An eigenfunction approach to multivariate and high-dimensional survival analysis

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Survival analysis is the study of so-called time-to-event data, and typically multivariate survival data arise either as parallel data, where one tries to observe times to events of interest (such as disease onset, death, failure) for several related individuals (such as family members, organs of human body, components of a system), or serial data, where a series of successive events are tracked (such as stages of a disease), or possibly as a combination of these two schemes. Often time to event data is combined with high-dimensional covariates, such as information on genetic markers available from gene expression data. Two major sources of complication in time to event data are random censoring, whereby certain events cannot be observed due to insufficient followup, and random truncation, whereby an event can be observed only if it occurs after a threshold event (such as a disease occuring after age at beginning of study).***For univariate survival data (i.e., when only one individual or component is being tracked) subject to random censoring and truncation, there exist two famous, and by now classical, estimation procedures called Kaplan-Meier and Lynden-Bell estimators, respectively. These estimates are widely applicable since they were derived without any parametric model assumption. Moreover, a host of statistical procedures for such data, such as studying effects of covariates on the time to event, can be developed based on these estimators. Unfortunately, no satisfactory multivariate versions of the latter were available until now. Consequently, statisticians were forced to rely on restrictive model assumptions in order to analyze censored or truncated multivariate survival data. For instance, effects of covariates on survival times were studied under the assumption that the former were not subject to censoring or truncation.***Recently, my co-author and I have constructed a multivariate Kaplan-Meier estimator under random censoring, which shares all the nice practical and theoretical properties of its univariate counterpart. In fact, our method is dimension-free, in that one needs to solve the same equation in any dimension, and it produces the classical univariate estimator when applied to univariate data. The main goal of the present project is precisely to extend the procedure to multivariate censoring-cum-truncation and a more complicated semi-censoring model, both of which are of great importance in survival studies. Just like the univariate Kaplan-Meier and Lynden-Bell estimators our new method too has the potential to generate a wealth of inference procedures for multivariate survival data, and we propose to explore them systematically. In particular, we shall study high-dimensional covariates in a much more general set-up than has been attempted so far. In addition to opening up a new line of research, our project has immense possibilities for training graduate students.**