Signal-to-noise ratios, performace criteria, and transformations.

*(English)*Zbl 0721.62103For the analysis of designed experiments, G. Taguchi [Introduction to quality engineering: Designing quality into products and processes (1986)] uses performance criteria that he calls signal-to-noise (SN) ratios. Three such criteria are here denoted by \(SN_ T\), \(SN_ L\), and \(SN_ S\). The criterion \(SN_ T\) was to be used in preference to the standard deviation for the problem of achieving, for some quality characteristic y, the smallest mean squared error about an operating target value. R. V. León, A. C. Shoemaker and R. N. Kacker [Technometrics 29, 253-285 (1987; Zbl 0664.62103)] showed how \(SN_ T\) was appropriate to solve this problem only when \(\sigma_ y\) was proportional to \(\mu_ y\). On that assumption, the same result could be obtained more simply by conducting the analysis in terms of log y rather than y. A more general transformation approach is here introduced for other, commonly met kinds of dependence between \(\sigma_ y\) and \(\mu_ y\) (including no dependence), and a lambda plot is presented that uses the data to suggest an appropriate transformation. The criteria \(SN_ L\) and \(SN_ S\) were for problems in which the objective was to make the response as large or as small as possible.

##### MSC:

62P30 | Applications of statistics in engineering and industry; control charts |

62K05 | Optimal statistical designs |