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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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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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  • $\begingroup$ I agree with you that statisticians have developed robust methods to handle time to event data vs. machine learning methods. Can you please let cite any head to head comparison between these two, so users can make an informed decision on which to use? $\endgroup$ – forecaster Mar 31 '16 at 19:48
  • $\begingroup$ I wish I had a bibliography of such comparisons. One good paper is citeulike.org/user/harrelfe/article/13467382 $\endgroup$ – Frank Harrell Apr 1 '16 at 2:59
  • $\begingroup$ Thank you for your comments. Could you explain a little more why "the term 'multivariate' is inappropriate in this context as there is only one Y." I thought multivariate refers to the variables (the baseline characteristics). Thanks again. $\endgroup$ – alize Apr 1 '16 at 7:45
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    $\begingroup$ Multivariate always should refer to the dimensionality of the dependent/response variable. Multivariate analysis is the part of statistics dealing with more than one dependent variable. When there are multiple predictors we use the terms multivariable model or multiple regression. $\endgroup$ – Frank Harrell Apr 1 '16 at 12:43
  • $\begingroup$ A fix for the citeulike entry above: bmcmedresmethodol.biomedcentral.com/articles/10.1186/… $\endgroup$ – Frank Harrell Apr 22 at 18:24
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You may might be interested in Random Survival Forests and the corresponding R package randomForestSRC:

http://www.ccs.miami.edu/~hishwaran/papers/randomSurvivalForests.pdf

https://cran.r-project.org/web/packages/randomForestSRC/

I believe the main limitation of the approach is that it doesn't deal with time varying predictors.

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  • $\begingroup$ I haven't heard about this one. I'd be very happy to give you +1 for this answer if you added some information about the general ideas behind this method. $\endgroup$ – Tim Mar 31 '16 at 10:23
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    $\begingroup$ I did a presentation of Lasso and RandoSurvivalForest in a R Users Conference in Spain this last year. Sorry, but the presentation is in Spanish, but it can be useful for R code and some general concepts: r-es.org/7jornadasR/ponencias/jesus_herranz.pdf $\endgroup$ – Jesus Herranz Valera Mar 31 '16 at 10:57
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    $\begingroup$ Note that if effects are mainly additive, random survival forests will easily be beat by additive regression methods. Random forests require much larger sample sizes than regression to perform reliably. $\endgroup$ – Frank Harrell Mar 31 '16 at 12:56
  • $\begingroup$ Further, if I remember it correctly, random survival forests don't rely on the proportional hazards assumption, so they might be a good fall back when this assumption is violated and the sample size is big enough. $\endgroup$ – Anton Mar 31 '16 at 16:05
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The majority of the linear models based in the likelihood function have extension to the Cox regression. For example, penalized regression models (lasso, rigde regression, elastic net) or partial least squares. In other hand, there are extensions from the classification trees to survival trees. This means all the ensemble methods based in trees are also extended to survival data in a natural way: random forest, bagging, gradient boosting machines, …. Finally, other methods, like support vector machines or neural networks have some theoretical versions for survival data, but there are difficult to apply in the practice.

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  • $\begingroup$ Thank you. Will look into LASSO. Is this a commonly used technique in clinical studies? $\endgroup$ – alize Mar 31 '16 at 10:08
  • $\begingroup$ I have used LASSO with the glmnet R package with genetic data in high-dimensionality problems successfully $\endgroup$ – Jesus Herranz Valera Mar 31 '16 at 10:55
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Any linear survival analysis method can be straightforwardly kernelised to generate a non-linear equivalent. I did something like this a while back for modelling the time-to-growth of microbial pathogens from spores in foods.

G. C. Cawley, N. L. C. Talbot, G. J. Janacek and M. W. Peck, Sparse Bayesian kernel survival analysis for modelling the growth domain of microbial pathogens, IEEE Transactions on Neural Networks, volume 17, number 2, pp. 471-481, March 2006. (www)

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