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.
- 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be withdrawn from the study or lost to follow-up.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ill tend to be withdrawn from the study),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 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.
- 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,
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.
- Special problems with small sample sizes:
-
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.
- Special problems with crossing hazard or
survival functions:
-
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).
- 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.
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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