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I would like to perform a time to event analysis on some biological example with animals. The event of interest is a terminal event X. But the animal may die from other reason before it happens. So I would like to compare three curves:

  1. Death and the terminal event X are equated. I can assign "1" to an animal regardless of the cause. I can use Kaplan-Meier.

  2. Death is censored. I am interested only in the event X. I can assign "0" to death. I can use Kaplan-Meier.

  3. Death is a competing risk. So I model the time to event X "corrected" for deaths. I am asked to use the Fine-Gray subdistribution hazard (I know it's under criticism, but I have to).

First question - do these models have any special name? I heard about crude, cause specific, and net survival, but still confuse them.

Each of the models can "generate" a survival curve to me. In R it's "survfit" I believe. Now, having three curves, telling me the probability of the event, I would like to compare them with a test, say, LogRank.

How to do that, knowing these are 3 separate models?

Or maybe it would be easier to compare median survival times? But it may happen the curve won't reach 50%. And I don't know any test that compares survival medians.

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What you described are different outcomes. You shouldn't do a hypothesis test to see if the survival curve of event X = the survival curve of (event X or death). They are different by definition. If there are not many deaths, the curves may look similar. The uncertainty in your curves may be much greater than the difference between the curves, which is what a hypothesis test would be looking at. But the curves are still different. Your null hypothesis is rejected by definition without ever doing a hypothesis test.

I focused on comparing outcomes 1 vs 2 above, but a similar argument holds for, say, comparing 2 and 3. Suppose you use the Fine and Gray model to estimate a cumulative incidence curve and compare it to 1 minus the Kaplan-Meier plot. You would be trying to estimate the cumulative incidence of event X in both cases (potentially badly with the 1-KM plot if the competing risk impactful). The null is true by the definition of what you are trying to estimate. If the curves looked different, you would simply focus on the method known to do a better job in your setting.

Rather than looking for statistical tests, in this case you are better off just plotting your curves with confidence bands so the difference between the curves can be seen in the light of the uncertainty of the estimates.

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  • $\begingroup$ Thank you very much. You enlightened me. If it is about "visual" comparing curves, would you use the pointwise confidence intervals - mostly reported - or the simultaneous confidence bands for the entire curve? Well, I think it answers two questions. Pointwise CIs allow me to do inference about that very time point, while the confidence bands will surely overlap, at least at the beginning. I was just asked to report these three models and "compare formally". I got lost how to do that "formally". Thank you once again. $\endgroup$ – humbleasker Nov 20 '19 at 22:41
  • $\begingroup$ By the way, if I had the same model, say with competing risk to fix our attention, but with different definitions of the event X, for instance 1) "X is defined as event if it occurs more than 10 days from some event Y" vs. 1) "X is defined as event if it occurs more than 13 days from some event Y", can I compare such curves (same model, a bit different definition of event, but still of the same type, just slightly different timing). $\endgroup$ – humbleasker Nov 20 '19 at 22:48
  • $\begingroup$ Happy to help! Personally, I like the confidence bands because a meaningful difference in the outcome definitions may only appear (relative to the uncertainty in the estimates) over a certain region of time. Regarding tweaking the definition of the "same" outcome, e.g. (X-Y)>10 days vs (X-Y)>20 days, the argument that the null is false by definition still holds. You can show how the two definitions differ by comparing number of events and comparing the curves, but it isn't something you would test. Showing the impact of tweaking the definition does often make great sensitivity analyses. $\endgroup$ – Robert Alan Greevy Jr PhD Nov 20 '19 at 22:56
  • $\begingroup$ Oh!I have never thought this way. I was looking at it as just "a curve", like a CDF, which can be compared with Kolmogorov-Smirnov or Anderson-Darling. So I expected, that if both definitions lead to similar numbers of events, so both hazard rates are very close (or even identical) then both curves will be similar, so also the comparison should reflect that. I only didn't know how to do it formally (what test to use). Now I think I get it. If I do typical analysis split by, say, sex, then the same definition holds for males and females. But here it's different, as I have different definitions? $\endgroup$ – humbleasker Nov 20 '19 at 23:03
  • $\begingroup$ I think I can also look at the confidence intervals for the median survival time, which will tell a little about the differences (assuming exponential curve). If the CIs don't overlap, there should be statistically significant difference. Otherwise - it's uknown, may be or may be not. Thank you! $\endgroup$ – humbleasker Nov 20 '19 at 23:10

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