eff.ridge {RXshrink}

Efficient Maximum Likelihood (ML) Shrinkage via the Shortest Piecewise Linear-Spline PATH

Description

Compute and display TRACEs for the p-paramater Shrinkage PATH passing through the (classical) Normal-theory Maximum Likelihood (ML) point-estimate of the Beta coefficient vector. The m-Extent of overall Optimal Shrinkage corresponding to this solution occurs at the only "interior" Knot on the Shrinkage Path and is also marked by the vertical dashed-line drawn on all 5-types of eff.ridge TRACE displays.

Usage

  eff.ridge(form, data, rscale = 1, steps = 20, ...)

Arguments

form

A regression formula [y~x1+x2+...+xp] 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 (after being "centered") to remove all "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.

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 = 20.)

...

Optional argument(s)

Details

Ill-conditioned and/or nearly multi-collinear 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 have unreasonable relative magnitudes or "wrong" numerical signs when the number of x-variables is 2 or more. Shrunken (Generalized Ridge Regression) estimates chosen to maximize their likelihood of reducing Mean Squared Error (MSE) Risk (expected Squared Error Loss) can be more stable and reasonable, numerically. On the other hand, because only OLS estimates are guaranteed 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 eff.ridge:

data

Name of the data.frame object specified as the second argument.

form

The regression formula is the first argument.

p

Number of regression x-predictor variables.

n

Number of complete observations after removal of all missing values.

r2

Numerical value of R-squared: proportion of variance explained.

s2

Numerical value of the residual mean square estimate of error.

prinstat

Listing of 5 summary statistics for each of p-Principal Axes.

rscale

Variable re-scaling code of 0, 1 or 2 used in calculations.

data

The data.frame containing all variables listed in the formula.

gmat

Orthogonal Matrix of Direction Cosines for Principal Axes.

coef

Matrix of shrinkage-ridge regression coefficient estimates.

rmse

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 an m-Extent for Shrinkage.

sext

Listing of summary statistics for all M-extents-of-shrinkage.

mStar

Optimal m-Extent of Shrinkage with delta[j] = dMSE[j] on TRACE displays.

mMSE

Minimum MSE Risk estimate.

mClk

Most Likely Observed Extent of Shrinkage: best multiple of (1/steps) <= p.

minC

Minimum Observed Value of Normal-theory -2*log(Likelihood-Ratio).

dMSE

Most Likely to be Optimal-values for Shrinkage: dMSE[j] for j in [1:p].

Author(s)

Bob Obenchain <[e-mail]>

References

Thompson JR. (1968) Some shrinkage techniques for estimating the mean. Journal of the American Statistical Association 63, 113-122. (The “cubic” estimator.)

Obenchain RL. (1978) Good and Optimal Ridge Estimators. Annals of Statistics 6, 1111-1121. doi:10.1214/aos/1176344314

Obenchain RL. (2021) The Efficient Shrinkage Path: Maximum Likelihood of Minimum MSE Risk. https://arxiv.org/abs/2103.05161

Obenchain RL. (2022) Efficient Generalized Ridge Regression. Open Statistics 3: 1-18. doi:10.1515/stat-2022-0108

Obenchain RL. (2022) RXshrink_in_R.PDF RXshrink package vignette-like document, Version 2.3. http://localcontrolstatistics.org

See Also

MLcalc, meff, correct.signs, MLtrue and RXpredict.

Examples

  data(longley2)
  form <- Employed~GNP+GNP.deflator+Unemployed+Armed.Forces+Population+Year
  rxefobj <- eff.ridge(form, data=longley2)
  rxefobj          # print shrinkage summary statistics...
  plot(rxefobj)    # 5 TRACEs on 1 plot...
  str(rxefobj)