This article was originally published on timgord.com on 2025-08-19.
A/E diagnostics are important but, if we have any mortality experience data, we should be using it to develop a model that takes account of that data, even if it’s nothing more than a simple how-much-heavier-or-lighter-is-the-mortality-of-this-population-than-average model. Otherwise, we’re not making full use of available information.
There are lots of possible approaches, including complex parametric formulas designed to capture all typically observed effects. But I promised concision and so in this article I’ll expound what I think is simultaneously one of the most powerful general approaches and one of the simplest. And the beauty of it is: we’ve already done most of the work.
To make this concrete, let’s assume that we have a mortality model \mu(\beta), where \beta is a vector of real parameters1 and that \mu(\beta) is differentiable by \beta.
A standard way to estimate \beta is to choose the value that maximises the log-likelihood, i.e.
\hat\beta = \underset{\beta}{\arg\max}\, L(\beta)\tag{5}
where L is the log-likelihood and the parameter \beta is passed through to the mortality \mu(\beta).
We saw in the previous article that the log-likelihood is
L=\text{A}w\log\mu-\text{E}w\tag{4}
where w\ge0 is an optional weighting variable.
We can try solving equation (5) by setting the derivative of L to zero, resulting in the vector differential equation
\frac{\partial L}{\partial\beta}(\hat\beta) = 0\tag{6}
Recalling that there is an implicit \mu(\beta) in the \text{E} operator, we can re-express the derivative as
\frac{\partial L}{\partial\beta}=\text{A}wX-\text{E}wX\tag{7}
where
X=\frac{1}{\mu}\frac{\partial \mu}{\partial\beta}\tag{8}
The proportional hazards model
The form of equation (7) begs the question: what if the vector X were an object in its own right, independent of \beta, i.e. \partial X/ \partial\beta=0?
Equation (8) would then be a simple first order differential equation in \mu, for which we can write down the solution as
\mu(\beta) = \mu^\text{ref}\exp\Big(\beta^\text{T}X\Big)\tag{9}
where
- \mu^\text{ref} is a mortality that does not depend on \beta, and
- \beta^\text{T}X means the ‘dot’ product of vectors \beta and X, i.e. \sum_j \beta_j X_j,
and don’t forget that \mu, \mu^\text{ref} and the components of the covariate vector X are all variables and therefore also have implicit fact (i) and time (t) arguments.
Equation (9) is the well-known proportional hazards model2, with the elements of X being the covariates and the elements of \beta the fitted covariate weights.
Some observations:
Let’s first note that we were led to equation (9) simply by writing the log-likelihood in terms of the \text{A} and \text{E} operators and the symmetry of equation (7).
The name ‘proportional hazards’ is not ideal because hazard3 rates combine using addition, i.e. \mu=\sum_i\mu^{(i)}, not multiplication, i.e. \mu=\prod_i e^{\beta_iX_i}. So the covariates do not relate to individual hazard rates; instead they are component effects used to build a model. The ultimate justification is the effectiveness – power and tractability – of the proportional hazards approach.
Finally, this is part of a bigger picture in which linear \log\mu4 models are ubiquitous, from Gompertz, arguably the world’s first realistic mortality model, via Lee-Carter5 to the CMI Mortality Projections Model.
We still need to solve equation (6), which, in general, we have to do numerically. The good news is that, provided we use a proportional hazards model, we can write down the first and second derivative of the log-likelihood in closed form. That in turn means we can use Newton–Raphson, which, in my experience is robust6 and beats most other numerical methods hands down7. The vector first derivative (from equation (7) above) and the matrix second derivative are
\begin{aligned}L'&=\text{A}wX-\text{E}wX\\\rule{0pt}{3.5ex}L''&=-\text{E}wXX^\text{T}\end{aligned}
where I have used ' to indicate \partial/\partial\beta.
At risk of repetition, note (a) the concision and (b) that everything can be expressed in terms of \text{A} and \text{E}, which means we’re re-using existing machinery for these calculations.
Information budget
Many expositions of the proportional hazards model do not include \mu^\text{ref}, i.e. a given ‘background’ hazard rate, but for mortality analysis this is often optimal. For instance:
If you’re analysing DB pension plan mortality experience over the last ten years, you probably don’t want to be trying to calibrate a mortality trends model8 at the same time, which you can avoid by putting your pre-existing mortality trends assumption into \mu^\text{ref}.
I’d suggest going further and modelling variation from a reasonable default mortality (including trends) so that you inherit a priori sensible behaviour9 from that default.
At the most extreme, if you have a postcode mortality model to hand, then use that as \mu^\text{ref}10.
In general, you want to spend the information budget provided by the experience data on fitting the unknowns you don’t know as opposed to spending it on refitting things you likely already do. The proportional hazards model makes this straightforward.
One model to rule them all
In my experience, the proportional hazards model is all you need in practice. The richness available from the infinite range of possible covariates, the sheer tractability of the approach, the straightforwardness of using a prior model and the interpretability of the results provide enough firepower to tackle any real world mortality modelling problem.
There is a lot more to this, e.g. how does mortality vary between populations, do we require additional procedures to select covariates initially, and so on, which are questions I hope to answer in due course.
But, for now, let’s take stock:
- With the proportional hazards model, we have an excellent framework for creating mortality models.
- And, by maximising log-likelihood, we can calibrate those models with relative ease.
Next article: Suddenly AIC
The obvious next question is: how should we choose between different models? This will be the subject of the next article.