Examining life table results to detect assumption violations
Potential assumption violations may be masked by the
grouped nature of the data. If the individual (ungrouped) data measurements
are available, they can be examined for signs of
lack of independence or
lack of uniformity in the censoring.
However, when examining life table results, you should keep these
potential problems in mind, along with the possibility
of implicit factors not surfaced in the data.
The problems detectable from the life table results themselves
are generally related to problems due to lack of data.
Although the grouped nature of the data can mask
systematic patterns in the censoring,
you may be able to spot very strong patterns.
For example, if there are many values
censored earlier in the experiment rather
than later, there may have been a change
of conditions during the experiment.
(For example, one physician may have withdrawn
referred patients early on while other
doctors did not.) If there was a
relatively large number of censored
values in a single interval, then
the censorings may be related.
(For example, a physician transfers to
another hospital, and all referred
patients suddenly leave the study.)
A common problem with a survival analysis
experiment studying medical treatments
is that patients who do not do well
one or more of the treatments must be
withdrawn from the study, so that
sicker patients may be more likely to
have censored survival times.
One sign of potential problems with grouping
is that the number of intervals is either
so small that the assumption of constant
survival rate within each time interval
is unlikely to hold, or so large that
the number of subjects in an interval
drops to a small number. A common rule
of thumb is: If the
number of intervals can not be at
least 8 to 10 without creating intervals
with very small sample sizes, the
life table results may not be reliable.
If there are many censored values, the life table
table estimates become less reliable, and the
estimated variances may be considerably smaller
than the actual variances. If many subjects are
left alive at the end of the study, the study
may simply not have continued long enough to
give reliable estimates.
If many subjects
are censored at approximately the same time,
the possibility of a common cause should be
considered. This would violate the assumption
of independence of censoring
and survival times.
Small sample sizes tend to lead to small numbers
of subjects within an interval, exacerbating
the effects of grouping.
High censoring rates
also reduce the effective sample size. If the final
interval(s) of a study contain only a few subjects,
the life table estimates for those intervals are
not reliable, and should not be given much weight.
The graphs of the survival functions can point to
possible parametric models
for the life table survival data.
Because the life table data are grouped, a plot will
either be a step function or a piecewise linear graph
connecting values at the midpoint of each interval.
However, it is often possible to get an approximate
idea of the shape of the underlying curve.
If the (negative) exponential model is appropriate, the
graph of the log of the survival function
(or the cumulative hazard function, which is
-log(survival function)), against
time should look like a straight line passing
through the origin. If the Weibull distribution
is appropriate, a graph of the log of the log of
the survival function (or the log of the cumulative
hazard function) against the log of time should
look like a straight line.
If the plot of the hazard function against time
is a horizontal line (constant hazard), then
the survival distribution is likely to be
negative exponential.
A hazard function that starts at 0 at time 0,
increases to a maximum value and then decreases
(like an inverted bathtub) suggests the possibility
of a log-normal or log-logistic survival distribution.
A monotonically increasing hazard function may
suggest a Poisson survival function.
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