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)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.
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
Ulrike Groemping
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 Groemping 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.
lowerbound_AR(24, c(2,3,4,6),2)
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