Graphics for Shrinkage in "Simple" Linear Regression: Models with only p=1 X-variable. The default is to first [1] display four TRACES in one plot and then to [2] display the "Y on X" scatter-plot with Three Fitted Lines: the OLS fit is BLUE, the optimally Shrunken fit is "purple", and the most shrinkage with estimated relative risk not exceeding OLS is "red". Note: these 3 fitted lines can be quite close to each other whenever model lack-of-fit is small.
# S3 method for YonX
plot(x, trace = "all", … )Output list object of class "YonX".
One of EIGHT possible options: "all" to display 4 traces on the first plot, then the "YonX" scatter plot; "seq" to display 5 full-sized plots in sequence (in response to user prompts); "coef" to display only the estimated beta coefficient trace (a straight line); "rmse" to display only the (quadratic) estimated relative MSE risk trace; "spat" to display only the delta-factor trace (a straight line); "lglk" to display only the "-2 log(Likelihood Ratio)" trace; "YonX" to display only the Y-vs-X scatter plot with 3 fitted-lines: the OLS fitted line (BLUE), the "purple" Maximum-Likelihood (optimally biased) line, and the "red" line marking the most shrinkage with estimated MSE Relative Risk less than or equal to that of OLS; or "exev" to display only the Excess Eigenvalue trace that is redundant with the "rmse" trace.
Optional argument(s) passed on to plot().
NULL
The effects of Shrinkage on Simple Linear Regression models (p = 1) are, in reality, NOT easier to illustrate than the corresponding effects on Multiple Linear Regressiom models (p > 1). In both situations, alternative estimates of effects and risks abound. For example, the estimate chosen can be [1] Maximum Likelihood under Normal-theory, [2] Unbiased under Normal-theory or [3] have "Correct Range". [See Obenchain (1978), equations (3.3) to (3.5), and corresponding text.] When a graphic contains only a single curve, a "reality" is that the general "shape" of the curve (plus any highlighted "points" on that curve) should "look right" or, at the very least, "reasonable".
My choices among alternative estimates of (nonlinear) MSE risk were initially made roughly 30 years ago ...and have remained mostly unchanged (primarily) for consistency with earlier versions of RXshrink.
The NEW "rmse" TRACE for class "YonX" displays MSE Relative Risk estimates from the "qrsk" vector rather than the (traditional) "rmse" estimates for all p = 1 models. This allows MSE Relative Risk estimates to satisfy a Quadratic equation and give the CORRECT visual impression that Relative Risk is MINIMIZED at the ML "purple" point and dotted-line at m = (1-dMSE). Note that the Relative Risk then starts to increase for m > (1-dMSE) and returns to its initial starting level at m = 0 ["blue" point and dotted-line] when the m-Extent reaches m = 2*(1-dMSE) ["red" point and dotted-line.] Finally, when 0.5 < dMSE < 1, the Relative Risk then continues to increase, reaching its Maximum at m = 1. As argued in Obenchain(1978), the "Good" Shrinkage Range is 0 < m < 2*(1-dMSE), between the "blue" and "red" vertical dotted-lines on the "rmse" TRACE.
Obenchain RL. (1978) Good and Optimal Ridge Estimators. Annals of Statistics 6, 1111-1121. <doi:10.1214/aos/1176344314>
Obenchain RL. (2021) RXshrink_in_R.PDF RXshrink package vignette-like document, Version 2.0. http://localcontrolstatistics.org
# NOT RUN {
data(haldport)
form <- heat ~ p4caf
YXobj <- YonX(form, data=haldport)
plot(YXobj)
# }
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