Complete factorial design

The experimental design shown in Table 3.1 is actually a two-level factorial design completed with one experiment at the center of the experimental domain. Properties of factorial designs are discussed in Chapter 5. 

No of factors Fractional factorial designs Complete factorial designs Other designs of resolution V (Ry) ... 

A second factorial matrix (also with centre points), carried out to complete the factorial design (complete or fractional of resolution Ry) and determine the interactions. 

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. 

If we have three independent variables to vary, the complete 2 factorial design corresponds to the following 8 experiments ... 

For the complete vapor-phase oxidation of methane over a palladium alumina catalyst, conversion-space-time data were taken at 350°C and 1 atm total pressure the fractional factorial design of Table X (Hll) specified the settings of the feed partial pressures of the reacting species. 

Note that even if the environmental variables are at two levels so that a 2 fractional factorial design can be run for the Taguchi environmental array, the complete crossed design has 9x8 = 72 runs, still more than the Box-Behnken design of Table 2.12, while providing estimates of fewer coefficients of the second-order model. 

According to the Hadamard matrix, a 22 factorial design was built. The complete linear models were fitted by regression for each response, reflecting the compression behaviour and dissolution kinetics. 

A 22 factorial design was built to obtain a complete linear model including only two parameters. 

Fig. 3—Release profiles from the factorial design. Table 9—Complete linear models...

The analysis of variance lends itself best to balanced factorial designs, whether complete, partially replicated, or otherwise modified. The concept of balance simplifies the calculations tremendously. There are ways of coping with missing data, unequal replication under various conditions, and even some lack of orthogonality in the design, but these methods seem to involve more calculation than the data may deserve. The analysis of variance is a procedure which makes it possible to compare the effects of the variables being studied, first independently of the effects of all other variables, and second in all possible combinations with one another. Sometimes the effect of a variable within a given level of another variable... 

Factorial design - si factors at two levels. The boxes marked X represent 1/4 replicate of the complete 2 64 factorial design ... 

For systems involving many variables and levels it is apparent that a complete factorial design leads to an excessive number of cases. However, it may be quite practical to make reasonable assumptions about the behavior of the model and, in so doing, to reduce the number of cases... 

When some of the possible data points are omitted in factorial designs they are known as fractionally or partially replicated designs. The choice of points to be omitted is of considerable importance. There is no single fractional replicate which is best for any given complete factorial design. Usually the experimenter will have some idea as to the expected effects. When this sort of intuition is available, a particular design may be developed to fit a particular problem. 

If a complete factorial design has three levels (low, middle and high) for each of three factors (A = temperature B = pH and C = reaction time), it is said to be a 3 X 3 X 3 or 3 CFD and 27 runs are required, each one corresponding to a particular combination of the factor levels. Furthermore, if we replicate each run k times, then the number of experiments needed is k x 3. ... 

This model allows us to estimate a response inside the experimental domain defined by the levels of the factors and so we can search for a maximum, a minimum or a zone of interest of the response. There are two main disadvantages of the complete factorial designs. First, when many factors were defined or when each factor has many levels, a large number of experiments is required. Remember the expression number of experiments = replicates x Oevels) " (e.g. with 2 replicates, 3 levels for each factor and 3 factors we would need 2 x 3 = 54 experiments). The second disadvantage is the need to use ANOVA and the least-squares method to analyse the responses, two techniques involving no simple calculi. Of course, this is not a problem if proper statistical software is available, but it may be cumbersome otherwise. 

From preliminary assays, the experimental error was estimated as 2.50%, expressed as percentage recovery. Note that the complete factorial design is a 2 , requiring 128 runs, whereas the Plackett-Burman design needs only 8 runs to estimate the effects. The responses to the 8 runs corresponding to the design matrix in Table 2.6 were as follows ... 

G. A. Zachariadis and J. A. Stratis, Optimisation of cold vapour atomic absorption spectrometric determination of mercury with and without amalgamation by subsequent use of complete and fractional factorial designs with univariate and modified simplex methods, J. Anal. At. Speetrom., 6(3), 1991, 239-245. 

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. 

Four experimental variables were selected sample pH, primary column type, secondary column type, and methanol concentration. By using each of the four variables at two levels, the complete arrangement of experimental runs became a2X2X2X2or24 factorial design requiring 16 runs. Table I represents the design matrix the high and... 

Complete factorial designs 2k require a minimum of 2k experiments, which... 

Second Experimental Matrix. For the technical reasons mentioned above, a complete factorial design 23 was chosen for investigating the effects of the three factors DOC0, Ti02 concentration, and temperature (Table 3). An additional experiment at the center of the experimental region (A = XA = X5 = 0 i.e., DOC0 = 2700 ppm, [Ti02] = 2.75 g L 1, tempera-... 

TABLE 3 Experimental Matrix for Investigating the Influence of Three Factors on the Ti02 Photocatalyzed Oxidative Degradation of Waste Water Pollutants in a Pilot Reactor Complete Factorial Design 23, Control Experiments 1 (Natural and Coded Variables are Indicated), and Values of the Experimental Response Y... 

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