## Summary

In many experimental situations, when a two-level factorial design is performed, the effects of higher order interactions are usually less important than the effects of lower order interactions (and effects of the same order are equally important). Hence, we should search for good fractional factorial designs with*N* observations allowing for the estimation of the effects of main factors and of some interactions under study. Obviously, a design is more preferable it if allows for several de-aliased estimates of the effects of interest. A goodness criterion of a design is its resolution (Box & Hunter (1961)). Since there is more than one design with the same resolution, to select the «best» subset of designs which have the same resolution (usually the maximum resolution), we may use the criterion of minimum aberration (Fries & Hunter (1980)). In experiments involving control and noise factors (Taguchi (1987)) neither the resolution nor the aberration criterion can guarantee a good statistical design. Here we introduce the degrees of freedom criterion in order to guarantee a good choice of the fractional design also in terms of the possibility to perform parametric tests with a good power and to obtain confidence intervals for the effects’ estimates. The two level orthogonal factorial designs are suitable for experiments with linear response, since they give the possibility to obtain uniformly optimal de-aliased estimates of the effects of interest. Here we consider the case where three and higher factor interactions are negligible and orthogonal designs of resolution IV and V are constructed; the necessary conditions for the existence of such designs are*N* ∈*0* mod 8, and*N* ∈*0* mod 16,*N* ≥16, respectively. Some cases when*N* ∈*0* mod 8,*N* ≥ 8, or*N* ∈*0* mod 16,*N* ≥ 16, are studied: at first we construct orthogonal designs of resolution IV and V for a given number of factors k, and then for a given number of runs*N* (in particular*N*=48, 80, 96, 112).

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Kounias, S., Salmaso, L. Orthogonal plans of resolution IV and V.
*J. Ital. Statist. Soc.* **7, **57 (1998). https://doi.org/10.1007/BF03178921

### Keywords

- Factorial designs
- Irregular fractions
- Minimum aberration
- Optimal moments
- Regular fractions