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