On the Treatment of Interrupted Spells and Initial Conditions in Event History Analysis

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Abstract
This article deals with some problems of event history analysis that arise in special types of sampling plans where spells in progress at time t0, the start date of the observation period, are sampled. Then, the available data are random samples of interrupted spells, or else may be spells that begin after the start date of the sample. For the interrupted spells, all relevant probability distributions of the elapsed duration until t0 and the duration of the spell after t0 have to account for the condition that the individual is in the state y0 in question at time t0. Relying on results of Ridder (1984) and Heckman and Singer (1986), we discuss the distribution of the “sampled” spells (i.e., conditional on the presence of the individual in state y0 at time t0), in particular the joint distribution of pre-interview and post-interview durations. In general, this distribution depends on the entry rate into the state y0. The distribution of the sampled spells for the special sampling plans differs from the corresponding duration distributions in the population. We then show that, sampling spells in progress, the distribution of the explanatory variables in the sample becomes informative for the parameters of interest. Next, we investigate the distribution of the sampled spells of individuals who enter or leave the state in question. This referred to as “sampling the flow.” We then discuss the special case of a time invariant entry rate. Appropriate likelihood contributions are given depending on which pre-interview, post-interview, and completed durations are observable. We also discuss a conditional likelihood approach due to Lancaster (1979) that does not require any assumptions on the entry rate. Next we derive the appropriate likelihood contributions of right censored observations and both left and right censored observations for the specific sampling plans considered here. Finally, we present the results of some simulation studies that show the usefulness of taking into account the observable part of spells with unknown origin in the estimation procedure.

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