Half-fraction

Experimental Design 1. Due to the large scale of each run, an experimental design with a minimum number of runs was employed to obtain the desired information. The one half fractional factorial in Table II was set up to determine the effect of each factor. 

Halcon Process, t-butyl hydroperoxide, 428 Half-fractional factorial design, analytical methods, 624... 

An alternative design would be to use a half-fraction of the design variables for each run of the chamber. Such a design, before randomization, is shown in Table 2.22. With this design the ABCDE five-factor interaction is confounded with the TxH whole-plot contrast. Under the assumption of negligible three-factor and higher-order interactions all main effects and two-factor interactions can be estimated as well as interactions between the design and the environmental variables. 

Let us first consider a half-fraction factorial design. Only half of the number of experiments needed for a full factorial are performed. For... 

SELECTION OF A HALF-FRACTION FACTORIAL DESIGN FROM A FULL FACTORIAL DESIGN FOR 4 FACTORS... 

In terms of absolute size, main effects tend to be larger than two-factor interactions, which in turn tend to be larger than three-factor interactions, and so on. In the half-fraction factorial design of Table 3.9 the main effects are expected to be significantly larger than the three-factor interactions with which they are confounded. As a consequence it is supposed that the estimate for the main effect and the interaction together is an estimate for the main effect alone. 

THE COLUMNS OF CONTRAST COEFFICIENTS FOR THE THREE-FACTOR INTERACTIONS OF THE HALF-FRACTION FACTORIAL DESIGN FOR FOUR FACTORS SELECTED FROM TABLE 3.8... 

Let us now consider how one selects the experiments from the full factorial to obtain a proper half-fraction factorial design. In practice, to construct a... 

A reflected half-fraction factorial design for three factors (2 ) was performed. The influence of the factors on the responses recovery (%), resolution between peaks R) and Ry-value was calculated. Approximate critical values were obtained using the method given by Youden and Steiner (Ecritical MSE)e)- The standard error was estimated from... 

The most commonly used designs are half-fractional designs and saturated fractional designs. 

Half-fractional designs are constructed by assuming that all interaction effects higher than first order can be assumed to be negligible. For a study... 

Half-fractional designs reduce the number of experiments by half for a two level design and can prove very efficient, however the number of experiments can still be prohibitive when a large number of factors require... 

Note that as the 2 full factorial designs needs 8 runs, it is also possible to select the other half fraction of the experiments, with the complementary signs in column C. In this case the generator would be denoted C = - AB. Furthermore, the product ABC renders a column, denoted I, with all elements showing positive signs. So we call 1 = ABC the "defining relation for this design. 

If it is suspected that the probability of significant three- and higher-order interactions is negligible, it will suffice to make 16 (those marked with an asterisk) instead of 32 experiments which is a considerable gain in resources and time. This experimental design is called a half-fractional factorial design [29],... 

The design in Table 1 is called the principal fraction of the 24 1 design, and the sign in the generator D = ABC is positive (that is, D = +ABC). Another one-half fraction could have been constructed by using D = —ABC. This design would have all of the levels in column I) of Table 5 reversed. The two one-half fractions can be concatenated to form the complete 24 factorial design. 

Several methods of analysis have been suggested in the literature and are discussed in the context of data from half of an experiment reported by Williams (1968) and analyzed by several authors. Twenty-three factors were varied in 28 runs and one continuous response was observed. The half-fraction analyzed by Lin (1993) is shown in Table 6, which incorporates the corrections noted by Box and Draper (1987) and Abraham et al. (1999). 

After a complete analysis of the first round of experimental data, the project team ran the second half fraction of the 24 factorial, and data similar to those in Table 5.8 were obtained. (Again, no significance should be attached to the order in which the observations in Table 5.8 are listed. It is not the order in which the experimental runs were made.)... 

Fig. 6.10. Two sets of four experirnems consiiiuling half fractions of a full factorial design.

Screening designs give information about the main effects in a minimum of experiments. Economy has a price. Some problems can occur. It can happen that the interaction term of two (strongly) significant factors also still is significant. Since this interaction effect can be confounded with a main effect one should be careful to consider the main effect responsible for a calculated effect [44[. This is what would happen if we were to apply a half-fraction design to the data of Table 6.1. Factor C is then confounded... 

This is one half-fraction of a complete three-variable factorial design and it is seen that the experiments correspond to Exp no 5, 2, 3, 8 in the complete factorial design. Fig. 6.1 shows how the experiments of the half-fraction are distributed in the space spanned by the three variables. 

Full factorial designs Such designs are the best choice when the number of variables is four, or less. A full four-variable factorial design gives estimates of all main effects and two-variable interaction effects, and also an estimate of the experimental enor variance. This is obtained firom the residual sum of squares after a least squares fit of a second-order interaction model, see (Example Catalytic hydrogenation, p. 112). A full factoral design should be used if individual estimates of the interaction effects are desired. Otherwise, it is recommended first to run a half fraction 2 " (I = 1234), and then run the complementary fraction, if necessary, (see Example Synthesis of a semicarbazide, p. 135). 

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