fold.design | R Documentation |
This function creates a foldover design for a 2-level fractional factorial. The purpose is to dealias (some) effects. Per default, all factors are folded upon, which makes the resulting design at least resolution IV. Different foldover versions can be requested.
fold.design(design, columns = "full", ...)
design |
a data frame of class design that contains a 2-level fractional factorial;
currently, |
columns |
indicates which columns to fold on; the default “full” folds on all columns,
i.e. swaps levels for all columns. |
... |
currently not used |
Foldover is a method to dealias effects in relatively small 2-level fractional factorial designs. The folded design has twice the number of runs from the original design, and an additional column “fold” that distinguishes the original runs from the mirror runs. This column should be used in analyses, since it captures a block effect on time (often the mirror runs are conducted substantially later than the original experiment).
Like most other software, this function conducts a full foldover per default,
i.e. the mirror portion reverses the levels of all factors. In terms of the
convenient -1/1 notation for factor levels, this can be written as
a multiplication with “-1” for the mirror portion of all factors.
Thus, all confounding relations involving
an odd number of factors (e.g. A=BC) are resolved, because the odd side of the
equation involves a minus for the mirror runs, and the even side does not
(since the minuses cancel each other). (These
confounding relations are replaced by even ones
for which the odd side of the equation is multiplied with minus the new mirror factor fold
.)
There are many situations, for which the default full foldover is not the best possible foldover fraction, cf. e.g. Li and Mee (2002). It is therefore possible to choose an arbitrary foldover fraction. For example, folding on one particular factor alone dealiases all confounding relations for that factor, folding on two particular factors dealiases all confounding relations of these two with others but not of these two together with others and so on.
Folding Plackett-Burman designs also removes the (partial) aliasing with 2-factor interactions for all main effects that are mirrored.
A data frame of class design with twice as many rows as design
and
the additional factor fold
(added as the last factor for folded pb
designs, as the first factor for splitplot designs,
and as the last base factor for other folded regular fractional
factorial designs).
Existing response values are of course preserved, and response values for the new mirror runs are NA.
The type in attribute design.info
is suffixed with “.folded”, and
nruns
(and, if applicable, nWPs
) is doubled,
nfactors
(and, if applicable, nfac.WP
)
is increased by one (for the factor fold, which
is a block factor and can also be treated as such, but will currently be treated as a fixed
(whole plot) factor by any automated analysis routine). The creator element receives a list entry for the fold columns.
For regular fractional factorials (design type starting with FrF2
), the generator element is adjusted
(the generators for all generated fold factors now involve the folding factor), and an existing
catlg.entry element is replaced by a new generators element. The aliased
element is
adapted to the new alias structure. Note that the fold factor enters as a new base factor and therefore
is added to the factor matrix after the first log2(nruns) factors. This implies that all factor
letters previously used for the generated factors are changed - for avoiding confusion it is always recommended to
work with factor names that are meaningful in a subject-matter sense.
Furthermore, for the regular fractional factorial designs,
the column run.no.in.std.order in attribute run.order
for the mirror portion of the design is
populated such that the base factors remain in the conventional order when ordered by
run.no.in.std.order (regardless whether or not they are included in the fold;
it is always possible to reorder runs such that the original base factors
together with the folding factor form the new base in standard order).
This function is still somewhat experimental.
Ulrike Groemping
Li, H. and Mee, R. (2002). Better foldover fractions for resolution III 2^(k-p) designs. Technometrics 44, 278–283. New York: Springer.
Mee, R. (2009). A Comprehensive Guide to Factorial Two-Level Experimentation. New York: Springer.
Montgomery, D.C. (2001). Design and Analysis of Experiments (5th ed.). Wiley, New York.
See also as pb
, FrF2
## create resolution III design
plan <- FrF2(8,5, factor.names=c("one","two","three","four","five"))
## add some resonse data
y <- c(2+desnum(plan)%*%c(2,3,0,0,0) +
1.5*apply(desnum(plan)[,c(1,2)],1,"prod") + rnorm(8))
## the "c()" makes y into a vector rather than a 1-column matrix
plan <- add.response(plan, y)
DanielPlot(lm(y~(.)^2,plan), alpha=0.2, half=TRUE)
## alias information
design.info(plan)
## full foldover for dealiasing all main effects
plan <- fold.design(plan)
design.info(plan)
## further data, shifted by -2
y <- c(y, desnum(plan)[9:16,1:5]%*%c(2,3,0,0,0) +
1.5*apply(desnum(plan)[9:16,c(1,2)],1,"prod") + rnorm(8))
plan <- add.response(plan, y, replace=TRUE)
linmod <- lm(y~(.)^2,plan)
DanielPlot(linmod, alpha=0.2, half=TRUE)
MEPlot(linmod)
IAPlot(linmod)
## fold on factor a only (also removes main effect aliasing here)
plan <- FrF2(8,5, factor.names=c("one","two","three","four","five"))
aliasprint(plan)
plan <- fold.design(plan, columns=1)
aliasprint(plan)
## fold a Plackett-Burman design with 11 factors
plan <- pb(12)
fold.design(plan)
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