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"), ...)An object of class list whose components are of class
formula.
Usually, the terms in a single formula have the same status in the
allocation of factors in the design. For example, all involve only
factors that were allocated, or all involve factors that were
recipients of allocated factors. The names of the components are
used to identify the
sources in the summary.pcanon object. They will also
be used to name the terms, sources and marginality
lists in the pcanon.object.
A data.frame contains the values of the factors and variables
that occur in formulae.
A logical indicating whether the terms should keep
their position in the expanded formula projector, or
reordered so that main effects precede two-factor
interactions, which precede three-factor interactions and
so on.
A logical indicating whether the projector for the
grand mean is to be included for each structure.
A character vector indicating the method for orthogonalizing a
projector to those for terms that occurred previously
in a single formula. Three options are available:
hybrid; differencing; eigenmethods.
The hybrid option is the most general and uses the
relationships between the projection operators for the
terms in the formula to decide which projectors to
substract and which to orthogonalize using eigenmethods.
The differencing option subtracts, from the current
projector, those previously orthogonalized projectors for
terms whose factors are a subset of the current projector's
factors. The eigemethods option recursively
orthogonalizes the projects using an eigenanalysis of
each projector with previously orthogonalized projectors.
If a single value is given, it is used for all formulae.
A character nominating the type of labels to be used in
labelling the projectors, and which will be used also
in the output tables, such the tables of the aliasing
in the structure. The two alternatives are terms and
sources. Terms have all factors/variables in it
separated by colons (:). Sources have factors/variables
in them that represent interactions separated by hashes
(#); if some factors are nested within others, the
nesting factors are surrounded by square brackets
([ and ]) and separated by colons (:).
If some generalized, or combined, factors have no marginal
terms, the constituent factors are separated by colons
(:) and if they interact with other factors in
the source they will be parenthesized.
A list that can be used to supply some or all of the
marginality matrices when it is desired to overwrite calculated
marginality matrices or when they are not calculated.
If the list is the same length as the
formulae list, they will be associated in
parallel with the components of formulae, irrespective of
the naming of the two lists. If the number of
components in marginlaity is less than the number of
components in formulae then both lists
must be named so that those in the marginality
list can be matched with those in the
formulae list.
Each component of the marginality list
must be either NULL or a square
matrix consisting of zeroes and ones that
gives the marginalites of the terms in the formula. It must have
the row and column names set to the terms from the expanded
formula, including being in the same order as these terms.
The entry in the ith row and jth column will be one if the
ith term is marginal to the jth term i.e. the column space of the
ith term is a subspace of that for the jth term and so the source
for the jth term is to be made orthogonal to that for the ith term.
Otherwise, the entries are zero. A row and column should not be
included for the grand mean even if grandMean is TRUE.
A logical indicating whether the marginality matrix,
when it is supplied, is to be checked against that computed
by pstructure.formula. It is ignored when
orthogonalize is set to eigenmethods.
A character vector nominating the efficiency
criteria to be included in the summary of aliasing
between terms within a structure. It can be
none, all or some combination of
aefficiency, mefficiency,
sefficiency, eefficiency, xefficiency,
order and dforthog -- for details see
efficiency.criteria.
If none, no summary is printed.
A logical indicating whether the aliasing between sources
is to be printed.
A character vector of the types of projectors to
omit from the returned pcanon object. If
pcanon is included in the vector then the
projectors in these objects will be replaced with a
numeric containing their degrees of freedom.
If combined is included in the vector
then the projectors for the combined decomposition will be
replaced with a numeric containing their degrees of
freedom. If none is included in the vector then no
projectors will be omitted.
further arguments passed to terms.
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.
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.
# NOT RUN {
## 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"))
# }
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