Solved – Cumulative Incidence vs. Kaplan Meier to estimate probability of failure

biostatisticskaplan-meiersurvival

To estimate the probability failure in medical sciences, it is not atypical to use 1-KM. However this does not account for competing risks, such as death by natural causes or causes unrelated with the disease, which preclude the event of interest. Thus 1-KM provides an inadequate measure, and cumulative incidence curves, such as the ones used in cmprsk (in R).

My question is that a lot of medical literature still reports KM curves or even 1-KM. Does this mean that the results reported in a lot of medical literature may be inaccurate (or more precisely over-estimated)? Or are there reasons why 1-KM would be preferred?

Furthermore, if there is a difference between 1-KM and the cumulative indices curve what other parts of your analysis are also effected (i.e. discrimination, calibration…)?

Best Answer

Let me state up front that I don't have answers for all of your questions. I'm not as strong on competing risks as simpler applications of survival analysis. So, I will just throw out a couple of pieces of information here that may be helpful. I suspect KM curves are more common because they are older and conceptually easier to understand (for both the researcher and the consumer of research). If the competing risks are truly independent, then I believe the KM estimates should be unbiased. That is, a plausible reason why people may prefer KM curves is that many people already understand them, and if those patients who had died due to other causes would have followed the same path as everyone else if they hadn't, the KM curves usefully illustrate what was learned from the study.

Regarding the question of whether there is over-estimation in the literature, one relevant fact, distinct from these issues, is that for practical purposes 'significance' is often required for publication. This guarantees that the literature is biased (specifically over-estimated), an issue known as the file drawer problem.

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Solved – Kaplan-Meier p-values

The p-value to which you are referring is result of the log-rank test or possibly the Wilcoxon. This test compares expected to observed failures at each failure time in both treatment and control arms. It is a test of the entire distribution of failure times, not just the median. The null hypothesis for the log-rank test for censured survival data is that the time-averaged hazard ratio for failure comparing treatment and control arms is 1.

It's worth mentioning that the power of this test isn't driven by the number of individuals at-risk in the various treatment arms or strata, but the number of failures observed. So, even if you have two fold sample size in one stratum, if many are censored before a failure is observed, it's not bizarre to see greater power in the other stratum --even if the KM curves look identical-- due to a larger number of failures.

If we reject the null hypothesis and find that subgroup2 has a significant difference in survival comparing treatment to control but subgroup1 does not have such a difference, then there is evidence of effect modification of treatment by subgroup. That suggests there is a difference in survival among those in subgroup2 but not subgroup1. As a sensitivity analysis, it would be useful to display the Kaplan Meier curves and possibly a smoothed estimate of the hazard ratio as a function of time. My guess is that, while the survival may be comparable in the median, it is the sequence of events in the first quartile of failure times that drives much of the inference and you see a quick drop-off in survival in one of the treatment arms for subgroup 2.

Solved – Is the cumulative incidence function just an inverted Kaplan-Meier survival curve

If you're talking about the cumulative incidence function that arises from a Kaplan-Meier estimator, then it's just $1 - S(t)$ where $t$ is time. In R:

library(survival)
fit <- survfit(Surv(time, status) ~ x, data = aml)
plot(fit) # the standard survival curve
plot(fit, fun="event") # the cumulative incidence curve

But, CIF is more commonly used in other contexts, for example in competing risk analysis using cause-specific Cox regression. There, the CIF is not necessarily equal to $1-S(t)$ because it takes into account the competing risk, estimating the risk of an event happening given survival up to time $t$ and also that a competing risk hasn't occurred up to time $t$.

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Solved – Kaplan-Meier p-values

Solved – Is the cumulative incidence function just an inverted Kaplan-Meier survival curve

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