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

联合建模是一种复杂的技术，它允许人们同时分析个体重复测量随时间的演变，以及这对特定感兴趣事件的时间的影响。通常，它被应用于医疗应用中，在这些应用中，随着时间的推移观察患者，目的是调查他们对治疗的反应如何以及为什么会发生变化，以及这如何影响他们的生存。由此可见，这些方法显然可以应用于大量的研究问题，从癌症研究到心脏病、糖尿病、中风等慢性病的分析，不一而足。由于这一优势，使用联合模型的研究出版物的数量在过去几十年中呈爆炸式增长。然而，尽管如此，只有有限的研究工作被用于调查这些模型的一个关键假设：这些模型中的随机项遵循正态分布假设。当纵向异常值存在时，这种普遍的正态假设受到不利影响。简单地去除这些异常值不仅会减少样本量，而且更重要的是，将排除通常指导生物医学科学创新的重要案例；它通常是对偏远病例的分析，它告诉我们更多关于疾病进展的信息。相反，这项研究将推进强大的联合建模技术，既限制了异常值的影响，提供更准确和精确的估计，并允许在识别这些异常值进行进一步探索的高精度。然而，这一研究领域还处于起步阶段，迄今为止在鲁棒关节建模方面的工作量目前有些有限。这是由于这些模型当前方法的潜在限制性假设，即异常值的影响是恒定的，不随时间变化。目前还没有现成的理论工具来处理这种情况，这是一种不受欢迎的情况，将通过本研究加以纠正。为此，我将开发一种新颖的方法，即时变异常值影响（TOI）方法，该方法将允许异常值被加权的程度随时间而变化。这样做，将允许使用这些技术来模拟更现实的场景，例如，模拟患者对开始一种新治疗的反应，考虑到他们需要时间来适应新治疗的事实，这可能导致从这些患者身上采取的孤立测量，或者从患者身上采取的所有测量都偏离了人口趋势。利用强健关节建模技术进行有限研究的另一个原因是缺乏可用的软件来拟合这些模型。直到最近几年，自2008年在R中引入JM软件包以来，该软件才可以用于适应标准的关节模型。这些联合建模软件包中的每一个都有随机项的正态分布假设，因此无法处理包含纵向异常值的数据分析，在异常值存在的情况下提供有偏差和不精确的估计。这个问题也将通过在R中开发一个软件包来缓解，该软件包将利用新开发的TOI方法进行鲁棒关节建模。

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.