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┌──────────────────────────────────────── Bob Obenchain, Ph.D.─────────┐
│ ╥───┐ ╖ ┌─ (tm) │
│ ║ │ ╙╖┌─┘ │
│ ╓─┐ ╓─┐ ╓── ─╥─ ╟─┬─┘ ╙┼╖ ╓─┐ ╖ ┌ ╓─┐ ╥─┐ ╥─┐ ╥ ┌ ┌ ╓─┐ ╥─┐ ╥─┐ │
│ ╙─┐ ║ │ ╟─ ║ ║ └┐ ┌─┘╙╖ ╙─┐ ╟─┤ ╟─┤ ╟┬┘ ╟─ ║ │ │ ╟─┤ ╟┬┘ ╟─ │
│ ╙─┘ ╙─┘ ╨ ╨ ╨ └ ─┘ ╙ ╙─┘ ╜ └ ╜ └ ╜└─ ╨─┘ ╙─┴─┘ ╜ └ ╜└─ ╨─┘ │
└─ 5261 Woodfield Drive North, Carmel, IN 46033, (317) 580-0144 voice ─┘
╔═════════════════════════════════════════════════════╗
║ softRX shareware's niche in the MS-DOS personal ║
║ computer market for statistical analysis software ║
╚═════════════════════════════════════════════════════╝
softRX shareware modules for generalized ridge regression are best
used to supplement (rather than replace) your favorite commercial
statistics package. Although softRX systems do provide
computational tools and interactive graphical displays that are
not presently available using any other statistical software,
softRX systems may not provide all of the "standard" features a
professional user of regression models would need to keep at
his/her fingertips. For example, RXridge.EXE cannot test general
linear hypotheses, form confidence intervals, measure leverage of
individual regressor combinations, or reveal lack-of-fit via
residuals. [This increased functionality is present in softRX's
RelaxR.EXE module, but RelaxR lacks graphics and its user
interface is quite un-friendly and old-fashioned by today's
standards.]
Starting ??? and Stopping !!! :
===============================
softRX shareware modules help users make well informed, objective
decisions about the two most important questions that arise in
each application of shrinkage methodology, Obenchain(1981):
Should I start ridging? ...and
If I do start, where do I stop?
An extensive simulation study, Gibbons(1981), demonstrated that
the two-parameter ridge approach (Q=>shape and MCAL=>extent of
shrinkage) implemented in softRX shareware systematically provides
"valuable information as to whether the regression situation is
favorable or unfavorable" and "offers potential improvement [in
mean squared error] over one-parameter ridge estimators."
Informal, Graphical Approach:
=============================
One's motivation for performing ridge regression analyses could be
to simply gain exploratory, data analytic insights. One may wish
to "see", in a highly graphical way, just how ill-conditioned
(statistically and/or numerically) one's regression formulation
really is. Usage of the ridge TRACE displays provided by softRX
shareware in this mode is explained in Obenchain(1984).
Classical (fixed coefficient) Approach:
=======================================
Ridge shrinkage does NOT cause classical confidence intervals /
regions to either shift in location or change their size or shape,
Obenchain(1977). Of course, Normal distribution theory confidence
hyper-ellipsoids will be highly oblate when regressors are nearly
multicollinear; the confidence intervals for certain linear
combinations of regression coefficients may be extremely wide! As
a result, an appropriate classical strategy would be, first, to
form confidence intervals using your favorite commercial
statistics package. Then use softRX shareware to identify
reasonable point estimates which, although within these intervals,
are pulled away from the interval centroid (at the least squares
point estimate) toward the shrinkage origin (usually "zeroed out"
coefficients) by an amount likely to reduce mean-squared-error.
Bayesian Approach:
==================
Ridge estimates can be viewed as posterior means that result from
combining observed sample information with one's prior information
about regression coefficients. The primary distinction between the
classical and Bayesian interpretations lies not in the resulting
family of point estimates but rather in their implied variability.
Bayes estimates incorporate "added information" from the prior
distribution as well as sample infromation from the available data;
the theory obscures any differences between these two sources of
(possibly conflicting) information by inflating the variance of point
estimates. softRX systems follow the "empirical Bayes" tradition of
viewing ridge coefficients as Bayes estimates and yet still using
classical variance formulas. Specifically, RXridge.EXE computes the
"empirical Bayes" likelihood criterion of Efron and Morris(1977) to
help monitor shrinkage extent. To perform full-blown Bayesian
analyses, you will need to use a software system like BRAP [Zellner,
Finnegan and Carlos(1988)] or XLISP-STAT [Tierney(1990,1992).]
Random Coefficient Approach:
============================
A specific amount of uniform shrinkage yields Best Linear Unbiased
Predictions, Henderson(1975) and Robinson(1991). Commercial
statistics packages for analysis of "mixed" linear models have
traditionally restricted attention to "variance component" models that
allow only class variables (that define "cells") to be declared
random. This restriction is beginning to disappear; for example, SAS
proc mixed [Wolfinger, Tobias and Sall(1991)], BMDP5V [Schluster
(1988)], and GLMM [Blouin and Saxton(1990)] each also allow continuous
variables to have random coefficients. RXridge.EXE computes the
random coefficient likelihood criterion of Golub, Heath, and
Wahba(1979) and Shumway(1982) that is maximized at Henderson's BLUPs.
Summary:
========
softRX shareware modules for generalized ridge regression have
something unique to offer every type of regression modeler. But
they are unlikely to satisfy ALL of the computing needs of a
professional statistician.
REFERENCES
Blouin, D. C. and Saxton, A. M. (1990). "General Linear Mixed Models
GLMM (Version 1.1)" Department of Experimental Statistics,
Louisiana State University, Baton Rouge, LA 70803.
Gibbons, D. G. (1981), "A Simulation Study of Some Ridge Estimators,"
Journal of the American Statisitical Association, 76,
131-139.
Henderson, C. R. (1975), "Best Linear Unbiased Estimation and
Prediction Under a Selection Model," Biometrics, 31, 423-447.
Obenchain, R. L. (1977), "Classical F-tests and Confidence Regions for
Ridge Regression," Technometrics, 19, 429-439.
Obenchain, R. L. (1981), Comment on "A Critique of Some Ridge
Regression Methods," Journal of the American Statisitical
Association, 75, 95-96.
Obenchain, R. L. (1984), "Maximum Likelihood Ridge Displays,"
Communications in Statistics, Theory and Methods 13, 227-240.
Robinson, G. K. (1991), "That BLUP is a Good Thing: The Estimation
of Random Effects" (with discussion). Statistical Science 6,
15-51.
Schluster, M. D. (1988). "Unbalanced repeated measures models with
structured covariance matrices." BMDP Statistical Software
Manual, Vol.2, 1081-1114. Berkeley: University of California
Press.
Tierney, L. (1990). Lisp-Stat. New York: John Wiley and Sons.
Tierney, L. (1992). "Statistical computing and Dynamic graphics using
Lisp-Stat." School of Statistics, University of Minnesota.
Wolfinger, R., Tobias, R., and Sall, J. (1991). "Mixed models: a
future direction." SUGI 16 Proceedings (SAS Users Group
International, 16th Annual Conference, New Olreans, Louisiana)
1380-1388.
Zellner, A., Finnegan, F. and Carlos, S. (1988). PCBRAP (Version 2.1
Beta Test), Bayesian Regression Analysis Package. H.G.B.
Alexander Research Foundation, 205D Rosenwald Hall, Graduate
School of Business, University of Chicago, Chicago, IL 60637.