Stopping the clock on the Poisson process

"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 a model's predictions and empirical reality. Likewise, the tiniest non-zero probability of an impossible event ought to invite us to dig deeper. Any puzzle about aggregated data is often clarified by looking at individual data. Our most basic assumption is that, if that the hazard rate at age \(x\) is \(\mu\), then the probability that a person alive at age \(x\) will die in time \(dt\) is:

\[\Pr[{\rm Death\ in\ }dt|{\rm Alive\ at\ }x]=\mu dt+o(dt)\qquad (1)\]

(see also equation (4) in Stephen's earlier blog.) This is so fundamental that it is the equivalent, if you like, of the physicist's model of the electron. It is also the fundamental assumption underlying a Poisson process with parameter \(\mu\), if we replace 'death' with 'the process jumps'. For \(i=1,\ldots,n\) let \(\tilde{N}_i(t)\) be a Poisson process with parameter \(\mu\). Suppose all the \(\tilde{N}_i(t)\) are mutually independent, and define \(\tilde{N}(t)=\sum_{i=1}^n \tilde{N}_i(t)\) (also a Poisson process). Then the following are true for any time \(t\ge 0\) chosen in a non-random way:

• \(\tilde{N}_i(t)\) is a Poisson random variable with parameter \(t \mu\), a non-random quantity.
• \(\tilde{N}_i(t)\) can take any non-negative integer value.
• The total time exposed-to-risk is \(nt\), a non-random quantity.
• The total number of jumps, \(\tilde{N}(t)\), is a Poisson random variable with parameter \(nt\mu\), a non-random quantity. So \(\Pr[\tilde{N}(t)>n]>0\); compare this with \(\Pr[D_x>n]> 0\) above.

Note the emphasis given to non-random in the above. Replace the time \(t\) with a random variable and the resulting process and associated random variables are no longer Poisson.

To make the link between these Poisson processes and a survival model, we would like to define \(\tilde{N}_i(t)\) to be the number of times the \(i^{\rm th}\) of \(n\) individuals has died by time \(t\), but to prevent each process from jumping more than once. A neat way to do this is to define an indicator process, \(Y_i(t)\), as follows:

\[Y_i(t) = \begin{cases} \mbox{1 if the \(i^{\rm th}\) individual is alive just before time \(t\)} \qquad \mbox{(2)} \\ \mbox{0 otherwise} \end{cases}\]

and to replace the constant hazard rate \(\mu\) with the hazard process \(Y_i(t)\mu\). This has the following effect:

• While the \(i^{\rm th}\) individual is alive, the hazard rate is 'switched on' and the individual is at risk of dying.
• As soon as the \(i^{\rm th}\) individual dies, the hazard rate is 'switched off' and they are no longer at risk of dying (again).
• The resulting process, denoted by \(N_i(t)\), can take only the values 0 or 1. That makes it a suitable model for the mortality of an individual.
• \(N_i(t)\) is not a Poisson process, it is a Poisson process 'stopped' after the first jump, a different thing altogether. As a result the total number of observed deaths, \(N(t) = \sum_{i=1}^n N_i(t)\), cannot exceed \(n\), and is not a Poisson random variable.
• The time spent exposed-to-risk by the \(i^{\rm th}\) individual up to time \(t\) is then \(E_i^c = \int_0^t Y_i(s)ds\). This is a random variable, as is the total exposed-to-risk, \(E_x^c\). However, if we treat \(E_x^c\) as non-random, as we did at the start of this blog, then \(D_x\) does have a Poisson distribution, but the cost of this assumption is that \(\Pr[D_x>n]>0\).

The mean time between jumps of a Poisson process with parameter \(\mu\) is \(1/\mu\); if \(\mu\) is very small, then the probability of any of the original \(\tilde{N}_i(t)\) jumping more than once is also very small. This is why the Poisson distribution is usually a good approximation for the number of deaths \(D_x\) in a survival model. But, like the physicist, we cannot ignore that tiny gap between model and reality. Further progress depends on studying the 'stopped' processes \(N_i(t)\), not the Poisson processes \(\tilde{N}_i(t)\), and certainly not the aggregate Poisson process \(\tilde{N}(t)\).

The indicator process \(Y_i(t)\) is important in its own right, and it appears in several places in our forthcoming book Modelling Mortality with Actuarial Applications. As just one example, if we tweak definition (2) slightly, so that \(Y_i(t) = 1\) if the \(i^{\rm th}\) individual is alive and under observation just before time \(t\), we have included both left-truncation and right-censoring in the model.