YonX {RXshrink}

Maximum Likelihood (ML) Shrinkage in Simple Linear Regression

Description

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.

Usage

  YonX(form, data, delmax = 0.999999)

Arguments

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.)

Details

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.

Value

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.

Author(s)

Bob Obenchain <[e-mail]>

References

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

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

correct.signs and MLtrue

Examples

  data(haldport)
  form <- heat ~ p4caf
  YXobj <- YonX(form, data=haldport)
  YXobj
  plot(YXobj)