HLM T (tau) matrix by method of moments, with EB shrinkage

T_hat = S - V_bar (sample covariance of per-group OLS coefficients minus average sampling covariance), clipped to positive semidefinite; per-group multivariate reliability Lambda_j = T (T + V_j)^{-1} and empirical Bayes composites beta*_j = Lambda_j beta_hat_j + (I - Lambda_j) gamma_hat.

Usage

morie_hlmgr(betas, V = NULL)

Arguments

betas

Matrix (J x q) of per-group OLS coefficients.

V

Optional list of J sampling covariance matrices.

Value

A list with elements tau, gamma, reliabilities, shrunken, s_total, J, method.

References

Raudenbush, S. W. and Bryk, A. S. (2002). Hierarchical Linear Models, 2nd ed. Sage, Ch. 3.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
morie_hlmgr(V)
#> $tau
#>      [,1]
#> [1,]    6
#> 
#> $gamma
#> [1] 4.5
#> 
#> $reliabilities
#> $reliabilities[[1]]
#>      [,1]
#> [1,]    1
#> 
#> $reliabilities[[2]]
#>      [,1]
#> [1,]    1
#> 
#> $reliabilities[[3]]
#>      [,1]
#> [1,]    1
#> 
#> $reliabilities[[4]]
#>      [,1]
#> [1,]    1
#> 
#> $reliabilities[[5]]
#>      [,1]
#> [1,]    1
#> 
#> $reliabilities[[6]]
#>      [,1]
#> [1,]    1
#> 
#> $reliabilities[[7]]
#>      [,1]
#> [1,]    1
#> 
#> $reliabilities[[8]]
#>      [,1]
#> [1,]    1
#> 
#> 
#> $shrunken
#>      [,1]
#> [1,]    1
#> [2,]    2
#> [3,]    3
#> [4,]    4
#> [5,]    5
#> [6,]    6
#> [7,]    7
#> [8,]    8
#> 
#> $s_total
#>      [,1]
#> [1,]    6
#> 
#> $J
#> [1] 8
#> 
#> $method
#> [1] "HLM T matrix, MoM (R&B 2002 Eqs. 3.28, 3.57)"
#>