Higher Factorial Experiments

It will be noted that the four factor experiment is even more efficient relative to the classical design than the three factor experiment, which achieved double the accuracy for the main effects. 

The four factor experiment requires 2 = 16 runs, and its main ects are given as the comparison of the average of 8 runs with the average of 8. The nearest equivalent classical experiment is to carry out the five combinations 

Finally, it will have been noted that the larger factorial experiments require a fair number of experiments, e.g. a2 or2x2x2x2 experiment requires 2 = 16 runs. This number may be larger than can be comfortably put into a block. Methods have been worked out which evade these difficulties and enable us to put these 16 runs into 2 blocks of 8 runs or into 4 blocks of 4 runs. These are discussed in Chapter XIV. 

Statistics, like every other science, lias its own words and symbols of special meaning. These may be confusing to the newcomer, but are almost essential for the normal use of its methods. 

An important concept is that of population, broadly speaking a large number of individuals from a particular source. We frequently wish to estimate the mean value of a property of a whole set of individuals from a small number of them this part withdrawn from the whole we refer to as a sample.  

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Fowlie and Bulman [43] have carried out a detailed study of the extraction of anthracene and benzo[tf]pyrene from soil. They carried out a replicated [24] factorial experiment using Soxhlet extraction and Polytron techniques. Soxhlet extraction followed by thin layer chromatography gave higher recoveries of the two polyaromatic hydrocarbons. 

Finally, the problem was resolved by irradiating standards and mixtures of standards in a factorial experiment. The experiment design was a full factorial experiment with three variables, mercury, selenium, and ytterbium, at two levels with replication and with a center point added to test higher order effects. The pertinent information on treatments and levels of variables are shown in Table VII. 

One problem with a complete factorial experiment such as this is that the number of experiments required rises rapidly with the number of factors for k factors at two levels with two replicates for each combination of levels, 2 experiments are necessary, e.g. for five factors, 64 experiments. When there are more than three factors some economy is possible by assuming that three-way and higher-order interactions are negligible. The sums of squares corresponding to these interactions can then be combined to give an estimate of the residual sum of squares, and replicate... 

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. 

After some experimentation, it can be determined that the basic factors are A, B, C, and D, while the dependent factor is E. The generator for this experiment can be written as E = ABCD. The complete defining relationship is then 1 = ABCDE. Based on the above analysis, it can be concluded that this is a A-fractional factorial experiment with a resolution of V, that is, 2y . It should be noted here that not all of the parameters can be estimated since they will be confounded with others. Without going into the details here, all of the zero-, first-, and second-order interactions are estimable. They will be confounded with various higher-order interactions. 

For fractional factorial experiment, it is useful if higher-order interactions are known to be unimportant. 

Some problem situations, such as aberrant values (or outliers) and nonlinear response functions, lead to artifacts that make the interpretation of the results of factorial experiments not always straightforward. An aberrant value leads to many artificially high interaction effects. When this happens, and certainly when it happens with third- and higher-order interactions, one should suspect that an aberrant value is present. 

Using a "home made" aneroid calorimeter, we have measured rates of production of heat and thence rates of oxidation of Athabasca bitumen under nearly isothermal conditions in the temperature range 155-320°C. Results of these kinetic measurements, supported by chemical analyses, mass balances, and fuel-energy relationships, indicate that there are two principal classes of oxidation reactions in the specified temperature region. At temperatures much lc er than 285°C, the principal reactions of oxygen with Athabasca bitumen lead to deposition of "fuel" or coke. At temperatures much higher than 285°C, the principal oxidation reactions lead to formation of carbon oxides and water. We have fitted an overall mathematical model (related to the factorial design of the experiments) to the kinetic results, and have also developed a "two reaction chemical model". 

The key step is to determine the errors associated with the effect of each variable and each interaction so that the significance can be determined. Thus, standard errors need to be assigned. This can be done by repeating the experiments, but it can also be done by using higher-order interactions (such as 123 interactions in a 24 factorial design). These are assumed negligible in their effect on the mean but can be used to estimate the standard error. Then, calculated... 

Full factorial designs can be fractionated by the exclusion of experiments designed to identify higher order effects and such reduced designs are known as fractional factorial designs. 

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 last twenty years of the last millennium are characterized by complex automatization of industrial plants. Complex automatization of industrial plants means a switch to factories, automatons, robots and self adaptive optimization systems. The mentioned processes can be intensified by introducing mathematical methods into all physical and chemical processes. By being acquainted with the mathematical model of a process it is possible to control it, maintain it at an optimal level, provide maximal yield of the product, and obtain the product at a minimal cost. Statistical methods in mathematical modeling of a process should not be opposed to traditional theoretical methods of complete theoretical studies of a phenomenon. The higher the theoretical level of knowledge the more efficient is the application of statistical methods like design of experiment (DOE). 

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