estimability (version 1.5.1)

nonest.basis: Estimability Tools

Description

This documents the functions needed to test estimability of linear functions of regression coefficients.

Usage

nonest.basis(x, ...)
# S3 method for default
nonest.basis(x, ...)
# S3 method for qr
nonest.basis(x, ...)
# S3 method for matrix
nonest.basis(x, ...)
# S3 method for lm
nonest.basis(x, ...)
# S3 method for svd
nonest.basis(x, tol = 5e-8, ...)
legacy.nonest.basis(x, ...)
all.estble
is.estble(x, nbasis, tol = 1e-8)

Value

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 NAs); 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.

Arguments

x

For nonest.basis, an object of a class in methods("nonest.basis"). Or, in is.estble, a numeric vector or matrix for assessing estimability of sum(x * beta), where beta is the vector of regression coefficients.

nbasis

Matrix whose columns span the null space of the model matrix. Such a matrix is returned by nonest.basis.

tol

Numeric tolerance for assessing rank or nonestimability. For determining rank, singular values less than tol times the largest singular value are regarded as zero. For determining estimability with a nonzero \(x\), \(\beta'x\) is assessed by whether or not \(||N'x||^2 < \tau ||x'x||^2\), where \(N\) and \(\tau\) denote nbasis and tol, respectively.

...

Additional arguments passed to other methods.

Author

Russell V. Lenth <[e-mail]>

Details

Consider a linear model \(y = X\beta + E\). If \(X\) is not of full rank, it is not possible to estimate \(\beta\) uniquely. However, \(X\beta\) is uniquely estimable, and so is \(a'X\beta\) 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 function legacy.nonest.basis is the original default method in early versions of the estimability package. It may be called with x being either a matrix or a qr object, and after obtaining the R matrix, it uses an additional QR decomposition of t(R) to obtain the needed basis. (The current nonest.basis method for qr objects is instead based on the singular-value decomposition of R, and requires much simpler code.)

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.

References

Monahan, John F. (2008) A Primer on Linear Models, CRC Press. (Chapter 3)

Examples

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)