Determine a transformation B of the rows of a matrix L
such that B %*% L is estimable.
A practical example is in jointly testing a set of contrasts L
in a linear model, and we need to restrict to the subspace spanned by
the rows of L that are estimable.
Arguments
- L
A matrix of dimensions k by p
- nbasis
A k by b matrix whose columns form a basis for non-estimable linear functions – such as is returned by
nonest.basis- tol
Numeric tolerance for assessing nonestimability. See
is.estble.
Value
An r by p matrix M = B %*% L
whose rows are all orthogonal to the columns of
nbasis. The matrix B is attached as attr(M, "B").
Note that if any rows of L were non-estimable, then r
will be less than k. In fact, if there are no estimable functions
in the row space of L, then r = 0.
Details
We require B such that all the rows of M = B %*% L
are estimable, i.e. orthogonal to the columns of nbasis.
Thus, we need B %*% L %*% nbasis to be zero, or equivalently,
t(B) must be in the null space of t(L %*% nbasis).
This can be found using nonest.basis.
Examples
### Find a set of estimable interaction contrasts for a 3 x 4 design
### with two empty cells.
des <- expand.grid(A = factor(1:3), B = factor(1:4))
des <- des[-c(5, 12), ] # cells (2,2) and (3,4) are empty
X <- model.matrix(~ A * B, data = des)
N <- nonest.basis(X)
L <- cbind(matrix(0, nrow = 6, ncol = 6), diag(6))
# i.e., give nonzero weight only to interaction effects
estble.subspace(L, N)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
#> [1,] 0 0 0 0 0 0 0 0 1 0 0 0
#> [2,] 0 0 0 0 0 0 0 0 0 1 0 0
#> [3,] 0 0 0 0 0 0 0 0 0 0 1 0
#> [4,] 0 0 0 0 0 0 0 -1 0 0 0 0
#> attr(,"B")
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0 0 1 0 0 0
#> [2,] 0 0 0 1 0 0
#> [3,] 0 0 0 0 1 0
#> [4,] 0 -1 0 0 0 0
# Tougher demo: create a variation where all rows of L are non-estimable
set.seed(1921.0923) # Franklin Graybill's birthday
LL <- matrix(rnorm(36), ncol = 6) %*% L
estble.subspace(LL, N)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
#> [1,] 0 0 0 0 0 0 -7.024565e-17 1.0316251 0.2402223
#> [2,] 0 0 0 0 0 0 -1.914267e-16 0.9679828 0.3871052
#> [3,] 0 0 0 0 0 0 3.613173e-16 -0.8807992 -0.4495615
#> [4,] 0 0 0 0 0 0 4.003655e-16 0.1534469 -1.8498715
#> [,10] [,11] [,12]
#> [1,] 1.12000141 -0.67713307 5.497174e-17
#> [2,] -0.04158445 -1.04336175 9.151698e-17
#> [3,] -1.22043728 0.89366353 -2.091758e-16
#> [4,] 0.63353316 0.01685268 -1.332937e-17
#> attr(,"B")
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.2368168 0.44441284 0.81010361 0.2315774 0.1846316 -0.04929783
#> [2,] -0.7367046 -0.04759893 0.17253661 0.5347192 -0.3145821 0.20086111
#> [3,] -0.4882847 -0.14729815 0.15061904 -0.3305079 0.7689707 0.12901096
#> [4,] 0.3357518 -0.18734622 -0.01314464 0.1720998 0.1026899 0.90101930