1. University of Warwick, Department of Sociology, 2012/13 SO 201: SSAASS (Surveys and Statistics) (Richard Lampard) Survival Analysis/Event History Analysis: Proportional Hazards Models (Week 17)

2. Definitions... Event history = Series of episodes/spells of time spent in discrete states. (Can be thought of in terms of transition times, or durations/waiting times) Survival analysis = Duration of time to a permanent transition into a different state.

3. Variables... (in a survival analysis) Dependent variable: Survival (to a given duration) Independent variables: An individual’s chance of ‘survival’ can be affected both by background characteristics which remain constant over time and by time-varying factors.

4. A complication... Censoring = When an individual is still in the original state and their final survival duration is thus as yet unknown. Censored cases need to be included in analyses to avoid biases. (Where censoring is related to the event history process in some way, biases can still occur).

5. Looking at the risk of not surviving over time Hazard rate = probability of event occurring between time t and time t+1. Hazard function [ h(t) ] = frequency curve for times at which events occur.

6. A key form of model... Proportional hazards model: log h(t) = log h 0 (t) + b 1 x 1 + b 2 x 2 +... The model is proportional in the sense that the hazard functions have the same shape but differ in magnitude, i.e. the hazard rate for two individuals differs by a constant multiplicative factor (e.g. the hazard for the first is consistently twice the hazard for the second).

7. The pros of not being parametric... Unlike some other models used for analyzing duration data, proportional hazards models do not require a researcher to specify a form for the hazard curve (technically speaking, they are semi- parametric rather than parametric). These models are thus well-suited to situations where a researcher is more interested in the difference between the hazards for different groups than in the way in which hazards vary with duration.

8. a.k.a. Cox regression The classic paper which introduced proportional hazards models (and highlights their relationship with a standard demographic tool: the ‘life table’ - see Hinde 1998: Ch. 4; Newell, 1988: Ch. 6) is: Cox, D.R. 1972. ‘Regression models and life tables’ (with discussion), Journal of the Royal Statistical Society (Series B), 34.2: 187-220. According to the Web of Science (Social Sciences Citation Index) this has been cited 26,919 times...

9. But... Marsh and Elliott (2009) suggest that there are some advantages to discrete time hazard models, in which a form can be specified for the hazard function. Discrete time models are also easier to integrate more than one time-dependent covariate into.

10. Similarities to logistic regression For a short discussion of proportional hazards alongside a discussion of logistic regression see: Rose, D. and Sullivan, O. 1996. Introducing Data Analysis for Social Scientists (2nd Edition). Buckingham: Open University Press. [Chapter 12].

11. Applications to marital formation and marital dissolution Lampard, R. 1994. ‘An Examination of the Relationship between Marital Dissolution and Unemployment’. In Gallie, D., Marsh, C. and Vogler, C. (eds) Social Change and the Experience of Unemployment. Oxford: OUP. [264-298]. Lampard, R. and Peggs, K. 1999. ‘Repartnering: the relevance of parenthood and gender to cohabitation and remarriage among the formerly married’, British Journal of Sociology, 50.3: 443-465.

12. But it isn’t just me! (Examples...) South, S.J. and Lloyd, K.M. 1995. ‘Spousal alternatives and marital dissolution’, American Sociological Review, 60.1: 21-35. Reinhold, S. 2010. ‘Reassessing the Link Between Premarital Cohabitation and Marital Instability’, Demography 47.3: 719-733. Aryal, T.R. 2007. ‘Age at first marriage in Nepal: Differentials and Determinants’, Journal of Biosocial Science 39.5: 693-706.

13. A useful textbook? Kleinbaum, D. 2011. Survival Analysis: A Self- Learning Text (3 rd edition). New York: Springer- Verlag. Other texts listed in the module reading list are quite technical...