pstructure.object that includes the orthogonalized projectors for the terms in a formulaConstructs a pstructure.object that includes a
set of mutually orthogonal projectors, one for each
term in the formula. These are used to specify a structure,
or an orthogonal decomposition of the data space.
There are three methods available for
orthogonalizing the projectors corresponding to the terms
in the formula: differencing, eigenmethods
or the default hybrid method.
It is possible to use this function to find out what sources are associated with the terms in a model and to determine the marginality between terms in the model. The marginality matrix can be saved.
# S3 method for formula
pstructure(formula, keep.order = TRUE, grandMean = FALSE,
orthogonalize = "hybrid", labels = "sources",
marginality = NULL, check.marginality = TRUE,
omit.projectors = FALSE,
which.criteria = c("aefficiency","eefficiency","order"),
aliasing.print = TRUE, data = NULL, ...)An object of class formula from which the terms will be
obtained.
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 in the set produced.
A character vector indicating the method for orthogonalizing a
projector to those for terms that occurred previously
in the 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 projectors
using an eigenanalysis of each projector
with previously orthogonalized projectors.
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 square matrix that can be used to supply the
marginality matrix when it is desired to overwrite
the calculated marginality matrix or when it is not
being calculated. It should consist 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. 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
within the structure is to be printed.
A data frame contains the values of the factors and variables
that occur in formula.
further arguments passed to terms.
Firstly, the primary projector
\(\mathbf{X(X'X)^-X'}\), where X is the design
matrix for the term, is calculated for each term. Then each
projector is made orthogonal to terms aliased with it, either
by differencing, eigenmethods or the default
hybrid method.
Differencing relies on
comparing the factors involved in two terms, one previous to the
other, to identify whether to subtract the orthogonalized projector
for the previous term from the primary projector of the other. It
does so if factors/variables for the previous term are a subset of
the factors/variablesfor for the other term. This relies on ensuring that all
projectors whose factors/variables are a subset of the current
projector occur before it in the expanded formula. It is checked that
the set of matrices are mutually orthogonal. If they are not then
a warning is given. It may happen that differencing does not produce
a projector, in which case eigenmethods must be used.
Eigenmethods forces each projector to be orthogonal
to all terms previous to it in the expanded formula. It uses
equation 4.10 of James and Wilkinson (1971), which involves
calculating the canonical efficiency factors for pairs of primary
projectors. It produces a
table of efficiency criteria for partially aliased terms. Again,
the order of terms is crucial. This method has the disadvantage that
the marginality of terms is not determined and so sources names are set
to be the same as the term names.
The hybrid method is the most general and uses the relationships
between the projection operators for the terms in the formula
to decide which projectors to subtract and which to orthogonalize using
eigenmethods. If \(\mathbf{Q}_i\) and \(\mathbf{Q}_j\) are
two projectors for two different terms, with \(i < j\), then
if \(\mathbf{Q}_j\mathbf{Q}_i \neq \mathbf{0}\) then have to orthogonalize \(\mathbf{Q}_j\) to \(\mathbf{Q}_i\).
if \(\mathbf{Q}_j\mathbf{Q}_i = \mathbf{Q}_j\) then, if \(\mathbf{Q}_i = \mathbf{Q}_j\), they are equal and \(\mathbf{Q}_j\) will be removed from the list of terms; otherwise they are marginal and \(\mathbf{Q}_i\) is subtracted from \(\mathbf{Q}_j\).
if have to orthogonalize and \(\mathbf{Q}_j\mathbf{Q}_i = \mathbf{Q}_i\) then \(\mathbf{Q}_j\) is aliased with previous terms and will be removed from the list of terms; otherwise \(\mathbf{Q}_i\) is partially aliased with \(\mathbf{Q}_j\) and \(\mathbf{Q}_j\) is orthogonalized to \(\mathbf{Q}_i\) using eigenmethods.
Again, the order of terms is crucial in this process.
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.
proj2.efficiency, proj2.combine, proj2.eigen,
projs.2canon 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)
## manually obtain projectors for units
Q.G <- projector(matrix(1, nrow=24, ncol=24)/24)
Q.B <- projector(fac.meanop(PBIBD2.lay$Block) - Q.G)
Q.BP <- projector(diag(1, nrow=24) - Q.B - Q.G)
## manually obtain projector for trt
Q.T <- projector(fac.meanop(PBIBD2.lay$trt) - Q.G)
##compute intrablock efficiency criteria
effic <- proj2.efficiency(Q.BP, Q.T)
effic
efficiency.criteria(effic)
##obtain projectors using pstructure.formula
unit.struct <- pstructure(~ Block/Unit, data = PBIBD2.lay)
trt.struct <- pstructure(~ trt, data = PBIBD2.lay)
##obtain combined decomposition and summarize
unit.trt.p2canon <- projs.2canon(unit.struct$Q, trt.struct$Q)
summary(unit.trt.p2canon, which = c("aeff","eeff","order"))
# }
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