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Survival Analysis With Generalized Additive Models : Part I (background and rationale)

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After a really long break, I’d will resume my blogging activity. It is actually a full circle for me, since one of the first posts that kick started this blog, matured enough to be published in a peer-reviewed journal last week. In the next few posts I will use the R code included to demonstrate the survival fitting capabilities of Generalized Additive Models (GAMs) in real world datasets. The first post in this series will summarize the background, rationale and expected benefits to be realized by adopting GAMs from survival analysis.

In a nutshell, the basic ideas of the GAM approach to survival analysis are the following:

  1. One approximates the integral defining the survival function as a discrete sum using a quadrature rule
  2. One evaluates the likelihood at the nodes of the aforementioned quadrature rule
  3. A regression model is postulated for the log-hazard rate function
  4. As a result of 1-3 the survival regression problem is transformed into a Poisson regression one
  5. If penalized regression is used to fit the regression model, then GAM fitting software may be used for survival analysis

Ideas along the lines 1-4 have been re-surfacing in the literature ever since the Proportional Hazards Model was described. The mathematical derivations justifying Steps 1-4 are straightforward to follow and are detailed in the PLoS paper. The corresponding derivations for the Cox model are also described in a previous post.

Developments such as 1-4 were important in the first 10 years of the Cox model, since there were no off-the-shelf implementations of the partial (profile) likelihood approach. This limited the practical scope of proportional hazards modeling and set off a parallel computational line of research in how one could use other statistical software libraries to fit the Cox model.  In fact, the first known to the author application of a proportional model for the analysis of a National Institute of Health (NIH) randomized controlled trial used a software implementing a Poisson regression to calculate the hazard ratio. The trial was the NCDS trial that examined adequacy indices for the hemodialysis prescription (the description of the software was published 6 months prior to the clinical paper).  Many of these efforts were computationally demanding and died off as the Cox model was implemented in the various statistical packages after the late 80s and semi-parametric theory took off and provide a post-hoc justification for many of the nuances implicit in the Cox model.  Nevertheless, one can argue that in the era of the modern computer, no one really needs the Cox model. This technical report and the author’s work on a real world, complex dataset provides the personal background for my research on GAM approaches for survival data.

The GAM (or Poisson GAM, PGAM as called in the paper) is an extension of these old ideas (see the literature survey here and here). In particular, PGAM models the quantities that are modeled semi-parametrically (e.g. the baseline hazard) in the Cox model with parametric, flexible functions that are estimated by penalized regressio. One of the first applications of penalized regression for survival analysis is the Fine and Gray spline model, which is however not a PGAM model.  There are specific benefits to be realized from penalizing the Poisson regression and adopting GAMs  in the context of survival analysis:

These benefits follow directly from the  mixed model equivalence between semi-parametric, penalized regression and Generalized Mixed Linear Models. An excellent, survey may be found here, while Simon Wood’s book in the GAM implementation of the mgcv package in R contains a concise presentation of these ideas.

As it stands the method presented has no software  implementation similar to the survival package in R. Even though we provide R code to run the examples in the paper, the need for the various functions may not be entirely clear. Hence the next series of posts will go over the code and the steps required to fit the PGAM using the R programming language.


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