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I should make prediction on survival data, using the random Forest method. My question is: should I follow the same approach as in logistic regression? taking into account only the status variable or whether I should take into account the delay to the event? Are there any specific R functions for survival analysis other than randomForest? Or could I use this function for survival analysis as well? I've seen a function called ranger() that seems to do random forest on survival data, but I haven't understood much of it.

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The phrase "survival analysis" is usually taken to involve time-to-event data, not just event outcomes at a certain time. Otherwise, you're just throwing away potentially important time-to-event data. Also, survival analysis is well equipped to handle censored event times, for example individuals who still haven't experienced the event by the end of data collection. Depending on the study design, that might not be handled well by logistic regression.

If you have time-to-event data, use proper survival analysis. The CRAN survival task view points out some implementations of random-forest survival analysis.

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