Interaction between variables

With the one-variable-at-a-time approach, it is difficult to determine the amount of interaction between variables. Response surface methodology, since it looks at all the variables at the same time, can calculate the interaction 

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The practice of estabHshing empirical equations has provided useflil information, but also exhibits some deficiencies. Eor example, a single spray parameter, such as may not be the only parameter that characterizes the performance of a spray system. The effect of cross-correlations or interactions between variables has received scant attention. Using the approach of varying one parameter at a time to develop correlations cannot completely reveal the tme physics of compHcated spray phenomena. Hence, methods employing the statistical design of experiments must be utilized to investigate multiple factors simultaneously. 

Fig. 2. Situation with interactions between variables, where in (a) an increase in temperature is beneficial for preparation A but does not make any difference for preparation B, and (b) an increase in temperature raises time to mpture for preparation A but decreases it for preparation B.

What is the most meaningful way to express the controllable or independent variables For example, should current density and time be taken as the experimental variables, or are time and the product of current density and time the real variables affecting response Judicious selection of the independent variables often reduces or eliminates interactions between variables, thereby leading to a simpler experiment and analysis. Also inter-relationships among variables need be recognized. For example, in an atomic absorption analysis, there are four possible variables air-flow rate, fuel-flow rate, gas-flow rate, and air/fuel ratio, but there are really only two independent variables. 

Each value in the hnal column of the table consuiicted above is now divided by 4, which is the number of additions or subuactions made in each column. The results of tlris division show the numerical effects of each variable and the interaction between variables. The value opposite tire second row shows the effect of the temperature, the third shows the effect of the slag phase composition, and, the hfth the effect of the metal composition. The interaction terms then follow the symbols of each row, dre fourth showing the effect of... 

Kehat and Shacham ( 6) used split fraction models to estimate the Jacobian when the Newton-Raphson method is used to solve Equation (1). The authors concluded that their method is very efficient for systems with more than one tear stream and when there is only a weak interaction between variables in the tear stream. 

If the regression, however, is to provide the maximum amount of information on precision, differences between sets of data, and interaction between variables, it will be necessary to use the statistical technique of regression... 

Statistical experimental design was used to study the effects of blowing agents, processing aids and fdlers in rigid PVC foam formulations. This technique provided an alternative approach to the classical experimental method of changing one variable at a time. It provided information about interactions between variables and could be used to help to predict an optimum formulation. 24 refs. 

The EVOP philosophy is so natural that it can be (and undoubtedly has been) practiced without relying on statistics. However, the addition of a few simple statistical techniques can help avoid the economic penalties accompanying the production of off-specification product. By running many cycles in each phase, the small, virtually undetectable effects on the product in the individual runs can be evaluated, using statistics. Another advantage of statistical analysis is that interactions between variables are readily identified. 

Interactions between Variables. Obviously there are strong interactions among the variables mentioned above. As an example, consider... 

The dynamic model presented herein builds on that reported previously (I) by incorporating the interactions between volatile acids, pH, alkalinity, gas production rate, and gas composition. The model is developed from material balances on the biological, liquid, and gas phases of a continuous-flow, complete mixing reactor. Appropriate relationships such as yield constants, an inhibition function, Henry s law, charge balances, and ionization equilibria are used to express the interactions between variables. The inputs and outputs for the reactor and the reactions considered are illustrated in Figure 2. 

The perturbation H causes resonant interaction between variables ... 

There were no significant first or second order interactions between variables that effected either the relative filtration times or the plutonium concentrations in the filtrate. 

Each measure of an analysed variable, or variate, may be considered independent. By summing elements of each column vector the mean and standard deviation for each variate can be calculated (Table 7). Although these operations reduce the size of the data set to a smaller set of descriptive statistics, much relevant information can be lost. When performing any multivariate data analysis it is important that the variates are not considered in isolation but are combined to provide as complete a description of the total system as possible. Interaction between variables can be as important as the individual mean values and the distributions of the individual variates. Variables which exhibit no interaction are said to be statistically independent, as a change in the value in one variable cannot be predicted by a change in another measured variable. In many cases in analytical science the variates are not statistically independent, and some measure of their interaction is required in order to interpret the data and characterize the samples. The degree or extent of this interaction between variables can be estimated by calculating their covariances, the subject of the next section. 

The parameters should measure the influence of the corresponding variables, i.e. the slopes of the surface, flj, as measures of linear dependencies of the variables the twists of the surface, By, as measures of interactions between variables, and the curvatures, fly, as measures or non-linear influences of the variables, and nothing else This calls for careful spacing of the variable settings in the experimental domain to determine the series of experiments used to estimate the parameters, i.e. the experimental design. These aspects will be treated in detail in the following chapters Chapters 5-7, which deal with screening experiments based on linear and second order interaction models, and Chapter 12, which describes quadratic models for optimization. 

Step 8 If necessary, add more runs to the design matrix to eliminate aliases or confounding patterns. For example, if a fractional factorial experiment shows evidence of interactions between variables, it may be necessary to run the full factorial to determine which interactions are truly important. 

Gives information about the interaction between variables. 

To analyse the complex interaction between variables, the SDTF rigorously compared downwind deposit data against predictions from sophisticated... 

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