If the populations from which data to be analyzed by a life table were sampled
violate one or more of the life table 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 life table may not give useful estimates for survival.
Some small violations may have little practical effect on the analysis, while
other violations may render the life table results uselessly incorrect or
uninterpretable. In particular, lengthy
time intervals and small
sample sizes may increase the effect of assumption violations. Heavy
censoring may also affect the reliability of the life table 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probability that a subject will be withdrawn from the study or lost to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.
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withdrawn from the study), or too low (if subjects who will survive longer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.
The life table 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), and
that the censored values in an interval are uniformly distributed
throughout the interval. The estimates calculate the equivalent number of
subjects exposed (at risk) in an interval by assuming that censored subjects
were, on the average, at risk for half the interval. If subjects tend to be
censored more toward the beginning of an interval, then this estimate of then
number of subjects at risk is too high, and the survival estimate for that
interval will be too low. If the survival rate changes during the course of an
interval, then the survival estimates for that interval will not be reliable
or informative.
Any estimation procedure that relies on grouped data is vulnerable to
distortion from the grouping algorithm. The intervals for a life table should
be chosen before the data are collected, so that the interval boundaries will
be independent of the observed data.
The wider (longer) a time interval, the less likely it is that the
assumption of a constant survival rate throughout the interval will be
reasonable. A common rule of thumb is that there should be at least 8 to 10
intervals. If there are many censored values, it is particularly important
that the number of time intervals not be too small.
On the other hand, an interval with very few subjects in it will not have
reliable variance estimates for the survival functions, and the calculated
variance will tend to underestimate the true variance. If there are few
subjects left alive in the final intervals of a study, then the variance
estimates for those intervals should not be given as much credence as those
for earlier intervals with more patients.
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), making the life table
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,
and the estimated variances become poorer approximations, perhaps considerably
smaller than the actual variances. On the other hand, with high levels of
censoring, it is also important to avoid having only a small number of
intervals.
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 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, nor patterns within
the time intervals. Obviously, this sort of graph can only be constructed when
the individual values are known.
If the sample size is small, it becomes particularly difficult to create
time intervals that have enough subjects in them to provide reliable estimates
of the survival functions and their variances while still being short enough
to justify the assumption of a constant survival rate within each interval. A
small sample size also 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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