My vague understanding is that machine learning methods are based on classification labels. How about a survival type of problem? That is to say, not only "have event" or "have no event", but also "time to event".

In statistics, we can perform e.g. Cox PH regression, but we can then only combine the multiple baseline characteristics in a linear manner (multivariable Cox analysis). If we want to look at a more advanced way to combine them, such as nonlinear, kernel-based, etc., is there corresponding machine learning methods which takes time-to-event into account?

Thanks for any comments.

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#### Best Answer

It is a mistake to assume that the Cox proportional hazards model makes simple assumptions such as linearity. All regression models have been extended for decades using regression splines, tensor interaction splines, and other approaches to allow for great flexibility in the low- to mid-dimensional case. As others have said, penalization is instrumental in handling higher-dimensional cases. [One problem is how to scale nonlinear terms when using penalization and regression splines simultaneously.]

Note also that the term 'multivariate' is inappropriate in this context as there is only one $Y$.

More to the original question, one of the amazing things about statistics is the ability of statistical approaches to extend models in various ways based on sound principles. Faraggi and Simon (*Statistics in Medicine*, 1995) did just that to develop a likelihood function for obtaining an artificial neural network Cox model.

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