arrays: Orthogonal arrays in the package

Description

Orthogonal arrays in the package

Usage

L18
L36
L54
L4.2.3
L8.2.4.4.1
L9.3.4
L12.2.11
L12.2.2.6.1
L12.2.4.3.1
L16.2.8.8.1
L16.4.5
L18.3.6.6.1
L20.2.19
L20.2.2.10.1
L20.2.8.5.1
L24.2.11.4.1.6.1
L24.2.12.12.1
L24.2.13.3.1.4.1
L24.2.20.4.1
L25.5.6
L27.3.9.9.1
L28.2.12.7.1
L28.2.2.14.1
L28.2.27
L32.2.16.16.1
L32.4.8.8.1
L36.2.1.3.3.6.3
L36.2.10.3.1.6.2
L36.2.10.3.8.6.1
L36.2.13.3.2.6.1
L36.2.13.6.2
L36.2.16.9.1
L36.2.18.3.1.6.1
L36.2.2.18.1
L36.2.2.3.5.6.2
L36.2.20.3.2
L36.2.27.3.1
L36.2.3.3.2.6.3
L36.2.3.3.9.6.1
L36.2.35
L36.2.4.3.1.6.3
L36.2.8.6.3
L36.2.9.3.4.6.2
L36.3.12.12.1
L36.3.7.6.3
L40.2.19.4.1.10.1
L40.2.20.20.1
L40.2.25.4.1.5.1
L40.2.36.4.1
L44.2.15.11.1
L44.2.2.22.1
L44.2.43
L45.3.9.15.1
L48.2.24.24.1
L48.2.31.6.1.8.1
L48.2.33.3.1.8.1
L48.2.40.8.1
L48.4.12.12.1
L49.7.8
L50.5.10.10.1
L52.2.16.13.1
L52.2.2.26.1
L52.2.51
L54.3.18.18.1
L54.3.20.6.1.9.1
L56.2.27.4.1.14.1
L56.2.28.28.1
L56.2.37.4.1.7.1
L56.2.52.4.1
L60.2.15.6.1.10.1
L60.2.17.15.1
L60.2.2.30.1
L60.2.21.10.1
L60.2.23.5.1
L60.2.24.6.1
L60.2.30.3.1
L60.2.59
L63.3.12.21.1
L64.2.32.32.1
L64.2.5.4.10.8.4
L64.2.5.4.17.8.1
L64.4.14.8.3
L64.4.16.16.1
L64.4.7.8.6
L64.8.9
L68.2.18.17.1
L68.2.2.34.1
L68.2.67
L72.2.10.3.13.4.1.6.3
L72.2.10.3.16.6.2.12.1
L72.2.10.3.20.4.1.6.2
L72.2.11.3.17.4.1.6.2
L72.2.11.3.20.6.1.12.1
L72.2.12.3.21.4.1.6.1
L72.2.14.3.3.4.1.6.6
L72.2.15.3.7.4.1.6.5
L72.2.17.3.12.4.1.6.3
L72.2.18.3.16.4.1.6.2
L72.2.19.3.20.4.1.6.1
L72.2.27.3.11.6.1.12.1
L72.2.27.3.6.6.4
L72.2.28.3.2.6.4
L72.2.30.3.1.6.4
L72.2.31.6.4
L72.2.34.3.3.4.1.6.3
L72.2.34.3.8.4.1.6.2
L72.2.35.3.12.4.1.6.1
L72.2.35.3.5.4.1.6.2
L72.2.35.4.1.18.1
L72.2.36.3.2.4.1.6.3
L72.2.36.3.9.4.1.6.1
L72.2.36.36.1
L72.2.37.3.1.4.1.6.3
L72.2.37.3.13.4.1
L72.2.41.4.1.6.3
L72.2.42.3.4.4.1.6.2
L72.2.43.3.1.4.1.6.2
L72.2.43.3.8.4.1.6.1
L72.2.44.3.12.4.1
L72.2.46.3.2.4.1.6.1
L72.2.46.4.1.6.2
L72.2.49.4.1.9.1
L72.2.5.3.3.4.1.6.7
L72.2.51.3.1.4.1.6.1
L72.2.53.3.2.4.1
L72.2.6.3.3.6.6.12.1
L72.2.6.3.7.4.1.6.6
L72.2.60.3.1.4.1
L72.2.68.4.1
L72.2.7.3.4.4.1.6.6
L72.2.7.3.7.6.5.12.1
L72.2.8.3.12.4.1.6.4
L72.2.8.3.8.4.1.6.5
L72.2.9.3.12.6.3.12.1
L72.2.9.3.16.4.1.6.3
L72.3.24.24.1
L75.5.8.15.1
L76.2.19.19.1
L76.2.2.38.1
L76.2.75
L80.2.40.40.1
L80.2.51.4.3.20.1
L80.2.55.8.1.10.1
L80.2.61.5.1.8.1
L80.2.72.8.1
L80.4.10.20.1
L81.3.27.27.1
L81.9.10
L84.2.14.6.1.14.1
L84.2.2.42.1
L84.2.20.21.1
L84.2.20.3.1.14.1
L84.2.22.6.1.7.1
L84.2.27.6.1
L84.2.28.7.1
L84.2.33.3.1
L84.2.83
L88.2.43.4.1.22.1
L88.2.44.44.1
L88.2.56.4.1.11.1
L88.2.84.4.1
L90.3.26.6.1.15.1
L90.3.30.30.1
L92.2.2.46.1
L92.2.21.23.1
L92.2.91
L96.2.12.4.20.24.1
L96.2.17.4.23.6.1
L96.2.18.4.22.12.1
L96.2.19.3.1.4.23
L96.2.26.4.23
L96.2.39.3.1.4.14.8.1
L96.2.43.4.12.6.1.8.1
L96.2.43.4.15.8.1
L96.2.44.4.11.8.1.12.1
L96.2.48.48.1
L96.2.71.6.1.16.1
L96.2.73.3.1.16.1
L96.2.80.16.1
L98.7.14.14.1
L99.3.13.33.1
L100.2.16.5.3.10.3
L100.2.18.5.9.10.1
L100.2.2.50.1
L100.2.22.25.1
L100.2.29.5.5
L100.2.34.5.3.10.1
L100.2.4.10.4
L100.2.40.5.4
L100.2.5.5.4.10.3
L100.2.51.5.3
L100.2.7.5.10.10.1
L100.2.99
L100.5.20.20.1
L100.5.8.10.3
L104.2.100.4.1
L104.2.51.4.1.26.1
L104.2.52.52.1
L104.2.65.4.1.13.1
L108.2.1.3.33.6.2.18.1
L108.2.1.3.35.6.3.9.1
L108.2.10.3.31.6.1.18.1
L108.2.10.3.33.6.2.9.1
L108.2.10.3.40.6.1.9.1
L108.2.107
L108.2.12.3.29.6.3
L108.2.13.3.30.6.1.18.1
L108.2.13.6.3
L108.2.15.6.1.18.1
L108.2.17.3.29.6.2
L108.2.18.3.31.18.1
L108.2.18.3.33.6.1.9.1
L108.2.2.3.35.6.1.18.1
L108.2.2.3.37.6.2.9.1
L108.2.2.3.42.18.1
L108.2.2.54.1
L108.2.20.3.34.9.1
L108.2.21.3.1.6.2
L108.2.22.27.1
L108.2.27.3.33.9.1
L108.2.3.3.16.6.8
L108.2.3.3.32.6.2.18.1
L108.2.3.3.34.6.3.9.1
L108.2.3.3.39.18.1
L108.2.3.3.41.6.1.9.1
L108.2.34.3.29.6.1
L108.2.4.3.31.6.2.18.1
L108.2.4.3.33.6.3.9.1
L108.2.40.6.1
L108.2.8.3.30.6.2.18.1
L108.2.9.3.34.6.1.18.1
L108.2.9.3.36.6.2.9.1
L108.3.36.36.1
L108.3.37.6.2.18.1
L108.3.39.6.3.9.1
L108.3.4.6.11
L108.3.44.9.1.12.1
L112.2.104.8.1
L112.2.56.56.1
L112.2.75.4.3.28.1
L112.2.79.8.1.14.1
L112.2.89.7.1.8.1
L112.4.12.28.1
L116.2.115
L116.2.2.58.1
L116.2.23.29.1
L117.3.13.39.1
L120.2.116.4.1
L120.2.28.10.1.12.1
L120.2.30.6.1.20.1
L120.2.59.4.1.30.1
L120.2.60.60.1
L120.2.68.4.1.6.1.10.1
L120.2.70.3.1.4.1.10.1
L120.2.70.4.1.5.1.6.1
L120.2.74.4.1.15.1
L120.2.75.4.1.10.1
L120.2.75.4.1.6.1
L120.2.79.4.1.5.1
L120.2.87.3.1.4.1
L121.11.12
L124.2.123
L124.2.2.62.1
L124.2.22.31.1
L125.5.25.25.1
L126.3.20.6.1.21.1
L126.3.21.42.1
L126.3.23.6.1.7.1
L126.3.24.14.1
L128.2.3.4.11.8.13
L128.2.3.4.18.8.10
L128.2.3.4.25.8.7
L128.2.4.4.15.8.9.16.1
L128.2.4.4.22.8.6.16.1
L128.2.4.4.29.8.3.16.1
L128.2.4.4.36.16.1
L128.2.4.4.8.8.12.16.1
L128.2.5.4.10.8.11.16.1
L128.2.5.4.17.8.8.16.1
L128.2.5.4.24.8.5.16.1
L128.2.5.4.31.8.2.16.1
L128.2.5.4.8.8.14
L128.2.6.4.12.8.10.16.1
L128.2.6.4.19.8.7.16.1
L128.2.6.4.26.8.4.16.1
L128.2.6.4.33.8.1.16.1
L128.2.6.4.5.8.13.16.1
L128.2.15.8.1
L128.2.64.64.1
L128.4.32.32.1
L128.8.16.16.1
L132.2.131
L132.2.15.6.1.22.1
L132.2.18.3.1.22.1
L132.2.18.6.1.11.1
L132.2.2.66.1
L132.2.22.33.1
L132.2.27.11.1
L132.2.42.6.1
L135.3.27.45.1
L135.3.32.9.1.15.1
L136.2.132.4.1
L136.2.67.4.1.34.1
L136.2.68.68.1
L136.2.83.4.1.17.1
L140.2.139
L140.2.17.10.1.14.1
L140.2.2.70.1
L140.2.21.7.1.10.1
L140.2.22.35.1
L140.2.25.5.1.14.1
L140.2.27.5.1.7.1
L140.2.34.14.1
L140.2.36.10.1
L140.2.38.7.1
L144.12.7
L144.2.103.8.1.18.1
L144.2.111.6.1.24.1
L144.2.113.3.1.24.1
L144.2.117.8.1.9.1
L144.2.136.8.1
L144.2.16.3.3.6.6.24.1
L144.2.44.3.11.12.2
L144.2.72.72.1
L144.2.74.3.4.6.6.8.1
L144.2.75.3.3.4.1.6.6.12.1
L144.2.76.3.12.6.4.8.1
L144.2.76.3.7.4.1.6.5.12.1
L144.3.48.48.1
L144.4.11.12.2
L144.4.36.36.1
L256.2.19
L2048.2.63
L32.2.9
L32.2.16
L32.2.4.4.2
L40.2.6.5.1
L48.2.9.3.1
L48.2.7.6.1
L48.2.4.3.1.4.1
L54.2.1.3.5
L64.2.12.4.2
L64.2.8.4.3
L64.2.7.8.1
L64.2.6.4.4
L64.4.6
L72.2.12.3.2
L72.2.4.3.1.6.1
L80.2.12.5.1
L80.2.6.4.1.5.1
L96.2.7.3.1
L96.2.20.4.2
L96.2.5.4.2.6.1
L128.2.6.4.2
L128.2.20.4.3
L128.2.28.4.2
L128.2.8.8.2
L192.2.36.4.3
L243.3.20
L256.2.24.8.2
L256.2.52.4.3
L256.4.17
L372.2.40.8.2
L512.2.56.8.2
L729.3.12
L729.3.14
L4096.4.12

Arguments

Value

All arrays are matrices of class oa, with all colums coded as integers from 1 to the number of levels. Attributes origin and comment are sometimes available.

Warning

For designs with only 2-level factors, it is usually more wise to use package FrF2. Exceptions: The three arrays by Mee (2009; cf. section “Details” above) are very useful for 2-level factors.

Make sure you understand the implications of using an orthogonal main effects array for experimentation. In particular, for some arrays there is a very severe risk of obtaining biased main effect estimates, if there are some interactions between experimental factors. The documentations for generalized.word.length and function oa.design contain examples that illustrate this remark.

Details

All arrays are guaranteed to have orthogonal main effects.

When being fully populated with experimental factors, most of the arrays are guaranteed to work well only under the ASSUMPTION that there are NO INTERACTIONS; for arrays documented in oacat, known exceptions are noted in the comment attribute of the array. Exceptions are, for example, arrays L128.2.15.8.1, L256.2.19 and L2048.2.63, which have been taken from Mee (2009, chapter 8) and are resolution V in the 2-level factors (but are not regular arrays, there is partial aliasing between higher order effects). Further stronger arrays have been added since version 0.28 of the package, and are documented in oacat3.

Populating a main effects array with fewer than the maximum number of factors can result in a reasonable design even in the presence of interactions. The degree of confounding can be checked using various functions based on generalized.word.length, and some optimization of column allocation is possible with the column option of function oa.design. Such investigations of a designs properties work well for smaller designs but may be resource-wise prohibitive for larger designs / numbers of factors.

The array names indicate the number of runs and the numbers of factors: The first portion of each array name (starting with L) indicates number of runs, each subsequent pair of numbers indicates a number of levels together with the frequency with which it occurs. For example, L18.3.6.6.1 is an 18 run design with six factors with 3 levels each and one factor with 6 levels.

It is possible to obtain an overview about available arrays for a certain purpose by using function show.oas, based on the data frames oacat or oacat3, which hold entries for most arrays and their numbers of factors (exceptions: L18, L36 and L54 are Taguchi arrays explicitly given, which are listed in oacat in an isomorphic but not identical form ). Data frame oacat additionally holds entries for further attrays that can be constructed from the above-listed explicitly available arrays as “child arrays”, following so-called “lineage” recipes.

The source for most parent arrays as listed in oacat as well as for the lineages for the child arrays is Warren Kuhfelds (2009) collection; the Taguchi arrays L18, L36 and L54 are available in addition (not listed in oacat), and the Mee 2009 resolution V arrays mentioned above are for historical reasons still listed in oacat. All stronger parent arrays (strength > 2, resolution > III) are listed in oacat3. The arrays from oacat3 have been pulled together from several sources, as documented in the origin attribute of the respective array; all the sources are listed in the references below.

It is also possible to combine arrays with each other by so-called expansive replacement (expansive.replace), using the nesting process described by Warren Kuhfeld.

References

Agrawal, V. and Dey, A. (1983). Orthogonal resolution IV designs for some asymmetrical factorials. Technometrics 25, 197--199.

Brouwer, A. Small mixed fractional factorial designs of strength 3. https://www.win.tue.nl/~aeb/codes/oa/3oa.html#toc1 accessed March 1 2016

Brouwer, A., Cohen, A.M. and Nguyen, M.V.M. (2006). Orthogonal arrays of strength 3 and small run sizes. Journal of Statistical Planning and Inference 136, 3268--3280.

Eendebak, P. and Schoen, E. Complete Series of Orthogonal Arrays. http://pietereendebak.nl/oapage/ accessed March 1 2016

Hedayat, A.S., Sloane, N.J.A. and Stufken, J. (1999) Orthogonal Arrays: Theory and Applications, Springer, New York.

Kuhfeld, W. (2009). Orthogonal arrays. Website courtesy of SAS Institute https://support.sas.com/techsup/technote/ts723b.pdf and references therein.

Mee, R. (2009). A Comprehensive Guide to Factorial Two-Level Experimentation. New York: Springer.

Nguyen, M.V.M. (2005). Journal of Statistical Planning and Inference 138, 220--233.

Nguyen, M.V.M. (2008). Some new constructions of strength 3 mixed orthogonal arrays. Journal of Statistical Planning and Inference 138, 220--233.

Sloane, N. Orthogonal Arrays. http://neilsloane.com/oadir/ accessed March 1 2016