RXridge {RXshrink}R Documentation

Maximum Likelihood Shrinkage in Regression

Description

Identify and display TRACEs for the Q-shaped shrinkage path, including the M-extent of shrinkage along that path, that are most likely under normal distribution theory to yield optimal reducions in MSE Risk.

Usage

  RXridge(form, data, rscale = 1, Q = "qmse", steps = 8, nq = 21,
              qmax = 5, qmin = -5, omdmin = 9.9e-13)

Arguments

form A regression formula [y~x1+x2+...] suitable for use with lm().
data Data frame containing observations on all variables in the formula.
rscale One of three possible choices (0, 1 or 2) for rescaling of variables as they are being "centered" to remove non-essential ill-conditioning: 0 implies no rescaling; 1 implies divide each variable by its standard error; 2 implies rescale as in option 1 but re-express answers as in option 0.
Q Shape parameter that controls the curvature of the shrinkage path through regression-coefficient likelihood space (default = "qmse" implies use the value found most likely to be optimal.) Use Q = 0 to specify Hoerl-Kennard "ordinary" ridge regression.
steps Number of equally spaced values per unit change along the horizontal M-extent-of-shrinkage axis for estimates to be calculated and displayed in TRACES (default = 8.)
nq Number of equally spaced values on the lattice of all possible values for shrinkage Q-shape between the "qmin" and "qmax" parameter settings (default = 21.)
qmax Maximum allowed Q-shape (default = +5.)
qmin Minimum allowed Q-shape (default = -5.)
omdmin Strictly positive minimum allowed value for one-minus-delta (default = 9.9e-013.)

Details

Illconditioned and/or nearly multicollinear regression models are unlikely to produce Ordinary Least Squares (OLS) regression coefficient estimates that are very close, numerically, to their unknown true values. Specifically, OLS estimates can then tend to have "wrong" numerical signs and/or unreasable relative magnitudes, while shrunken (generalized ridge) estimates chosen to maximize their likelihood of reducing Mean Squared Error (MSE) Risk (expected loss) can be much more stable and reasonable, numerically. On the other hand, because only OLS estimates are quaranteed to be minimax when risk is matrix valued (truly multivariate), no guarantee of an actual reduction in MSE Risk is necessarily associated with shrinkage.

Value

An output list object of class RXridge:

form The regression formula specified as the first argument.
data Name of the data.frame object specified as the second argument.
p Number of regression predictor variables.
n Number of complete observations after removal of all missing values.
r2 Numerical value of R-square goodness-of-fit statistic.
s2 Numerical value of the residual mean square estimate of error.
prinstat Listing of principal statistics.
crlqstat Listing of criteria for maximum likelihood selection of path Q-shape.
qmse Numerical value of Q-shape most likely to be optimal.
qp Numerical value of the Q-shape actually used for shrinkage.
coef Matrix of shrinkage-ridge regression coefficient estimates.
risk Matrix of MSE risk estimates for fitted coefficients.
exev Matrix of excess MSE eigenvalues (ordinary least squares minus ridge.)
infd Matrix of direction cosines for the estimated inferior direction, if any.
spat Matrix of shrinkage pattern multiplicative delta factors.
mlik Listing of criteria for maximum likelihood selection of M-extent-of-shrinkage.
sext Listing of summary statistics for all M-extents-of-shrinkage.

Author(s)

Bob Obenchain <wizbob@att.net>

References

Goldstein M, Smith AFM. (1974) Ridge-type estimators for regression analysis. J. Roy. Stat. Soc. B 36, 284-291. (2-parameter shrinkage family.)

Burr TL, Fry HA. (2005) Biased Regression: The Case for Cautious Application. Technometrics 47, 284-296.

Obenchain RL. (2005) Shrinkage Regression: ridge, BLUP, Bayes, spline and Stein. Electronic book-in-progress (200+ pages.) http://members.iquest.net/~softrx/.

Obenchain RL. (2009) RXshrink-R.PDF ../R_HOME/library/RXshrink

See Also

RXtrisk and RXtsimu.

Examples

  data(longley2)
  form <- GNP~GNP.deflator+Unemployed+Armed.Forces+Population+Year+Employed
  rxrobj <- RXridge(form, data=longley2)
  rxrobj
  names(rxrobj)
  plot(rxrobj)

[Package RXshrink version 1.0-4 Index]