## Actuarial exceptionalism

In an earlier posting I listed some actuarial terms and their statistical equivalents (and later a short list of statistical terms and their equivalents in other fields). Using different expressions for the same concept is an unfortunate barrier to understanding across disciplines. However, sometimes different disciplines need to use tools and methodologies in very different ways. In this posting we look at why actuaries approach survival models in a different way to statisticians, especially those who routinely use survival models in medical trials.

The first difference between actuarial work and other disciplines begins with the data. In a medical trial there is usually one record per person, as the data have been recorded at the level of the individual. Actuaries, however, have to contend with the problem of duplicates: the basic unit of administration is the policy or benefit, and people are free to have more than one of these. Indeed, wealthier people have a tendency to have more policies than others. Before even fitting the first model, actuaries therefore have to deduplicate.

The second difference is that statisticians working in medical trials often simply want to know if there is a difference between two populations, say with and without a particular treatment. This kind of hypothesis testing does not require a model of the shape of the mortality curve, and simply establishing whether one group has a higher survival rate than another will often suffice. In survival-model work, the Cox proportional hazard model does not require the estimation of the so-called *baseline hazard* in order to answer the hypothesis being tested. Actuarial work is quite different: it is essential to model the shape of the mortality curve in order to get the timings of cashflows correct.

Another major area of difference is *left truncation*, the situation where information is unavailable below a lower limit. Medical statisticians rarely need to deal with left truncation, as they are modelling the time lived since treatment began — the lower limit is therefore zero and there is no left truncation. In contrast, actuaries always need to deal with left truncation, as they are modelling the lifetime of the individuals — the lower limit is also zero (birth), but lives enter actuarial investigations at adult ages. An illustration of the different requirements of medical statisticians and actuaries is the book *Modelling Survival Data in Medical Research* by Collett (2003) — "truncation" doesn't even appear in the index, as it is not something medical statisticians usually have to worry about.

However, left truncation is a challenge for actuaries wanting to use standard software packages such as SAS and R. These packages fit survival models by transforming the observed lifetimes and then using the GLM-fitting algorithm. This is a convenient result for the software developer, but it comes at a cost: the loss of the ability to handle left-truncated data for most lifetime distributions. Medical statisticians don't view this as a great loss, so such software is still well suited to their needs. For actuaries looking to build survival models, however, it is fatal loss of functionality. It is for this reason that actuaries typically have to write their own software to fit survival models, rather than relying on standard statistical packages. Actuarial needs are sometimes genuinely exceptional!

**References**

Collett, D. (2003) Modelling survival data in medical research, *Chapman and Hall*, ISBN 1-58488-325-1

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