Variable interaction

Once the results have been collected they may be analysed to show the effects of each variable separately, T, S, M, the interaction of two variables, T-S, T-M, S-M, and tlrree variable interaction, T-S-M. These interaction terms measure the effect of the change in one variable on die results for the others, and T-S-M, tire ternary interaction indicates the dependence of the results for one variable of tire simultaneous change of the others. 

As an example, an experimental program interested in investigating the effect of three parameters would only need to complete eight experiments to estimate the eight coefficients in Eqn (2). However, to complete the same analysis for seven factors, 128 experiments are necessary. The number of model parameters for seven variables can be broken down according to the different variable interactions, as done in Table 11.1. 

It is also apparent that the composite pad properties of interest are affected by a large number of process and structural variables. Interaction effects make analysis of property variability particularly difficult. As a consequence,... 

This analysis allows the user to account for variable interaction that is another level of sophistication. Two terms need clarification—... 

Generally, formula variables interact with each other. 

This crossover theory has been repeatedly tested with regard to MC simulations of the 3D lattice gas with variable interaction range. For example, a recently developed MC algorithm [318] allows the ratio of t/Na to be varied over eight orders of magnitude to cover the full crossover region [319]. The crossover theory gives an excellent representation of these data [320]. 

The effects of the culturing variables, interaction coefficients (95%), correlation matrix for estimated parameters, respective confidence inter-... 

Response Surface Methodology (RSM) was used to investigate the effects of temperature, pH and relative concentration on the quantity of selected volatiles produced from rhamnose and proline. These quantities were expressed as descriptive mathematical models, computed via regression analysis, in the form of the reaction condition variables. The prevalence and importance of variable interaction terms to the computed models was assessed. Interaction terms were not important for models of compounds such as 2,5-dimethyl-4-hydroxy-3(2H)-furanone which are formed and degraded through simple mechanistic pathways. The explaining power of mathematical models for compounds formed by more complex routes such as 2,3-dihydro-(lH)-pyrrolizines suffered when variable interaction terms were not included. 

The reaction of rhamnose and proline produces many pleasant aromas which are important to the food industry among them are bready, cracker-like and roasted aromas. These aromas are greatly influenced by changes in reaction conditions such as temperature, pH and the relative concentration of reactants (l.,2). The effects of changes in reaction conditions may be additive or synergistic. Research was undertaken to determine the prevalence and importance of synergies (variable interactions) between common reaction condition parameters for the reaction of rhamnose and proline. 

A variable interaction or synergy occurs when the effect on the response caused by one variable can be changed by varying the level of a second variable. RSM provides an estimate of the effect of a single variable at selected fixed conditions of the other variables. If the variables do act additively, the factorial (experimental design) does the job with more precision. If the variables do not act additively, the factorial, unlike the one-factor-at-a-time design, can detect and estimate interactions that measure the nonadditivity (5). 

C(r)pH are derived from the independant variables (temperature (T), rhamnose concentration C(r) and pH. Thus the model is composed of a constant, 3 linear, 3 quadratic and 3 variable interaction terms. The models were refined by eliminating those terms which were not statistically significant. The resulting mathematical equations may be graphically represented as a response surface as shown in Figure 1. 

Mathematical Models. Secondary variable interactions quantify the synergies which are common in food chemistry. These interactions cannot be computed from pooled primary variable/sequential design studies and interpolations from such pooled data would lack the information given by the secondary interaction terms. Prob > t is an estimate of the relative importance of each model term. Terms with the lowest Prob > t could well be the driving force of the reaction processes accounting for the quantity of the volatiles found. From Table IV, about 25% of the model terms present at >0.05 Prob > t are seen to be interaction terms. 

Thus reaction flavor generation may well be suitably investigated by systematic changes in all of the reaction condition variables in addition to the one-variable-at-a-time approach which is commonly employed. Such systematic change requires a suitable design such as the central composite factorial design which was used for the estimation of both primary, quadratic and variable interaction terms. 

Variable interaction terms do not aid in the understanding of DMHF content within the experimental space studied because the primary variable effects are very strong. This is reasonable for a compound which is both easily formed and readily degraded. Variable interaction terms are more important in understanding the formation of 2,3-dihydro-lH-pyrrolizines. These compounds are formed through more complicated mechanistic pathways. Where the interaction terms are important, a 17% and 35% improvement in model fit as expressed by R-Square value was obtained when the interaction terms are considered. 

In summary, reaction flavors are complex systems and are strongly influenced by changes in reaction conditions. As the number of reaction condition variables increases so does the possibility of variable interactions or synergies. Without appropriate experimental designs it is not possible to assess the contribution of each variable singly and in concert. Response Surface Methodology is another tool for understanding the effects of reaction conditions on Maillard type flavors. 

Distillation separates the components of a mixture on the basis of their boiling points and on the difference in the compositions of the liquids and their vapors. The product purity of a distillation process is maintained by the manipulation of the material and energy balances. Difficulties in maintaining that purity arise because of dead times, nonlinearities, and variable interactions. 

Gu and Wahba (1993) used a smoothing-spline approach with some similarities to the method described in this chapter, albeit in a context where random error is present. They approximated main effects and some specified two-variable interaction effects by spline functions. Their example had only three explanatory variables, so screening was not an issue. Nonetheless, their approach parallels the methodology we describe in this chapter, with a decomposition of a function into effects due to small numbers of variables, visualization of the effects, and an analysis of variance (ANOVA) decomposition of the total function variability. 

Each estimated marginal effect leads to the estimate of the corresponding corrected effect in (12) or (13). This is done recursively the estimated main effects are corrected first, followed by the two-variable interaction effects. 

Table 1. Estimated main effects and two-variable interaction effects accounting for more than 1 % of the total variance of the predictor the variable names are defined in Table 2 suffix .n or .s indicates the northern region or the southern region, respectively...

From the estimated marginal effects, it is straightforward to compute estimates of the corrected main effects (12), the two-variable interaction effects (13), and so on. The ANOVA contributions on the right-hand side of (14) for these low-order effects involve correspondingly low-dimension integrals. 

Very often, electrofocused proteins show patterns of multiple bands in places where only a single band is expected. This phenomenon is called microheterogeneity. Early discussions of microheterogeneity attempted to explain it in terms of denaturation or of variable interactions between carrier... 

We pointed out earlier in this chapter that simplicity is the key to controllability. What we mean by simplicity is that the list of dominant variables should be kept as short as possible. It is easy to construct reactors where the number of dominant variables far exceed the number of manipulated variables or where the dominant variables interact in a complicated way. 

These are each entered into the program at given levels in such a nanner as to cover a specific range for the variable. Next, possible effects from independent variable interactions and curvilinear effects on the variables are described. Last, one or more dependent variables to be acted upon are entered. 

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