Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples

Function for augmenting an existing fractional factorial with a star portion in case of a late decision for a sequential procedure.

1 2 | ```
ccd.augment(cube, ncenter = 4, columns="all", block.name="Block.ccd",
alpha = "orthogonal", randomize=TRUE, seed=NULL, ...)
``` |

`cube` |
design generated by function |

`ncenter` |
integer number of center points,
or vector with two numbers, the first for the cube and the second for
the star portion of the design. |

`block.name` |
name of block factor that distinguishes (at least) between blocks; even for unblocked cubes, the ccd design has a cube and a star point block |

`alpha` |
“orthogonal”, “rotatable”, or a number that indicates the position of the star points; the number 1 would create a face-centered design. |

`randomize` |
logical that indicates whether or not randomization should occur |

`seed` |
NULL or a vector of two integer seeds for random number generation in randomization |

`...` |
reserved for future usage |

`columns` |
not yet implemented; it is intended to later allow to add star points for only some factors of a design (after eliminating the others as unimportant in a sequential process), and columns will be used to indicate those |

The statistical background of central composite designs is briefly described
under `CentralCompositeDesigns`

.

Function `ccd.augment`

augments an existing 2-level fractional factorial
that should already have been run with center points and should have resolution V.

In exceptional situations, it may be useful to base a ccd on a resolution IV design
that allows estimation of all 2-factor interactions of interest. Thus, it can be
interesting to apply function `ccd.augment`

to a cube
based on the `estimable`

functionality of function `FrF2`

in cases where a resolution V cube is not feasible.
Of course, this does not allow to estimate the aliased 2-factor interactions
and therefore generates a warning.

The function returns a data frame of S3 class `design`

with attributes attached. The data frame itself is in the original data scale.
The data frame `desnum`

attached as attribute `desnum`

is the original data frame
returned by package `rsm`

. The attribute `design.info`

is a list of various design properties.
The element `type`

of that list is the character string `ccd`

.
Besides the elements present in all class `design`

objects,
there are the elements quantitative (vector with `nfactor`

TRUE entries),
and a `codings`

element usable in the coding functions available in the rsm
package, e.g. `coded.data`

.

Note that the row names and the standard order column in the `run.order`

attribute of ccd designs based on
estimability requirements (cf. also the details section) are not in conventional order
and should not be used as the basis for any calculations. The same is true for
blocked designs, if the blocking routine `blockpick.big`

was used.

Since R version 3.6.0, the behavior of function `sample`

has changed
(correction of a biased previous behavior that should not be relevant for the randomization of designs).
For reproducing a randomized design that was produced with an earlier R version,
please follow the steps described with the argument `seed`

.

This package is still under (slow) development. Reports about bugs and inconveniences are welcome. `ccd.augment`

is based on version 1 of package rsm.

Ulrike Groemping

Box, G.E.P., Hunter, J.S. and Hunter, W.G. (2005, 2nd ed.). *Statistics for Experimenters*.
Wiley, New York.

Box, G.E.P. and Wilson, K.B. (1951). On the Experimental Attainment of Optimum Conditions.
*J. Royal Statistical Society*, **B13**, 1-45.

NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/pri/section3/pri3361.htm, accessed August 20th, 2009.

Myers, R.H., Montgomery, D.C. and Anderson-Cook, C.M. (2009). *Response Surface Methodology.
Process and Product Optimization Using Designed Experiments*. Wiley, New York.

See also `ccd.design`

, `FrF2`

,
`lhs-package`

, `rsm`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | ```
## purely technical examples for the sequential design creation process
## start with a fractional factorial with center points
plan <- FrF2(16,5,default.levels=c(10,30),ncenter=6)
## collect data and add them to the design
y <- rexp(22)
plan <- add.response(plan,y)
## assuming that an analysis has created the suspicion that a second order
## model should be fitted (not to be expected for the above random numbers):
plan.augmented <- ccd.augment(plan, ncenter=4)
## add new responses to the design
y <- c(y, rexp(14)) ## append responses for the 14=5*2 + 4 star points
r.plan.augmented <- add.response(plan.augmented, y, replace=TRUE)
## for info: how to analyse results from such a desgin
lm.result <- lm(y~Block.ccd+(.-Block.ccd)^2+I(A^2)+I(B^2)+I(C^2)+I(D^2)+I(E^2),
r.plan.augmented)
summary(lm.result)
## analysis with function rsm
rsm.result <- rsm(y~Block.ccd+SO(A,B,C,D,E), r.plan.augmented)
summary(rsm.result) ## provides more information than lm.result
loftest(rsm.result) ## separate lack of fit test
## graphical analysis
## (NOTE: purely for demo purposes, the model is meaningless here)
## individual contour plot
contour(rsm.result,B~A)
## several contour plots
par(mfrow=c(1,2))
contour(rsm.result,list(B~A, C~E))
## many contourplots, all pairs of some factors
par(mfrow=c(2,3))
contour(rsm.result,~A+B+C+D)
``` |

```
Loading required package: FrF2
Loading required package: DoE.base
Loading required package: grid
Loading required package: conf.design
Attaching package: 'DoE.base'
The following objects are masked from 'package:stats':
aov, lm
The following object is masked from 'package:graphics':
plot.design
The following object is masked from 'package:base':
lengths
Loading required package: rsm
Call:
lm.default(formula = y ~ Block.ccd + (. - Block.ccd)^2 + I(A^2) +
I(B^2) + I(C^2) + I(D^2) + I(E^2), data = r.plan.augmented)
Residuals:
Min 1Q Median 3Q Max
-0.9493 -0.2943 -0.0580 0.2310 2.3250
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.041e+00 3.284e+00 -1.230 0.2388
Block.ccd2 5.722e-02 3.205e-01 0.179 0.8608
A 7.395e-03 1.098e-01 0.067 0.9473
B 7.862e-02 1.098e-01 0.716 0.4859
C 4.434e-02 1.098e-01 0.404 0.6925
D 1.635e-01 1.098e-01 1.489 0.1587
E 1.620e-01 1.098e-01 1.475 0.1624
I(A^2) 8.582e-04 1.355e-03 0.633 0.5367
I(B^2) 2.770e-04 1.355e-03 0.204 0.8410
I(C^2) 2.140e-04 1.355e-03 0.158 0.8768
I(D^2) -1.242e-03 1.355e-03 -0.917 0.3749
I(E^2) -1.063e-03 1.355e-03 -0.785 0.4456
A:B -1.065e-03 2.344e-03 -0.454 0.6565
A:C 4.487e-03 2.344e-03 1.915 0.0762 .
A:D 4.214e-05 2.344e-03 0.018 0.9859
A:E -4.314e-03 2.344e-03 -1.841 0.0869 .
B:C -1.112e-03 2.344e-03 -0.474 0.6425
B:D -3.841e-03 2.344e-03 -1.639 0.1235
B:E 1.708e-03 2.344e-03 0.729 0.4782
C:D -3.689e-04 2.344e-03 -0.157 0.8772
C:E -3.942e-03 2.344e-03 -1.682 0.1147
D:E -9.052e-04 2.344e-03 -0.386 0.7051
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9375 on 14 degrees of freedom
Multiple R-squared: 0.6336, Adjusted R-squared: 0.08391
F-statistic: 1.153 on 21 and 14 DF, p-value: 0.4002
Call:
rsm(formula = y ~ Block.ccd + SO(A, B, C, D, E), data = r.plan.augmented)
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.0405e+00 3.2840e+00 -1.2304 0.23883
Block.ccd2 5.7225e-02 3.2050e-01 0.1785 0.86085
A 7.3948e-03 1.0983e-01 0.0673 0.94727
B 7.8619e-02 1.0983e-01 0.7158 0.48586
C 4.4339e-02 1.0983e-01 0.4037 0.69252
D 1.6353e-01 1.0983e-01 1.4890 0.15866
E 1.6195e-01 1.0983e-01 1.4747 0.16244
A:B -1.0649e-03 2.3436e-03 -0.4544 0.65652
A:C 4.4869e-03 2.3436e-03 1.9145 0.07621 .
A:D 4.2136e-05 2.3436e-03 0.0180 0.98591
A:E -4.3144e-03 2.3436e-03 -1.8409 0.08692 .
B:C -1.1119e-03 2.3436e-03 -0.4744 0.64251
B:D -3.8413e-03 2.3436e-03 -1.6391 0.12347
B:E 1.7078e-03 2.3436e-03 0.7287 0.47820
C:D -3.6893e-04 2.3436e-03 -0.1574 0.87716
C:E -3.9421e-03 2.3436e-03 -1.6820 0.11472
D:E -9.0518e-04 2.3436e-03 -0.3862 0.70513
A^2 8.5822e-04 1.3551e-03 0.6333 0.53672
B^2 2.7697e-04 1.3551e-03 0.2044 0.84099
C^2 2.1399e-04 1.3551e-03 0.1579 0.87677
D^2 -1.2420e-03 1.3551e-03 -0.9166 0.37488
E^2 -1.0635e-03 1.3551e-03 -0.7848 0.44565
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Multiple R-squared: 0.6336, Adjusted R-squared: 0.08391
F-statistic: 1.153 on 21 and 14 DF, p-value: 0.4002
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
Block.ccd 1 0.0280 0.02802 0.0319 0.8608
FO(A, B, C, D, E) 5 7.3952 1.47905 1.6830 0.2033
TWI(A, B, C, D, E) 10 12.0459 1.20459 1.3707 0.2865
PQ(A, B, C, D, E) 5 1.8033 0.36066 0.4104 0.8337
Residuals 14 12.3034 0.87882
Lack of fit 6 2.0809 0.34682 0.2714 0.9353
Pure error 8 10.2225 1.27781
Stationary point of response surface:
A B C D E
13.98258 20.68867 17.93291 23.74228 21.05230
Eigenanalysis:
eigen() decomposition
$values
[1] 0.004707318 0.001366771 -0.001698885 -0.002507125 -0.002824384
$vectors
[,1] [,2] [,3] [,4] [,5]
A 0.6221270 -0.23784018 0.67513990 0.23428113 -0.2137489
B -0.2827130 -0.75653721 0.10192956 -0.52926351 -0.2391973
C 0.5472055 -0.22080628 -0.71658674 0.05176566 -0.3682859
D 0.1122690 0.56743113 0.13900783 -0.60524357 -0.5289373
E -0.4700992 0.01870804 0.03125294 0.54405242 -0.6940367
Df Sum Sq Mean Sq F value Pr(>F)
Model residual 14 12.3034 0.87882
Lack of fit 6 2.0809 0.34682 0.2714 0.9353
Pure error 8 10.2225 1.27781
```

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