Function to add center points to a 2-level fractional factorial
This function adds center points to a 2-level fractional factorial design. All factors must be quantitative!
a data frame of class design that contains a 2-level fractional factorial
(regular or non-regular);
the number of center points to be added to each block
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);
currently not used
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
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
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),
distribute positions for placing center points are as evenly
placed throughout (each block of) the design as possible.
ncenter is not a multiple of
some center point groups have one more center point than others. If
is one or two only, the beginning and (for two) the end of (each block of) the design have one more center point,
ncenter%%distribute extra center points are randomized over the center point positions.
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
A data frame of class design with
ncenter center point runs per block
(or per replication block) added to the
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
The design is identifiable as a design with center points by the
.center to the
type element of attribute
and the elements
ncenter are added
(with the updated
nruns being their sum). The element
also added to the
design.info, in order to support steepest ascent/descent
analysis from the center point.
This function is still somewhat experimental.
Montgomery, D.C. (2001). Design and Analysis of Experiments (5th ed.). Wiley, New York.
See also as
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## 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
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