Statistical Demography

At a Glance Projects Publications Team


Multiple Time Scales in Survival and Event-History Models (Dissertation)

Angela Carollo, Jutta Gampe, Hein Putter (Leiden University Medical Center, Netherlands); in Collaboration with Paul H.C. Eilers (Erasmus University Rotterdam, Netherlands)

Detailed Description

In survival and event-history analysis, the occurrence of one or several events is studied, and the occurrence is described by hazard rates. Standard models consider hazards that vary along one main time scale. In demography, this time scale is usually the age of the individual.

However, in many applications multiple time scales are relevant for the occurrence of an event. For example, marital fertility is influenced by the age of the mother and the time since marriage, or mortality from chronic diseases varies with the age of the patient and the duration of illness. Traditional models select one of the time scales as dominant and incorporate other time scales as a time-varying covariate that changes at some discrete time points, commonly in a proportional hazards setting.

This project develops an approach that treats all time scales on an equal footing and, in the case of two time scales, models the hazard as a bivariate function over both scales. The two-dimensional hazard is assumed to vary smoothly over the time axes, but otherwise the interplay between the time scales is unrestricted. The new model thus avoids the need to single out one of the time scales as dominant and provides a tool to study even complex interactions between the time domains.

We employ penalized splines (P-splines) in order to estimate the smooth bivariate hazard. The approach allows to incorporate standard observations schemes, such a right-censoring and left-truncation. Further, the model is extended to hazard regression where a two-dimensional baseline hazard is modulated by covariates in a proportional hazards setting. It can be applied to the analysis of a single event, but was also extended to competing risks.

Efficient computation is essential in such a complex setting, and we devised algorithms and software that allows to analyze even large datasets conveniently. We developed an R-package to make the approach accessible to a wider audience.

Research Keywords:

Statistics and Mathematics


Carollo, A.; Eilers, P. H. C.; Gampe, J.:
Software. GitHub. unpublished. (2023)
Carollo, A.; Eilers, P. H. C.; Putter, H.; Gampe, J.:
arXiv e-prints 2305.09342. unpublished. (2023)    
Carollo, A.; Putter, H.; Eilers, P. H. C.; Gampe, J.:
SocArxiv papers. unpublished. (2023)    
Carollo, A.; Putter, H.; Eilers, P. H. C.; Gampe, J.:
In: Proceedings of the 36th International Workshop on Statistical Modelling (IWSM), Trieste, Italy, 18–22 July 2022, 123–128. Trieste: EUT Edizioni Università di Trieste. (2022)    
Carollo, A.; Gampe, J.:
In: Proceedings of the 34th International Workshop on Statistical Modelling, Guimarães, Portugal, July 7-12, 2019, volume II, 96–100. Statistical Modelling Society. (2020)
Carollo, A.; Putter, H.; Eilers, P. H. C.; Gampe, J.:
In: Proceedings of the 35th International Workshop on Statistical Modelling: July 20-24, 2020, Bilbao, Basque Country, Spain, 31–34. Bilbao: Servicio Editorial de la Universidad del País Vasco / Euskal Herriko Unibertsitateko Argitalpen Zerbitzua. (2020)    
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