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Here is a plot of death registration frequencies by age for the UK in 1974.

UK deaths by single year of age, 1974

I see distributions like this quite often: there is some event (e.g. death) which happens either close to birth, or according to some other reasonably well-behaved distribution.

I don't have much exposure to time-to-event/survival analysis/actuarial statistics, but it feels like this must be something that has been explored a dozen times over in a subfield that isn't mine. I've tried searching the literature/Google, but I really don't know what terms I'm supposed to be searching for.

My first inclination would be to model this as a mixture model with two components (perhaps Gaussian, perhaps Poisson). This makes intuitive sense – each unit is drawn from one of two populations with some probability and either experiences the event postnatally or over a longer time horizon. Is this sensible? Is there some other well-established set of models for this I simply haven't come across?

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For time-to-event models, it is useful to think in terms of the hazard function. Here, the hazard of death is high at low ages and at high ages, something that is sometimes called a "bathtub-shape". If you need a parametric survival model/distribution that can reflect such a shape, then one option is the generalized gamma distribution - see e.g. Cox, C., Chu, H., Schneider, M. F., & Muñoz, A. (2007). Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Statistics in medicine, 26(23), 4352-4374.

This is of course assuming that you need some parametric distribution that can reflect such data. Alternatively, if you want to do some regression modeling of to compare some groups of people, you could potentially use Cox regression and avoid assumptions about the hazard function. In case the hazards for some effects are not proportional over time similar semi-parametric approaches with time-varying covariates are possible extensions of Cox regression - or one could just calculate (and plot) Kaplan-Meier estimates of the survival function for each group of people (or the overall population). Whether this makes sense in what you aim to do depends on what you are trying to do.

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