The functions serve the calculation of lower bounds for the worst case confounding. lowerbound_AR is intended for direct use, lowerbounds and lowerbound_chi2 are internal functions.
lowerbound_AR(nruns, nlevels, R, crit = "total")
lowerbounds(nruns, nlevels, R)
lowerbound_chi2(nruns, nlevels)positive integer, the number of runs
vector of positive integers, the numbers of levels for the factors
positive integer, the resolution of the design; if it is uncertain whether resolution R is feasible, this should be checked by function oa_feasible before applying any of the lower bound functions.
"total" or "worst"; if "total",
a bound for the overall A_R (sum of the results from lowerbounds) is calculated;
otherwise, a bound for the largest individual contribution from an R factor set is calculated
lowerbound_AR returns a lower bound for the number of words of length R (either total or worst case),
lowerbounds returns a vector of lower bounds for individual R factor sets on a different scale (division by nruns^2 needed for transforming this into the contributions to words of length R),
and function lowerbound_chi2 returns a lower bound on the \(\chi^2\) value which can be used as a quality criterion for supersaturated designs.
Note: if the specified resolution R is not feasible (necessary conditions can be checked with function oa_feasible), any bound(s) returned will be meaningless.
Function lowerbounds provides (integral) bounds on \(n^2 A_R\) (with \(n\)=nruns) according to Groemping and Xu (2014) Theorem 5 for all R factor sets. If the number of runs permits a design with resolution larger than R, the value(s) will be 0. For resolution at least III, the result of function lowerbound_AR is the sum (crit="total") or maximum (crit="worst") of these individual bounds, divided by the square of the number of runs.
For resolution II and crit="total", function lowerbound_chi2 implements the lower bound B on chi^2 which was provided in Lemma 2 of Liu and Lin (2009). For supersaturated resolution II designs, this bound is is usually sharper than the one obtained on the basis of Gr<U+00F6>mping and Xu (2014). Due to the relation between \(A_2\) and \(\chi^2\) that is stated in Groemping (2017) (summands of \(A_2\) are an nth of a \(\chi^2\), with \(n\)=nruns), this bound can be easily transformed into a bound for \(A_2\); this relation is also used to slightly sharpen the bound B itself: \(n^2 \cdot A_2\) must be integral, which implies that B can be replaced by ceiling(nruns*B)/nruns, which is applied in function lowerbound_chi2. Function lowerbound_AR increases the lower bound on \(A_2\) accordingly, if lowerbound_chi2 provides a sharper bound than the sum of the elements returned by functioni lowerbounds.
Groemping, U. and Xu, H. (2014). Generalized resolution for orthogonal arrays. The Annals of Statistics 42, 918-939.
Groemping, U. (2017). Frequency tables for the coding-invariant quality assessment of factorial designs. IISE Transactions 49, 505-517.
Liu, M.Q. and Lin, D.K.J. (2009). Construction of Optimal Mixed-Level Supersaturated Designs. Statistica Sinica 19, 197-211.
See also oa_feasible.
# NOT RUN {
lowerbound_AR(24, c(2,3,4,6),2)
# }
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