## Less is More: when weakness is a strength

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 life died at age \(x_i+t_i\), and \(d_i = 0\) otherwise (i.e. if our observation was right-censored at age \(x_i+t_i\)) the probability of this observation can be written compactly as:

\[L_i = \exp\left(-\int_0^{t_i}\mu_{x_i+s}ds\right)\mu^{d_i}_{x_i+t_i}.\qquad (2)\]

See Chapter 5 of our new book, *Modelling Mortality with Actuarial Applications*, where this expression is derived. Notably, it requires neither an assumption about any particular formula for \(\mu_{x+t}\), nor any statistical model for the number of deaths.

- Want to fit a model to a portfolio with data for each individual life? Then (2) gives you the contribution of the \(i^{\rm th}\) life to the likelihood.
- Have a particular parametric model in mind (Gompertz, Makeham, Perks, G-M family, etc.)? No problem, just plug your assumptions into (2) and reach for your favourite maximisation routine. This also requires no statistical model for the number of deaths.
- Want to estimate the hazard rate at single years of age? (useful for checking model fit, or fitting a generalised linear model, or projecting mortality). Then aggregate the observations made between ages \(x\) and \(x+1\), resulting in \(d_x\) observed deaths and \(E^c_x\) person-years of time exposed to the risk of death. Then from (2) we can show that the random number of deaths, \(D_x\) (of which \(d_x\) is the observed value) is, to a very good approximation, a Poisson random variable with parameter \(E^c_x\mu_{x+\frac{1}{2}}\). So now we
*do* have a statistical model for the number of deaths but, as shown in an earlier blog, it is not an *a priori* assumption, but is a property that emerges from (1).
- Want to incorporate a vector of covariates \(z_i\) for the \(i^{\rm th}\) life? (For example \(z_i\) could describe sex, smoking status or benefit amount.) Either
*stratify* the data, fitting a separate hazard-rate model for each value of the covariates, or *model* the data, defining a hazard rate, \(\mu_{z,x+t}\), as a function of the covariates as well as age, and fitting a model to all the data (see Chapter 7 of our book). In either case, putting the hazard rates into (2) gives the likelihood.
- Need to model more than just 'alive' or 'dead'? Assume that movements between any two of a number of
*states* (e.g. healthy, sick, dead) are governed by a hazard rate analogous to (1). This leads to *Markov multiple-state models* (see Chapters 14–17 of our book).

All in all, a respectable outcome starting from such a weak assumption as (1). But perhaps this is not so surprising when we remember that (1) is also the assumption underlying the *Poisson process*, which seems to be one of nature's most fundamental models.

**References**

Macdonald, A. S., Richards. S. J. and Currie, I. D. (2018). Modelling Mortality with Actuarial Applications, Cambridge University Press, Cambridge.

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