nonest.basis | R Documentation |
This documents the functions needed to test estimability of linear functions of regression coefficients.
nonest.basis(x, ...) ## Default S3 method: nonest.basis(x, ...) ## S3 method for class 'qr' nonest.basis(x, ...) ## S3 method for class 'matrix' nonest.basis(x, ...) ## S3 method for class 'lm' nonest.basis(x, ...) ## S3 method for class 'svd' nonest.basis(x, tol = 5e-8, ...) all.estble is.estble(x, nbasis, tol = 1e-8)
x |
For |
nbasis |
Matrix whose columns span the null space of the model matrix. Such a matrix is returned by |
tol |
Numeric tolerance for assessing rank or nonestimability.
For determining rank, singular values less than |
... |
Additional arguments passed to other methods. |
Consider a linear model y = Xβ + E. If X is not of full rank, it is not possible to estimate β uniquely. However, Xβ is uniquely estimable, and so is a'Xβ for any conformable vector a. Since a'X comprises a linear combination of the rows of X, it follows that we can estimate any linear function where the coefficients lie in the row space of X. Equivalently, we can check to ensure that the coefficients are orthogonal to the null space of X.
The nonest.basis
method for class 'svd'
is not really functional as a method because there is no "svd"
class (at least in R <= 4.2.0). But the function nonest.basis.svd
is exported and may be called directly; it works with results of svd
or La.svd
. We require x$v
to be the complete matrix of right singular values; but we do not need x$u
at all.
The default
method does serve as an svd
method, in that it only works if x
has the required elements of an SVD result, in which case it passes it to nonest.basis.svd
.
The matrix
method runs nonest.basis.svd(svd(x, nu = 0))
. The lm
method runs the qr
method on x$qr
.
The constant all.estble
is simply a 1 x 1 matrix of NA
. This specifies a trivial non-estimability basis, and using it as nbasis
will cause everything to test as estimable.
When X is not full-rank, the methods for nonest.basis
return a basis for the null space of X. The number of rows is equal to the number of regression coefficients (including any NA
s); and the number of columns is equal to the rank deficiency of the model matrix. The columns are orthonormal. If the model is full-rank, then nonest.basis
returns all.estble
. The matrix
method uses X itself, the qr
method uses the QR decomposition of X, and the lm
method recovers the required information from the object.
The function is.estble
returns a logical value (or vector, if x
is a matrix) that is TRUE
if the function is estimable and FALSE
if not.
Russell V. Lenth <russell-lenth@uiowa.edu>
Monahan, John F. (2008) A Primer on Linear Models, CRC Press. (Chapter 3)
require(estimability) X <- cbind(rep(1,5), 1:5, 5:1, 2:6) ( nb <- nonest.basis(X) ) SVD <- svd(X, nu = 0) # we don't need the U part of UDV' nonest.basis.svd(SVD) # same result as above # Test estimability of some linear functions for this X matrix lfs <- rbind(c(1,4,2,5), c(2,3,9,5), c(1,2,2,1), c(0,1,-1,1)) is.estble(lfs, nb) # Illustration on 'lm' objects: warp.lm1 <- lm(breaks ~ wool * tension, data = warpbreaks, subset = -(26:38), contrasts = list(wool = "contr.treatment", tension = "contr.treatment")) zapsmall(warp.nb1 <- nonest.basis(warp.lm1)) warp.lm2 <- update(warp.lm1, contrasts = list(wool = "contr.sum", tension = "contr.helmert")) zapsmall(warp.nb2 <- nonest.basis(warp.lm2)) # These bases look different, but they both correctly identify the empty cell wcells = with(warpbreaks, expand.grid(wool = levels(wool), tension = levels(tension))) epredict(warp.lm1, newdata = wcells, nbasis = warp.nb1) epredict(warp.lm2, newdata = wcells, nbasis = warp.nb2)
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