Failure Time Models for Thinned Crime Commission Data

Abstract

In studying the effects of correctional treatments and other variables on criminal behavior, researchers often use time to arrest, instead of time until a crime is committed, as the criterion variable. We examine conditions under which inferences can validly be made from the effects of predictors on the time to the first arrest to their effects on the time to the first crime committed. A nonhomogeneous Poisson process and a renewal process are considered as possible models for crime commission. An individual's crimes are assumed to result in arrest independently, with some fixed probability less than one. Multiplicative regression models for the time to the first crime are invariant with respect to thinning for either a Poisson or a renewal process. Proportional hazards models for the time to the first crime are invariant under thinning for a Poisson process but not for a general renewal process. Some limit results for thinned point processes yield an interpretation of proportional hazard effects as multiplicative effects on the expected number of crimes committed.

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