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Biometrika Advance Access originally published online on November 25, 2007
Biometrika 2008 95(1):75-92; doi:10.1093/biomet/asm078
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© 2007 Biometrika Trust

Articles

Predicting future responses based on possibly mis-specified working models

Tianxi Cai

Department of Biostatistics, Harvard University, Boston, Massachusetts 02115, U.S.A. tcai{at}hsph.harvard.edu

Lu Tian

Department of Preventive Medicine, Northwestern University, Chicago, Illinois 60611, U.S.A. lutian{at}northwestern.edu

Scott D. Solomon

Cardiovascular Division, Brigham & Women's Hospital, Boston, Massachusetts 02115, U.S.A. ssolomon{at}rics.bwh.harvard.edu

L.J. Wei

Department of Biostatistics, Harvard University, Boston, Massachusetts 02115, U.S.A. wei{at}sdac.harvard.edu

Received for publication 1 July 2006. Revision received 1 May 2007.
  Abstract

Under a general regression setting, we propose an optimal unconditional prediction procedure for future responses. The resulting prediction intervals or regions have a desirable average coverage level over a set of covariate vectors of interest. When the working model is not correctly specified, the traditional conditional prediction method is generally invalid. On the other hand, one can empirically calibrate the above unconditional procedure and also obtain its crossvalidated counterpart. Various large and small sample properties of these unconditional methods are examined analytically and numerically. We find that the K-fold crossvalidated procedure performs exceptionally well even for cases with rather small sample sizes. The new proposals are illustrated with two real examples, one with a continuous response and the other with a binary outcome.

Key Words: Heterogeneous regression • K-fold crossvalidation • Mis-specified regression model • Optimal prediction region • Prediction error rate


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