Don't fear the integral!

Actuaries denote with \({}_tp_x\) the probability that a life alive aged exactly \(x\) years will survive a further \(t\) years or more.  The most basic result in survival analysis is the following relationship with the instantaneous mortality hazard, \(\mu_x\):

\[{}_tp_x = e^{-H_x(t)}\qquad(1)\]

where \(H_x(t)\) is the integrated hazard:

\[H_x(t) = \int_0^t\mu_{x+s}ds\qquad(2).\]

It is at this point that cortisol levels spike for some people, often because it is a long time since they last saw an integral.  In the case of actuaries, this can lead them to seek safety in the comfort of standard tables based on \(q_x\).  However, even if you have forgotten everything you ever knew about integration, equation (2) poses no hurdle for mortality work.  The reason is that \(H_x(t)\) is just a function, and Richards (2008, Table 5) has a list of explicit formulae for the seven most common mortality models used by actuaries.  For example, assume that you want to integrate the Gompertz mortality hazard \(\mu_x=\exp(\alpha+\beta x)\).  Table 5 of Richards (2008) has this formula to use:

\[H_x(t) = \frac{(e^{\beta t}-1)}{\beta}e^{\alpha+\beta x}\qquad(3).\]

Thus, the integrated hazard between age 60 and 80 with \(\alpha=-12\) and \(\beta=0.12\) is 0.6874017.  No integration is required because it has already been done.  For the more adventurous, Richards (2012, Table 2) lists formulae for \(H_x(t)\) for a further ten models.

But what if you have an unusual mortality model that isn't listed?  Then \(H_x(t)\) can be approximated numerically using R's \(\tt integrate\) function.  The following snippet of R code illustrates this using the same Gompertz model as an example:

dAlpha = -12.0
dBeta = 0.12
GompertzHazard = function(x)
{
  return(exp(dAlpha+dBeta*x))
}
integrate(GompertzHazard, 60, 80)

If you paste the above into an R console you will see output like this:

0.6874017 with absolute error < 7.6e-15

which is the same result using the closed-form expression in equation (3).  Use \(\tt help(integrate)\) within your R console to find out how to fine-tune your control over this function.  And if your IT department won't let you use R, then there are some surprisingly accurate approximations available for integration; see also Appendix D of Macdonald et al (2018).

So, with explicit formulae freely available for most mortality models - and accurate approximations for all others - you can use survival models without ever having to work out a single integral yourself.  Unless you really want to, that is.