MLcalc {RXshrink}

Calculate Efficient Maximum Likelihood (ML) point-estimates for a Linear Model that are either Unbiased (OLS) or Most Likely to be Optimally Biased under Normal-distribution theory.

Description

Compute MSE risk-optimal point-estimates of Beta-Coefficients and their Relative MSE risks. Much of the code for this function is identical to that of eff.ridge(), which computes multiple points along the "Efficient" Shrinkage Path. MLcalc() restricts attention to only two points: [1] the Unbiased OLS (BLUE) vector and [2] the Most Likely to be Optimally Biased [Minimum MSE Risk] vector of estimates.

Usage

  MLcalc(form, data, rscale = 1)

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 (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.

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 then tend to have "wrong" numerical signs and/or unreasonable relative magnitudes, while shrunken (generalized ridge) estimates chosen to Maximize their Likelihood of reducing Mean Squared Error (MSE) Risk (expected squared-error loss) can be more stable 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 expected reduction in MSE Risk is necessarily associated with "Optimal" Generalized Ridge Regression shrinkage.

Value

An output list object of class MLcalc:

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.

gmat

Orthogonal Matrix of Direction Cosines for Principal Axes [1:p, 1:p].

beta

Numerical shrinkage-ridge regression coefficient estimates [1:2, 1:p].

rmse

Numerical MSE risk estimates for fitted coefficients [1:2, 1:p].

dMSE

Numerical delta-factors for shrinking OLS components [1:p].

ys

Numerical rescaling factor for y-outcome variable [1, 1].

xs

Numerical rescaling factors for given x-variables [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. (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 2.1. http://localcontrolstatistics.org

See Also

eff.ridge, MLboot, eff.aug