1\name{oacat}
2\alias{oacat}
3\alias{oacat3}
4%- Also NEED an '\alias' for EACH other topic documented here.
5\title{
6Data Frames That List Available Orthogonal Arrays
7}
8\description{
9These data frames hold the lists of available orthogonal arrays,
10except for a few structurally equivalent additional arrays known
11as Taguchi arrays (L18, L36, L54). Arrays in
12in oacat are mostly from the Kuhfeld collection,
13those in oacat3 from some other sources.
14}
15\usage{
16oacat
17oacat3
18}
19
20\details{
21   The data frames hold a list of orthogonal arrays, as described in Section \dQuote{value}.
22   Inspection of these arrays can be most easily done with function \code{\link{show.oas}}.
23   Some of the listed arrays are directly accessible through their names (\dQuote{parent} arrays,
24   also listed under \code{\link{arrays}}) or
25   are full factorials the construction of which is obvious. Others
26   can be constructed as \dQuote{child} arrays from the parent and full factorial
27   arrays, using a so-called \code{lineage} which is also included as a column
28   in data frame \code{oacat}. Most of the listed arrays have been taken
29   from Kuhfeld 2009. Exceptions: The three arrays \code{L128.2.15.8.1},
30   \code{L256.2.19} and \code{L2048.2.63}) have been taken from Mee 2009; these
31   are irregular resolution IV or V arrays for which all main effects can be
32   orthogonally estimated even in the presence of interactions, or even all 2fis
33   can be orthogonally estimated, provided there are no higher order effects.
34
35   Note that most of the arrays in \code{oacat}, per default, are guaranteed to
36   orthogonally estimate
37   all main effects, \bold{provided all higher order effects are negligible}
38   (again, the Mee arrays are an exception). This can be
39   a very severe limitation, of course, and arbitrary strong biases can distort the
40   estimates even of main effects, if this assumption is violated.
41   It is therefore strongly recommended to inspect
42   the quality of an orthogonal array quite closely before deciding to use it
43   for experimentation. Some functions for inspecting arrays are provided in the
44   package (cf. \code{\link{generalized.word.length}}).
45
46   The data frame \code{oacat3} contains stronger arrays that have at least the main
47   effects unconfounded with two-factor interactions. If only these are of interest,
48   function \code{\link{show.oas}} can be restricted to strong arrays
49   by option \code{Rgt3=TRUE}. Function \code{\link{oa.design}} will use a strong
50   array, if possible. It may also be worthwhile to check whether expansive replacement
51   of a strong array with a full factorial can yield a suitable strong array
52   (for an example, see function \code{\link{expansive.replace}}); this is not
53   automatically checked and can only be done by the user.
54}
55\value{
56   The data frames contain the columns \code{name}, \code{nruns}, \code{lineage}
57   and further columns \code{n2} to \code{n72}; furthermore, some columns with
58   calculated metrics are included. \code{name} holds the name of the
59   array, \code{nruns} its number of runs, and \code{lineage} the way the array can
60   be constructed from other arrays, if applicable. The columns \code{n2} to \code{n72}
61   each contain the number of factors with the respective number of levels.
62
63   The logical columns \code{ff}, \code{regular.strict} and \code{regular} indicate a
64   full factorial and a regular design in the strict or weak sense, respectively
65   (strict: all ARFT entries are 0 or 1, defined as \dQuote{R^2 regular} in Groemping and Bailey (2016);
66   weak: all SCFT entries are 0 or 1, defined as \dQuote{CC regular} in
67   Groemping and Bailey (2016)). For R^2 regularity, it suffices to check all full resolution factor sets,
68   i.e., sets of j factors with resolution j; for CC regularity, this is conjectured to be also true.
69   The entries in column \code{regular} are based on that conjecture (and for some larger designs,
70   even those checks were not completed);
71   thus, designs denominated as CC regular might prove otherwise if the conjecture
72   proves wrong, and for larger designs also for unchecked full resolution factor sets of higher dimensions).
73
74   Column \code{SCones} contains the number of worst case (=1)
75   squared canonical correlations for the number of R factor subsets, with
76   R the resolution; if this number is 0, main effects can be considered
77   to have partial confounding only with any interactions of up to R-1 factors.
78   \code{GR}, \code{GRind}, \code{maxAR}
79   and \code{maxSC} contain the generalized resolution in two versions,
80   the maximum average R^2 and the maximum squared canonical correlation.
81
82   \code{dfe} contains the error degrees of freedom of a main effects model,
83   if all columns of the array are populated; if this is 0, the design is saturated.
84   \code{A3} to \code{A8} contain the numbers of words of lengths 3 to 8.
85   More information on these metrics can be found in
86   \code{\link{generalized.word.length}} and the literature therein.
87
88   The design names also indicate the number of runs and the numbers of factors:
89   The first portion of each array name (starting with L) indicates the number of runs,
90   each subsequent pair of numbers indicates a number of levels together with the frequency with which it occurs.
91   For example, \code{L18.2.1.3.7} is an 18 run design with one factor with
92   2 levels and seven factors with 3 levels each.
93
94   The columns \code{gmarule} and \code{sgmarule} refer to the implementation of
95   known rules from the literature that certain subsets of array columns have
96   generalized minimum aberration (Butler 2005); if such a subset is requested,
97   there is no message of caution even if the array columns are used with
98   \code{column="order"} instead of optimizing the selection. Currently,
99   only the rules from Butler (2005) are implemented; hopefully, more rules will be added
100   in the future.
101
102   The column \code{lineage} deserves particular attention for \code{oacat} (always empty for \code{oacat3}):
103   it is an empty string, if the design is directly available and can be accessed via its name, or if the design
104   is a full factorial (e.g. L6.2.1.3.1). Otherwise, the lineage entry is structured as follows:
105   It starts with the specification of a parent array, given as \code{levels1~no of factors; levels2~no of factors;}.
106   After a colon, there are one or more replacements, each enclosed in brackets; within each pair of brackets,
107   the left-hand side of the exclamation mark shows the to-be-replaced factor, the right-hand side the
108   replacement array that has to be used for replacing the levels of such a factor one or more times. For example,
109   the lineage for \code{L18.2.1.3.7} is \code{3~6;6~1;:(6~1!2~1;3~1;)}, which means that the parent array in
110   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
111   with the full factorial with one 2 level factor and one 3 level factor.
112}
113\author{
114   Ulrike Groemping, with contributions by Boyko Amarov
115}
116\section{Warning}{
117   For designs with only 2-level factors, it is usually more wise to
118   use package \pkg{\link[FrF2:FrF2-package]{FrF2}}. Exceptions: The three arrays by
119   Mee (2009; cf. section \dQuote{Details} above) are very useful for 2-level factors.
120
121   Most of the orthogonal arrays from \code{oacat},
122   especially when using all columns for experimentation,
123   are guaranteed to orthogonally estimate all main effects,
124   \bold{provided all higher order effects are negligible}.
125
126   Make sure you understand the implications of using an orthogonal main effects design
127   for experimentation. In particular, for some designs there is a very severe
128   risk of obtaining biased main effect estimates, if there are some interactions between
129   experimental factors. The documentation for \code{\link{generalized.word.length}} and
130   examples section below that illustrate this remark.
131   Cf. also the instructions in section \dQuote{Details}).
132}
133\references{
134  Agrawal, V. and Dey, A. (1983). Orthogonal resolution IV designs for some asymmetrical factorials.
135  \emph{Technometrics} \bold{25}, 197--199.
136
137  Brouwer, A. Small mixed fractional factorial designs of strength 3. \url{https://www.win.tue.nl/~aeb/codes/oa/3oa.html#toc1} accessed March 1 2016
138
139  Brouwer, A., Cohen, A.M. and Nguyen, M.V.M. (2006). Orthogonal arrays of strength 3 and small run sizes. \emph{Journal of Statistical Planning and Inference} \bold{136}, 3268--3280.
140
141  Butler, N.A. (2005). Generalised minimum aberration construction results for symmetrical orthogonal arrays. \emph{Biometrika} \bold{92}, 485 -- 491.
142
143  Eendebak, P. and Schoen, E. Complete Series of Orthogonal Arrays. \url{http://pietereendebak.nl/oapage/} accessed March 1 2016
144
145  Groemping, U. and Bailey, R.A. (2016). Regular fractions of factorial arrays. In:
146  \emph{mODa 11 -- Advances in Model-Oriented Design and Analysis}.
147    Cham: Springer International Publishing.
148
149  Kuhfeld, W. (2009). Orthogonal arrays. Website courtesy of SAS Institute \url{http://support.sas.com/techsup/technote/ts723.html}.
150
151  Mee, R. (2009). \emph{A Comprehensive Guide to Factorial Two-Level Experimentation}.
152    New York: Springer.
153
154  Nguyen, M.V.M. (2005). \emph{Journal of Statistical Planning and Inference} \bold{138},
155    220--233.
156
157  Nguyen, M.V.M. (2008). Some new constructions of strength 3 mixed orthogonal arrays. \emph{Journal of Statistical Planning and Inference} \bold{138},
158    220--233.
159
160  Sloane, N. Orthogonal Arrays. \url{http://neilsloane.com/oadir/} accessed March 1 2016
161
162  }
163\examples{
164   head(oacat)
165
166   sapply(oacat3$name, function(nn) unlist(attributes(get(nn))[c("origin", "comment")]))
167
168}
169\seealso{
170    \code{\link{oa.design}} for using the designs from \code{oacat} in design creation\cr
171    \code{\link{show.oas}} for inspecting the available arrays from \code{oacat}\cr
172    \code{\link{generalized.word.length}} for inspection functions for array properties\cr
173    \code{\link{arrays}} for a list of orthogonal arrays which are directly accessible
174    within the package
175}
176
177\keyword{ array }
178\keyword{ design }
179