Maximum Likelihood (ML) Shrinkage in Simple Linear Regression
Compute and display Normal-theory ML Shrinkage statistics when a y-Outcome Variable is regressed upon a SINGLE x-Variable (i.e. p = 1). This illustration is usefull in regression pedagogy. The OLS (BLUE) estimate is a scalar in these simple cases, so the MSE optimal Shrinkage factor, dMSE, is also a scalar less than +1 and greater than 0 when cor(y,x) differs from Zero. The corresponding m-Extent of Optimal Shrinkage is marked by the "purple" vertical dashed-line on all YonX() TRACE Diagnostics.
YonX(form, data, delmax = 0.999999)
form: A regression formula [y ~ x] suitable for use with lm().data: Data frame containing observations on both variables in the formula.delmax: Maximum allowed value for Shrinkage delta-factor that is strictly less than 1. (default = 0.999999, which prints as 1 when rounded to fewer than 6 decimal places.)Since only a single x-Variable is being used, these "simple" models are (technically) NOT "Ill-conditioned". Of course, the y-Outcome may be nearly multi-collinear with the given x-Variable, but this simply means that the model then has low "lack-of-fit". In fact, the OLS estimate can never have the "wrong" numerical sign in these simple p = 1 models! Furthermore, since "risk" estimates are scalar-valued, no "exev" TRACE is routinely displayed; its content duplicates information in the "rmse" TRACE. Similarly, no "infd" TRACE is displayed because any "inferior direction" COSINE would be either: +1 ("upwards") when an estimate is decreasing, or -1 ("downwards") when an estimate is increasing. The m-Extent of shrinkage is varied from 0.000 to 1.000 in 1000 "steps" of size 0.001.
An output list object of class YonX: - data: Name of the data.frame object specified as the second argument.
form: The regression formula specified as the first argument to YonX() must have only ONE right-hand-side X-variable in calls to YonX().
p: Number of X-variables MUST be p = 1 in YonX().
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 for error.
prinstat: Vector of five Principal Statistics: eigval, sv, b0, rho & tstat.
yxnam: Character Names of "Y" and "X" data vectors.
yvec: "Y" vector of data values.
xvec: "X" vector of data values.
coef: Vector of Shrinkage regression Beta-coefficient estimates: delta * B0.
rmse: Vector of Relative MSE Risk estimates starting with the rmse of the OLS estimate.
spat: Vector of Shrinkage (multiplicative) delta-factors: 1.000 to 0.000 by -0.001.
qrsk: Vector of Quatratic Relative MSE Risk estimates with minimum at delta = dMSE.
exev: Vector of Excess Eigenvalues = Difference in MSE Risk: OLS minus GRR.
mlik: Normal-theory Likelihood ...for Maximum Likelihood estimation of Shrinkage m-Extent.
sext: Listing of summary statistics for all M-extents-of-shrinkage.
mUnr: Unrestricted optimal m-Extent of Shrinkage from the dMSE estimate; mUnr = 1 - dMSE.
mClk: Most Likely Observed m-Extent of Shrinkage: best multiple of (1/steps) <= 1.
minC: Minimum Observed Value of CLIK Normal-theory -2*log(Likelihood-Ratio).
minE: Minimum Observed Value of EBAY (Empirical Bayes) criterion.
minR: Minimum Observed Value of RCOF (Random Coefficients) criterion.
minRR: Minimum Relative Risk estimate.
mRRm: m-Extent of the Minimum Relative Risk estimate.
mReql: m-Extent where the "qrsk" estimate is first >= the observed OLS RR at m = 0.
Phi2ML: Maximum Likelihood estimate of the Phi-Squared noncentrality parameter of the F-ratio for testing H: true beta-coefficient = zero.
Phi2UB: Unbiased Phi-Squared noncentrality estimate. This estimate can be negative.
dALT: This Maximim Likelihood estimate of Optimal Shrinkage has serious Downward Bias.
dMSE: Best Estimate of Optimal Shrinkage Delta-factor from the "Correct Range" adjustment to the Unbiased Estimate of the NonCentrality of the F-ratio for testing Beta = 0.
Obenchain RL. (1978) Good and Optimal Ridge Estimators. Annals of Statistics
6 , 1111-1121. tools:::Rd_expr_doi("10.1214/aos/1176344314")
Obenchain RL. (2022) Efficient Generalized Ridge Regression. Open Statistics
3 : 1-18. tools:::Rd_expr_doi("10.1515/stat-2022-0108")
Obenchain RL. (2022) RXshrink_in_R.PDF RXshrink package vignette-like document, Version 2.1. http://localcontrolstatistics.org
Bob Obenchain wizbob@att.net
correct.signs and MLtrue
data(haldport) form <- heat ~ p4caf YXobj <- YonX(form, data=haldport) YXobj plot(YXobj)