Longitudinal ageing model


  • David Steinsaltz, University of Oxford

  • Martin Kolb, University of Oxford


Partners and collaborators

  • Professor James Carey, UC Davis

  • Professor Steven Evans, UC Berkeley

  • Professor Shripad Tuljapurkar, Stanford University

  • Instituto Nacional de Astrofísica, Optica y Electrónica (INAOE) in Puebla, Mexico


David Steinsaltz


Operational definitions of ageing have proved elusive. It is presumed that there is a senescence state, correlated with but also distinct from calendar age, which drives the observable phenomena of ageing.

New experimental techniques have been made possible at a much higher level of resolution in studying physiological and behavioural changes in standard model systems for ageing. These experiments have already yielded masses of data for which there are currently no statistical tools that are designed to tease the signal of progressive ageing from the momentary random fluctuations.

Markov models are frequently applied in theoretical studies of ageing. There has been little opportunity to validate the applicability of a Markov model of ageing based on real data. Such validation - or demonstration of the inadequacy of such models - would have important ramifications on the theoretical side.

Statistical methods have also been lacking for effectively understanding the link between ageing, population growth, and environment for wild populations subject to environmental fluctuations. Some of the same statistical tools being developed may be applicable to studying the evolution of ageing in natural populations.

Aims and objectives

The overall aim of this project was to improve statistical methodology for analysing longitudinal ageing experiments with simple model organisms.

Validated model and fitting method

We aimed to prove appropriate theorems and carry out simulation studies which showed that Gaussian approximation should provide appropriate estimates of the parameters in these models. Had this planned approach turn out to be inadequate, we planned to work on developing new estimation methods, perhaps using new innovations in quasi-likelihood approaches.

Analysis of ongoing fruitfly experiments

Data was slowly coming in from the experiments in Mexico. We used these to refine our models, particularly as regards the best way of including daily cycles and the most appropriate model of mortality. We fitted the data to our model, yielding validated estimates of the senescence process in these flies. These results fed back into the design of later rounds of experiments, and also provided a jumping-off point for analysis of the variability and predictability of senescence in C. capitata and related species.

Other applications of Markov switching model methodology

There are other data sets and other problems connected with ageing, to which related methods could be applied. We developed mathematical tools for analysing the evolution of ageing in populations subject to randomly fluctuating environments, and statistical tools for analysing age- and stage-structured data from wild populations. We also hoped to analyse newly available data on progressive calcification in rats with kidney failure.


Research methods

Electronic Behaviour Monitoring System

This apparatus was designed and built at the Instituto Nacional de Astrofísica, Optica y Electrónica (INAOE) in Puebla, Mexico. Each of the three EBMS is capable of monitoring nine individual flies arranged in a 3 x 3 cage configuration (ie 27 flies in total).

The operational steps include the following:

  1. A single newly-enclosed fly is placed in each of the nine cages within a system.

  2. The digitised images of these individual flies are captured on a micro-second scale over a predetermined interval (eg 10, 20; 30 seconds; 1, 2, 3 mins) and stored.

  3. The system records, on a scale of micro-seconds, the activity and behaviour of individual fruit flies throughout their adult lives including movement in space

  4. Position records are combined with multiple digital pictures to classify the fly's behaviour (resting, moving in place, walking, flying, feeding, and drinking) and its location within the cage.

Markov switching model

Our plan was to adapt Markov switching models to these longitudinal data. We assumed the existence of a hidden diffusion process specifying an implicit physiological "age" or senescent condition for the organism.

The random development of this hidden variable drives the levels of markovian switching rates which specify probabilities of transitions between behavioural states, the transitions that comprise the data. The hidden variable serves as the predictor variable in a log-odds model for the switching rates.

Gaussian approximation and Kalman filter

One of the key problems in fitting these models is the difficulty of computing likelihoods efficiently enough to be able to optimise them. We used a Gaussian approximation to the Markov likelihood, which enabled us to apply efficient Kalman filter techniques to compute approximate likelihoods.

Semiparametric mortality modelling

One key open question concerned the functional form linking senescence state with mortality rate. We developed techniques to model senescence as a proportional hazards factor, allowing us to dispense with preselected functional forms.


  1. Effective software for fitting longitudinal behavioural data with mortality to Markov switching models.

  2. Strengthened links with experimentalists and other statisticians in the US.

  3. Training a postdoctoral researcher in statistical and mathematical methods relevant to the study of ageing.

Policy implications

Key policy and practice implications of the research

The project was expected to yield improved statistical approaches to longitudinal studies of ageing in model organisms.

  • Hidden Markov model optimised for continuous monitoring experiments.

  • Recommendations for new studies of senescence in a variety of organisms.

  • Improved understanding of the rate and variability of ageing.

  • Operational definition of ageing.

  • Evaluation of theoretical Markov models of ageing, and links to evolutionary models.

  • Improved methods for evaluating and planning studies of ageing in the wild, and impact of changing environments on age- and stage-structured populations.

Key non-academic user groups that will be targeted

Understanding from this project made its way into presentations and writing on ageing for the general public.