estimable.2fis {FrF2} | R Documentation |
This help page documents the statistical and algorithmic details of requesting 2-factor interactions to be estimable in FrF2
The option estimable
allows to specify 2-factor interactions (2fis) that
have to be estimable in the model. Per default, it is assumed that a resolution IV
model is intended, as it is normally not reasonable to allow main effects to be
aliased with other 2-factor interactions in this situation. There are two types of
estimability that are distinguished by the setting of option clear
in
function link{FrF2}
.
Let us first consider the situation of designs of at least resolution IV.
With option clear=TRUE
, FrF2
searches for
a model for which all main effects and all 2fis given in estimable
are
clear of aliasing with any other 2fis. This is a weaker requirement than resolution V,
because 2fis outside those specified in estimable
may be aliased with
each other. But it is much stronger than what is done in case of clear=FALSE
:
For the latter, FrF2
searches for a design that has a distinct column in
the model matrix for each main effect and each interaction requested
in estimable
.
Users can explicitly permit that resolution III designs are included in the
search of designs for which the specified 2fis are estimable (by the res3=TRUE
option).
In case of clear=TRUE
, this leads to the somewhat
strange situation that main effects can be aliased with 2fis from outside
estimable
while 2fis from inside estimable
are not aliased with
any main effects or 2fis.
With clear=TRUE
, the algorithms compares the requirement set to
catalogued sets of clear 2fis by a graph isomorphism algorithm from R-package
igraph. The search is quite fast in this case.
With clear=FALSE
, the algorithm loops through the eligible designs from
catlg.select
from good to worse (in terms of MA) and, for each design, loops
through all eligible permutations of the experiment factors from perms
.
If perms
is omitted, the permutations are looped through in lexicographic
order starting from 1:nfac or perm.start
. Especially in this case,
run times of the search algorithm can be very long.
The max.time
option allows to limit this run time.
If the time limit is reached, the final situation (catalogued design and
current permutation of experiment factors) is printed so that the user can
decide to proceed later with this starting point (indicated by catlg.select
for the catalogued design(s) to be used and perm.start
for the current
permutation of experiment factors).
Note that - according to the structure of the catalogued designs and the lexicographic
order of checking permutations - the initial order of the factors has a strong influence
on the run time for larger or unlucky problems. For example, consider
an experiment in 32~runs and 11~factors, for six of which the pairwise interactions are to be estimable
(Example 1 in Wu and Chen 1992). estimable
for this model can be specified as
formula("~(F+G+H+J+K+L)^2")
OR
formula("~(A+B+C+D+E+F)^2")
.
The former runs a lot faster than the latter (I have not yet seen the latter finish
the first catalogued design, if perms
is not specified).
The reason is that the latter needs more permutations of the experiment factors than
the former, since the factors with high positions
change place faster and more often than those with low positions.
For this particular design, it is very advisable to constrain the permutations of the experiment factors to the different subset selections of six factors from eleven, since permutations within the sets do not change the possibility of accomodating a design. The required permutations for the second version of this example can be obtained e.g. by the following code:
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]))
Handing perms.full to the procedure using the perms
option makes the second version of the
requested interaction terms fast as well, since up to almost 40 Mio permutations of experiment
factors are reduced to at most 462. Thus, whenever possible,
one should try to limit the permutations necessary in case of clear=FALSE
.
In order to support relatively comfortable creation of distinct designs of some frequently-used types
of required interaction patterns, the function compromise
has been
divised: it supports creation of the so-called compromise designs of classes 1 to 4.
The list it returns also contains a component perms.full
that can be used as input
for the perms
option.
Please contact me with any suggestions for improvements.
Ulrike Groemping
Addelman, S. (1962). Symmetrical and asymmetrical fractional factorial plans. Technometrics 4, 47-58.
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.
Ke, W., Tang, B. and Wu, H. (2005). Compromise plans with clear two-factor interactions. Statistica Sinica 15, 709-715.
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.
See Also FrF2
for regular fractional factorials and
catlg
for the Chen, Sun, Wu catalogue of designs
and some accessor functions
########## 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) ## 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) ## End(Not run) ## 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 )