oacat: Data Frames That List Available Orthogonal Arrays
In DoE.base: Full Factorials, Orthogonal Arrays and Base Utilities for DoE Packages
oacat R Documentation
Data Frames That List Available Orthogonal Arrays
Description
These data frames hold the lists of available orthogonal arrays,
except for a few structurally equivalent additional arrays known
as Taguchi arrays (L18, L36, L54). Arrays in
in oacat are mostly from the Kuhfeld collection,
those in oacat3 from some other sources.
Usage
oacat
oacat3
Details
The data frames hold a list of orthogonal arrays, as described in Section “value”.
Inspection of these arrays can be most easily done with function show.oas.
Some of the listed arrays are directly accessible through their names (“parent” arrays,
also listed under arrays) or
are full factorials the construction of which is obvious. Others
can be constructed as “child” arrays from the parent and full factorial
arrays, using a so-called lineage which is also included as a column
in data frame oacat. Most of the listed arrays have been taken
from Kuhfeld 2009. Exceptions: The three arrays L128.2.15.8.1,
L256.2.19 and L2048.2.63) have been taken from Mee 2009; these
are irregular resolution IV or V arrays for which all main effects can be
orthogonally estimated even in the presence of interactions, or even all 2fis
can be orthogonally estimated, provided there are no higher order effects.
Note that most of the arrays in oacat, per default, are guaranteed to
orthogonally estimate
all main effects, provided all higher order effects are negligible
(again, the Mee arrays are an exception). This can be
a very severe limitation, of course, and arbitrary strong biases can distort the
estimates even of main effects, if this assumption is violated.
It is therefore strongly recommended to inspect
the quality of an orthogonal array quite closely before deciding to use it
for experimentation. Some functions for inspecting arrays are provided in the
package (cf. generalized.word.length).
The data frame oacat3 contains stronger arrays that have at least the main
effects unconfounded with two-factor interactions. If only these are of interest,
function show.oas can be restricted to strong arrays
by option Rgt3=TRUE. Function oa.design will use a strong
array, if possible. Since package version 1.2, oacat3 contains arrays
that were obtained via expansive replacement (indicated in the lineage
column). It is important to note that this automatic replacement is not optimized
in any way; in some cases it may be worthwhile to check whether a better array
can be produced with different level choices or by expanding not the first but
a different column of the parent array
(for an example, see function expansive.replace); this is not
automatically checked and can only be done by the user.
Value
The data frames contain the columns name, nruns, lineage
and further columns n2 to n72; furthermore, some columns with
calculated metrics are included. name holds the name of the
array, nruns its number of runs, and lineage the way the array can
be constructed from other arrays, if applicable. The columns n2 to n72
each contain the number of factors with the respective number of levels.
The logical columns ff, regular.strict and regular indicate a
full factorial and a regular design in the strict or weak sense, respectively
(strict: all ARFT entries are 0 or 1, defined as “R^2 regular” in Groemping and Bailey (2016);
weak: all SCFT entries are 0 or 1, defined as “CC regular” in
Groemping and Bailey (2016)). For R^2 regularity, it suffices to check all full resolution factor sets,
i.e., sets of j factors with resolution j; for CC regularity, this is conjectured to be also true.
The entries in column regular are based on that conjecture (and for some larger designs,
even those checks were not completed);
thus, designs denominated as CC regular might prove otherwise if the conjecture
proves wrong, and for larger designs also for unchecked full resolution factor sets of higher dimensions).
Column SCones contains the number of worst case (=1)
squared canonical correlations for the number of R factor subsets, with
R the resolution; if this number is 0, main effects can be considered
to have partial confounding only with any interactions of up to R-1 factors.
GR, GRind, maxAR
and maxSC contain the generalized resolution in two versions,
the maximum average R^2 and the maximum squared canonical correlation.
dfe contains the error degrees of freedom of a main effects model,
if all columns of the array are populated; if this is 0, the design is saturated.
A3 to A8 contain the numbers of words of lengths 3 to 8.
More information on these metrics can be found in
generalized.word.length and the literature therein.
The design names also indicate the number of runs and the numbers of factors:
The first portion of each array name (starting with L) indicates the number of runs,
each subsequent pair of numbers indicates a number of levels together with the frequency with which it occurs.
For example, L18.2.1.3.7 is an 18 run design with one factor with
2 levels and seven factors with 3 levels each.
The columns gmarule and sgmarule refer to the implementation of
known rules from the literature that certain subsets of array columns have
generalized minimum aberration (Butler 2005); if such a subset is requested,
there is no message of caution even if the array columns are used with
column="order" instead of optimizing the selection. Currently,
only the rules from Butler (2005) are implemented; hopefully, more rules will be added
in the future.
The column lineage deserves particular attention:
it is an empty string, if the design is directly available and can be accessed via its name, or if the design
is a full factorial (e.g. L6.2.1.3.1). Otherwise, the lineage entry is structured as follows:
It starts with the specification of a parent array, given as levels1~no of factors; levels2~no of factors;.
After a colon, there are one or more replacements, each enclosed in brackets; within each pair of brackets,
the left-hand side of the exclamation mark shows the to-be-replaced factor, the right-hand side the
replacement array that has to be used for replacing the levels of such a factor one or more times. For example,
the lineage for L18.2.1.3.7 is 3~6;6~1;:(6~1!2~1;3~1;), which means that the parent array in
18 runs with six 3 level factors and one 6 level factor has to be used, and the 6 level factor has to be replaced
with the full factorial with one 2 level factor and one 3 level factor.
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.
Many of the orthogonal arrays from oacat,
especially when using all columns for experimentation,
are guaranteed to orthogonally estimate all main effects,
provided all higher order effects are negligible.
Make sure you understand the implications of using an orthogonal main effects design
for experimentation. In particular, for some designs there is a very severe
risk of obtaining biased main effect estimates, if there are some interactions between
experimental factors. The documentation for generalized.word.length and
examples section below that illustrate this remark.
Cf. also the instructions in section “Details”).
Author(s)
Ulrike Groemping, with contributions by Boyko Amarov
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.
Butler, N.A. (2005). Generalised minimum aberration construction results for symmetrical orthogonal arrays. Biometrika 92, 485 – 491.
Eendebak, P. and Schoen, E. Complete Series of Orthogonal Arrays. http://www.pietereendebak.nl/oapackage/series.html accessed March 1 2016
Groemping, U. and Bailey, R.A. (2016). Regular fractions of factorial arrays. In:
mODa 11 – Advances in Model-Oriented Design and Analysis.
Cham: Springer International Publishing.
Groemping, U. and Fontana, R. (2019). An Algorithm for Generating Good Mixed
Level Factorial Designs. Computational Statistics and Data Analysis 137, 101–114.
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
See Also
oa.design for using the designs from oacat in design creation
show.oas for inspecting the available arrays from oacat
generalized.word.length for inspection functions for array properties
arrays for a list of orthogonal arrays which are directly accessible
within the package
Examples
head(oacat)
sapply(oacat3$name[which(oacat3$lineage=="")],
function(nn) unlist(attributes(get(nn))[c("origin", "comment")]))
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| oacat | R Documentation |
Data Frames That List Available Orthogonal Arrays
Description
These data frames hold the lists of available orthogonal arrays, except for a few structurally equivalent additional arrays known as Taguchi arrays (L18, L36, L54). Arrays in in oacat are mostly from the Kuhfeld collection, those in oacat3 from some other sources.
Usage
oacat
oacat3
Details
The data frames hold a list of orthogonal arrays, as described in Section “value”.
Inspection of these arrays can be most easily done with function show.oas.
Some of the listed arrays are directly accessible through their names (“parent” arrays,
also listed under arrays) or
are full factorials the construction of which is obvious. Others
can be constructed as “child” arrays from the parent and full factorial
arrays, using a so-called lineage which is also included as a column
in data frame oacat. Most of the listed arrays have been taken
from Kuhfeld 2009. Exceptions: The three arrays L128.2.15.8.1,
L256.2.19 and L2048.2.63) have been taken from Mee 2009; these
are irregular resolution IV or V arrays for which all main effects can be
orthogonally estimated even in the presence of interactions, or even all 2fis
can be orthogonally estimated, provided there are no higher order effects.
Note that most of the arrays in oacat, per default, are guaranteed to
orthogonally estimate
all main effects, provided all higher order effects are negligible
(again, the Mee arrays are an exception). This can be
a very severe limitation, of course, and arbitrary strong biases can distort the
estimates even of main effects, if this assumption is violated.
It is therefore strongly recommended to inspect
the quality of an orthogonal array quite closely before deciding to use it
for experimentation. Some functions for inspecting arrays are provided in the
package (cf. generalized.word.length).
The data frame oacat3 contains stronger arrays that have at least the main
effects unconfounded with two-factor interactions. If only these are of interest,
function show.oas can be restricted to strong arrays
by option Rgt3=TRUE. Function oa.design will use a strong
array, if possible. Since package version 1.2, oacat3 contains arrays
that were obtained via expansive replacement (indicated in the lineage
column). It is important to note that this automatic replacement is not optimized
in any way; in some cases it may be worthwhile to check whether a better array
can be produced with different level choices or by expanding not the first but
a different column of the parent array
(for an example, see function expansive.replace); this is not
automatically checked and can only be done by the user.
Value
The data frames contain the columns name, nruns, lineage
and further columns n2 to n72; furthermore, some columns with
calculated metrics are included. name holds the name of the
array, nruns its number of runs, and lineage the way the array can
be constructed from other arrays, if applicable. The columns n2 to n72
each contain the number of factors with the respective number of levels.
The logical columns ff, regular.strict and regular indicate a
full factorial and a regular design in the strict or weak sense, respectively
(strict: all ARFT entries are 0 or 1, defined as “R^2 regular” in Groemping and Bailey (2016);
weak: all SCFT entries are 0 or 1, defined as “CC regular” in
Groemping and Bailey (2016)). For R^2 regularity, it suffices to check all full resolution factor sets,
i.e., sets of j factors with resolution j; for CC regularity, this is conjectured to be also true.
The entries in column regular are based on that conjecture (and for some larger designs,
even those checks were not completed);
thus, designs denominated as CC regular might prove otherwise if the conjecture
proves wrong, and for larger designs also for unchecked full resolution factor sets of higher dimensions).
Column SCones contains the number of worst case (=1)
squared canonical correlations for the number of R factor subsets, with
R the resolution; if this number is 0, main effects can be considered
to have partial confounding only with any interactions of up to R-1 factors.
GR, GRind, maxAR
and maxSC contain the generalized resolution in two versions,
the maximum average R^2 and the maximum squared canonical correlation.
dfe contains the error degrees of freedom of a main effects model,
if all columns of the array are populated; if this is 0, the design is saturated.
A3 to A8 contain the numbers of words of lengths 3 to 8.
More information on these metrics can be found in
generalized.word.length and the literature therein.
The design names also indicate the number of runs and the numbers of factors:
The first portion of each array name (starting with L) indicates the number of runs,
each subsequent pair of numbers indicates a number of levels together with the frequency with which it occurs.
For example, L18.2.1.3.7 is an 18 run design with one factor with
2 levels and seven factors with 3 levels each.
The columns gmarule and sgmarule refer to the implementation of
known rules from the literature that certain subsets of array columns have
generalized minimum aberration (Butler 2005); if such a subset is requested,
there is no message of caution even if the array columns are used with
column="order" instead of optimizing the selection. Currently,
only the rules from Butler (2005) are implemented; hopefully, more rules will be added
in the future.
The column lineage deserves particular attention:
it is an empty string, if the design is directly available and can be accessed via its name, or if the design
is a full factorial (e.g. L6.2.1.3.1). Otherwise, the lineage entry is structured as follows:
It starts with the specification of a parent array, given as levels1~no of factors; levels2~no of factors;.
After a colon, there are one or more replacements, each enclosed in brackets; within each pair of brackets,
the left-hand side of the exclamation mark shows the to-be-replaced factor, the right-hand side the
replacement array that has to be used for replacing the levels of such a factor one or more times. For example,
the lineage for L18.2.1.3.7 is 3~6;6~1;:(6~1!2~1;3~1;), which means that the parent array in
18 runs with six 3 level factors and one 6 level factor has to be used, and the 6 level factor has to be replaced
with the full factorial with one 2 level factor and one 3 level factor.
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.
Many of the orthogonal arrays from oacat,
especially when using all columns for experimentation,
are guaranteed to orthogonally estimate all main effects,
provided all higher order effects are negligible.
Make sure you understand the implications of using an orthogonal main effects design
for experimentation. In particular, for some designs there is a very severe
risk of obtaining biased main effect estimates, if there are some interactions between
experimental factors. The documentation for generalized.word.length and
examples section below that illustrate this remark.
Cf. also the instructions in section “Details”).
Author(s)
Ulrike Groemping, with contributions by Boyko Amarov
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.
Butler, N.A. (2005). Generalised minimum aberration construction results for symmetrical orthogonal arrays. Biometrika 92, 485 – 491.
Eendebak, P. and Schoen, E. Complete Series of Orthogonal Arrays. http://www.pietereendebak.nl/oapackage/series.html accessed March 1 2016
Groemping, U. and Bailey, R.A. (2016). Regular fractions of factorial arrays. In: mODa 11 – Advances in Model-Oriented Design and Analysis. Cham: Springer International Publishing.
Groemping, U. and Fontana, R. (2019). An Algorithm for Generating Good Mixed Level Factorial Designs. Computational Statistics and Data Analysis 137, 101–114.
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
See Also
oa.design for using the designs from oacat in design creation
show.oas for inspecting the available arrays from oacat
generalized.word.length for inspection functions for array properties
arrays for a list of orthogonal arrays which are directly accessible
within the package
Examples
head(oacat)
sapply(oacat3$name[which(oacat3$lineage=="")],
function(nn) unlist(attributes(get(nn))[c("origin", "comment")]))
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