xbeta
Duration model

Survival analysis concerns statistical models of the time until the first occurance of some event. In a biological setting, the event in question might be the death of an organism. In mechanics, it could be the failure of a machine. In economics, it could be entering the workforce.

Model

The survivor function is the fundamental part of such models, denoting the probability that the time of death is later than some time $t$:

where $T$ is a random variable denoting the time of death. Let $f(t)$ denotes the pdf of $T$ and $F(t)$ denote the cdf of $T$. We can see that $S$ must be non-increasing since

and $f$ is nonnegative.

Another fundamental object is the hazard function of $T$, $\lambda (t)$, which gives the rate at which the probability of death is changing at some time $t$:

Using the definition of conditional probability and the cdf, we can write

The cumulative hazard function of $T$, $\Lambda (t)$, is defined as

$\Lambda (t)$ is also called the integrated hazard. It follows that $\Lambda (t)=-\ln S(t)$.

Common Distributions

Some commonly used survivor functions (survival time distributions) include:

• The exponential distribution:

• The Weibull distribution:

  $$F(t)=1-e^{-\gamma t^{\alpha }}\text{or}\lambda (t)=\gamma \alpha t^{\alpha -1}.$$F(t) = 1 - e^{-\gamma t^\alpha} \quad \text{or} \quad \lambda(t) = \gamma \alpha t^{\alpha - 1}.

  There is positive or negative duration dependence as $\alpha \gtrless 1$.

Literature

• Cox, D. R. (1959). The analysis of exponentially distributed life-times with two types of failure. Journal of the Royal Statistical Society: Series B 21, 411–421.

• Cox, D. R. (1962). Renewal Theory. London: Methuen.

• Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society: Series B 34, 187–202.

• Tsiatis, A. (1975). A Nonidentifiability Aspect of the Problem of Competing Risks. Proceedings of the National Academy of Sciences 72, 20–22.

• Peterson, A. V. (1976). Bounds for a joint distribution function with fixed sub-distribution functions: Application to competing risks. Proceedings of the National Academy of Science 73, 11–13.

• Heckman, J. J. and Singer (1984) A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data. Econometrica 52, 271–320.

• Kiefer N. M. (1988). Economic Duration Data and Hazard Functions. Journal of Economic Literature 26, 646–679.

• Heckman, J. J. and B. E. Honoré (1989). The identifiability of the competing risks model. Biometrika 76, 325–330.

• Han A. K. and J. A. Hausman (1990). Flexible Parametric Estimation of Duration and Competing Risk Models. Journal of Applied Econometrics 5, 1–28.

• Honoré, B. E. (1990). Simple Estimation of a Duration Model with Unobserved Heterogeneity. Econometrica 58, 453–473.

• Ridder, G. (1990). The non-parametric identification of generalized accelerated failure time models. Review of Economic Studies 57, 167–182.

• Honoré, B. E. (1993). Identification results for duration models with multiple spells. Review of Economic Studies 60, 241–246.

• Ferrall, C. (1997). Unemployment Insurance Eligibility and the School-to-Work Transition in Canada and the United States. Journal of Business & Economic Statistics 15, 115–129.

• Honoré, B. E. and E. Kyriazidou (2000). Panel data discrete choice models with lagged dependent variables. Econometrica 68, 839–874.

• Van den Berg, G. J. (2001). Duration Models: Specification, Identification, and Multiple Durations. J. J. Heckman and E. Leamer, editors, Handbook of Econometrics, Volume 5, 3381–3460. Amsterdam: North Holland.

• Abbring, J. H., and G. J. Van den Berg (2003). The Identifiability of the Mixed Proportional Hazards Competing Risks Model. Journal of the Royal Statistical Society: Series B 65, 701–710.

• D’Addio, A. C. and B. E. Honoré (2006). Duration dependence and timevarying variables in discrete time duration models. Unpublished manuscript, Princeton University.

• Honoré, B. E. and A. Lleras-Muney (2006). Bounds in Competing Risks Models and the War on Cancer. Econometrica 74, 1675–1698.

• Brinch, C. N. (2007). Nonparametric identification of the mixed hazards model with time-varying covariates. Econometric Theory 23, 349–354.

• Frederiksen, A., B. E. Honoré, and L. Hu (2007). Discrete time duration models with group-level heterogeneity. Journal of Econometrics 141, 1014–1043.

• Khan, S. and E. Tamer (2007). Partial rank estimation of duration models with general forms of censoring. Journal of Econometrics 136, 251–280.