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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.

## Usage

create_G_mat(p = 3)

create_T2_mat(p = 3)

create_Beta_mat(p = 3)

## Arguments

p

(integer > 0) The number of items.

## Value

A matrix. Additionally, we may inspect the attributes regarding the ordering of the pairwise components of the $$G$$ matrix.

## Author

Myrsini Katsikatsou (create_Beta_mat())

## 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