Does your data violate Kaplan-Meier assumptions?

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If the populations from which data for a Kaplan-Meier estimation were sampled violate one or more of the Kaplan-Meier assumptions, the results of the analysis may be incorrect or misleading. For example, if the assumption of independence of censoring times is violated, then the estimates for survival may be biased and unreliable. If there are factors unaccounted for in the analysis that affect survival and/or censoring times, then the Kaplan-Meier calculations may not give useful estimates for survival.

Some small violations may have little practical effect on the analysis, while other violations may render the Kaplan-Meier results uselessly incorrect or uninterpretable. In particular, small sample sizes may increase the effect of assumption violations. Heavy censoring may also affect the reliability of the Kaplan-Meier estimates.

Potential assumption violations include:

  • Implicit factors:
  • Lack of independence within a sample is often caused by the existence of an implicit factor in the data. For example, if we are measuring survival times for cancer patients, diet may be correlated with survival times. If we do not collect data on the implicit factor(s) (diet in this case), and the implicit factor has an effect on survival times, then we in effect no longer have a sample from a single population, but a sample that is a mixture drawn from several populations, one for each level of the implicit factor, each with a different survival distribution. Implicit factors can also affect censoring times, by affecting the probability that a subject will be withdrawn from the study or lost to follow-up. For example, younger subjects may tend to move away (and be lost to follow-up) more frequently than older subjects, so that age (an implicit factor) is correlated with censoring. If the sample under study contains many younger people, the results of the study may be substantially biased because of the different patterns of censoring. This violates the assumption that the censored values and the noncensored values all come from the same survival distribution. Stratification can be used to control for an implicit factor. For example, age groups (such as under 50, 51-60, 61-70 and 71 or older) can be used as strata to control for age. This is similar to using blocking in analysis of variance. The goal is to have each group/stratum combination's subjects have the same survival distribution.
  • Lack of independence of censoring:
  • If the pattern of censoring is not independent of the survival times, then survival estimates may be too high (if subjects who are more ill tend to be withdrawn from the study), or too low (if subjects who will survive longer tend to drop out of the study and are lost to follow-up). If a loss or withdrawal of one subject could tend to increase the probability of loss or withdrawal of other subjects, this would also lead to lack of independence between censoring and the subjects. The estimates for the survival functions and their variances rely on independence between censoring times and survival times. If independence does not hold, the estimates may be biased, and the variance estimates may be inaccurate. An implicit factor not accounted for by stratification may lead to a lack of independence between censoring times and observed survival times.
  • Lack of uniformity within a time interval:
  • The Kaplan-Meier estimates for the survival functions and for their standard errors rely on the assumptions that the probability of survival is constant within each interval (although it may change from interval to interval), where the interval is the time between two successive noncensored survival times. If the survival rate changes during the course of an interval, then the survival estimates for that interval will not be reliable or informative.
  • Many censored values:
  • A study may end up with many censored values, from having large numbers of subjects withdrawn or lost to follow-up, or from having the study end while many subjects are still alive. Large numbers of censored values decrease the equivalent number of subjects exposed (at risk) at later times, making the Kaplan-Meier estimates less reliable than they would be for the same number of subjects with less censoring. Moreover, if there is heavy censoring, the survival estimates may be biased (because the assumption that all censored survival times occur immediately after their censoring times may not be reasonable and may not allow for a good estimate), and the estimated variances become poorer approximations, perhaps considerably smaller than the actual variances. A high censoring rate may also indicate problems with the study: ending too soon (many subjects still alive at the end of the study), or a pattern in the censoring (many subjects withdrawn at the same time, younger patients being lost to follow-up sooner than older ones, etc.) If the last observation is censored, the Kaplan-Meier estimate of survival can not reach 0.
  • Patterns in plots of data:
  • If the assumptions for the censoring and survival distributions are correct, then a plot of either the censored or the noncensored values (or both together) against time should show no particular patterns, and the patterns should be similar across the various groups.
  • Special problems with small sample sizes:
  • The time intervals in a Kaplan-Meier calculation are determined by the distinct noncensored survival times. These means that the smaller the sample size is, the longer the intervals will be, raising the question of whether the assumption of a constant survival probability within each interval is appropriate. A small sample size makes it more difficult to detect possible dependencies between censoring and survival, or the presence of implicit factors. If the number of subjects exposed (at risk) in an interval or the number of subjects that survived to the beginning of that interval is small, the variance estimates for the survival functions will tend to underestimate the actual variance. This situation is most likely to occur for later intervals, when most subjects have either died or been censored, so that the variance estimates for later intervals are generally less reliable than those for earlier intervals.

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