Description Usage Arguments Details Value Author(s) References Examples

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

1 2 3 4 5 6 7 8 9 10 11 | ```
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, ...)
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 nonestimability. For nonzero |

`...` |
Additional arguments, currently ignored. |

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 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 <[email protected]>

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
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))
``` |

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