FrF2 (version 2.1)

fold.design: Function to create a foldover for 2-level fractional factorials

Description

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.

Usage

fold.design(design, columns = "full", ...)

Arguments

design

a data frame of class design that contains a 2-level fractional factorial; currently, design must neither be blocked nor a long version parameter design

columns

indicates which columns to fold on; the default “full” folds on all columns, i.e. swaps levels for all columns. A specific fold on certain columns can be requested giving a character vector of factor names or a numeric vector of factor positions. See the details section for some statistical comments.

…

currently not used

Value

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).

Details

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.

References

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

See also as pb, FrF2

Examples

# NOT RUN {
  ## 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)
   
# }