If the populations from which data for a survival test were sampled violate one
or more of the survival test 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 results for the survival test may be biased and
unreliable. If there are factors
unaccounted for in the analysis that affect survival and/or censoring times,
then the survival test may give inappropriate results unless the data are
stratified to reflect the factor(s).
Some small violations may have little practical effect on the analysis, while
other violations may render the survival test results uselessly incorrect or
uninterpretable. In particular, small
sample sizes may increase the effect of assumption violations. Heavy
censoring or crossing
hazard or survival functions may also affect the reliability of the survival
tests.
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probability that a subject will be withdrawn from the study or lost to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 survival tests rely on independence between censoring times and
survival times. If independence does not hold, the results 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.
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, reducing the
effective sample sizes.
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.)
The survival tests perform better when the censoring is not too heavy, and,
in particular, when the pattern of censoring is similar across the different
groups.
The Gehan-Breslow, Mantel-Cox, and Tarone-Ware survival tests are all
based on chi-square statistics, and thus rely on asymptotic theory. If the
sample sizes are too small, this reliance may not be appropriate.
The Gehan-Breslow, Mantel-Cox, and Tarone-Ware survival tests are not
particularly good at detecting differences in survival functions when the hazard
or survival
functions are not parallel (that is, if their graphs cross).
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.
A Kaplan-Meier plot, plots of the life table survival functions, plots of
the life table hazard functions for each sample will demonstrate whether the
survival functions or hazard functions cross (are non-parallel). If the
functions do cross then none of the three tests (Gehan-Breslow, Mantel-Cox, or
Tarone-Ware) will be particularly good at detecting differences between the
survival functions.
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