designAnatomy {dae} | R Documentation |

Computes the canonical efficiency factors for the joint decomposition of two or more structures or sets of mutually orthogonally projectors (Brien and Bailey, 2009; Brien, 2017; Brien, 2019), orthogonalizing projectors in a set to those earlier in the set of projectors with which they are partially aliased. The results can be summarized in the form of a decomposition table that shows the confounding between sources from different sets. For examples of its use also see the vignette daeDesignNotes.pdf.

```
designAnatomy(formulae, data, keep.order = TRUE, grandMean = FALSE,
orthogonalize = "hybrid", labels = "sources",
marginality = NULL, check.marginality = TRUE,
which.criteria = c("aefficiency","eefficiency","order"),
aliasing.print = FALSE,
omit.projectors = c("pcanon", "combined"), ...)
```

`formulae` |
An object of class |

`data` |
A |

`keep.order` |
A |

`grandMean` |
A |

`orthogonalize` |
A |

`labels` |
A |

`marginality` |
A Each component of the |

`check.marginality` |
A |

`which.criteria` |
A |

`aliasing.print` |
A |

`omit.projectors` |
A |

`...` |
further arguments passed to |

For each formula supplied in `formulae`

, the set of projectors is
obtained using `pstructure`

; there is one projector
for each term in a formula. Then `projs.2canon`

is used
to perform an analysis of the canonical relationships between two sets
of projectors for the first two formulae. If there are further formulae,
the relationships between its projectors and the already established
decomposition is obtained using `projs.2canon`

. The core
of the analysis is the determination of eigenvalues of the product of
pairs of projectors using the results of James and Wilkinson (1971).
However, if the order of balance between two projection matrices is
10 or more or the James and Wilkinson (1971) methods fails to produce
an idempotent matrix, equation 5.3 of Payne and Tobias (1992) is used
to obtain the projection matrices for their joint decompostion.

Chris Brien

Brien, C. J. (2017) Multiphase experiments in practice: A look back.
*Australian & New Zealand Journal of Statistics*, **59**, 327-352.

Brien, C. J. (2019) Multiphase experiments with at least one later
laboratory phase . II. Northogonal designs.
*Australian & New Zealand Journal of Statistics*, accepted for publication.

Brien, C. J. and R. A. Bailey (2009). Decomposition tables for
multitiered experiments. I. A chain of randomizations.
*The Annals of Statistics*, **36**, 4184 - 4213.

James, A. T. and Wilkinson, G. N. (1971) Factorization of the residual
operator and canonical decomposition of nonorthogonal factors in the
analysis of variance. *Biometrika*, **58**, 279-294.

Payne, R. W. and R. D. Tobias (1992). General balance, combination of
information and the analysis of covariance.
*Scandinavian Journal of Statistics*, **19**, 3-23.

`designRandomize`

, `designLatinSqrSys`

, `designPlot`

,

`pcanon.object`

, `p2canon.object`

,
`summary.pcanon`

, `efficiencies.pcanon`

,
`pstructure`

,

`projs.2canon`

, `proj2.efficiency`

, `proj2.combine`

,
`proj2.eigen`

, `efficiency.criteria`

,
in package dae,

`eigen`

.

`projector`

for further information about this class.

```
## PBIBD(2) from p. 379 of Cochran and Cox (1957) Experimental Designs.
## 2nd edn Wiley, New York
PBIBD2.unit <- list(Block = 6, Unit = 4)
PBIBD2.nest <- list(Unit = "Block")
trt <- factor(c(1,4,2,5, 2,5,3,6, 3,6,1,4, 4,1,5,2, 5,2,6,3, 6,3,4,1))
PBIBD2.lay <- designRandomize(allocated = trt,
recipient = PBIBD2.unit,
nested.recipients = PBIBD2.nest)
##obtain combined decomposition and summarize
unit.trt.canon <- designAnatomy(formulae = list(unit=~ Block/Unit, trt=~ trt),
data = PBIBD2.lay)
summary(unit.trt.canon, which.criteria = c("aeff","eeff","order"))
summary(unit.trt.canon, which.criteria = c("aeff","eeff","order"), labels.swap = TRUE)
## Three-phase sensory example from Brien and Payne (1999)
## Not run:
data(Sensory3Phase.dat)
Eval.Field.Treat.canon <- designAnatomy(formulae = list(
eval= ~ ((Occasions/Intervals/Sittings)*Judges)/Positions,
field= ~ (Rows*(Squares/Columns))/Halfplots,
treats= ~ Trellis*Method),
data = Sensory3Phase.dat)
summary(Eval.Field.Treat.canon, which.criteria =c("aefficiency", "order"))
## End(Not run)
```

[Package *dae* version 3.2-13 Index]