add.center: Function to add center points to a 2-level fractional...

In FrF2: Fractional Factorial Designs with 2-Level Factors

Function to add center points to a 2-level fractional factorial

Description

This function adds center points to a 2-level fractional factorial design. All factors must be quantitative!

Usage

add.center(design, ncenter, distribute=NULL, ...)

Arguments

design	a data frame of class design that contains a 2-level fractional factorial (regular or non-regular); design must neither be a split-plot nor a long version parameter design. For function add.center, the design must not contain center points yet, while it has to contain center points for function iscube. For function add.center, blocked and replicated (or repeated measurement) designs must be in the original run order (column run.no in run.order attribute in ascending order), as the algorithm relies on the related runs being grouped as expected. An error is thrown, if this condition is violated.
ncenter	the number of center points to be added to each block
distribute	the number of positions over which to distribute the center points within each block; note that the center points are not randomized but placed evenly throughout the (hopefully randomdomized) design (but see also the details section); if distribute is NULL, center points are all added at the end for non-randomized designs and are distributed as evenly as possible to beginning, middle and end of the experiment for randomized designs. distribute must neither be larger than ncenter nor than the number of runs of the design plus one.
...	currently not used

Details

Function add.center adds center points to 2-level fractional factorial designs. Instead of using this function directly, center points should usually be added directly with calls to functions FrF2 or pb. These make use of function add.center for this purpose.

Center points are added to designs for three main reasons: they provide a repeated benchmark run that can alert the experimenter to unplanned changes in experimental conditions, they provide an independent estimate of experimental error, and finally they provide a possibility for checking whether a first order model is sufficient. Especially for the first purpose, package FrF2 follows the recommendation in Montgomery (2001, p.275). To distinguish them from the center points, the original fractional factorial runs are called “cube points”.

Addition of center points does not affect estimates for main effects and interactions. The difference between the averages of cube points and center points gives an indication whether quadratic terms might be needed in the model.

For blocked designs and properly replicated designs, ncenter center points are added to each (replication) block. In case of repeated measurements, center points are also measured repeatedly.

Center points are distributed as evenly as possible over the distribute selected positions throughout each block. distribute=1 always adds all center points at the end of each block. If distribute > 1, (each block of) the design starts and ends with a (group of) center point(s), and the distribute positions for placing center points are as evenly placed throughout (each block of) the design as possible.
If ncenter is not a multiple of distribute, some center point groups have one more center point than others. If ncenter%%distribute is one or two only, the beginning and (for two) the end of (each block of) the design have one more center point, otherwise the ncenter%%distribute extra center points are randomized over the center point positions.

Function iscube from package DoE.base provides a logical vector that is TRUE for cube points and FALSE for center points, which allows to use of simple functions for “clean” 2-level fractional factorials like MEPlot.

Value

A data frame of class design with ncenter center point runs per block (or per replication block) added to the design (and its desnum and run.order attributes). The run.no.in.std.order column of run.order is “0” for the center points.

Existing response values for cube runs are preserved, and response values for the new center point runs are NA. Note, however, that center points should be added BEFORE running the experiment in order to benefit from all their useful properties; this should best be done within functions pb or FrF2.

The design is identifiable as a design with center points by the suffix .center to the type element of attribute design.info, and the elements ncube and ncenter are added (with the updated nruns being their sum). The element coding is also added to the design.info, in order to support steepest ascent/descent analysis from the center point.

Note

This function is still somewhat experimental.

Author(s)

Ulrike Groemping

References

Montgomery, D.C. (2001). Design and Analysis of Experiments (5th ed.). Wiley, New York.

See Also

See also as pb, FrF2

Examples

  ## purely technical example 
  plan <- FrF2(8,5, factor.names=c("one","two","three","four","five"))
  add.center(plan, 6)
  add.center(plan, 6, distribute=1)
  add.center(plan, 6, distribute=6)
  add.center(plan, 6, distribute=4)
  
  ## very artificial analysis example
    plan <- FrF2(8,4, factor.names=list(one=c(0,10),two=c(1,3),three=c(25,32),four=c(3.7,4.8)))
  ## add some response data
    y <- c(2+desnum(plan)%*%c(2,3,0,0) +
       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)
  ## analysing this design provides an impression
    MEPlot(lm(y~(.)^2, plan))
    IAPlot(lm(y~(.)^2, plan))
    DanielPlot(lm(y~(.)^2,plan), half=TRUE, alpha=0.2)
  ## tentative conclusion: factors one and two do something
  ## wonder whether the model with one and two and their interaction is sufficient
  ## look at center points (!!! SHOULD HAVE BEEN INCLUDED FROM THE START,
  ##      but maybe better now than not at all)
  ## use distribute=1, because all center points are run at the end
    planc <- add.center(plan, 6, distribute=1)
      ## conduct additional runs for the center points
        y <- c(y, c(2+desnum(planc)[!iscube(planc),1:4]%*%c(2,3,0,0) +
             1.5*apply(desnum(planc)[!iscube(planc),][,c(1,2)],1,"prod") + rnorm(6)))
  ## add to the design
    planc <- add.response(planc, y, replace=TRUE)
  ## sanity check: repeat previous analyses for comparison, with the help of function iscube()
    MEPlot(lm(y~(.)^2, planc, subset=iscube(planc)))
    IAPlot(lm(y~(.)^2, planc, subset=iscube(planc)))
    DanielPlot(lm(y~(.)^2, planc, subset=iscube(planc)), half=TRUE, alpha=0.2)
  ## quick check whether there a quadratic effect is needed: is the cube indicator significant ?
    summary(lm(y~(.)^2+iscube(planc), planc))
        ## (in this unrealistic example, the quadratic effect is dominating everything else;
        ## with an effect that strong in practice, it is likely that
        ## one would either have expected a strong non-linearity before conducting the experiment, 
        ## OR that the effect is not real but the result of some stupid mistake
  ## alternatively, the check can be calculated per hand (cf. e.g. Montgomery, Chapter 11):
    (mean(planc$y[iscube(planc)])-mean(planc$y[!iscube(planc)]))^2*8*6/(8+6)/var(y[!iscube(planc)])
    ## must be compared to the F-quantile with 1 degree of freedom
    ## is the square of the t-value for the cube indicator in the linear model
   

FrF2 documentation built on Sept. 20, 2023, 9:08 a.m.