Computes the canonical efficiency factors for the joint
decomposition of two structures or sets of mutually orthogonally
projectors (Brien and Bailey, 2009), orthogonalizing projectors in the
list to those earlier in the
list of projectors with
which they are partially aliased. The results can be summarized in the
form of a skeleton ANOVA table.
Two loops, one nested within the other, are performed. The first cycles
over the components of
Q1 and the nested loop cycles over the
Q2. The joint decomposition of the two projectors
in each cycle, one from
Q1[[i]]) and the other
Q2[[j]]) is obtained using
In particular, the nonzero canonical efficiency factors for the joint
decomposition of the two projectors is obtained. The nonzero canonical
efficiency factors are the nonzero eigenvalues of
Q1[[i]] %*% Q2[[j]] %*% Q1[[i]] (James and Wilkinson, 1971).
An eigenvalue is regarded as zero if it is less than
daeTolerance, which is initially set to
.Machine$double.eps ^ 0.5 (about 1.5E-08). The function
set.daeTolerance can be used to change
However, a warning occurs if any pair of Q2 projectors (say
Q2[[k]]) do not have adjusted orthgonality
with respect to any Q1 projector (say
Q1[[i]]), because they are
partially aliased. That is, if
Q2[[j]] %*% Q1[[i]] %*% Q2[[k]]
is nonzero for any pair of different Q2 projectors and any
Q1 projector. When it is nonzero, the projector for the later term in
the list of projectors is orthogonalized to the projector that is
earlier in the list. A list o such projectors is returned in the
aliasing component of the
Brien, C. J. and R. A. Bailey (2009). Decomposition tables for multitiered experiments. I. A chain of randomizations. The Annals of Statistics, 36, 4184 - 4213.
James, A. T. and Wilkinson, G. N. (1971) Factorization of the residual operator and canonical decomposition of nonorthogonal factors in the analysis of variance. Biometrika, 58, 279-294.
projector for further information about this class.
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## PBIBD(2) from p. 379 of Cochran and Cox (1957) Experimental Designs. ## 2nd edn Wiley, New York PBIBD2.unit <- list(Block = 6, Unit = 4) PBIBD2.nest <- list(Unit = "Block") trt <- factor(c(1,4,2,5, 2,5,3,6, 3,6,1,4, 4,1,5,2, 5,2,6,3, 6,3,4,1)) PBIBD2.lay <- designRandomize(allocated = trt, recipient = PBIBD2.unit, nested.recipients = PBIBD2.nest) ##obtain projectors using pstructure unit.struct <- pstructure(~ Block/Unit, data = PBIBD2.lay) trt.struct <- pstructure(~ trt, data = PBIBD2.lay) ##obtain combined decomposition and summarize unit.trt.p2canon <- projs.2canon(unit.struct$Q, trt.struct$Q) summary(unit.trt.p2canon)
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