Dataset: each row is a patient. Categorical covariates $x$ (often with many categories) include patient information, one binary outcome variable indicating survival.

Background: I fitted a probabilistic model to the data described above in order to calculate the probability of survival. I then calculated with the fitted model the probability of survival for each sample $P(survival|x)$ in the dataset and plotted the distribution of this estimated probability. The distribution has many peaks (see attached picture).

Question: I would like to "characterize" the peaks of the distribution. The outcome of this characterization* would be a statistical description of each peak. Is this achievable with existing statistical methods?

Example: The outcome of one such characterization would be

  • Peak 1: Samples in this peak have mostly diag_1_4 either equal to 4,5,6 and are men (not shown)
  • Peak 2: Samples in this peak have mostly diag_1_4 either equal to 0,8,NA and are women (not shown)
  • Peak 3: Samples in this peak have mostly diag_1_4 either equal to 1,2,3 and are unemployed (not shown)

My approach so far: For each covariate, I have stratified the distribution of estimated survival risk according to each level of the covariate (see attached picture). By visual inspection I could Identify some covariates that seemed to explain the observed peaks. In the attached picture one can see that the peaks seem to be explained by the diag_1_4 covariate.

* I know my description of "characterization" is vague. Probably part of the solution is to define exactly what I mean with "characterization". Any help with this definition is welcome!

enter image description here


I don't think it is really fruitful to characterize the peaks. The peaks are functions of the patient entry criteria. I would just show the distribution of predicted risks using a histogram with 100 bins. If on the other hand you want to get a simplified picture of what the predictive model is doing, you can decode the model by approximating it with a simpler model or with recursive partitioning (CART).

What is the time frame, i.e., the follow-up interval for survival? If it's more than a few times you'll probably need a time-to-event model and not a binary model. You can convert a time-to-event model to event probabilities at the back end.

  • $\begingroup$ Hi Frank, thank you for your answer. In reality we use a flexible survival model (piecewise constant hazard) but I thought that it is irrelevant for the purpose of my question. My question really applies to any probabilistic model, and I specified that it is used for survival analysis just for giving some practical context. $\endgroup$ – Gino_JrDataScientist May 4 '18 at 11:04
  • $\begingroup$ Also I do not quite understand what you mean by "patient entry criteria". All patients enter at time 0 and are observed until the event. No censoring. $\endgroup$ – Gino_JrDataScientist May 4 '18 at 11:09
  • $\begingroup$ I was referring to the characteristics of the patients who are recruited into the study. If sex is a big predictor and you change the number of males recruited the risk distribution may become more bimodal. $\endgroup$ – Frank Harrell May 4 '18 at 17:04
  • $\begingroup$ Ah I understand! We don't have that problem as our dataset covers the whole population of our country. $\endgroup$ – Gino_JrDataScientist May 5 '18 at 18:12
  • $\begingroup$ So I would say that our data is representative of the whole population, and because of this a characterization of the peaks does reflect the underlying risk factors. Wouldn't you agree? Thanks for suggesting using a CART for decoding the model! p.s.: I just realized you are that Harrell. Thank you very much very taking your time to answer to me! $\endgroup$ – Gino_JrDataScientist May 8 '18 at 14:42

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