R: Function to provide regular Fractional Factorial 2-level...

FrF2 {FrF2}

R Documentation

Function to provide regular Fractional Factorial 2-level designs

Description

Regular fractional factorial 2-level designs are provided. Apart from obtaining the usual minimum aberration designs in a fixed number of runs, it is possible to request highest number of free 2-factor interactions instead of minimum aberration or to request the smallest design that fulfills certain requirements (e.g. resolution V with 8 factors).

Usage

FrF2(nruns = NULL, nfactors = NULL, factor.names = if (!is.null(nfactors)) {
        if (nfactors <= 50) Letters[1:nfactors] else 
                             paste("F", 1:nfactors, sep = "")} else NULL, 
        default.levels = c(-1, 1), ncenter=0, center.distribute=NULL, 
        generators = NULL, design = NULL, 
        resolution = NULL, select.catlg=catlg, 
        estimable = NULL, clear = TRUE, method="VF2", sort="natural", 
        ignore.dom = !isTRUE(all.equal(blocks,1)), 
        useV = TRUE, firsthit=FALSE, res3 = FALSE, max.time = 60, 
        perm.start=NULL, perms = NULL, 
        MaxC2 = FALSE, replications = 1, repeat.only = FALSE, 
        randomize = TRUE, seed = NULL, alias.info = 2, 
        blocks = 1, block.name = "Blocks", block.old=FALSE, 
        force.godolphin=alias.block.2fis,
        bbreps=replications, wbreps=1, 
        alias.block.2fis = FALSE, 
        hard = NULL, check.hard=10, WPs=1,nfac.WP=0,
        WPfacs=NULL, check.WPs = 10, ...)
FrF2.currentlychecked()

Arguments

`nruns`	Number of runs, must be a power of 2 (4 to 4096), if given. The number of runs can also be omitted. In that case, if `resolution` is specified, the function looks for the smallest design of the requested resolution that accomodates `nfactors` factors. If the smallest possible design is a full factorial or not catalogued, the function stops with an error. If `generators` is specified, `nruns` is required. If estimable is specified and `nruns` omitted, `nruns` becomes the size of the smallest design that MIGHT accomodate the effects requested in `estimable`. If this run size turns out to be too low, an error is thrown. In that case, explicitly choose `nruns` as twice the run size given in the error message and retry.
`nfactors`	is the number of 2-level factors to be investigated. It can be omitted, if it is obvious from `factor.names`, a specific catalogued design given in `design`, `nruns` together with `generators`, or `estimable`. If `estimable` is used for determining the number of factors, it is assumed that the largest main effect position number occurring in `estimable` coincides with `nfactors`. For blocked designs, block generator columns are not included in `nfactors`, except if the user explicitly specifies 2-level block generation factors as part of the fractional factorial design (e.g. a factor shift with levels morning and afternoon). For automatically-generated split-plot designs (cf. details section), `nfactors` simply is the number of all factors (whole plot and split plot together). If `nfac.WP < log2(WPs)`, the algorithm will add (an) artificial plot generation factor(s). For manually-specified split-plot designs (through options `generators` or `design` together with `WPfacs`), the user must specify at least `log2(WPs)` split plot factors, i.e. `nfac.WP >= log2(WPs)` is required (and must, if necessary, be achieved by adding `log2(WPs) - nfac.WP` extra independent generator columns for the whole plot structure, which have to be counted in `nfac.WP` and `nfactors`).
`factor.names`	a character vector of `nfactors` factor names or a list with `nfactors` elements; if the list is named, list names represent factor names, otherwise default factor names are used; the elements of the list are EITHER vectors of length 2 with factor levels for the respective factor OR empty strings. For each factor with an empty string in factor.names, the levels given in `default.levels` are used; Default factor names are the first elements of the character vector `Letters`, or the factors position numbers preceded by capital F in case of more than 50 factors.
`default.levels`	default levels (vector of length 2) for all factors for which no specific levels are given
`ncenter`	number of center points per block; `ncenter > 0` is permitted, if all factors are quantitative and the design is not a split-plot design
`center.distribute`	the number of positions over which the center points are to be distributed for each block; if NULL (default), center points are distributed over end, beginning, and middle (in that order, if there are fewer than three center points) for randomized designs, and appended to the end for non-randomized designs. for more detail, see function `add.center`, which does the work.
`generators`	There are `log2(nruns)` base factors the full factorial of which spans the design (e.g. 3 for 8 runs). The generators specify how the remaining factors are to be allocated to interactions of these. `generators` can be a list of vectors with position numbers of base factors (e.g. c(1,3,4) stands for the interaction between first, third and fourth base factor) a vector of character representations of these interactions, e.g. “ACD” stands for the same interaction as above a vector of columns numbers in Yates order (e.g. 13 stands for ACD). Note that the columns 1, 2, 4, 8, etc., i.e. all powers of 2, are reserved for the base factors and cannot be used for assigning additional factors, because the design would become a resolution II design. For looking up which column number stands for which interaction, type e.g. `names(Yates)[1:15]` for a 16 run design. In all cases, preceding the respective entry with a minus sign (e.g. -c(1,3,4), “-ACD”, -13) implies that the levels of the respective column are reversed. WARNING: Minus signs do not cause an error, but neither have an effect in case of automatic assignment of split-plot designs or hard-to-change columns.
`design`	is a character string specifying the name of a design listed in the catalogue specified as `select.catlg`, which is usually the catalogue `catlg`
`resolution`	is the arabic numeral for the requested resolution of the design. `FrF2` looks for a design with at least this resolution. Option `resolution` does not work, if `estimable`, `blocks` or `WPs` are specified, and neither if `nruns` is given. A design with resolution III (resolution=3) confounds main effects with 2-factor interactions, a design with resolution IV confounds main effects with three-factor interactions or 2-factor interactions with each other, and designs with resolution V or higher are usually regarded as very strong, because all 2-factor interactions are unconfounded with each other and with main effects.
`select.catlg`	specifies a catalogue of class `catlg` from which an adequate design is selected and adapted. The specified catalogue is used for design construction, unless `generators` explicitly constructs a non-catalogued design. The default `catlg` is adequate for most applications. If a specific different catalogue of designs is available, this can be specified here. Specification of a smaller subset of designs is useful, if `estimable` has been given and `clear=FALSE`, for restricting the search to promising designs. Names of catalogues from package FrF2.catlg128 can be given here without prior loading of that package; loading of the package and the selected catalogue will then happen automatically, provided the package is installed (for version >=1.2 of package FrF2.catlg128; for earlier versions, the suitable catalogue has to be manually loaded using the `data()` command).
`estimable`	indicates the 2-factor interactions (2fis) that are to be estimable in the design. Consult the specific help file (`estimable.2fis`) for details of two different approaches of requesting estimability, as indicated by the status of the `clear` option. `estimable` cannot be specified together with `splitplot`, `generators` or `design`. `estimable` can be a numeric matrix with two rows, each column of which indicates one interaction, e.g. column 1 3 for interaction of the first with the third factor OR a character vector containing strings of length 2 with capital letters from `Letters` (cf. package DoE.base) for the first 25 factors and small letters for the last 25 (e.g. `c("AB","BE")` OR a formula that contains an adequate model formula, e.g. `formula("~A+B+C+D+E+(F+G+H+J+K+L)^2")` for a model with (at least) eleven factors. The names of the factors used in the formula can be the same letters usable in the character vector (cf. above, A the first factor, B the second etc.), or they can correspond to the factor names from `factor.names`.
`clear`	logical, indicating how estimable is to be used. See `estimable.2fis`.
`method`	character string indicating which subgraph isomorphism search routine of package igraph is used (`graph.subisomorphic.vf2` or `graph.subisomorphic.lad`). The default `"VF2"` uses VF2 algorithm by Cordella et al. (2001), which was the only available algorithm before version 1.7 of package FrF2. The alternative `"LAD"` uses the LAD algorithm by Solnon (2010), which was reported to be substantially faster than VF2 especially for some notoriously difficult VF2 cases (see Gr?mping 2014b). This option is relevant for `estimable` with `clear=TRUE` only. NOTE: The resulting design may be different for different settings of this option!
`sort`	character string indicating how the estimability requirement and the candidate design clear 2fis are handed to the subgraph isomorphism search routine of package igraph. The default `"natural"` leaves them in unchanged order (like in FrF2 versions up to 1.6). `sort="high"` and `sort="low"` sort both requirement set and candidate design graph according to vertex degrees (high first or low first). This option is relevant for `estimable` with `clear=TRUE` only. It has been added, because pre-sorting of vertices sometimes speeds up the search by several orders of magnitude especially for the `VF2` method. NOTE: The resulting design may be different for different settings of this option!
`ignore.dom`	logical, default FALSE for unblocked designs, TRUE for blocked designs; if TRUE, `estimable` ignores the dominating attribute of the catalogue entries; can be useful for searching a blocked design with estimable 2fis from a reduced catalogue
`useV`	NULL or logical; relevant for designs with `blocks` and `estimable` only; if TRUE, function `link{colpick}` is used (default), otherwise function `link{colpickIV}`; if set to NULL, `colpick` is used for fractions with resolution at least V and function `link{colpickIV}` for resolution IV fractions; `useV=FALSE` does the subgraph isomorphism check once only, at the expense of missing out opportunities; it may be worth trying `useV=NULL` or `useV=FALSE` for resolution IV situations, for which the search takes very long with the default
`firsthit`	logical; relevant for designs with `blocks` and `estimable` only; if FALSE, the function tries to find a design with as many as possible clear 2fis, otherwise the function stops at the first possible blocking; in case of resource problems, setting `firsthit` to TRUE may be helpful
`res3`	logical; if TRUE, `estimable` includes resolution III designs into the search for adequate designs; otherwise resolution IV and higher designs are included only.
`max.time`	maximum time for design search as requested by `estimable`, in seconds (default 60); introduced for `clear=FALSE` situations because the search can take a long time in complicated or unlucky situations; set max.time to Inf if you want to force a search over an extended period of time; however, be aware that it may still take longer than feasible (cf. also `estimable.2fis`)
`perm.start`	used with `estimable` specified, and `clear=FALSE`. Provides a start permutation for permuting experiment factors (numeric vector). This is useful for the case that a previous search was not (yet) successful because of a time limit, since the algorithm notifies the user about the permutation at which it had to stop.
`perms`	used with `estimable` specified, and `clear=FALSE`. Provides the matrix of permutations of experiment factors to be tried; each row is a permutation. For example, for an 11-factor design with the first six factors and their 2fis estimable, it is only relevant, which of the eleven factors are to be allocated to the first six experiment factors, and these as well as the other five factors can be in arbitrary order. This reduces the number of required permutations from about 40 Mio to 462. It is recommended to use `perms` whenever possible, if `clear=FALSE`, since this dramatically improves performance of the algorithm. It is planned to automatically generate perms for certain structures like compromise designs in the (not so near) future.
`MaxC2`	is a logical and defaults to FALSE. If TRUE, maximizing the number of clear 2-factor interactions takes precedence over minimizing aberration. Resolution is always considered first. `MaxC2` is ignored when using the `estimable` option. `MaxC2=TRUE` is not a recommended choice.
`replications`	positive integer number. Default 1 (i.e. each row just once). If larger, each design run is executed replication times. If `repeat.only`, repeated measurements are carried out directly in sequence, i.e. no true replication takes place, and all the repeat runs are conducted together. It is likely that the error variation generated by such a procedure will be too small, so that average values should be analyzed for an unreplicated design. Otherwise (default), the full experiment is first carried out once, then for the second replication and so forth. In case of randomization, each such blocks is randomized separately. In this case, replication variance is more likely suitable for usage as error variance (unless e.g. the same parts are used for replication runs although build variation is important).
`repeat.only`	logical, relevant only if replications > 1. If TRUE, replications of each run are grouped together (repeated measurement rather than true replication). The default is `repeat.only=FALSE`, i.e. the complete experiment is conducted in `replications` blocks, and each run occurs in each block.
`randomize`	logical. If TRUE, the design is randomized. This is the default. In case of replications, the nature of randomization depends on the setting of option `repeat.only`.
`seed`	optional seed for the randomization process In R version 3.6.0 and later, the default behavior of function `sample` has changed. If you work in a new (i.e., >= 3.6.-0) R version and want to reproduce a randomized design from an earlier R version (before 3.6.0), you have to change the RNGkind setting by `RNGkind(sample.kind="Rounding")` before running function `FrF2`. It is recommended to change the setting back to the new recommended way afterwards: `RNGkind(sample.kind="default")` For an example, see the documentation of the example data set `VSGFS`.
`alias.info`	can be 2 or 3, gives the order of interaction effects for which alias information is to be included in the `aliased` component of the `design.info` element of the output object.
`blocks`	is EITHER the number of blocks into which the experiment is subdivided OR a character vector of names of independent factors that are used as block constructors OR a vector of Yates column numbers OR a list of generators similar to list entries for `generators`. In the latter case, the differences to `generators` are that numbers/letters refer to the factors of the experiment and not to column numbers of the Yates matrix that numbers/letters can refer to all `nfactors` factors rather than the log2(nruns) base factors only, that one single number is always interpreted as the number of blocks rather than a column reference, that individual numbers are allowed in a list (i.e. individual factors specified in the experiment can be used as block factors) and that no negative signs are allowed. If `blocks` is a single number, it must be a power of 2. A blocked design can have at most `nruns-blocks-1` treatment factors, but should usually have fewer than that. If the experiment is randomized, randomization happens within blocks. In case of many blocks, units should also be randomized to blocks wherever possible! For the statistical and algorithmic background of blocked designs, see `block`.
`block.name`	name of the block factor, default “Blocks”
`block.old`	logical; if TRUE, blocking behavior of FrF2 version 1.7.2 is activated
`force.godolphin`	logical; if TRUE, blocking is forced to be done with the Godolphin method (using function `colpick`, see `block`), even if the default would have been to use function `blockpick`. The Godolphin method was introduced with package version 2.0, and since package version 2.3, the default for `alias.block.2fis=TRUE` is to force the use of the Godolphin method.
`bbreps`	between block replications; these are always taken as genuine replications, not repeat runs; default: equal to `replications`; CAUTION: you should not modify `bbreps` if you do not work with blocks, because the program code uses it instead of `replications` in some places
`wbreps`	within block replications; whether or not these are taken as genuine replications depends on the setting of `repeat.only`
`alias.block.2fis`	logical indicating whether blocks may be aliased with 2fis (default: `FALSE`); it will often be necessary to modify this option, because there is otherwise no solution.
`hard`	gives the number of hard to change factors. These must be the first factors in `factor.names`. Implementation is via a non-randomized split-plot design with as few as possible whole plots (number of possible whole plots is determined via left-adjustment) and as few as possible non-hard factors within the whole plot structure (by applying split-plot after left-adjustment). Observations within whole plots are randomized, whole plots themselves are not randomized (contrary to split plot designs). From the statistical point of view, randomisation of whole plots is strongly preferrable (cf. `splitplot` for a brief discussion of the difference). If this appears feasible, you may want to explicitly handle the situation by treating the hard to change factors as whole-plot factors via `WPs` and `nfac.WP`.
`check.hard`	is the number of candidate designs from the catalogue specified in `select.catlg` that are checked for making hard-to-change factors change as little as possible. The default is 10 - if too many changes are needed, a larger choice might help find a design with fewer level changes (but will also take longer run time and will find a worse design in terms of resolution / aberration). If you want to use the best design and do not want to compromise the confounding structure for ease-of-change reasons, set check.hard to 1.
`WPs`	is the number of whole plots and must be a power of 2. If `WPs > 1`, at least one of `nfac.WP` or `WPfacs` must be given. If `WPs = 1`, all settings for split-plot related options are ignored. For statistical and algorithmic information on treatment of split-plot designs see the separate help file `splitplot`.
`nfac.WP`	is the number of whole plot factors and must be smaller than `WPs`. The `nfac.WP` whole plot factors are counted within `nfactors`. Per default, the first `nfac.WP` factors are the whole plot factors. If a design is provided and whole plot factors are manually provided (design or generators option together with `WPfacs`), `nfac.WP` can be omitted (i.e. remains 0). If given, it must coincide with the length of `WPfacs`. If `nfac.WP` is given without `WPfacs`only, `generators` must not be given. If `nfac.WP` is omitted (i.e. remains 0) and `WPs` > 1, an error is thrown, because the situation is a block rather than a split-plot situation (and either it was forgotten to specify the number of whole plot factors, or blocks should be specified).
`WPfacs`	is per default NULL. In this case, the first `nfac.WP` factors are considered whole plot factors (and are, if necessary, automatically supplemented by additional whole plot constructor factors). If `WPfacs` is given, it must specify at least `log2(WPs)` whole plot factors. A custom design must be specified with options `design` or `generators`, and the requested factors in `WPfacs` must indeed create a split-plot structure with `WPs` whole plots. If the number of whole plots created by the whole plot factors differs from `WPs` or factors other than the specified factors from `WPfacs` would in fact become whole plot factors as well, an error is thrown. `WPfacs` can be any of a vector or list of factor position numbers OR a character vector of factor position letters from `Letters` OR a character vector with entries “F” followed by factor position number OR a character vector of factor names (this takes precedence over factor letters, i.e. if the factor names were B, A, C, D, and E, factor letter entries in the `nfac.WP` are interpreted as factor names, not position letters). It is not possible to specify additional whole plot generators from interaction effects manually through `WPfacs`. Rather, all whole plot factors - even artificial ones needed only to increase the number of plots - need to be included in the design factors.
`check.WPs`	is the number of potential split-plot designs that are compared by function `splitpick` w.r.t. resolution of the whole plot portion of the design. This option is effective, if `nfac.WP>k.WP` (i.e. bad resolution possible) and nfac.WP not larger than half the number of plots (i.e. resolution better than III is possible). The default is 10 - if not satisfied with the structure of the whole plot factors, a larger choice might help find a better design (but also take longer run time).
`...`	currently not used

Details

Per default, the function picks the best design from the default design catalogue catlg (a list object of class catlg).

Alternatively, the user can explicitly specify a design through accessing a specific catalogued design using the design option or specifying non-catalogued generators via the generators option.

Apart from generation of simple fractional factorial designs based on catalogued or non-catalogued generators, function FrF2 allows specification of blocked designs and split-plot designs, as well as specification of a set of 2fis that are required to be estimable. The implementation of these possibilities is explained in the separate help files block, splitplot and estimable.2fis. If you consider to use option hard, it may also be worth while to look at the splitplot help file.

Function FrF2 is still under development, although most features are now included, and the principle structure of inputs and outputs should not change much any more. Please contact me with any suggestions for improvements.

Function FrF2.currentlychecked is meant as a diagnostic tool, when searching for designs with option estimable and clear=TRUE. If the search takes very long, it can be interrupted (CAUTION: in some igraph versions, interrupting the search may crash R). After a successful interruption, and FrF2.currentlychecked() returns a character string with the name of the design that was checked at the time of interruption.

Value

Function FrF2 returns a data frame of S3 class design that has attached attributes that can be accessed by functions desnum, run.order and design.info.

The data frame itself contains the design with levels coded as requested. If no center points have been requested, the design columns are factors with contrasts -1 and +1 (cf. also contr.FrF2); in case of center points, the design columns are numeric.

The following attributes are attached to it:

desnum

Design matrix in -1/1 coding

run.order

a three column data frame;
the first column (run.no.in.std.order) contains the run number in standard order, possibly including a block number and in that case also a number for the original position within the block,
the second column (run.no) contains the actual run number as randomized,
the third column contains (run.no.std.rp) contains the content of the first column accumulated with a replication identifier, if applicable.
A few remarks on the run number in standard order are needed here:
In blocked and split plot designs, the run number in standard order refers to a row ordering with the first base factor changing slowest, different from the usual order with the first base factor changing fastest. Also note that the run number in standard order may not refer to the base columns one would naturally expect for designs created with blocking, estimable 2fis or split plot designs; the elements map and/or orig.fac.order of the attribute design.info help identify which base factors drive the run number in standard order. (In case of option hard=TRUE, the remark on split plot designs applies, and the numbering refers to the special slow change matrix from Cheng et al. 1998.)
Before version 2 of package FrF2, blocking for large situations was internally done with function blockpick.big. This behavior can be reproduced using the argument block.old=TRUE; if blocking for a large design is not successful otherwise, block.old=TRUE might be worth a try. For blocked designs created internally with function blockpick.big, the run number in standard order is not easily related to the final design.

design.info

list with the entries

type: character string “full factorial”, “FrF2”, “FrF2.estimable”, “FrF2.generators”, “FrF2.blocked” or “FrF2.splitplot” depending on the type of design
nruns: number of runs (replications are not counted)
nfactors: number of factors; since version 0.97, this is also true for designs of type FrF2.blocked (nfactors is now equal to ntreat) and for designs of type FrF2.splitplot, where nfactors is now the sum of nfac.WP and nfac.SP.
ntreat: for designs of type FrF2.blocked only;
number of treatment factors
nfac.WP: for designs of type FrF2.splitplot only;
number of whole plot factors (including extra factors that may have been added for whole plot construction); these are the first factors in the design data frame
nfac.SP: for designs of type FrF2.splitplot only;
number of split-plot factors
nlevels: for designs of type full factorial only;
vector with number of levels for each factor (of course, all the nfactors entries are “2” for FrF2)
factor.names: list named with (treatment) factor names and containing as entries vectors of length two each with coded factor levels
FrF2.version: version number of package FrF2, supporting correct usage of FrF2-specific functionality in functions summary and generators methods for class design
nblocks: for designs of type FrF2.blocked only;
number of blocks
block.gen: vector of columns of the Yates matrix for generating block factors in case of automatic block generation
OR list of (vectors of) factor numbers in case the blocks argument specifies certain factor combinations for blocking; ;
for designs of type FrF2.blocked only
blocksize: for designs of type FrF2.blocked only;
size of each block (without consideration of wbreps)
nWPs: for designs of type FrF2.splitplot only;
number of whole plots
plotsize: for designs of type FrF2.splitplot only;
size of each plot (without consideration of repeat.only replications if applicable)
orig.fac.order: designs of type FrF2.splitplot only;
factor order of the design before reshuffling of factors in order to accomodate the split plot request; the run number in standard order can only be interpreted together with this information
catlg.entry: for designs of type FrF2 only;
list with one element, which is the entry of catlg on which the design is based
generators: for designs of type FrF2.generators only;
character vector of generators in the form D=ABC etc.
base.design: for designs of type FrF2.blocked or FrF2.splitplot only;
gives a character string that contains the name of the base design in the catalogue or the column numbers of generating columns in Yates matrix; in case of automatic block generation, the exclusion or inclusion of k.block in the number of design factors / generators indicates whether the design was generated using function blockpick or blockpick.big.
aliased.with.blocks: for designs of type FrF2.blocked only;
treatment effects that are aliased with block main effects, up to 2fis or 3fis, depending on the choice of alias.info
aliased: alias structure of main effects, 2fis and possibly 3fis, depending on the choice of alias.info; For non-blocked and non-split-plot designs, aliased is itself a list of the two or three components main, fi2, and optionally fi3, given in terms of factor letters from Letters (up to 50~factors) or F1, F2, and so forth (more than 50~factors). For blocked and split-plot designs, aliased is a single list with an entry for each column of the Yates matrix that accomodates aliased low-order effects, and entries are in terms of factor names.)
replication: option setting in call to FrF2
repeat.only: option setting in call to FrF2
bbreps: for designs of type FrF2.blocked only; number of between block replications
wbreps: for designs of type FrF2.blocked only; number of within block replications;
repeat.only indicates whether these are replications or repetitions only
map: the mapping relation between factors in the base design and experimental factors, after using option estimable and for split-plot designs
clear: option setting in call to FrF2, in case of estimable
res3: option setting in call to FrF2, in case of estimable
randomize: option setting in call to FrF2
seed: option setting in call to FrF2
creator: call to function FrF2 (or stored menu settings, if the function has been called via the R commander plugin RcmdrPlugin.DoE)
ncube: number of cube points per block, in case center points have been requested
ncenter: number of center points per block, in case center points have been requested

Warning

Since R version 3.6.0, the behavior of function sample has changed (correction of a biased previous behavior that should not be relevant for the randomization of designs). For reproducing a randomized design that was produced with an earlier R version, please follow the steps described with the argument seed.

Author(s)

Ulrike Groemping

References

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Cheng, C.-S., Martin, R.J., and Tang, B. (1998). Two-level factorial designs with extreme numbers of level changes. Annals of Statistics 26, 1522-1539.

Cheng, C.-S. and Tsai, P.-W. (2009). Optimal two-level regular fractional factorial block and split-plot designs. Biometrika 96, 83-93.

Cheng, S.W. and Wu, C.F.J. (2002). Choice of optimal blocking schemes in 2-level and 3-level designs. Technometrics 44, 269-277.

Cordella, L.P., Foggia, P., Sansone, C. and Vento, M. (2001). An improved algorithm for matching large graphs. Proc. of the 3rd IAPR TC-15 Workshop on Graphbased Representations in Pattern Recognition, 149–159.

Godolphin, J. (2021). Construction of Blocked Factorial Designs to Estimate Main Effects and Selected Two-Factor Interactions. J. Royal Statistical Society B 83, 5-29. doi: 10.1111/rssb.12397.

Groemping, U. (2012). Creating clear designs: a graph-based algorithm and a catalogue of clear compromise plans. IIE Transactions 44, 988–1001. Early preprint available at http://www1.bht-berlin.de/FB_II/reports/Report-2010-005.pdf.

Groemping, U. (2014a). R Package FrF2 for Creating and Analyzing Fractional Factorial 2-Level Designs. Journal of Statistical Software, 56, Issue 1, 1-56. https://www.jstatsoft.org/v56/i01/.

Groemping, U. (2014b). A Note on Dominating Fractional Factorial Two-Level Designs With Clear Two-Factor Interactions. Technometrics 56, 42–45.

Groemping, U. (2021). An algorithm for blocking regular fractional factorial 2-level designs with clear two-factor interactions. Computational Statistics and Data Analysis 153, 1-18. doi: 10.1016/j.csda.2020.107059. Preprint at Report 3/2019.

Huang, P., Chen, D. and Voelkel, J.O. (1998). Minimum-Aberration Two-Level Split-Plot Designs. Technometrics 40, 314-326.

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

Solnon, C. (2010). AllDifferent-based Filtering for Subgraph Isomorphism. Artificial Intelligence 174, 850–864.

Sun, D.X., Wu, C.F.J. and Chen, Y.Y. (1997). Optimal blocking schemes for 2^n and 2^{n-p} designs. Technometrics 39, 298-307.

Wu, C.F.J. and Chen, Y. (1992) A graph-aided method for planning two-level experiments when certain interactions are important. Technometrics 34, 162-175.

Examples

## maximum resolution minimum aberration design with 4 factors in 8 runs
FrF2(8,4)
## the design with changed default level codes
FrF2(8,4, default.level=c("current","new"))
## the design with number of factors specified via factor names 
      ## (standard level codes)
FrF2(8,factor.names=list(temp="",press="",material="",state=""))
## the design with changed factor names and factor-specific level codes
FrF2(8,4, factor.names=list(temp=c("min","max"),press=c("low","normal"),
     material=c("current","new"),state=c("new","aged")))
## a full factorial
FrF2(8,3, factor.names=list(temp=c("min","max"),press=c("low","normal"),
     material=c("current","new")))
## a replicated full factorial (implicit by low number of factors)
FrF2(16,3, factor.names=list(temp=c("min","max"),press=c("low","normal"),
     material=c("current","new")))
## three ways for custom specification of the same design
FrF2(8, generators = "ABC")
FrF2(8, generators = 7)
FrF2(8, generators = list(c(1,2,3)))
## more than one generator
FrF2(8, generators = c("ABC","BC"))
FrF2(8, generators = c(7,6))
FrF2(8, generators = list(c(1,2,3),c(2,3)))
## alias structure for three generators that differ only by sign
design.info(FrF2(16,generators=c(7,13,15),randomize=FALSE))$aliased
design.info(FrF2(16,generators=c(7,-13,15),randomize=FALSE))$aliased
design.info(FrF2(16,generators=c(-7,-13,-15),randomize=FALSE))$aliased
## finding smallest design with resolution 5 in 7 factors
FrF2(nfactors=7, resolution=5)
## same design, but with 12 center points in 6 positions
FrF2(nfactors=7, resolution=5, ncenter=12, center.distribute=6)


## maximum resolution minimum aberration design with 9 factors in 32 runs
## show design information instead of design itself
design.info(FrF2(32,9))
## maximum number of free 2-factor interactions instead of minimum aberration
## show design information instead of design itself
design.info(FrF2(32,9,MaxC2=TRUE))

## usage of replication
## shows run order instead of design itself
run.order(FrF2(8,4,replication=2,randomize=FALSE))
run.order(FrF2(8,4,replication=2,repeat.only=TRUE,randomize=FALSE))
run.order(FrF2(8,4,replication=2))
run.order(FrF2(8,4,replication=2,repeat.only=TRUE))


## Not run: 
## examples below do work, but are repeated in the 
## respective method's separate help file and are therefore prevented 
## from running twice

########## automatic blocked designs ###################
## from a full factorial ##
FrF2(8,3,blocks=2)
## with replication
run.order(FrF2(8,3,blocks=2,wbreps=2))
run.order(FrF2(8,3,blocks=2,wbreps=2,repeat.only=TRUE))
run.order(FrF2(8,3,blocks=2,bbreps=2))
run.order(FrF2(8,3,blocks=2,bbreps=2,wbreps=2))

## automatic blocked design with fractions
FrF2(16,7,blocks=4,alias.block.2fis=TRUE,factor.names=c("MotorSpeed", 
      "FeedMode","FeedSizing","MaterialType","Gain","ScreenAngle","ScreenVibLevel"))
## isomorphic non-catalogued design as basis, using Godolphin approach
FrF2(16,gen=c(7,11,14),blocks=4,alias.block.2fis=TRUE)
## isomorphic non-catalogued design as basis, not using Godolphin approach
## (different design of comparable quality in this case)
FrF2(16,gen=c(7,11,14),blocks=4,alias.block.2fis=TRUE, force.godolphin=FALSE)
## FrF2 uses blockpick.big and ignores the generator
FrF2(64,gen=c(7,11,14),blocks=16,alias.block.2fis=TRUE, block.old=TRUE)
## FrF2 uses Godolphin approach, regardless of force.godolphin argument
##      because the setting is large
FrF2(64,gen=c(7,11,14),blocks=16,alias.block.2fis=TRUE)

########## manual blocked design ####################
### example that shows why order of blocks is not randomized
### can of course be randomized by user, if appropriate
FrF2(32,9,blocks=c("Day","Shift"),alias.block.2fis=TRUE, 
    factor.names=list(Day=c("Wednesday","Thursday"), Shift=c("Morning","Afternoon"),
        F1="",F2="",F3="",F4="",F5="",F6="",F7=""), default.levels=c("current","new"))

########## blocked design with estimable 2fis ####################
### all interactions of last two factors to be estimable clearly
### in 64 run design with blocks of size 4
### not possible with catalogue entry 9-3.1
FrF2(64, 6, blocks=16, factor.names=Letters[15:20], 
                 estimable=compromise(6,3)$requirement,
                 alias.block.2fis=TRUE, randomize=FALSE)
FrF2(design="9-3.2", blocks=16, alias.block.2fis=TRUE, 
    factor.names = list(C1="",C2="",C3="",C4="",C5="",C6="",C7="",
    N1=c("low","high"),N2=c("low","high")), 
    default.levels = c("current","new"),
    estimable=compromise(9, 8:9)$requirement)
FrF2(256, 13, blocks=64, alias.block.2fis=TRUE, 
    factor.names = list(C1="",C2="",C3="",C4="",C5="",C6="",C7="",C8="",
    N1=c("low","high")), 
    default.levels = c("current","new"),
    estimable=compromise(13, 1)$requirement)

########## hard to change factors ####################
## example from Bingham and Sitter Technometrics 19999
## MotorSpeed, FeedMode,FeedSizing,MaterialType are hard to change
BS.ex <- FrF2(16,7,hard=4,
     factor.names=c("MotorSpeed", "FeedMode","FeedSizing","MaterialType",
                  "Gain","ScreenAngle","ScreenVibLevel"), 
     default.levels=c("-","+"),randomize=FALSE)
design.info(BS.ex)
BS.ex
## NOTE: the design has 8 whole plots.
## If randomize=FALSE is used like here, the first hard-to-change factors 
## do not always change between whole plots. 
## A conscious and honest decision is required whether this is 
##    acceptable for the situation at hand!
## randomize=TRUE would cause more changes in the first four factors.

########## automatic generation for split plot ##########
## 3 control factors, 5 noise factors, control factors are whole plot factors
## 8 plots desired in a total of 32 runs
## Bingham Sitter 2003
BS.ex2a <- FrF2(32, 8, WPs=8, nfac.WP=3, 
      factor.names=c(paste("C",1:3,sep=""), paste("N",1:5,sep="")),randomize=TRUE)

## manual generation of this same design
BS.ex2m <- FrF2(32, 8, generators=c("ABD","ACD","BCDE"),WPs=8, WPfacs=c("C1","C2","C3"), nfac.WP=3, 
      factor.names=c(paste("C",1:3,sep=""),paste("N",1:5,sep="")),randomize=TRUE)

## design with few whole plot factors
## 2 whole plot factors, 7 split plot factors
## 8 whole plots, i.e. one extra WP factor needed
BSS.cheese.exa <- FrF2(32, 9, WPs=8, nfac.WP=2, 
      factor.names=c("A","B","p","q","r","s","t","u","v"))
design.info(BSS.cheese.exa)
## manual generation of the design used by Bingham, Schoen and Sitter
## note that the generators include a generator for the 10th spplitting factor
    ## s= ABq, t = Apq, u = ABpr and v = Aqr, splitting factor rho=Apqr
BSS.cheese.exm <- FrF2(32, gen=list(c(1,2,4),c(1,3,4),c(1,2,3,5),c(1,4,5),c(1,3,4,5)), 
      WPs=8, nfac.WP=3, WPfacs=c(1,2,10),
      factor.names=c("A","B","p","q","r","s","t","u","v","rho"))
design.info(BSS.cheese.exm)

########## usage of estimable ###########################
  ## design with all 2fis of factor A estimable on distinct columns in 16 runs
  FrF2(16, nfactors=6, estimable = rbind(rep(1,5),2:6), clear=FALSE)
  FrF2(16, nfactors=6, estimable = c("AB","AC","AD","AE","AF"), clear=FALSE)
  FrF2(16, nfactors=6, estimable = formula("~A+B+C+D+E+F+A:(B+C+D+E+F)"), 
       clear=FALSE)
            ## formula would also accept self-defined factor names
            ## from factor.names instead of letters A, B, C, ...
            
  ## estimable does not need any other input
  FrF2(estimable=formula("~(A+B+C)^2+D+E"))

  ## estimable with factor names 
  ## resolution three must be permitted, as FrF2 first determines that 8 runs 
  ##     would be sufficient degrees of freedom to estimate all effects 
  ##     and then tries to accomodate the 2fis from the model clear of aliasing in 8 runs
  FrF2(estimable=formula("~one+two+three+four+two:three+two:four"), 
       factor.names=c("one","two","three","four"), res3=TRUE)
  ## clear=FALSE allows to allocate all effects on distinct columns in the 
  ##     8 run MA resolution IV design
  FrF2(estimable=formula("~one+two+three+four+two:three+two:four"), 
       factor.names=c("one","two","three","four"), clear=FALSE)

  ## 7 factors instead of 6, but no requirements for factor G
  FrF2(16, nfactors=7, estimable = formula("~A+B+C+D+E+F+A:(B+C+D+E+F)"), 
       clear=FALSE)
  ## larger design for handling this with all required effects clear
  FrF2(32, nfactors=7, estimable = formula("~A+B+C+D+E+F+A:(B+C+D+E+F)"), 
       clear=TRUE)
  ## 16 run design for handling this with required 2fis clear, but main effects aliased
  ## (does not usually make sense)
  FrF2(16, nfactors=7, estimable = formula("~A+B+C+D+E+F+A:(B+C+D+E+F)"), 
       clear=TRUE, res3=TRUE)

## End(Not run)
## example for the sort option added with version 1.6-1
## and for usage of a catalogue from package FrF2.catlg128 (simplified with version 1.6-5)
  ## Not run: 
  estim <- compromise(17,15:17)$requirement  ## all interactions of factors 15 to 17 (P,Q,R)
  ## VF2 algorithm without pre-sorting of vertices
  ### CAUTION: in some igraph versions, the following may crash R ###
  FrF2(128, 17, estimable=estim, select.catlg=catlg128.17) 
                        ## very slow, interrupt with ESC key after a short while
                        ## !!! save all important work before, in case R crashes
  FrF2.currentlychecked()   ## displays the design that was currently checked 
                            ## should be 17-10.2407, if the interrupt was successful
  ## VF2 algorithm with pre-sorting of vertices
  FrF2(128, 17, estimable=estim, sort="high", select.catlg=catlg128.17)  ## very fast
  FrF2(128, 17, estimable=estim, sort="low", select.catlg=catlg128.17)  ## very fast
  ## LAD algorithm
  FrF2(128, 17, estimable=estim, method="LAD", select.catlg=catlg128.17)  ## very fast
  ## guaranteed to be MA clear design 
  ## only works, if package FrF2.catlg128 is installed
  
## End(Not run)

## example for necessity of perms, and uses of select.catlg and perm.start
## based on Wu and Chen Example 1
  ## Not run: 
  ## runs per default about max.time=60 seconds, before throwing error with 
  ##        interim results
  ## results could be used in select.catlg and perm.start for restarting with 
  ##       calculation of further possibilities
  FrF2(32, nfactors=11, estimable = formula("~(A+B+C+D+E+F)^2"), clear=FALSE)
  ## would run for a long long time (I have not yet been patient enough)
  FrF2(32, nfactors=11, estimable = formula("~(A+B+C+D+E+F)^2"), clear=FALSE, 
       max.time=Inf)
  ## can be easily done with perms, 
  ## as only different subsets of six factors are non-isomorphic
  perms.6 <- combn(11,6)
  perms.full <- matrix(NA,ncol(perms.6),11)
  for (i in 1:ncol(perms.6))
     perms.full[i,] <- c(perms.6[,i],setdiff(1:11,perms.6[,i]))
  FrF2(32, nfactors=11, estimable = formula("~(A+B+C+D+E+F)^2"), clear=FALSE, 
      perms = perms.full )
  
## End(Not run)

[Package FrF2 version 2.3-3 Index]