This help page documents the statistical and algorithmic details of blocking in FrF2
Blocking is done with the purpose to balance the design with respect to a factor that is known or strongly suspected to have an influence but is not in itself of interest, and it is usually assumed that block factors do not interact with experimental factors. Examples are batches of material that are not large enough to accomodate the complete experiment so that e.g. half the experiment is done on the first batch and the other half on the second batch (two blocks). The block factor should be orthogonal to the experimental factors, at least to their main effects. Per default, it is also requested that the block factor is orthogonal to the 2-factor interactions. This can be changed by the user, if no such design can be found.
Blocking is currently implemented for regular fractional factorial designs only.
There are two principal ways to handle blocked designs, manual definition
(i.e. the user specifies exactly which columns are to be used for which purpose) and automatic
definition. Each situation has its specifics. These are detailed below. For users with
not so much mathematical/statistical background, it will often be best to use the automatic way,
specifying the treatement factors of interest via nfactors or factor.names
and a single number for blocks or WPs.
Users with more mathematical background may want to use the manual definitions, perhaps
in conjunction with published catalogues of good block designs, or
after inspecting possibilities with functions blockpick or blockpick.big.
The user can start from a design with a number of factors and manually specify which factors or
interactions are to be used as block generators. If this route is chosen, blocks
can be a vector of factor names or factor letters, or of the same form as generators, except that
not only base factors but all factors can be used and single factors are permitted
(which would lead to resolution II designs if used in generators). For example,
block = Letters[c(2,4,5)]
or
block = list(2,4,5)
specify that the 2nd, 4th and 5th factor are to be used as block generators, while
block = c("Day","Shift")
indicates that the named factors “Day” and “Shift” specified in factor.names
are to be treated as blocking factors). In this case, the number of blocks is calculated,
and a new factor with the default name “Blocks” (in general the name chosen in
option block.name) is generated, which would for example contain as levels
the Day/Shift combinations. It is also possible to choose interaction effects rather than factors themselves
as block generators, e.g.
block = c("ABCD","EFGH")
or
block = list(c(1,2,3,4),c(5,6,7,8)) .
Finally, it is also possible to specify choice of blocks using a vector of Yates column numbers,
in order to be able to use catalogued blocking structures of this form, e.g. from Sitter, Chen and Feder
(1997).
The block main effects are defined by the k.block specified effect
and all interactions between them. The specified block effects are required to be independent from each other,
which implies that they generate 2^k.block blocks.
CAUTION: If the user manually generates a blocked design, it is his/her responsibility to ensure a
good choice of design (e.g. by using a catalogued design from Bisgaard 1994,
Sun, Wu and Chen 1997, Sitter, Chen and Feder (1997), or Cheng and Wu 2002).
If the user only specifies the number of blocks required for the experiment, function FrF2
automatically generates the blocks. For full factorial designs, function FrF2 uses
the Sun, Wu and Chen (1997) catalogue of blocked designs (implemented in function blockpick).
Otherwise, depending on the size of the problem,
function FrF2 uses function blockpick or function blockpick.big
for finding an appropriate allocation of block generator columns:
Smaller problems
(choose(nruns-1-nfactors,k.block) < 100000)
are treated with blockpick.
The search for an appropriate blocked design starts with the overall best unblocked design
(in terms of aberration or MaxC2, if requested).
If this best design does not yield an adequate blocking possibility, the search continues with
the next best design and so forth.
For the smaller problems, function blockpick
looks for k.block independent subsets among the eligible columns of the design.
(The eligible columns are all columns of the Yates matrix that are neither occupied
by treatment main effects nor by 2fis among treatments (if alias.block.2fis=FALSE,
which is the default), or all columns of the Yates matrix that are not occupied by treatment main effects
(if alias.block.2fis=TRUE). Note that no effort is made to avoid aliasing with 2-factor interactions,
if alias.block.2fis=TRUE is chosen.
For the larger problems,
function blockpick.big permutes the k~base factors of candidate designs with nfactors + k.block factors
in search of a design the first k.block~factors of which can be used for block construction. In the latter case, any
specification of design (via options design or generators) is ignored. Note that function
blockpick.big is not guaranteed to find an existing blocked design.
Sun, Wu and Chen (1997) provide a catalogue of blocked designs
with a few quality criteria, and they state that there is no single best design, but that the choice
depends on the situation. FrF2 always comes up with one specific solution design.
Comparisons to the catalogued designs in Sun, Wu and Chen (1997) have shown that
the designs found in FrF2 are often but not always isomorphic to the catalogued ones.
Differences do occur, especially if the base designs are resolution III, or if blockpick.big
has to be used. Users who want to be certain to use a “best” blocked design should manually
implement a specific catalogued design or inspect several solutions from functions blockpick
blockpick.big.
Please contact me with any suggestions for improvements.
Bisgaard, S. (1994a). Blocking generators for small 2\^(k-p) designs. J. Quality Technology 26, 288-294.
Chen, J., Sun, D.X. and Wu, C.F.J. (1993) A catalogue of 2-level and 3-level orthogonal arrays. International Statistical Review 61, 131-145.
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.
Sitter, R.R., Chen, J. and Feder, M. (1997). Fractional Resolution and Minimum Aberration in Blocked 2n-k Designs. Technometrics 39, 382--390.
Sun, D.X., Wu, C.F.J. and Chen, Y.Y. (1997). Optimal blocking schemes for 2\^p and 2\^(n-p) designs. Technometrics 39, 298-307.
See Also FrF2 for regular fractional factorials,
catlg for the Chen, Sun, Wu catalogue of designs
and some accessor functions,
and splitplot for the statistical aspects of split-plot designs.
# NOT RUN {
########## 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)
## isomorphic non-catalogued design as basis
FrF2(16,gen=c(7,11,14),blocks=4,alias.block.2fis=TRUE)
## FrF2 uses blockpick.big and ignores the generator
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"))
# }
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