Jtr: General Classes of Influence Measures

Description

These functions implement the general classes of influence measures for multivariate regression models defined in Barrett and Ling (1992), Eqn 2.3, 2.4, as shown in their Table 1.

Usage

Jtr(H, Q, a, b, f)
Jdet(H, Q, a, b, f)
COOKD(H, Q, n, p, r, m)
DFFITS(H, Q, n, p, r, m)
COVRATIO(H, Q, n, p, r, m)

Value

The scalar result of the computation.

Arguments

H: a scalar or $m \times m$ matrix giving the hat values for subset $I$
Q: a scalar or $m \times m$ matrix giving the residual values for subset $I$
a: the $a$ parameter for the $J^{d e t}$ and $J^{t r}$ classes
b: the $b$ parameter for the $J^{d e t}$ and $J^{t r}$ classes
f: scaling factor for the $J^{d e t}$ and $J^{t r}$ classes
n: sample size
p: number of predictor variables
r: number of response variables
m: deletion subset size

Author

Michael Friendly

Details

There are two classes of functions, denoted $J_{I}^{d e t}$ and $J_{I}^{t r}$ , with parameters $n, p, q$ of the data, $m$ of the subset size and $a$ and $b$ which define powers of terms in the formulas, typically in the set -2, -1, 0.

They are defined in terms of the submatrices for a deleted index subset $I$ , $H_{I} = X_{I} (X^{T} X)^{- 1} X_{I}$ $Q_{I} = E_{I} (E^{T} E)^{- 1} E_{I}$ corresponding to the hat and residual matrices in univariate models.

For subset size $m = 1$ these evaluate to scalar equivalents of hat values and studentized residuals.

For subset size $m > 1$ these are $m \times m$ matrices and functions in the $J^{d e t}$ class use $| H_{I} |$ and $| Q_{I} |$ , while those in the $J^{t r}$ class use $t r (H_{I})$ and $t r (Q_{I})$ .

The functions COOKD, COVRATIO, and DFFITS implement some of the standard influence measures in these terms for the general cases of multivariate linear models and deletion of subsets of size m>1, but they have not yet been incorporated into our main functions mlm.influence and influence.mlm.

References

Barrett, B. E. and Ling, R. F. (1992). General Classes of Influence Measures for Multivariate Regression. Journal of the American Statistical Association, 87(417), 184-191.