Time-dependent Robust Joint Modelling: Analysing a wealth of longitudinal outliers

Joint modelling is a sophisticated technique that allows one to simultaneously analyse the evolution, over time, of repeated measurements from individuals and the impact this has on the time to a particular event of interest. Commonly, it is applied to medical applications where patients are observed over time with the aim of investigating how and why their responses change to treatment and how this affects their survival. From this, it is evident that such approaches can be applied to a vast array of research questions, from cancer research to the analysis of chronic diseases such as heart disease, diabetes, stroke, to name but a few. As a result of this advantage, the volume of research publications utilising joint models has exploded in the last few decades.

Despite this, however, only limited research efforts have been directed at investigating one of the key assumptions of these models: that the random terms within these models follow normal distributional assumptions. This prevailing assumption of normality is detrimentally impacted when longitudinal outliers are present. Simple removal of these outliers will not only reduce sample size but, more importantly, would exclude important cases which commonly guide innovation in biomedical sciences; it is typically the analysis of outlying cases which tell us more about disease progression. Instead, this research will advance robust joint modelling techniques which both restrict the impact of outliers, providing more accurate and precise estimates to be obtained, and allow a high level of precision in the identification of such outliers for further exploration.

However, this research area is in its infancy with the volume of work to date on robust joint modelling being currently somewhat limited. This is due to the potentially restrictive assumptions of the current methodology for these models i.e. that the impact of outliers is constant, unchanging over time. There are no established theoretical tools for handling such a situation, an undesirable situation that will be rectified through this research. To do so, I will develop a novel methodology, the time-varying outlier impacts (TOI) approach, which will allow the degree at which outliers are down weighed to change over time. Doing so, will allow more realistic scenarios to be modelled using such techniques, for example, modelling patients reaction to starting a new treatment, accounting for the fact that it will take time for them to adjust to the new treatment, which could result in outlying measurements being taken from such patients or all measurements taken from the patient outlying from the trends of the population.

Another reason for limited research utilising robust joint modelling techniques is the lack of available software to fit such models. It has only been in recent years, since the introduction of the JM software package in R in 2008, that software has become available to fit standard joint models. Each of these joint modelling software packages have normal distributional assumptions for the random terms and thus cannot handle the analysis of data which contains longitudinal outliers, providing biased and imprecise estimates in the presence of outliers. This issue will also be alleviated through the work undertaken in this project through the development of a software package in R for robust joint modelling that will utilise the newly developed TOI approach.

The overall goal of this research is fundamental theoretical developments in robust joint modelling methodology. As such it will have wide and far reaching academic impacts, holding the potential to revolutionise this field of research through the removal of restrictive assumptions which have halted the utilisation of robust joint models in current literature. This academic impact will not only be felt within statistics but in the multitude of application areas that such research may be utilised, for example, renal research, as will be evidenced by the findings of the proposed work. Due to the ability of statistics to impact an array of applications in medicine, geostatistics, business analytics, astrostatistics, econometrics, environmental statistics and epidemiology, to name but a few, the EPSRC has deemed 'Statistics and applied probability' as a growth research area within the theme of 'Mathematical sciences', where, as stated by the EPSRC, "this research area provides economic, industrial and societal impact".

Examples of such societal impacts arise from the gains in understanding; for example, in renal research, patients' haemoglobin levels is an emerging biomarker whose volatility in the initial stages upon commencement of haemodialysis has great impacts on the risk of death of such patients. More generally, a significant impact of this work is the ability of robust joint models to identify individuals who are classified as outliers, patients who do not react the same to treatments as typical patients and thus are in need of more personalised treatment plans. Previous research suggests that such patients tend to have worse survival rates and therefore the enhancement of techniques able to identify such at-risk patients is a key societal impact. In the long term, this has the potential to have economic impact on the NHS as a method to enable personalised medicine to become a reality.

An additional important impact of this project would be the development of the skills of the personnel involved in the project. A concern expressed by the International Review of Mathematical Sciences (IRMS) 2010 is the fragility of UK statistics due to the shortage of researchers at various career stages. This project will help to address this issue giving both the PI and PDRA the opportunity to develop skills that will stand them in good stead for their future careers in the field, benefiting greatly from the expertise and guidance received from the research visits to and from our collaborators that this project will facilitate. Added value is given through the cross disciplinary nature of this project (statistics and medicine) which will further enhance the skills of the personnel involved.