estimability (version 1.3)

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

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 nonestimability. For nonzero \(x\), estimability of \(\beta'x\) is assessed by whether or not \(||N'x||^2 < \tau ||x'x||^2\), where \(N\) and \(\tau\) denote nbasis and tol, respectively.

…

Additional arguments, currently ignored.

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.

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 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

# NOT RUN {
require(estimability)

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

# 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(nonest.basis(warp.lm1))

warp.lm2 <- update(warp.lm1, 
    contrasts = list(wool = "contr.sum", tension = "contr.helmert"))
zapsmall(nonest.basis(warp.lm2))
# }