More than one kind of information

(Jul 19, 2018)

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:

\[ {\cal I}_{ij} = \frac{\partial^2 \ell}{\partial \alpha_i \partial \alpha_j}. \]

It is important because \(-{\cal I}^{-1}\) evaluated at the fitted maximum \((\hat{\alpha}_1, \ldots, \hat{\alpha}_n)\) approximates the variance-covariance matrix of…

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Tags: information, indicator process

Testing the tests

(Jul 1, 2018)

Examining residuals is a key aspect of testing a model's fit.  In two previous blogs I first introduced two competing definitions of a residual for a grouped count, while later I showed how deviance residuals were superior to the older-style Pearson residuals.  If a model is correct, then the deviance residuals by age should look like random N(0,1) variables.  In particular, they should be independent with no obvious pattern linking the residual at one age with the next, i.e. there should be no autocorrelation.

In this article we will look at three alternative test statistics for lag-1 autocorrelation, i.e. correlation with the neighbouring value.  Each test statistic is based on the Pearson correlation…

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Tags: deviance residuals, autocorrelation, Fisher transform

Socio-economic differentials: convergence and divergence

(Jun 18, 2018)

Many western countries, including the UK, have recently experienced a slowdown in mortality improvements.  This might lead to the conclusion that the age of increasing life expectancies is over.  But is that the case for everyone?  Or are there some groups in the UK who are still experiencing mortality improvements?  The short answer is that mortality rates are still falling for the least deprived half of the population in England, while mortality improvements since 2011 have been virtually zero for the most deprived third.  This has important consequences for reserving for pensions and annuities, so let us explore in a bit more detail.  The findings in this blog are based on some early results of research…

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Tags: mortality convergence, mortality improvements, concentration risk, basis risk

Getting animated about longevity

(Jun 7, 2018)

We'll be the first to admit that what we have here doesn't exactly provide Pixar levels of entertainment.  However, with the release of v2.7.9 users of the Projections Toolkit can now generate animations of fitted past mortality curves and their extrapolation into the future.  Such animations can help analysts understand the behaviour of a forecast, as well as being a particularly useful way of communicating with non-specialists.  Below is a selection of animations from a smoothed Lee-Carter model fitted to the data for males in England & Wales between ages 50 and 104.

Figure 1 shows the logarithm of the force of mortality in the data region (1971-2015) and the forecast region.  It shows how mortality is…

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Tags: survival curve, curve of deaths, mortality compression

Less is More: when weakness is a strength

(Jun 1, 2018)

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…

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Tags: survival models, Poisson distribution

(GDP)Renewing our mail-list

(May 25, 2018)

A short and simple administrative announcement ...

In common with many other organisations, we are celebrating the arrival of the EU General Data Protection Regulation (GDPR) by renewing our mailing list. We only use our mailing list for relatively infrequent communication about our blogs, research and software. We don't sell or pass on anyone's contact details.

In order to keep things simple, we are going to start from a clean slate. So, even if you had previously joined our mailing list, in this post-GDPR world, we're going to ask for you to reconfirm your desire to hear from us. If you don't do this, you won't receive mailshots from us again (but obviously can still find out what we are up to by visiting us here).

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Tags: GDPR, data protection

What's in a (file)name?

(May 14, 2018)

The upcoming EU General Data Protection Regulation places focus on the potential for personal data exposures to create a risk to the rights of natural persons. The best way to reduce such risk is to minimise the ability to identify individuals from the data you use in your analysis. Thankfully, not all data used for modelling runs the risk of identifying individuals. Group data, such as that used by Longevitas group count survival models, or the grouped death and exposure formats used within the Projections Toolkit service, are not personal data under the terms of the GDPR. Such data stands no risk of identifying individuals. However, individual data used within, and within Longevitas…

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Tags: GDPR, data protection

Functions of a random variable

(May 9, 2018)

Assume we have a random variable, \(X\), with expected value \(\eta\) and variance \(\sigma^2\).  Often we find ourselves wanting to know the expected value and variance of a function of that random variable, \(f(X)\).  Fortunately there are some workable approximations involving only \(\eta\), \(\sigma^2\) and the derivatives of \(f\).  In both cases we make use of a Taylor-series expansion of \(f(X)\) around \(\eta\):

\[f(X)=\sum_{n=0}^\infty \frac{f^{(n)}(\eta)}{n!}(X-\eta)^n\]

where \(f^{(n)}\) denotes the \(n^{\rm th}\) derivative of \(f\) with respect to \(X\).  For the expected value of \(f(X)\) we then have the following second-order approximation:

\[{\rm E}[f(X)] \approx f(\eta)+\frac{f''(\eta)}{2}\sigma^2\qquad(1)\]

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Tags: GLM, log link, logit link

The Karma of Kaplan-Meier

(May 7, 2018)

Our new book, Modelling Mortality with Actuarial Applications, describes several non-parametric estimators of two quantities:

  1. The survival function, \(S_x(t)\), defined as the probability that a person now aged \(x\) will survive at least \(t\) years (\({}_tp_x\) to actuaries), and
  2. The integrated hazard function, \(\Lambda_x(t) = \displaystyle\int_0^t\mu_{x+s}ds\).

The estimators of the above quantities are based on two items of data collected at the times of the observed deaths (denoted by \(t_1,t_2,\ldots,t_n\)):

  1. The number, \(d_{x+t_i}\), who died at time \(t_i\), and
  2. The number, \(l_{x+t_i^-}\), who were alive and under observation immediately before time \(t_i\) (which time we denote…

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Tags: Kaplan-Meier, Nelson-Aalen, Fleming-Harrington, product integral

Battle of the Bulge

(May 1, 2018)

[Regular visitors to our blog will have guessed from the title that this posting is about obesity.  If you landed here looking for WWII material, you want the other Battle of the Bulge.]

As winter gives way to spring here in the Northern Hemisphere, many New Year's Resolutions, will have faltered.  Along with stopping smoking, losing weight is one of the more common goals, not least because the fortnight preceding the New Year is usually one of considerable excess in eating and drinking (that goes for this author, too).  However, reducing obesity and improving health is back on the agenda, as these are the specific aims of the UK's tax on sugary drinks, which was introduced in April 2018.

Losing weight (or, more accurately,…

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Tags: BMI, obsesity, sugar tax

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