Angus Macdonald
Articles written by Angus Macdonald
The Mystery of the Non-fatal Deaths
In the course of a recent investigation, with my colleagues Dr Oytun Haçarız and Professor Torsten Kleinow, a key parameter was the mortality rate of persons suffering from Hypertrophic Cardiomyopathy (HCM), an inherited heart disorder characterized by thickening of the left ventricular muscle wall. It is quite rare, so precision is not to be expected, and indeed an annual mortality rate of 1% \((q_x=0.01)\), independent of age \(x\), is widely cited. I
A Model for Reporting Delays
In his recent blog Stephen described some empirical evidence in support of his practice of discarding the most recent six months' data, to reduce the effect of delays in reporting deaths. This blog demonstrates that the practice can also be justified theoretically in the survival modelling framework, although the choice of six months as the cut-off remains an empirical matter.
From magical thinking to statistical thinking
See You Later, Indicator
A recurring feature in my previous blogs, such as this one on information, is the indicator process:
\[Y^*(x)=\begin{cases}1\quad\mbox{ if a person is alive at age \(x^-\)}\\0\quad\mbox{ otherwise}\end{cases}\]
where \(x^-\) means immediately before age \(x\) (never mind the asterisk for now). When something keeps cropping up in any branch of mathematics or statistics, there are usually good reasons, and this is no exception. Here are some:
Right-Censoring Rules!
A fundamental assumption underlying most modern presentations of mortality modelling (see our new book) is that the future lifetime of a person now age \(x\) can be represented as a non-negative random variable \(T_x\). The actuary's standard functions can then be defined in terms of the distribution of \(T_x\), for example:
\[{}_tp_x = \Pr[ T_x > t ].\]
The long shadow of the life table
Smooth Models Meet Lumpy Data
Most of the survival models used by actuaries are smooth or piecewise smooth; think of a Gompertz model for the hazard rate, or constant hazard rates at individual ages. When we need a cumulative quantity, we use an integral, as in the cumulative hazard function, \(\Lambda_x(t)\):
\[ \Lambda_x(t) = \int_0^t \mu_{x+s} \, ds. \qquad (1) \]
More than one kind of information
This collection of blogs is called Information Matrix, and it is named after an important quantity in statistics. If we are fitting a parametric model of the hazard rate, with log-likelihood:
\[ \ell( \alpha_1, \ldots, \alpha_n ) \]
as a function of \(n\) parameters \(\alpha_1, \ldots, \alpha_n\), then the information matrix is the matrix of second-order partial derivatives of \(\ell\). That is, the matrix \({\cal I}\) with \(ij\)th component:
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:
\[{}_{dt}q_{x+t} = \mu_{x+t}dt + o(dt)\qquad (1)\]
The Karma of Kaplan-Meier
Our new book, Modelling Mortality with Actuarial Applications, describes several non-parametric estimators of two quantities: