correct.signs: Normal-Theory Maximum Likelihood Estimation of Beta...

In RXshrink: Maximum Likelihood Shrinkage using Generalized Ridge or Least Angle Regression

Normal-Theory Maximum Likelihood Estimation of Beta Coefficients with "Correct" Signs

Description

Obenchain(1978) discussed the risk of linear generalized ridge estimators in individual directions within p-dimensional X-space. While shrinkage to ZERO is clearly optimal for all directions strictly ORTHOGONAL to the true BETA, he showed that optimal shrinkage in the UNKNOWN direction PARALLEL to the true BETA is possible. This optimal BETA estimate is of the form k * X'y, where k is the positive scalar given in equation (4.2), page 1118. The correct.signs() function computes this estimate, B(=), that uses GRR delta-shrinkage factors proportional to X-matrix eigenvalues.

Usage

  correct.signs(form, data)

Arguments

form	A regression formula [y~x1+x2+...] suitable for use with lm().
data	Data frame containing observations on all variables in the formula.

Details

Ill-conditioned (nearly multi-collinear) regression models can produce Ordinary Least Squares estimates with numerical signs that differ from those of the X'y vector. This is disturbing because X'y contains the sample correlations between the X-predictor variables and y-response variable. After all, these variables have been "centered" by subtracting off their mean values and rescaled to vectors of length one. Besides displaying OLS estimates, the correct.signs() function also displays the "correlation form" of X'y, the estimated delta-shrinkage factors, and the k-rescaled beta-coefficients. Finally, the "Bfit" vector of estimates proportional to B(=) is displayed that minimizes the restricted Residual Sum-of-Squares. This restricted RSS of Bfit cannot, of course, be less than the RSS of OLS, but it can be MUCH less that the RSS of B(=) whenever B(=) shrinkage appears excessive.

Value

An output list object of class "correct.signs":

data	Name of the data.frame object specified as the second argument.
form	The regression formula specified as the first 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 (p by 5) from qm.ridge().
kpb	Maximum likelihood estimate of k-factor in equation (4.2) of Obenchain(1978).
bmf	Rescaling factor for B(=) to minimize the Residual Sum-of-Squares.
signs	Listing of five Beta coefficient statistics (p by 5): OLS, X'y, Delta, B(=) and Bfit.
loff	Lack-of-Fit statistics: Residual Sum-of-Squares for OLS, X'y, B(=) and Bfit.
sqcor	Squared Correlation between the y-vector and its predicted values. The two values displayed are for OLS predictions or for predictions using Bfit, X'y or B(=). These two values are the familiar R^2 coefficients of determination for OLS and Bfit.

Author(s)

Bob Obenchain <[e-mail]>

References

Obenchain RL. (1978) Good and Optimal Ridge Estimators. Annals of Statistics 6, 1111-1121. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/aos/1176344314")}

Obenchain RL. (2022) RXshrink_in_R.PDF RXshrink package vignette-like document, Version 2.2. http://localcontrolstatistics.org

See Also

eff.ridge, qm.ridge and MLtrue.

Examples

  data(longley2)
  form <- GNP~GNP.deflator+Unemployed+Armed.Forces+Population+Year+Employed
  rxcsobj <- correct.signs(form, data=longley2)
  rxcsobj
  str(rxcsobj)

RXshrink documentation built on Aug. 8, 2023, 1:09 a.m.