Abstract
The use of maximum likelihood methods in analysing times to failure in the presence of unobserved randomly changing covariates requires constrained optimization procedures. An alternative approach using a generalized version of the EM-algorithm requires smoothed estimates of covariate values. Similar estimates are needed in evaluating past exposures to hazardous chemicals, radiation or other toxic materials when health effects only become evident long after their use. In this paper, two kinds of equation for smoothing estimates of unobserved covariates in survival problems are derived. The first shows how new information may be used to update past estimates of the covariates' values. The second can be used to project the covariates' trajectory from the present to the past. If the hazard function is quadratic in form, both types of smoothing equation can be derived in a closed analytical form. Examples of both types of equation are presented. Use of these equations in the extended EM-algorithm, and in estimating past exposures to hazardous materials, are discussed. © 1997 by John Wiley & Sons, Ltd.
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