Factor interval of variation

Having selected the system response, we start choosing factors, levels of the factors and center point of the design (basic level or the null point). By factor we understand the controllable independent variable that corresponds to one possibility of influence on the object of research. A factor is considered defined if its name and domain of factors are determined. A factor may take several values in this field. The chosen factor values, both qualitative and quantitative, are called factor variation levels. Factor variation levels in the design of experiments are coded values. Under factor interval of variation we understand the difference between two factor levels, which in their coded form have value one. When selecting the factors one should pay attention to the conditions they must meet. 

Table 5.26 Fundamental level and intervals of variation ofthe factors (example 5.5.4.1)...

Having performed the initial selection of components, the factors of choice can be ratios between components of the polymerization mixture (e.g., the ratio between two functional monomers, the ratio between monomer and template, the ratio between functional monomers and crosslinker, the ratio between monomers and porogen, etc.). Physical parameters, such as temperature and pressure of polymerization, can also be screened. Remember to define the interval of variation for each factor (maximum and minimum). 

Domain of factors is marked O . The figure clearly shows that intervals of factor variations are part of the domain of factors when the optimization problem is being solved. This is necessary in order to realize movement towards optimum in this domain. The experiment domain is in the same figure marked by letter E". In studies with an objective of approximation or interpolation, that is mathematical modeling, the factor-variation intervals cover the whole of the domain of factors. For a two-factor experiment the upper level of factors X and X2 corresponds to values Xlmax,-and X2max, while the lower levels have values Xlmin, X2min. Domain of factors O is in that case called intcrpolational, and E the domain of extreme experiment. 

A check of statistical significance must be done for the calculated regression coefficients and a check of lack of fit for the regression model. Both checks are a subject of statistical analysis that will be elaborated in more detail in the next chapter. The check of the obtained regression model has shown that it is inadequate, so that we have to reduce variation intervals of factors and increase the number of design-point replications. 

In the next stage, two variation levels for varying factors in the experiment are determined for each factor. One variation level is called the lower and the other one the upper level. The number that, when added to the basic level gives the upper and when subtracted the lower level, is called the variation interval of the associated factor. To simplify the way of recording conditions of doing an experiment and processing experimental results, axes ratios are such that the value +1 corresponds to the upper, -1 to the lower and zero to the basic level. 

The second variant, an increase in variation intervals of insignificant factors with additional design points, is acceptable if movement to optimum appears to be inefficient. A change in variation intervals will require at least eight expensive design points. 

The variation interval of the first factor is 4% of the domain, that of the second and third 20% and the fourth 17%. It is known from the block diagram that if a variation interval does not exceed 10% it is narrow, and if it is not larger than 30% it is average. Requirements of the block diagram have been followed in the actual case. 

To clarify this situation, it suffices to increase the intervals of analyzed factors. When effects grow after this realization, then assumptions a) and b) are correct. When assumption c) is correct, there will be no increase in effect. Factors X4 and X5 have in the first series of trials had small effects. It is therefore necessary to increase their variation intervals in the second series. 

No factor is significant for the efficiency of scrubber, which may be explained either that real factors have not been included in the research or that variation intervals of analyzed factors are too small. 

The fundamental level of the factors and their variation intervals have been established and are given in Table 5.26. We accept that the factors domains cover the great curvature of the response surface. Consequently, a regression relationship with interaction effects is a priori acknowledged. 

The fundamental levels of the factors and the variation intervals are shown in Table 5.32. 

The data for this plot were calculated by DFT/FPT for 30° intervals of couplings with geometrical factors.154 For propane, the following expression was proposed ... 

By definition, the experimental unit is the smallest unit randomly allocated to a distinct level of a treatment factor. Note that if there is no randomization, there is no experimental unit and (in nearly all cases) no experiment. Although it is possible to perform experiments without randomization, it is difficult to do well, and risky unless the experimental system is very well understood (7). Randomization is important for several reasons. Randomization changes the sources of bias into sources of variation in general, a noisy assay is better than a biased assay. Further, randomization allows estimates of variation to represent variation in the population this in turn justifies statistical inference (standard errors, confidence intervals, etc.). A common practice in cell-culture bioassay is to rotate among a small collection of layouts rather than use random allocation. Whereas rotation among a collection of layouts is certainly better than a fixed layout, it is both possible and practical to use carefully structured randomization on a routine basis, particularly when using a robot. 

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