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Biostatistics Advance Access published online on September 12, 2006
Biostatistics, doi:10.1093/biostatistics/kxl024
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© The Author 2006. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org
Received May 1, 2006
Revised August 25, 2006
Accepted September 8, 2006

Article

Numerical equivalence of imputing scores and weighted estimators in regression analysis with missing covariates

C. Y. Wang 1 *, Shen-Ming Lee 2, and Ed Chao 3

1 Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, P. O. Box 19024, Seattle, WA 98109-1024, USA
2 Department of Statistics, Feng-Chia University, Taichung, Taiwan, ROC
3 Insightful Corporation, 1700 Westlake Ave N., Suite 500, Seattle, WA 98109, USA

* To whom correspondence should be addressed.
C. Y. Wang, E-mail: cywang{at}fhcrc.org


  Abstract

Imputation, weighting, direct likelihood and direct Bayesian inference (Rubin, 1976) are important approaches for missing data regression. Many useful semiparametric estimators have been developed for regression analysis of data with missing covariates or outcomes. It has been established that some semiparametric estimators are asymptotically equivalent, but it has not been shown that many are numerically the same. We applied some existing methods to a bladder cancer case-control study and noted that they were the same numerically when the observed covariates and outcomes are categorical. To understand the analytical background of this finding, we further show that when observed covariates and outcomes are categorical, some estimators are not only asymptotically equivalent but are actually numerically identical. That is, although their estimating equations are different, they lead numerically to exactly the same root. This includes a simple weighted estimator, an augmented weighted estimator and a mean-score estimator. The numerical equivalence may elucidate the relationship between imputing scores and weighted estimation procedures.

Keywords: Estimating equation; Inverse selection probability; Ignorable missingness; Missing at random.
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