Factorial design model matrix

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. 

Assume a constrained factor space of -5 < jc, +5, -5 < jcj +5. Assume the full two-factor model with interaction, y, = Po + PiJCi, + 2 21 + Pn ii i + "ii- Assume a 2 factorial design. How should the four design points be placed to maximize the determinant of the (X X) matrix Demonstrate with a few calculations. 

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. 

The exact structure of each of the functions/ x),..., fk(x) depends on the transformation or factor coding used. For example, the F matrix for a three-level full factorial design for two process variables and a second-order model is shown in Table 8.4. 

If the mattix I) is a square matrix, the estimated values of y are identical with the observed values y. The model provides an exact fit to die data, and there are no degrees of freedom remaining to determine die lack-of-fit. Under such circumstances diere will not be any replicate information but, nevertheless, the values of b can provide valuable information about the size of different effects. Such a situation might occur, for example, in factorial designs (Section 2.3). The residual error between die observed and fitted data will be zero. This does not imply that the predicted model exactly represents die underlying data, simply that the number of degrees of freedom is insufficient for determination of prediction errors. In all other circumstances there is likely to be an error as die predicted and observed response will differ. 

Fractional factorial designs are constructed from the model matrix X of a 2k p complete factorial design using the orthogonal columns of X to define the variable settings in 2k p experiments. 

From this conclusion follows, that a factorial design can be used to fit a response surface model to account for main effects and interaction effects. In the concluding section of this chapter is discussed how the properties of the model matrix X influence the quality of the estimated parameters in multiple regression. It is shown that factorial design have optimum qualities. 

The determinant (X X) = S which is the minimum value. The eigenvalues of the dispersion matrix are all equal and the variance of all estimated model parameters are / 8. The parameters are independently estimated and the dispersion matrix is a diagonal matrix (the covariances of the models parameters are zero). This means that parameters estimated from a two-level factorial design are independently estimated, with equal and maximum precision. 

Example-. A two-variable factorial design has the following complete model matrix... 

Another example-. We can study seven variables in a 2 fractional factorial design. This design is defined from the model matrix of a 7 factorial design, see Fig. 6.2. 

A 2 " fractional factorial design (I = 12345) was used to estimate the model parameters. The design has a Resolution V and the desired parameters can be estimated free from confoundings with each other. The design matrix and the yields (%), y, obtained are given in Table 6.7. 

The first term is known as the sum of squares, model (SSM). The second term is known as the sum of squares, residual (SSR), and the final term is known as the sum of squares, total (SST). Equation 3.16 is true for any least squares solution whatsoever. However, SSM can be apporhoned by component (i.e., Uq, a-, U2,. ..) only for so-called orthogonal models or data sets—that is, those that generate diagonal X X matrices (as is the case for factorial designs). Equation 3.16 also has a matrix formulation ... 

Table 3.3 Model Matrix, X, of a Complete 2 Factorial Design Matrix...

In the case of a 2-level factorial design the different columns of the model (effects) matrix correspond to the linear combinations for calculating the corresponding effects. 

The coefficients in the model equation 3.4 may be estimated as before, as linear combinations or contrasts of the experimental results, taking the columns of the effects matrix as described in section III.A.5 of chapter 2. Alternatively, they may be estimated by multi-linear regression (see chapter 4). The latter method is more usual, but in the case of factorial designs both methods are mathematically equivalent. 

We split the full factorial design in two, in the simplest possible way, and observe what information can be obtained and what information is lost in the process. Consider what the situation would be if only the second half of the design, had been carried out. The reader may examine the resulting design and model matrix, that is experiments 9 to 16, by covering up rows 1-8 of table 3.12. 

Assume first of all that we have solved the problem We call the 3 selected variables A, B, and C. The full factorial design is therefore the design whose model matrix as shown below in table 3.26. AB, AC, BC, ABC represent the interactions. 

Exercise 5.6. The fit of a statistical model to the results of a factorial design, which we discussed in Chapter 3, can also be done by the least-squares method, that is, by solving matrix equation b = (X X) X y. Consider Eq. (3.11), corresponding to the 2 factorial... 

Earlier in this chapter we discussed the results of a full 3 factorial design, used to study the synthesis of poljrpyrrole in an EPDM rubber matrix. This design, which requires 27 different level combinations, resulted in the models given by Eqs. (6.10)-(6.12). Now, we shall see that essentially the same information could be obtained from a 13-run Box-Behnken design. Table 6.15 contains the reaction jdeld and Young s... 

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