Calculate the maximum likelihood estimates

This function fits the model using maximum likelihood. It takes an optional pattern matrix \(P\) as in (6.51), which specifies which \(\beta _{ij}\)'s are zero.

bothsidesmodel.mle(x, y, z = diag(qq), pattern = matrix(1, nrow = p, ncol = l))

Arguments

x

An \(N \times P\) design matrix.

y

The \(N \times Q\) matrix of observations.

z

A \(Q \times L\) design matrix

pattern

An optional \(N \times P\) matrix of 0's and 1's indicating which elements of \(\beta\) are allowed to be nonzero.

Value

A list with the following components:

Beta

The least-squares estimate of \(\beta\).

SE

The \(P \times L\) matrix with the \(ij\)th element being the standard error of \(\hat{\beta}_{ij}\).

T

The \(P \times L\) matrix with the \(ij\)th element being the \(t\)-statistic based on \(\hat{\beta}_{ij}\).

Covbeta

The estimated covariance matrix of the \(\hat{\beta}_{ij}\)'s.

df

A \(p\)-dimensional vector of the degrees of freedom for the \(t\)-statistics, where the \(j\)th component contains the degrees of freedom for the \(j\)th column of \(\hat{\beta}\).

Sigmaz

The \(Q \times Q\) matrix \(\hat{\Sigma}_z\).

Cx

The \(Q \times Q\) residual sum of squares and crossproducts matrix.

ResidSS

The dimension of the model, counting the nonzero \(\beta _{ij}\)'s and components of \(\Sigma _z\).

Deviance

Mallow's \(C_p\) Statistic.

Dim

The dimension of the model, counting the nonzero \(\beta _{ij}\)'s and components of \(\Sigma_z\)

AICc

The corrected AIC criterion from (9.87) and (aic19)

BIC

The BIC criterion from (9.56).

See also

bothsidesmodel.chisquare, bothsidesmodel.df, bothsidesmodel.hotelling, bothsidesmodel.lrt, and bothsidesmodel.

Examples

data(mouths)
x <- cbind(1, mouths[, 5])
y <- mouths[, 1:4]
z <- cbind(1, c(-3, -1, 1, 3), c(-1, 1, 1, -1), c(-1, 3, -3, 1))
bothsidesmodel.mle(x, y, z, cbind(c(1, 1), 1, 0, 0))
#> $Beta
#>           [,1]       [,2] [,3] [,4]
#> [1,] 24.937126  0.8268033    0    0
#> [2,] -2.271745 -0.3504386    0    0
#> 
#> $SE
#>           [,1]       [,2] [,3] [,4]
#> [1,] 0.5205837 0.09051471    0    0
#> [2,] 0.7935186 0.13797033    0    0
#> 
#> $T
#>           [,1]      [,2] [,3] [,4]
#> [1,] 47.902240  9.134463    0    0
#> [2,] -2.862876 -2.539956    0    0
#> 
#> $Covbeta
#>              [,1]         [,2]         [,3]         [,4]
#> [1,]  0.271007398  0.005297044 -0.273389373 -0.005343601
#> [2,]  0.005297044  0.008192913 -0.005343601 -0.008264923
#> [3,] -0.273389373 -0.005343601  0.629671752  0.012307409
#> [4,] -0.005343601 -0.008264923  0.012307409  0.019035812
#> 
#> $df
#> [1] 23
#> 
#> $SigmaR
#>           Age8    Age10    Age12    Age14
#> Age8  5.119199 2.440902 3.610510 2.522243
#> Age10 2.440902 3.927948 2.717514 3.062349
#> Age12 3.610510 2.717514 5.979798 3.823461
#> Age14 2.522243 3.062349 3.823461 4.617984
#> 
#> $Deviance
#> [1] 220.9863
#> 
#> $Dim
#> [1] 14
#> 
#> $AICc
#> [1] 258.7863
#> 
#> $BIC
#> [1] 267.128
#>