Nonparametric Bayesian Lomax delegate racing for survival analysis with competing risks
We propose Lomax delegate racing (LDR) to explicitly model the mechanism of survival under competing risks and to interpret how the covariates accelerate or decelerate the time to event. LDR explains non-monotonic covariate effects by racing a potentially infinite number of sub-risks, and consequently relaxes the ubiquitous proportional-hazards assumption which may be too restrictive. Moreover, LDR is naturally able to model not only censoring, but also missing event times or event types. For inference, we develop a Gibbs sampler under data augmentation for moderately sized data, along with a stochastic gradient descent maximum a posteriori inference algorithm for big data applications. Illustrative experiments are provided on both synthetic and real datasets, and comparison with various benchmark algorithms for survival analysis with competing risks demonstrates distinguished performance of LDR.
The paper proposes a survival regression method for competing risks based on generalizing exponential distributions by using scale mixtures with gamma variables, giving Lomax distributions for the competing event times. This is further extended so that each competing event time can be divided into latent causes, basically using the same racing "trick" (taking minimum of the latent times) and gamma process. This allows some non-linearity in the covariate associations to the event times or hazards, since min-function is non-linear. A Gibbs sampling algorithm for full posterior inference and a MAP estimation algorithm are detailed in the Appendix. Experiments on synthetic data and real data are presented, showing good performance compared to some alternatives.  The paper introduces an interesting parametric method for competing risk survival regression. Overall, it is well-written and clear. I'm not sure if it brings anything drastically new to survival analysis, but the methodology seems reasonable and possibly useful. The way to decompose a risk into latent causes seems especially interesting contribution.  Main comments for possible improvements of the paper: (1) It's not really true that one can't tell how covariates affect the survival in Gaussian process models, although, of course, with flexible non-linear models and/or high-dimensional covariates it gets more difficult. For low-dim. example, see Joensuu et al. Risk of recurrence of gastrointestinal stromal tumour after surgery: an analysis of pooled population-based cohorts, Lancet Oncology, 2012. It would have been nice to have GP model(s) included in the results. (2) It's not very clear what is the benefit of the Lomax distribution for the sub-event times rather than any other parametric assumption (except over the exponential model, which has only a single parameter and leads to a constant baseline hazard model). It it mainly computational convenience in the availability of Gibbs sampling scheme? (How important is this when we now have probabilistic programming languages that are usually relatively efficient for (at least well-identifiable continuous parameter) regression models?) (3) The relation of the proposed approaches to previous literature could be more clearly delineated. Is is so that the Exponential racing is previously published but Lomax racing (and Lomax delegate racing) is original to this paper? No previous approaches to find latent causes of survival are discussed. (4) The interpretation of latent components is not easy/unambiguous as evident in the text as they are varyingly discussed as accounting for non-linearities or for subtypes of a competing risk.  Minor comments: * Non-monotonic covariate effects don't necessarily break the PH assumption, but rather the linearity assumption? * I found the discussion around Equation 3 somewhat confusing; it tries to explain the data augmentation way of computing missing information/censored data, but didn't seem very clear. * Intuitively (so could be wrong...), with a model that takes min. over linear components, one would think that two equally weighted components would be reasonable in the Synthetic data 2, since the risk functions are symmetric? * typo: p.4, line 169: "Lamax" * Is the code available or will it be made available?  Update after author feedback: This seems an interesting contribution for competing risks analysis and the author seem to acknowledge the main concerns of the reviewers in their response. (As a clarification to my point (4) above, if I remember correctly I meant something like that it's probably not really identifiable from data if there is "real" non-linearity of covariate effects or if the non-linearity arises from a combination of multiple linear latent causes. This might be important when interpreting the model inferences.)