The derived limited information test statistics involves some design matrices which act as transformations from the larger \(2^p\) response pattern space to the lower order univariate and bivariate marginals.
create_G_mat()returns the \(\tilde R \times R\) indicator matrix to obtain all pairwise components.create_T2_mat()returns the \(p(p+1)/2 \times 2^p\) indicator matrix \(T_2\) to pick out the unviariate and bivariate moments from the response patterns.create_Beta_mat()returns the \(4p \times p(p+1)/2\) design matrix \(\Beta\) described in the manuscript (used to express parameters in terms of residuals).
Note that ordering is similar to the ordering in create_resp_pattern().
These design matrices currently only apply to binary data. See technical
documents for more details.
Value
A matrix. Additionally, we may inspect the attributes regarding the ordering of the pairwise components of the \(G\) matrix.
Examples
create_G_mat(p = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] -1 -1 -1 -1 -1 -1 0 0
#> [2,] 0 0 1 1 0 0 0 0
#> [3,] 0 0 0 0 1 1 0 0
#> [4,] 1 1 0 0 0 0 0 0
#> [5,] -1 -1 -1 -1 -1 0 -1 0
#> [6,] 0 1 0 1 0 0 0 0
#> [7,] 0 0 0 0 1 0 1 0
#> [8,] 1 0 1 0 0 0 0 0
#> [9,] -1 -1 -1 0 -1 -1 -1 0
#> [10,] 0 1 0 0 0 1 0 0
#> [11,] 0 0 1 0 0 0 1 0
#> [12,] 1 0 0 0 1 0 0 0
#> attr(,"pairwise")
#> [,1] [,2] [,3]
#> [1,] 1 1 2
#> [2,] 2 3 3
create_T2_mat(p = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] 1 1 1 1 0 0 0 0
#> [2,] 1 1 0 0 1 1 0 0
#> [3,] 1 0 1 0 1 0 1 0
#> [4,] 1 1 0 0 0 0 0 0
#> [5,] 1 0 1 0 0 0 0 0
#> [6,] 1 0 0 0 1 0 0 0
create_Beta_mat(p = 3)
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -1 -1 0 1 0 0
#> [2,] 1 0 0 -1 0 0
#> [3,] 0 1 0 -1 0 0
#> [4,] 0 0 0 1 0 0
#> [5,] -1 0 -1 0 1 0
#> [6,] 1 0 0 0 -1 0
#> [7,] 0 0 1 0 -1 0
#> [8,] 0 0 0 0 1 0
#> [9,] 0 -1 -1 0 0 1
#> [10,] 0 1 0 0 0 -1
#> [11,] 0 0 1 0 0 -1
#> [12,] 0 0 0 0 0 1