Possible alternatives if your data violate life table assumptions
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.
In such cases, stratification
of the data or, if the individual values are available,
using a nongrouped test
may provide a better analysis.
The best cures for some problems--running an experiment
longer or doing more aggressive follow-up to avoid
a large proportion of censored values, or
using a large enough sample size to lessen the
problems of lengthy time intervals or the
effects of grouping--are outside the scope of
statistical analysis per se.
Alternative procedures:
Stratification: Dividing
the sample into homogeneous subsamples
Stratification involves dividing a sample into
subsamples based on one or more characteristics
of the population. For example, a sample may
be stratified by gender. This gives multiple
subsamples, each of which can be analyzed
separately.
If the survival function is different for
the different strata, then the characteristic
used for stratification may be an
implicit factor,
and the separate analysis
for each individual subsample may be more
informative than an analysis of the entire sample.
Stratification may also reveal
correlations between censoring and strata.
A potential drawback with stratification is that one or
more of the subsamples may be small in size, leading to
problems with the reliability of the estimates.
Also, the results for each subsample are generalizable
to only a part of the sample population.
If a specific survival distribution can be assumed
based on previous knowledge, then that assumption can
be used to make survival estimates.
Elandt-Johnson and Johnson
and Lawless
discuss methods of fitting parametric survival models to grouped data.
As it happens, assuming an (negative) exponential survival distribution
for grouped data leads to the usual actuarial life-table estimates.
If the individual survival values (noncensored and
censored) are available, then nongrouped methods can
may more complete use of the information available.
This is an obvious way to deal with the
effects of grouping.
A common nonparametric method with individual data is the
Kaplan-Meierproduct-limit
estimate for the survival function.
A specific functional (parametric) form for the survival
distribution function, such as the Weibull distribution
or the negative exponential distribution,
or the Cox proportional hazards model,
can also be fitted to individual data, if a particular
distribution makes sense a priori.
(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.)
Elandt-Johnson and Johnson,
Lawless,
and Lee
discuss these methods.
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