Factorial design linear models

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... 

Fit the linear model Y = b0 + bxxx + b2x2 using the preceding table. Report the estimated coefficients b0, bx, and b2. Was the set of experiments a factorial design ... 

Historically, factorial designs were introduced by Sir R. A. Fisher to counter the then prevalent idea that if one were to discover the effect of a factor, all other factors must be held constant and only the factor of interest could be varied. Fisher showed that all factors of interest could be varied simultaneously, and the individual factor effects and their interactions could be estimated by proper mathematical treatment. The Yates algorithm and its variations are often used to obtain these estimates, but the use of least squares fitting of linear models gives essentially identical results. 

The linear model most commonly fit to the data from 2 factorial designs is... 

A section has been added to Chapter 1 on the distinction between analytic vs. enumerative studies. A section on mixture designs has been added to Chapter 9. A new chapter on the application of linear models and matrix least squares to observational data has been added (Chapter 10). Chapter 13 attempts to give a geometric feel to concepts such as uncertainty, information, orthogonality, rotatability, extrapolation, and rigidity of the design. Finally, Chapter 14 expands on some aspects of factorial-based designs. 

According to the Hadamard matrix, a 22 factorial design was built. The complete linear models were fitted by regression for each response, reflecting the compression behaviour and dissolution kinetics. 

A 22 factorial design was built to obtain a complete linear model including only two parameters. 

In order to improve Y6 and according to the linear model previously found for this response, it was decided to fix Xx and X2 at -1 (8 kN) and +1 (400-800 pm) levels respectively then, a 22 factorial design was built with the two other significant parameters X3 andX4 at the upper levels. Table 7 gives the experimental and physical units of this factorial design. 

Fig. 3—Release profiles from the factorial design. Table 9—Complete linear models...

Designing a stability study is based on a factorial design of experiments where a systemic procedure is used to determine the effect on the response variable of various factors and factor combinations. A linear model is used to represent the relationship between the factors and factor combinations with the response variable. Once the experimental design is established, the assays are conducted and stability data are saved to finally estimate the shelf life period. 

A process having properties dependent on four factors has been tested. A full factorial experiment and optimization by the method of steepest ascent have brought about the experiment in factor space where only two factors are significant and where an inadequate linear model has been obtained. To analyze the given factor space in detail, a central composite rotatable design has been set up, as shown in Table 2.152. 

In the case of constraints on proportions of components the approach is known, simplex-centroid designs are constructed with coded or pseudocomponents [23]. Coded factors in this case are linear functions of real component proportions, and data analysis is not much more complicated in that case. If upper and lower constraints (bounds) are placed on some of the X resulting in a factor space whose shape is different from the simplex, then the formulas for estimating the model coefficients are not easily expressible. In the simplex-centroid x 23 full factorial design or simplex-lattice x 2n design [5], the number of points increases rapidly with increasing numbers of mixture components and/or process factors. In such situations, instead of full factorial we use fractional factorial experiments. The number of experimental trials required for studying the combined effects of the mixture com-... 

In simple linear models (first order polynomials) the adequacy can be tested by another simple procedure. A /-test is performed between the mean result, y0, from the experiments on the central point and the mean, y, of response values from factorial design ... 

In practice, augmentation can be performed after the experimenter has completed a full factorial design and found a linear model to be inadequate. A possible reason is that the true response function may be second order. Instead of starting a completely new set of experiments, we can use the results of the previous design and perform an additional set of measurements at points having one or more zero coordinates. All of the data collected can be used to tit a second-order model. 

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... 

Pan, G. and Taam, W. (2002). On generalized linear model method for detecting dispersion effects in unreplicated factorial designs. Journal of Statistical Computation and Simulation, 72, 431-450. 

Two level factorial designs are primarily useful for exploratory purposes and calibration designs have special uses in areas such as multivariate calibration where we often expect an independent linear response from each component in a mixture. It is often important, though, to provide a more detailed model of a system. There are two prime reasons. The first is for optimisation - to find the conditions that result in a maximum or minimum as appropriate. An example is when improving die yield of synthetic reaction, or a chromatographic resolution. The second is to produce a detailed quantitative model to predict mathematically how a response relates to die values of various factors. An example may be how the near-infrared spectrum of a manufactured product relates to the nature of the material and processing employed in manufacturing. 

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)... 

The calculations involved in the evaluation of factorial designs by using sign tables rely on an unrealistic assumption, viz. that each experiments was conducted exactly as was specified by the design. For instance, that the temperature was adjusted exactly to its high and low levels. In practice, this is never obtained in synthetic chemistry. The variables can be adjusted fairly close to the specified levels, but over the series of experiments, there will always be small differences between the runs. It is therefore better, and more honest, to use the settings actually used in the evaluation of the experiments. For this, the appropriate tool is multiple linear regression which is used to fit response surface models to the experimental data. This technique is described in the next section. 

A factorial design can be used to fit a response surface model to the experimental results. In this case, the effects will be the corresponding model parameters. To achieve this, the factors are scaled through a linear transformation to design variables, x, as was described in section 3.4.2, see also Fig. 5.2. 

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