Factorial design linear terms

ANOVA in these chapters also, back when it was still called Statistics in Spectroscopy [16-19] although, to be sure, our discussions were at a fairly elementary level. The experiment that Philip Brown did is eminently suitable for that type of computation. The experiment was formally a three-factor multilevel full-factorial design. Any nonlinearity in the data will show up in the analysis as what Statisticians call an interaction term, which can even be tested for statistical significance. He then used the wavelengths of maximum linearity to perform calibrations for the various sugars. We will discuss the results below, since they are at the heart of what makes this paper important. 

A potential concern in the use of a two-level factorial design is the implicit assumption of linearity in the true response function. Perfect linearity is not necessary, as the purpose of a screening experiment is to identify effects and interactions that are potentially important, not to produce an accurate prediction equation or empirical model for the response. Even if the linear approximation is only very approximate, usually sufficient information will be generated to identify important effects. In fact, the two-factor interaction terms in equation (1) do model some curvature in the response function, as the interaction terms twist the plane generated by the main effects. However, because the factor levels in screening experiments are usually aggressively spaced, there can be situations where the curvature in the response surface will not be adequately modeled by the two-factor interaction... 

Two level full factorial designs (also sometimes called saturated factorial designs), as presented in this section, take into account all linear terms, and all possible k way interactions. The number of different types of terms can be predicted by the binomial theorem [given by k /(k — m) m for mth-order interactions and k factors, e.g. there are six two factor (=m) interactions for four factors (= )]. Hence for a four factor, two level design, there will be 16 experiments, the response being described by an equation with a total of 16 terms, of which... 

The smallest possible fractional factorial three factor design consists of four experiments, used to estimate the three linear terms and the intercept. Such as design will not provide estimates of the interactions, replicates or squared terms. 

As explained in Section 6.2 simple empirical models such as those of Eq. (6.1) and Eq. (6.2) are usually applied. They can be easily generalized to more than two variables. Usually not all possible terms are included. For instance, when including three variables one could include a ternary interaction (i.e. a term in. vi.vi.vy) in Eq. (6.1) or terms with different exponents in Eq. (6.2). such as. vi.v , but in practice this is very unusual. The models are nearly always restricted to the terms in the individual variables and binary interactions for the linear models of Eq. (6.1), and additionally include quadratic terms for individual variables for the quadratic models of Eq. (6.2). To obtain the actual model, the coefficients must be computed. In the case of the full factorial design, this can be done by using Eq. (6.5) and dividing by 2 (see Section 6.4.1). In many other applications such as those of Section 6.4.3 there are more experiments than coefficients in the model. For instance, for a three-variable central composite design, the model of Eq. (6.2)... 

Let us now have a look at general screening experiments with many variables. Assume that k variables (xj, X2,..., xJ have been studied by a fractional factorial design and that a response surface model with linear and cross-product interaction terms has been determined. 

At the outset of an experimental study, the shape of the response surface is not known. A quadratic model will be necessary only if the response surface is curved. It was discussed in Chapters 5 and 6 how linear and second-order interaction models can be established from factorial and fractional factorial designs, and how such models might be useful in screening experiments. However, these models cannot describe the curvatures of the surface, and should there be indications of curvature, it would be convenient if a complementary set of experiments could be run by which an interaction model could be augmented with squared terms. 

A factorial or fractional factorial design, which is used to estimate the coefficients of the linear and the interaction terms. 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

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