Less is More: when weakness is a strength

(Jun 1, 2018)

A mathematical model that obtains extensive and useful results from the fewest and weakest assumptions possible is a compelling example of the art.  A survival model is a case in point.  The only material assumption we make is the existence of a hazard rate, \(\mu_{x+t}\), a function of age \(x+t\) such that the probability of death in a short time \(dt\) after age \(x+t\), denoted by \({}_{dt}q_{x+t}\), is:

\[{}_{dt}q_{x+t} = \mu_{x+t}dt + o(dt)\qquad (1)\]

(see Stephen's earlier blog on this topic).  It would be hard to think of a weaker mathematical description of mortality as an age-related process.  But from it much follows:

  • If we observe a life age \(x_i\) for a time \(t_i\), and define \(d_i = 1\) if the…

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Tags: survival models, Poisson distribution

Stopping the clock on the Poisson process

(Apr 12, 2018)

"The true nature of the Poisson distribution will become apparent only in connection with the theory of stochastic processes\(\ldots\)"

Feller (1950)

 

In a previous blog, we showed how survival data lead inexorably toward a Poisson-like likelihood. This explains the common assumption that if we observe \(D_x\) deaths among \(n\) individuals, given \(E_x^c\) person-years exposed-to-risk, and we assume a constant hazard rate \(\mu\), then \(D_x\) is a Poisson random variable with parameter \(E_x^c\mu\). But then \(\Pr[D_x>n]>0\). That is, an impossible event has non-zero probability, even if it is negligibly small. What is going on?

Physicists are ever alert to the tiniest difference between…

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Tags: Poisson distribution, survival model

Everything points to Poisson

(Jan 16, 2018)

One recurring theme in our forthcoming book, Modelling Mortality with Actuarial Applications, is the all-pervading role of likelihoods that suggest the lurking presence of a Poisson distribution.  A popular assumption in modelling hazard rates is that the number of deaths observed at any given age is a Poisson random variable, so perhaps that might explain it?  Surprisingly, it is the other way round - it is the very nature of the data in a survival model that leads inexorably to the Poisson distribution, even if we assume no such thing.

Stripped back to basics, we observe \(n\) individuals and record our observations as:

  • The length of time \(E_i\) that the \(i\)th person was observed and alive; and
  • An indicator…

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Tags: survival data, Poisson distribution

Groups v. individuals

(Sep 28, 2012)

We have previously shown how survival models based around the force of mortality, μx, have the ability to use more of your data.  We have also seen that attempting to use fractional years of exposure in a qx model can lead to potential mistakes. However, the Poisson distribution also uses μx, so why don't we use a Poisson model for the grouped count of deaths in each cell?  After all, a model using grouped counts sounds like it might fit faster.  In this article we will show why survival models constructed at the level of the individual are still preferable.

The first step when using the Poisson model is to decide on the width of the age interval.  This is necessary because the Poisson model for grouped counts…

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Tags: survival models, Poisson distribution, GLM

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