Abstract
The risk of many demographic events varies by both current state and duration in that state. However, the use of such semi-Markov models has been substantially constrained by data limitations. Here, a new specification of the semi-Markov transition probability matrix in terms of the underlying rates is provided, and a general procedure is developed to estimate semi-Markov probabilities and rates from adjacent population data.Multistate models recognizing marriage and divorce by duration in state are constructed for United States Females, 1995. The results show that recognizing duration in the married and divorced states adds significantly to the model’s analytical value. Extending the constant-α method to semi-Markov models, 2000–2005 U.S. population data and 1995 cross-product ratios are employed to estimate 2000–2005 duration-dependent transfer probabilities and rates.The present analyses provide new relationships between probabilities and rates in semi-Markov models. Extending the constant cross-product ratio estimation approach opens new sources of data and expands the range of data susceptible to state-duration analyses.
Highlights
Multistate models typically follow persons as they move from state to state over age and/or time, using the Markov assumption that the risk of movement depends only on a person’s current status
Demographers have long known that, in many situations, the duration or length of time a person has been in their current state, can substantially affect the risk of an interstate movement
Models that recognize both current state and duration in that state are known as semi-Markov models, and a substantial body of statistical, actuarial, and demographic literature has explored them
Summary
Multistate models typically follow persons as they move from state to state over age and/or time, using the Markov assumption that the risk of movement depends only on a person’s current status. We turn to applying the approaches presented here, first to use rates to calculate a state-duration life table, and second to estimate interstate transfer probabilities and rates using the constant-α method. The relative sizes of those published duration-specific rates were weighted by the initial state composition at each age interval in the extended life table. Estimating probabilities from adjacent populations using constant-α The approach here uses the cross-product ratios from the 1995 extended life table of the previous section to estimate duration-specific probabilities from marital status life table populations for United States Females, 2000–2005, at ages 30 to 35.
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