Information Matrix

More than a decade ago, we first posted on public health interventions proposed or implemented by the Scottish Government. A key focus area from those initiatives, alcohol mortality, has recently reported status, and the news does not seem encouraging. Despite the introduction of a minimum unit price (MUP) for alcohol in 2018, Scottish alcohol deaths reached a new high in 2024.

Stephen and I recently presented a pair of papers to the Institute and Faculty of Actuaries: Richards & Macdonald (2024) and Macdonald & Richards (2024).  In these papers we encourage actuaries to use continuous-time models in their work. But where does that leave discrete-time?

The World Health Organization (WHO) makes available a one-page checklist for use by surgical teams. The WHO claims that this checklist has made "significant reduction in both morbidity and mortality" and is "now used by a majority of surgical providers around the world".  For example, the checklist is used by surgical teams in NHS England.

In Richards & Macdonald (2024) we advocate that actuaries use the Kaplan-Meier estimate of the survival curve.  This is not just because it is an excellent visual communication tool, but also because it is a particularly useful data-quality check.

How should we describe a lifestyle change that doubles our likelihood of suffering a major traffic accident? Oddly,  evidence from Scotland suggests the answer is "worth making". Let me explain.

The short answer for mortality work is that your Poisson model is never truly Poisson. The longer answer is that the true distribution has a similar likelihood, so you will get the same answer from treating it like Poisson.  Your model is pseudo-Poisson, but not actually Poisson.

In a recent blog, I looked at the most fundamental unit of observation in a mortality study, namely an individual life. But is there such a thing as a fundamental unit of modelling mortality?  In Macdonald & Richards (2024) we argue that there is, namely an infinitesimal Bernoulli trial based on the mortality hazard.

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:

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

In Macdonald & Richards (2024), Angus and I continue our long-standing advocacy for using individual records for mortality analysis, rather than grouped counts of lives. One argument in our paper is that the individual life is the most irreducible unit of observation in mortality analysis.  After all, any group can be disaggregated into individuals, but further subdivision would just be dismemberment.

In a (much) earlier blog, Angus introduced the product-integral representation of the survival function:

\[{}_tp_x = \prod_0^t(1-\mu_{x+s}ds),\qquad(1)\]