Fractional factorial experiment

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. 

Full Factorial Experiments and Fractional Factorial Experiments... 

To obtain the mathematical model of the process, 1/4-replica of a full factorial experiment of type 2s has been realized. Design points-trials have been done in a completely random order. The Table 2.129 shows conditions and outcomes of doing a 26 2 fractional factorial experiment. 

To control the complex process of cooking by the sulfate process cellulose, by means of a computer, it is necessary to have a mathematical model of the process. To obtain this model for the process of cooking by the sulfate process cellulose from a mixture of soft and hardwood and deciduous trees, we have used a fractional factorial experiment. It included these seven factors xi consumption of active lye, % Na20 on completely dry wood X2 cooking temperature, °C ... 

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 ... 

The Method of steepest ascent, Table 2.190, was applied to outcomes of a fractional factorial experiment 2s"4, Table 2.95. The experiment included eight factors xrquan-tity of binder, % x2-quantity of linen fabrics, g/m2 x3-pressure of pressing, kp/cm2 X4-temperature, °C xs-time of thermic processing, min x6-time of pressing, min xytype of binder, [1] and xg-quantity of dibutylphtalate, %. The system response was the relative elongation at strain yu %. 

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-... 

Interpretation of a fractional factorial experiment always requires careful study of the results, engineering or scientific knowledge about the process being studied, and sometimes the judicious use of Occam s razor.1 Confirmation experiments... 

In Section 5, we introduced the dyestuffs experiment to illustrate the methods for screening for dispersion effects in unreplicated fractional factorial experiments. Typically we anticipate that smaller experiments will be used for screening. So, in this section, we analyze two sets of 16 runs that are extracted from the dyestuffs experiment and which constitute fractional factorials more typical of the actual size of screening experiments. 

Step 8 If necessary, add more runs to the design matrix to eliminate aliases or confounding patterns. For example, if a fractional factorial experiment shows evidence of interactions between variables, it may be necessary to run the full factorial to determine which interactions are truly important. 

Thus, for a control medium containing, for example, 2% glucose and 1% soya flour, example a. above would test 1, 2, and 4% glucose each at 0.5, 1, and 2% soya flour, i.e, a total of nine treatments. Determination of titer in each test medium allows response surfaces to be plotted from which the effects of high and low levels of each factor and their interactions can be readily identified As with the fractional factorial experiments described above, these experiments are not designed for absolute optimization but will rapidly identify trends leading to increased titer... 

This happens in two kinds of circumstances. The first may be involuntary, when the experiments are carried out sequentially, in two or several stages. This may be for reasons of economy, when a preliminary fractional factorial experiment has been carried out and a complementary design added some time later, to clarify the ambiguities in estimations of the main and interaction effects. The extrusion-spheronization experiment where a 2 design was followed by a second 2 foldover design is an example of this. 

Determining the resolution requires looking at the complete confotmding pattern for the given fractional factorial experiment and determining the term with the smallest number of variables multiplied together. 

Modular arithmetic denoted as x mod y, where x is the divisor and y is the dividend (or base), seeks to determine the remainder when x is divided by y, for example, 7 mod 2 will be equal to 1, since the remainder when 7 is divided by 2 is 1 (7 = 3x2+ ). When seeking to determine the confounding pattern in fractional factorial experiments and higher-order terms are encountered, then reduction of these terms is performed using Z-base modular arithmetic, where /, as before, is the number of levels in the design. 

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