Saturation designators

A Plackett-Burman design with N experiments can examine up to N-1 factors. This is a difference with fractional factorial designs. Some saturated fractional factorial designs however contain also N-1 factors (e.g. the design of Table 3.14) but this is not always the case. The saturated design for 5 factors, for example, is the 2 design. In this design only 5 factors are examined in 8 experiments. 

Although these saturated designs, assume interaction effects to be negligible and only estimate main effects, they have the feature known as confounding where higher order effects can overwrite the main effects. 

Saturated Designs Plackett-Burman Designs. Use in Screening and Robustness Studies... 

The coefficients of the fourth-order regression equation are calculated by Eq. (3.29) using the property of saturated design matrix. The regression equation in pseudocomponent variables has the form ... 

For k coefficients the underlying design is called a saturated design. 

The 27-4 design is a saturated design by this we mean that the number of factors, / = 7, is one less than the number of runs, n = <8. The alias relationships here are somewhat more complicated. Specifically, if we ignore all interactions of order three or higher, each main effect is aliased with a chain of three two-factor interactions ... 

Experiment with only n - 1 factors (a saturated design) ... 

Subsequently, Zahn (1969,1975ab) considered some variations on the iterative methods of Daniel (1959) and Birnbaum (1959), but his results were primarily empirical. The subjective use of half-normal plots remains a standard methodology for the analysis of orthogonal saturated designs, but the development of objective methods is progressing rapidly. 

The most influential method of analysis of orthogonal saturated designs yet proposed is the robust adaptive method of Lenth (1989). The quick and easy method that he proposed is based on the following estimator of the standard deviation, op, of the effect estimators Pi. This estimator is robust and adaptive , concepts which are explained in detail after the following description of the method. 

From empirical comparisons of various proposed methods of analysis of orthogonal saturated designs (Hamada and Balakrishnan, 1998 Wang and Voss, 2003), Lenth s method can be shown to have competitive power over a variety of parameter configurations. It remains an open problem to prove that the null case is the least favourable parameter configuration. 

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. 

Like many methods of analysis of orthogonal saturated designs proposed in the literature, the critical values for Lenth s method are obtained in the null case (all A zero), assuming this is sufficient to control the Type I error rates. This raises the question can one establish analytically that Lenth s and other proposed methods do indeed provide the claimed level of confidence or significance under standard model assumptions The rest of this chapter concerns methods for which the answer is yes. ... 

The first confidence interval for the analysis of orthogonal saturated designs that provided strong control of the error rate was established by Voss (1999). His confidence interval for fa excludes fa from the computation of the standard error and is obtained using the random variable... 

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