Mean orthogonal design

Step 1 Choose a balanced (mean orthogonal) design at random label the design matrix D. 

Decide how many concentration levels are required for each compound ( = /), typically four or five. Mutually orthogonal designs are only possible if the number of concentration levels is a prime number or a power of a prime number, meaning that they are possible for 3, 4, 5, 7, 8 and 9 levels but not 6 or 10 levels. 

Taguchi methods, and the other fractional factorials that we studied, have in common the idea of performing multivariate studies by means of orthogonal designs. Even if the researcher does not choose the ideal one, just by using them she will have more chances of succeeding than if she uses traditional univariate methods. 

The fact that the dispersion matrix is diagonal means that there is no covariance among the coefficients, and therefore all of them can be computed independently from each other (it is an orthogonal design). 

This means that in the set of matrices constituting any one irreducible representation any set of corresponding matrix elements, one from each matrix, behaves as the components of a vector in /i-dimensional space such that all these vectors are mutually orthogonal, and each is normalized so that the square of its length equals hllh This-interpretation of 4.3-1 will perhaps be more obvious if we take 4.3-1 apart into three simpler equations, each of which is contained within it. We shall omit the explicit designation of complex conjugates for simplicity, but it should be remembered that they must be used... 

When used as the dispersion formula for the phonons and polaritons in orthorhombic crystals, the symbols in Eq. (11.22) have the following meaning r)= 1,2,3 designates the three directions of the principal orthogonal axes. sv are the direction cosines of the normalized wave vector s = k/k with respect to the three principal axes of the crystal. If the unit vectors in the directions of these three principal axes are designated eue2,e3, one can write... 

Expression (2.136) is rarely used in practice, since uneven replication of a trial destroys orthogonality of a design of an experimental matrix. Sometimes, in the case of identical number of replications, a variance is determined which is typical of the mean error (yu) after replicated trials ... 

Consider now robustness. If the estimators A are computed from independent response variables then, as noted in Section 1, the estimators have equal variances and are usually at least approximately normal. Thus the usual assumptions, that estimators are normally distributed with equal variances, are approximately valid and we say that there is inherent robustness to these assumptions. However, the notion of robust methods of analysis for orthogonal saturated designs refers to something more. When making inferences about any effect A, all of the other effects At (k i) are regarded as nuisance parameters and robust means that the inference procedures work well, even when several of the effects ft are large in absolute value. Lenth s method is robust because the pseudo standard error is based on the median absolute estimate and hence is not affected by a few large absolute effect estimates. The method would still be robust even if one used the initial estimate 6 of op, rather than the adaptive estimator 6L, for the same reason. 

The coefficients of the non-independent variables may also be calculated directly for these orthogonal experimental designs by taking, for example, the mean degradation of all experiments with lactose and subtracting the mean of all 9 experiments. 

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