Lifespan AI - Project M1: Normalizing Flows for Lifespan Health Data

健康科学使用统计模型来量化健康状况和结果，因为它随着时间的推移而演变，并受到风险因素和/或治疗的影响。因此，不确定性的量化是统计学在方法论上的重要贡献。由于趋势和不确定性最好地用分布来描述，所有变量和特征在所考虑的所有时间点上的联合分布模型提供了最完整的统计图景。这允许-至少在形式上-统计可以提供的所有结论，而不是不可检验的假设。这种方法不是直接的，当数据是高维的时候，需要机器学习方法。寿命数据方面出现了进一步的挑战，这些数据具有不同的规模，是在各个具体的时间点衡量的，并且来自只有部分重叠的不同数据源，或者来自变量集随时间变化的研究队列。在这个项目中，我们通过开发利用可逆残差神经网络和使用基本分布的广义线性混合模型(GLMM)的归一化流量来应对这些挑战。GLMMS的使用对健康科学家特别有吸引力，因为它们经常在应用中使用。基于可逆残差网络的流量正常化的主要优势是提供了联合分布的分析表达式，可以很好地用于预期的统计和科学结论。我们方法的关键思想是通过GLMM的非线性变换开始对给定其他变量的每个变量的条件分布进行建模，并通过随机选择的平均顺序因式分解将它们组合到全局联合分布中，这可以通过顺序减少我们在条件分布中条件的变量集来实现(“反向边际化”)。在推导出一种方法来拟合具有完整观测的联合模型之后，我们将开发一个算法来更新具有不完整观测的模型并通过附加变量对其进行扩展。然后，联合分布将被用来获得科学上有趣的条件分布的(总体正规化)估计，从中可以推导出点和区间预测。我们还将开发解释整体(黑盒)模型的方法，并调查其内部和外部有效性。虽然联合模型已经考虑了任意的不确定性(通过模拟分布)，但我们将制定出也考虑认知不确定性的方法，即量化模型拟合和解释预测区间中的模型不确定性的方法。新方法将应用和说明IDEFICS/I的数据和变量。家庭队列、Nako健康研究和Gepard数据都是由BIPS收集和/或管理的。

Health sciences use statistical models to quantify health status and outcome as it evolves over time and is influenced by risk factors and/or treatments. Hereby the quantification of uncertainty is a crucial methodological contribution of statistics. Since trend and uncertainty are best described in terms of distributions, a model for the joint distribution of all variables and features over all time points under consideration provides the most complete statistical picture. This permits – at least formally – all conclusions statistics can provide without untestable assumptions.Such an approach is not straightforward and requires machine learning methods when the data are high dimensional. Further challenges arise in lifespan data, where the data are of different scales, are measured at individual specific time points and come from different data sources with only partial overlap or from study cohorts whose variable sets change over time. In this project, we tackle these challenges by the development of normalizing flows that utilise invertible residual neural networks and use generalised linear mixed models (GLMM) for the base distributions. The use of GLMMs is particularly attractive for health scientists because they are frequently used in applications. Normalizing flows based on invertible residual networks have the major advantage of providing analytical expressions for the joint distribution that can be well utilised for the anticipated statistical and scientific conclusions. The key idea of our method is to start modelling the conditional distribution of each variable given the other variables by nonlinear transformations of GLMMs and to combine them to an global joint distribution by an average of randomly selected sequential factorisations which can be achieved by sequentially reducing the set of variables we condition on in the conditional distributions (“reverse marginalisations”). After deriving an approach to fit a joint model with complete observations, we will develop an algorithm for updating the model with incomplete observations and extending it by additional variables. The joint distribution will then be utilised to obtain (overall regularised) estimates of scientifically interesting conditional distributions from which point and interval predications can be derived. We will also develop methods to interpret the overall (black box) model and to investigate its internal and external validity. While the joint model already accounts for aleatoric uncertainty (by modelling distributions) we will work out methods to account also for epistemic uncertainty, i.e. approaches for quantifying the model fit and to account for the model uncertainty in prediction intervals. The new methods will be applied to and illustrated with data and variables from the IDEFICS/I.Family cohort, the NAKO Health Study and GePaRD data which are all collected and/or managed by BIPS.