pb {FrF2}R Documentation

Function to generate non-regular fractional factorial screening designs

Description

The function generates Plackett-Burman designs and in some cases other screening designs in run numbers that are a multiple of 4. These designs are particularly suitable for screening a large number of factors, since interactions are not fully aliased with one main effect each but partially aliased. (The designs in 8 and 64 runs are exceptions from this rule.)

Usage

pb(nruns, nfactors = nruns - 1, factor.names = if (nfactors <= 50) 
     Letters[1:nfactors] else paste("F", 1:nfactors, sep = ""), 
     default.levels = c(-1, 1), ncenter=0, center.distribute=NULL, 
     boxtyssedal = TRUE, n12.taguchi = FALSE, 
     replications = 1, repeat.only = FALSE, 
     randomize = TRUE, seed = NULL, ...)

pb.list

Arguments

nruns number of runs, must be a multiple of 4
nfactors number of factors, default is nruns - 1, and it is recommended to retain this default. It is possible to specify factor names for fewer factors, and the remaining columns will be named e1, e2, ... They are useful for representing error in effects plots.
factor.names a character vector of factor names (length up to nfactors) 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; ncenter > 0 is permitted, if all factors are quantitative
center.distribute the number of positions over which the center points are to be distributed ; 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.
boxtyssedal logical, relevant only for nruns=16. If FALSE, the geometric (=standard) 16 run plan is used. If TRUE, the proposal by Box and Tyssedal is used instead, which has the advantage (for screening) of aliasing each interaction with several main effects, like the other Plackett-Burman designs.
n12.taguchi logical, relevant only for nruns=12. If TRUE, the 12 run design is given in Taguchi order.
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.
seed optional seed for the randomization process
... currently not used

Details

pb stands for Plackett-Burman. Plackett-Burman designs (Plackett and Burman 1946) are generally used for screening many variables in relatively few runs, when interest is in main effects only, at least initially. Different from the regular fractional factorial designs created by function FrF2, they do not perfectly confound interaction terms with main effects but distribute interaction effects over several main effects. The designs with number of runs a power of 2 are an exception to this rule: they are just the resolution III regular fractional factorial designs and are as such not very suitable for screening because of a high risk of very biased estimates for the main effects of the factors. Where possible, these are therefore replaced by different designs (cf. below).

For most run numbers, function pb uses Plackett-Burman designs, and simply fills columns from left to right. The generating rows for these designs can be found in the list pb.list (a 0 entry indicates that the design is constructed by a different method, e.g. doubling).

For 12 runs, the isomorphic design by Taguchi can be requested. For 16 runs, the default is to use the designs suggested by Box and Tyssedal (2001), which up to 14 factors do not suffer from perfect aliasing. For 32 runs, a cyclic design with generating row given in Samset and Tyssedal (1999) is used. For 64 runs, the 32 run design is doubled. For 92 runs, a design is constructed according to the Williamson construction with matrices A, B, C and D from Hedayat and Stufken (1999), p. 160.

So far, designs up to 96~runs are covered. More and different ones may follow, since the package is currently under intensive development.

Usage of the 8 run design for more than 4 factors is discouraged, as it completely aliases main effects with individual two-factor interactions. It is recommended to use at least the 12 run design instead for screening more than 4 factors.

Value

Value is a data frame of S3 class design and 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 three column data frame, first column contains the run number in standard order, second column the run number as randomized, third column the run number with replication number as postfix; useful for switching back and forth between actual and standard run number
design.info list with entries
type
character string “pb”, except for 8~runs with up to 4~factors, for which a type “FrF2” design is output
nruns
number of runs (replications are not counted)
nfactors
number of factors; not for designs of type FrF2.blocked, where ntreat takes this role
ntreat
for designs of type FrF2.blocked only;
number of treatment factors
factor.names
list named with (treatment) factor names and containing as entries vectors of length two each with coded factor levels
replication
option setting in call to pb
repeat.only
option setting in call to pb
randomize
option setting in call to pb
seed
option setting in call to pb
creator
call to function pb (or stored menu settings, if the function has been called via the R commander plugin RcmdrPlugin.DoE)

Warning

With version 1.0-5 of package FrF2, design generation for the designs based on doubling has changed (internal function double.des). This affects designs for 40,56,64,88,96 runs. The benefit is that these designs are now resolution IV, if they are used with a number of factors up to half the number of runs, i.e. the main effects for such situations can be estimated without being confounded by potential two-factor interactions.

The downside is that the versions from 1.0-5 onwards will not be able to exactly reproduce pb designs in these numbers of runs that have been created with an earlier version.

Author(s)

Ulrike Groemping

References

Box, G.E.P. and Tyssedal, J. (2001) Sixteen Run Designs of High Projectivity for Factor Screening. Communications in Statistics - Simulation and Computation 30, 217-228.

Hedayat, A.S., Sloane, N.J.A. and Stufken, J. (1999) Orthogonal Arrays: Theory and Applications, Springer, New York.

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

Plackett, R.L.; Burman, J.P. (1946) The design of optimum multifactorial experiments. Biometrika 33, 305-325.

Samset, O.; Tyssedal, J. (1999) Two-level designs with good projection properties. Technical Report 12, Department of Mathematical Sciences, The Norwegian University of Science and Technology, Norway.

Williamson, J. (1946) Determinants whose elements are 0 and 1. American Mathematical Monthly 53, 427-434.

See Also

See Also FrF2 for regular fractional factorial designs

Examples

   pb(12,randomize=FALSE)
   pb(12,randomize=FALSE,n12.taguchi=TRUE)
   pb(20,seed=29869)
   pb(16,factor.names=list(A="",B="",C="",D=c("min","max"),
          E="",F="",G="",H="",J=c("new","old")))
   pb(8,default.levels=c("current","new"))
   test <- pb(40) ## design created by doubling the 20 run design
   pb(12, ncenter=6) ## 6 center points with default placement

[Package FrF2 version 1.0-5 Index]