Interaction between factors

When an analytical method is being developed, the ultimate requirement is to be able to determine the analyte(s) of interest with adequate accuracy and precision at appropriate levels. There are many examples in the literature of methodology that allows this to be achieved being developed without the need to use complex experimental design simply by varying individual factors that are thought to affect the experimental outcome until the best performance has been obtained. This simple approach assumes that the optimum value of any factor remains the same however other factors are varied, i.e. there is no interaction between factors, but the analyst must be aware that this fundamental assumption is not always valid. 

As an analytical method becomes more complex, the number of factors is likely to increase and the likelihood is that the simple approach to experimental design described above will not be successful. In particular, the possibility of interaction between factors that will have an effect on the experimental outcome must be considered and factorial design [2] allows such interactions to be probed. 

Factorial design One method of experimental design that allows interactions between factors to be investigated, i.e. whether changing one experimental variable changes the optimum value of another. 

The ET cover cannot be tested at every landfill site so it is necessary to extrapolate the results from sites of known performance to specific landfill sites. The factors that affect the hydrologic design of ET covers encompass several scientific disciplines and there are numerous interactions between factors. As a consequence, a comprehensive computer model is needed to evaluate the ET cover for a site.48 The model should effectively incorporate soil, plant, and climate variables, and include their interactions and the resultant effect on hydrology and water balance. An important function of the model is to simulate the variability of performance in response to climate variability and to evaluate cover response to extreme events. Because the expected life of the cover is decades, possibly centuries, the model should be capable of estimating long-term performance. In addition to a complete water balance, the model should be capable of estimating long-term plant biomass production, need for fertilizer, wind and water erosion, and possible loss of primary plant nutrients from the ecosystem. 

When analyzing results of factorial experiments we talk about main effects and interaction effects. Main effects are factor effects and they are the difference of averaged response for two levels (+1 -1) for the associated factor. In case response difference for two levels of factor Xj is the same irrespetive of on which level factor X2 (excluding experimental error), one may say that there exists no interaction between factors X and X2 or that the interaction is XjX2=0. This statement may be graphically presented. Figures 2.34 and 2.35 show interaction between factors X2 and X2, and Fig. 2.36 indicates that such an interaction is nonexistent. 

The mathematical way of such analysis is the following Single factor loadings atj are the same as correlation coefficients between the original feature i and the factor j. It is, therefore, possible to define the interaction between factors from the factor solution... 

For a two-level factorial design, only two excipients can be selected for each factor. However, for the filler-binder, a combination of brittle and plastic materials is preferred for optimum compaction properties. Therefore, different combinations such as lactose with MCC or mannitol with starch can count as a single factor. Experimental responses can be powder blend flowability, compactibility, blend uniformity, uniformity of dose unit, dissolution, disintegration, and stability under stressed storage conditions. The major advantage of using a DOE to screen prototype formulations is that it allows evaluation of all potential factors simultaneously, systematically, and efficiently. It helps the scientist understand the effect of each formulation factor on each response, as well as potential interaction between factors. It also helps the scientist identify the critical factors based on statistical analysis. DOE results can define a prototype formulation that will meet the predefined requirements for product performance stability and manufacturing. 

The third common constraint is based on a priori knowledge of the three-way profiles. In this case, the known relative concentrations of the standards, or the known spectral profiles of one or more components, can be fixed as part of the solution. In the Tucker3 model, it is common to restrict some of the potential interactions between factors when they are known not to exist. Constraint values, again, lend themselves to careful selection, as the scaling of the factors must still be taken into account. 

As mentioned before, people tend intuitively to turn to the one-variable-at-a-time technique for its conceptual simplicity, and ignore the possible interaction between independent variables. A good example of the interaction between factors is that between enzyme concentration (E) and reaction temperature (T). Assuming E and T are the chosen factors for optimization, one possible interaction will be that T tends to influence the way E affects the conversion yield and vice versa. Since reaction temperature increased, enzyme activity was suppressed than at low temperature and the rate of enzyme-catalysis is affected by temperature this will inevitably affect conversion yield of the product. Should the interaction be minor or negligible, a one-factor-at-a-time search will give a satisfactory result. 

Another advantage of the factorial design is that interaction between factors can be estimated. To determine the interaction between A and B, it is necessary to average the effect of C. That is, in Fig. 17-3, the top and bottom planes are averaged. This results in... 

Third, there is the one-factor-at-a-time method in which the experimenter varies first one factor to find the best value, then another. Its disadvantages are that it cannot be used for multiple responses and that it will not work when there are strong interactions between factors. 

It is assumed that there are no interactions between factors that is to say, the effect of a given excipient on stability does not depend on what other excipients are found in the formulation. (The same reasoning applies to other kinds of factors or responses.) This can only be an approximation however, if it should be necessary to take interactions into account, many more experiments would be needed, and it would probably be necessary to limit the number of levels for each factor to two for the number of experiments to be manageable. 

Quite wide limits are generally chosen for screening quantitative factors. They are then often narrowed for more detailed quantitative study of the influence of factors where interactions between factors them are taken into account and for determining a predictive model for optimization. 

Whereas the purpose of a screening study is to determine which of a large number of factors have an influence on the formulation or process, that of a factor study is to determine quantitatively the influence of the different factors together on the response variables. The number of levels is usually again limited to two, but sufficient experiments are carried out to allow for interactions between factors. 

Evidently, for the 2 design, the 16 triple and higher interactions are not determined. In fact, they are confounded with the calculated effects. Thus, the estimate of the interaction between factors one and two includes the triple interaction between the other three factors. Because the latter is assumed negligible, this does not usually matter. 

The design must enable estimation of the first-order effects, preferably free from interference by the interactions between factors other variables. It should also allow testing for the fit of the model and, in particular, for the existence of curvature of the response surface (center points). Two-level factorial designs may be used for this (shown earlier). 

Zinc. The Zn data were analysed in a similar way to the B data (Table 4.5.7). Again there was a significant effect of transect. Only one (that between transect and time) of the possible interactions between factors had a significant effect on the concentration of Zn, and this is shown in the table. The concentration of nitrate measured with the YSI probe has a non-negligible association with the concentration of Zn. This association was not observed with B. 

If there are fewer factors than columns then some key interaction conditions can also be estimated. Simply do not assign any treatment to the interaction(s) of interest. For example, if in Table 4 there were only six factors in the design, then one of the columns could be used to focus on an interaction term of particular interest. Suppose an interaction between factors 1 and 2 was suspected, then one could leave the column associated with factor 4 unassigned, as it is confounded with the interaction between factors 1 and 2. Instead, factors 4 to 6 would be assigned to columns 5, 6, and... 

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