nonest.basis: Estimability Tools

In estimability: Tools for Assessing Estimability of Linear Predictions

Estimability Tools

Description

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

Usage

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,
  tol = 5e-08,
  rank = attr(x, "rank"),
  pivot = attr(x, "pivot"),
  ...
)

## S3 method for class 'lm'
nonest.basis(x, ...)

## S3 method for class 'svd'
nonest.basis(x, tol = 5e-08, rank = NULL, pivot = NULL, ...)

legacy.nonest.basis(x, ...)

is.estble(x, nbasis, tol = 1e-08)

all.estble

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.
...	Additional arguments passed to other methods.
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.
rank	Optional integer value. If not NULL, it is used in place of tol to decide the rank of the model matrix
pivot	Optional integer vector of length equal to the number of columns in the model matrix. If non-NULL, it is assumed that this specifies the order of columns in x relative to the order of predictors in the model matrix, and the returned non-estimable basis is permuted accordingly.
nbasis	Matrix whose columns span the null space of the model matrix. Such a matrix is returned by nonest.basis.

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. Because estimability constraints are entirely determined by X, these methods apply to a rich class of models that includes linear models, generalized linear models, mixed models, and models with more general correlated-error structures (e.g., spatial models, time series models).

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.

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.

Use with Cholesky decompositions

If the chol function is called with pivot = TRUE, the returned matrix is suitable to provide to the matrix method of nonest.basis, and the returned basis is correctly pivoted without specifying pivot explicitly.

This works because the Cholesky decomposition R of X'X (as constructed by chol()) satisfies X'X = R'R, implying that the rows of X'X are linear combinations of the rows of R; and they are both of equal rank - so R and X'X have the same row space.

Examples

require(estimability)

X <- cbind(rep(1,5), 1:5, 5:1, 2:6)
( nb <- nonest.basis(X) )

# via the singular-value decomposition
SVD <- svd(X, nu = 0)    # we don't need the U part of UDV'
nonest.basis.svd(SVD)    # same result as above

# Via the Cholesky decomposition of X'X
ch <- chol(t(X) %*% X, pivot = TRUE) |> suppressWarnings()
( nb.chol <- nonest.basis(ch) )

# 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)
is.estble(lfs, nb.chol) # same results even though nb.chol is not identical to 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))

# different parameterization of the same model:
warp.lm2 <- update(warp.lm1,
    contrasts = list(wool = "contr.sum", tension = "contr.helmert"))
zapsmall(warp.nb2 <- nonest.basis(warp.lm2))

# These bases depend of the parameterizations, 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)

estimability documentation built on June 26, 2026, 9:08 a.m.