Experimental factorial experiments

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. 

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. 

Adhesion on HLORIN -type fibers, has been studied as a function of five process factors. The names of factors, with their variation levels, are shown in Table 2.32. Matrix 23 of full factorial experiment has been used in constructing random balance matrix. The design matrix by the method of random balance with experimental results is shown in Table 2.33. Note that each design point was repeated 20 to 50 times due to high non reproducibility of the system. 

Although this direct method is more adequate for the given example, because the number of the values that are not available are smaller than the sum of rows and columns, the constant method has also been demonstrated for the case of comparison. It should be noted that both methods are generally used in two-way classification such as designs of completely randomized blocks, Latin squares, factorial experiments, etc. Once the values that are not available are estimated, the averages of individual blocks and factor levels are calculated and calculations by analysis of variance done. The degree of freedom is thereby counted only with respect to the number of experimental values. Results of analysis of variance for this example are... 

When analyzing results of factorial experiments we talk about main effects and interaction effects. Main effects are factor effects and they are the difference of averaged response for two levels (+1 -1) for the associated factor. In case response difference for two levels of factor Xj is the same irrespetive of on which level factor X2 (excluding experimental error), one may say that there exists no interaction between factors X and X2 or that the interaction is XjX2=0. This statement may be graphically presented. Figures 2.34 and 2.35 show interaction between factors X2 and X2, and Fig. 2.36 indicates that such an interaction is nonexistent. 

Based on previous testing of the research subject, the design of the full factorial experiment 23 with one replication to determine experimental error has been chosen. To eliminate the influence of systematic error in doing the experiment, the sequence of doing design point-trials, in accord with theory of design of experiments, has been completely random. The outcomes are given in Table 2.107. 

An increase in the number of replicated trials causes a decrease in reproducibility variance or experimental error as well as in the associated variances of regression coefficients. Design points-trials can be replicated in all points of the experiment or in some of them. An upgrade of the design of experiment may be realized by a shift from fractional to full factorial experiment, a switch to bigger replica (from 1/6 to 1 /2 replica), a switch to second-order design (when the optimum region is dose by), etc. 

The property of the method of steepest ascent lies in the fact that movement along the gradient of a function must be preceded by a local description of the response surface by means of full or fractional factorial experiments [49]. It has been demonstrated that by processing FUFE or FRFE experimental outcomes we may obtain a mathematical model of a research subject in the form of a linear regression ... 

In the case of constraints on proportions of components the approach is known, simplex-centroid designs are constructed with coded or pseudocomponents [23]. Coded factors in this case are linear functions of real component proportions, and data analysis is not much more complicated in that case. If upper and lower constraints (bounds) are placed on some of the X resulting in a factor space whose shape is different from the simplex, then the formulas for estimating the model coefficients are not easily expressible. In the simplex-centroid x 23 full factorial design or simplex-lattice x 2n design [5], the number of points increases rapidly with increasing numbers of mixture components and/or process factors. In such situations, instead of full factorial we use fractional factorial experiments. The number of experimental trials required for studying the combined effects of the mixture com-... 

Ruggedness is measured by imposing small variations on the experimental parameters and recording the results. Let us suppose we have developed an. AAS method and we want to test its robustness. The following seven parameters have been identified as possible sources of variation amount of water, reaction time, distillation rate, distillation time, n-Heptane, Aniline, and status of the reagent. For short, we will call these 7 variables or factors as AoW, Rt, Dr, Dt, n-H, Anl, SoR and perform a partial factorial experiment consisting of 8 experiments. The partial factorial experiment is described in Table 2.2. 

If the experimental runs are completely randomized, then randomization theory (see Hinkelmann and Kempthorne, 1994) tells us that least squares gives us unbiased estimators of any pre-chosen set of n — 1 linearly independent contrasts among the n combinations of factor levels (treatments). In most factorial experiments the pre-chosen treatment contrasts would be main effects and, perhaps, interactions. However, in supersaturated designs there is no rational basis for choosing a set of n — 1 contrasts before the analysis. Any model selection method will lead to selection biases, perhaps large biases, in the estimators of effects. If a2 is assumed known, then we can test the null hypothesis that all n treatment populations have equal means. This would not be of great interest, because even if this null hypothesis were true it would not imply that all main effects are zero, only that a particular set of n - 1 linear combinations of treatment means are zero. Of course, in practice, a2 is not known. 

Sukigara et al. [78] designed a factorial experiment by using two factors (electric field and concentration). For a quadratic model, experimentsmust be performed for at least three levels of each factor. These levels are best chosen equally spaced. The two factors (silk concentration and electric field) and three levels resulted in nine possible combinations of factor settings. A schematic of the experimental design is shown in Figure 28(A) and (B). 

The experimental research of a process with k factors and one response can be carried out considering all the combinations of the k factors with each factor at both levels. Thus, before starting the experimental research, we have a plan of the experiments which, for the mentioned conditions, is recognized as a complete factorial experiment (CFE) or plan. The levels of each of the various factors establish the frontiers of the process-investigated domain. 

The example shown above, introduces the necessity for a statistical investigation of the response surface near its great curvature domain. We can establish the proximity of the great curvature domain of the response surface by means of more complementary experiments in the centre of the experimental plan (xj = 0,X2 = 0,...Xij = 0). In these conditions, we can compute y, which, together with Pq (computed by the expression recommended for a factorial experiment... 

In this Chapter a number of miscellaneous aspects of the use of the analysis of variance will be discussed. The next five sections contain matter which might be borne in mind in planning and analysing the results of factorial experiments, Section (g) is an account of a useful experimental device (the Latin Square), and the subsequent sections describe various applications of the analysis of variance. 

Factorial experiments consists of a systematic variation of two or more process variables at a time. For a two-level experiment, each variable is set to a high or a low value according to a standard pattern. An experimental run is conducted for each possible combination of variable settings. Selection of the low and high levels for each variable is important for obtaining meaningful results. If the levels selected are too close... 

The factorial experiments above can be illustrated geometrically as shown in Fig. 5.1. The experimental domain is a cube spanned by the factor axes. The experiments are located at the "comers" of the cube. 

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