Find an estimable subspace
estble-subspace.RdDetermine 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.
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.
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.
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
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 3.791214e-16 0.94444860 -2.2476951
#> [2,] 0 0 0 0 0 0 3.499401e-16 0.79383782 0.2564956
#> [3,] 0 0 0 0 0 0 -3.144207e-16 -0.74158401 0.4100229
#> [4,] 0 0 0 0 0 0 -5.640383e-16 0.05621727 1.2228212
#> [,10] [,11] [,12]
#> [1,] -0.3746606 -2.2247074 -7.309463e-17
#> [2,] 0.8589392 -0.5328160 2.575030e-16
#> [3,] -1.0361710 -0.5532579 5.432394e-17
#> [4,] 1.1290187 -0.8990313 1.707290e-16
#> attr(,"B")
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -0.6542234 0.3710275 0.54920429 0.2615446 0.14314236 0.20930771
#> [2,] -0.3512571 -0.8812953 0.07252168 0.2916776 -0.09453098 -0.02580085
#> [3,] 0.1293034 -0.1892932 0.12285602 -0.1427821 0.95331535 -0.05619815
#> [4,] 0.3123951 -0.1644018 0.21154095 -0.1148613 -0.06632638 0.90168701