Administrative

Time to event

For many real-world problems, a model needs to account for uncertainty in event timing

• Recidivism
• Career length
• Life expectancy
• Time to residency
• Regime length
• Protest event length
• …

Time to (re)arrest

Duration vs. Hazard

Modeling duration

Choose an outcome distribution (e.g. Gamma, Weibull, exponential, log-normal, …) and follow the same strategy as we have for other GLM models:

Problem:
What do we do with people that were not arrested during the year? (censored data)

Censored data

Option 1 (bad)

We could drop observations for whom we do not know whether or when they were arrested. This is a bad idea and will almost certainly lead to biased results (see lecture on missing data)

N = 114 (out of 432)

Censored data

Option 2 (better)

A better approach is to treat the arrest timing for those who were not arrested during the 52 week sample period as missing data for which we have a definite lower limit.

There are many robust ways to deal with this sort of censored data. In a Bayesian context the most common approach is treat the missing values as parameters with strong priors

Censored data

Dropping censored observations (bad)

Modeling censoring process (better)

Hazard & survival functions

Proportional hazard model

Discrete version of Cox Proportional Hazard Model: