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.
- 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be withdrawn from the study or lost to follow-up.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ill tend to be withdrawn from the study),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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