Orthogonal arrays in the package
## strength 5 / resolution VI
L243.3.6
L384.2.4.3.1.4.2
L729.3.12
L1024.4.6
L2187.3.14
L4096.4.12
L6561.3.28 ## strength 4 / resolution V
L81.3.5
L96.2.7.3.1
L128.2.6.4.2
L192.2.3.3.1.4.2
L192.2.2.4.2.6.1
L243.3.11
L256.2.19
L256.4.5
L576.2.2.3.1.4.2.6.1
L625.5.6
L729.3.14
L1024.4.11
L2048.2.63
L2187.3.27
L2401.7.8
L4096.4.21
L4096.8.9
L6561.3.41
L6561.9.10
## strength 3 / resolution IV
L27.3.4
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
L48.2.3.3.1.4.1
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
L81.3.8
L81.3.10
L96.2.20.4.2
L96.2.5.4.2.6.1
L125.5.6
L128.2.20.4.3
L128.2.28.4.2
L128.2.15.8.1
L128.2.8.8.2
L192.2.1.3.1.4.3
L192.2.36.4.3
L243.3.20
L256.2.24.8.2
L256.2.52.4.3
L256.4.17
L256.4.85
L343.7.8
L384.2.40.8.2
L512.2.56.8.2
L512.8.9
L576.3.1.4.3.6.1
L729.3.56
L729.9.10
L1024.4.41
L2187.3.112
L4096.4.126
L6561.3.248
## strength 2 / resolution III
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
L144.2.1.3.2.4.2
L144.2.2.3.2.4.2
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
L144.12.7
L216.2.1.3.2.4.1.6.1
L243.3.121
L288.3.2.4.2.6.1
L432.2.1.3.3.4.2
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.
For designs with only 2-level factors, it is usually more wise to
use package FrF2. Exceptions: Three arrays by
Mee (2009), namely L128.2.15.8.1, L256.19, 2048.2.63, are very useful for 2-level factors.
When using a strength 2 array with only few error degrees of freedom (dfe in oacat), 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.
Ulrike Groemping
All arrays are guaranteed to have orthogonal main effects. The package holds arrays of resolution III (strength 2), tabulated in the catalogue oacat, and stronger arrays that are tabulated in the catalogue oacat3. Inspection of all arrays is possible via function show.oas.
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 arrays 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.
When being fully populated
with experimental factors, many of the strength 2 = resolution III arrays are guaranteed to work well only
under the ASSUMPTION that there are NO INTERACTIONS. Exceptions are, for example, arrays
L128.2.15.8.1 (the 2-level factors have resolution V / strength 4, as noted in the array's comment attribute) or L144.2.1.3.2.4.2 (the strength is almost 3, as can be seen from its GR value).
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 argument 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.
oacat3 was added with version 0.28 of the package, and version 1.2 substantially extended that collection. Contrary to the resolution III arrays, there are no automatically created children for the stronger arrays.
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. The “Examples” section shows how users can create custom expansions.
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://www.pietereendebak.nl/oapackage/series.html accessed March 1 2016
Groemping, U. and Fontana, R. (2019). An Algorithm for Generating Good Mixed Level Factorial Designs. Computational Statistics and Data Analysis 137, 101--114.
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.
MinT, the online database for optimal parameters of (t,m,s)-nets, (t,s)-sequences, orthogonal arrays, linear codes, and OOAs. Accessed August 2021. http://mint.sbg.ac.at/index.php.
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.
Pirsic, I. (2021). Personal communication regarding various specific generators from MinT.
Schuerer, R. and Schmid, W.Ch. (2010). MinT-Architecture and applications of the (t, m, s)-net and OOA database. Mathematics and Computers in Simulation 80(6), 1124-1132. https://doi.org/10.1016/j.matcom.2007.09.010.
Sloane, N. Orthogonal Arrays. http://neilsloane.com/oadir/ accessed March 1 2016
See also oacat, show.oas, generalized.word.length,
oa.design, FrF2, pb
## we want 729 runs with six 3-level factors and one 9-level factor
## with resolution higher than 3
show.oas(nruns=729, nlevels=c(3,3,3,3,3,3,9), Rgt3=TRUE)
## it can also be found if there is an OA with at least four 9-level factors
show.oas(nruns=729, nlevels=c(9,9,9,9), Rgt3=TRUE)
## create full factorial replacement matrix
threetimesthree <- as.matrix(expand.grid(1:3,1:3))
dim(threetimesthree)
## extract four nine-level columns,
## and expand the first three
L729.3.6.9.1 <-
expansive.replace(
expansive.replace(
expansive.replace(L729.9.10[,1:4],
threetimesthree),
threetimesthree),
threetimesthree)
class(L729.3.6.9.1) <- c("oa", "matrix")
oa.design(L729.3.6.9.1)
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