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.
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probability that a subject will be withdrawn from the study or lost to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.
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.
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.
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.
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.
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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