Factorial Design Models

The model that will be fit in factorial designs can be written abstractly as 

Therefore, based on the above discussion of the model to be determined, for a 2 -factorial design, the model can be written as 

This implies that a single row of the, 4-matrix can be written as 

The values to the factors are assigned so that all possible combinations of levels and factors are obtained. For a 2 factorial experiment (i.e. there are 2 levels with 3 factors), the regression matrix (,4) would look like this 

Consider a 2 full factorial experiment with two replicates and answer the following questions  

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The computation just outlined is easily extended to any number of factors. For a system with three factors, for example, a 2 factorial design can be used to determine the parameters for the empirical model described by the following equation... 

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

Curved one-factor response surface showing (a) the limitation of a 2 factorial design for modeling second-order effects and (b) the application of a 3 factorial design for modeling second-order effects. 

If the actual response is that represented by the dashed curve, then the empirical model is in error. To fit an empirical model that includes curvature, a minimum of three levels must be included for each factor. The 3 factorial design shown in Figure 14.13b, for example, can be fit to an empirical model that includes second-order effects for the factor. 

Four replicate measurements were made at the center of the factorial design, giving responses of 0.334, 0.336, 0.346, and 0.323. Determine if a first-order empirical model is appropriate for this system. Use a 90% confidence interval when accounting for the effect of random error. 

Because exceeds the confidence interval s upper limit of 0.346, there is reason to believe that a 2 factorial design and a first-order empirical model are inappropriate for this system. A complete empirical model for this system is presented in problem 10 in the end-of-chapter problem set. 

The following set of experiments provides practical examples of the optimization of experimental conditions. Examples include simplex optimization, factorial designs used to develop empirical models of response surfaces, and the fitting of experimental data to theoretical models of the response surface. 

Mathews and Rawlings (1998) successfully applied model-based control using solids hold-up and liquid density measurements to control the filtrability of a photochemical product. Togkalidou etal. (2001) report results of a factorial design approach to investigate relative effects of operating conditions on the filtration resistance of slurry produced in a semi-continuous batch crystallizer using various empirical chemometric methods. This method is proposed as an alternative approach to the development of first principle mathematical models of crystallization for application to non-ideal crystals shapes such as needles found in many pharmaceutical crystals. 

Based on the experimental data kinetic parameters (reaction orders, activation energies, and preexponential factors) as well as heats of reaction can be estimated. As the kinetic models might not be strictly related to the true reaction mechanism, an optimum found will probably not be the same as the real optimum. Therefore, an iterative procedure, i.e. optimization-model updating-optimization, is used, which lets us approach the real process optimum reasonably well. To provide the initial set of data, two-level factorial design can be used. 

This is rarely the case in engineering. Most of the time we do have some form of a mathematical model (simple or complex) that has several unknown parameters that we wish to estimate. In these cases the above designs are very straightforward to implement however, the information may be inadequate if the mathematical model is nonlinear and comprised of several unknown parameters. In such cases, multilevel factorial designs (for example, 3k or 4k designs) may be more appropriate. 

Step 1. Perform a series of initial experiments (based on a factorial design) to obtain initial estimates for the parameters and their covariance matrix for each of the r rival models. 

Using a "home made" aneroid calorimeter, we have measured rates of production of heat and thence rates of oxidation of Athabasca bitumen under nearly isothermal conditions in the temperature range 155-320°C. Results of these kinetic measurements, supported by chemical analyses, mass balances, and fuel-energy relationships, indicate that there are two principal classes of oxidation reactions in the specified temperature region. At temperatures much lc er than 285°C, the principal reactions of oxygen with Athabasca bitumen lead to deposition of "fuel" or coke. At temperatures much higher than 285°C, the principal oxidation reactions lead to formation of carbon oxides and water. We have fitted an overall mathematical model (related to the factorial design of the experiments) to the kinetic results, and have also developed a "two reaction chemical model". 

Because earlier experimental results and data analyses (3-10) had led us to anticipate the inadequacy of the simple approach considered above, we also planned and carried out (2) a second order factorial design of experiments and related data analysis. Mathematical analysis (of the results of 11 experiments) based on the second order model showed that all of these results could be represented satisfactorily by an equation of the form... 

Note, however, there are two critical limitations to these "predicting" procedures. First, the mathematical models must adequately fit the data. Correlation coefficients (R ), adjusted for degrees of freedom, of 0.8 or better are considered necessary for reliable prediction when using factorial designs. Second, no predictions outside the design space can be made confidently, because no data are available to warn of unexpectedly abrupt changes in direction of the response surface. The areas covered by Figures 8 and 9 officially violate this latter limitation, but because more detailed... 

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 ... 

In the following discussion, we shall again separate the terms of a hyperbolic model and identify two parameters Cl and C2. As before, each of these two parameters will be a collection of terms, one of which is multiplied by conversion and one not multiplied by conversion. In previous formulations, however, we have oriented the discussion toward a familiar type of experimental design in kinetics conversion versus space-time data at several pressure levels. Consequently, the parameters Cx and C2 were defined to exploit this data feature. Another type of design that is becoming more common is a factorial design in the feed-component partial pressures. 

These equations might be thought to apply for a high concentration of reactant A, at a given concentration of catalyst C, and at isothermal conditions. To describe measurements on the concentration of F as a function of time at each of the initial conditions of the 24 factorial design of Table XIII, we might tentatively entertain the integrated model ... 

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... 

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