systematically to obtain much needed information. The information can be utilized for optimization purpose. A valid optimization strategy would permit the adjustment of those manipulable variables, which influence the objective (Naik, Reddy, and Misra 2005). The statistical design of experiments was used when the effect of several factors were to be studied in order to determine the main and interaction effects. The effect of a variable is the change in response produced by varying the level of the factor. When the effect of a factor depends on the level of another factor, the two factors are said to interact (Naik, Reddy, and Misra 2004). The success of flotation depends on the selection of suitable parameters. The optimization of these parameters necessitates many tests. The total number of experiments required can be reduced by employing a factorial designed series using the Yates technique (Yates 1976).
Artificial neural networks approach for prediction of CIELab values for yarn after dyeing and finishing process
Published in The Journal of The Textile Institute, 2023
Cenk Şahin, Onur Balcı, Melek Işık, İlker Gökenç
To increase dyed yarn quality, it is crucial to address all factors affecting color, shown in Table 1, and determine their levels. One of the most challenging aspects of this study is the Color-Factor Number-Levels equation. This situation becomes even more complicated when these factors give different results individually or when combined. A representative sample of the entire area must be selected according to factors (parameters) and their levels. A full factorial experiment, which becomes more costly as the number of factors and levels increase, was designed to demonstrate all factors’ interactions and the analysis of variance (ANOVA) method was used for the analysis of the experimental data (Montgomery, 2017).