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 unr.ridge(), which computes multiple points along the "Unrestricted" shrinkage-path. MLcalc() restricts attention to only 2 points: Unbiased OLS (BLUE) and the optimally Biased estimate with minimum MSE.
MLcalc(form, data, rscale = 1, delmax = 0.999999)A regression formula [y~x1+x2+...] suitable for use with lm().
Data frame containing observations on all variables in the formula.
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.
Maximum allowed value for Shrinkage delta-factors that is strictly less than 1. (default = 0.999999, which prints as 1 when rounded to fewer than 6 decimal places.)
An output list object of class MLcalc:
Name of the data.frame object specified as the second argument.
The regression formula specified as the first argument.
Number of regression predictor variables.
Number of complete observations after removal of all missing values.
Numerical value of R-square goodness-of-fit statistic.
Numerical value of the residual mean square estimate of error.
Listing of principal statistics.
Orthogonal Matrix of Direction Cosines for Principal Axes [1:p, 1:p].
Numerical shrinkage-ridge regression coefficient estimates [1:2, 1:p].
Numerical MSE risk estimates for fitted coefficients [1:2, 1:p].
Numerical delta-factors for shrinking OLS components [1:p].
Numerical rescaling factor for y-outcome variable [1, 1].
Numerical rescaling factors for given x-variables [1:p].
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 much 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 Generalized Ridge Regression shrinkage.
Thompson JR. (1968) Some shrinkage techniques for estimating the mean. Journal of the American Statistical Association 63, 113-122. (The ``cubic'' estimator.)
Obenchain RL. (2020) Ridge TRACE Diagnostics. https://arxiv.org/abs/2005.14291
Obenchain RL. (2020) RXshrink_in_R.PDF RXshrink package vignette-like file. http://localcontrolstatistics.org