A
method to obtain the estimates for parameters and the size of fragile
population with their confidence intervals in mixed populations
of the fragile and durable samples, i.e., in the trunsored model,
along with those in the truncated model, is introduced. The confidence
intervals for the estimates in the trunsored model are compared
with those in the truncated model. The maximum likelihood estimates
for the parameters in the underlying probability distribution in
both models are exactly the same when all the samples have the
same censoring time, and consequently the confidence interval for
the parameters are also the same. The estimate for the number of
fragile samples in the trunsored model is the same as that in the
truncated model when the failure data are the same; however, the
confidence interval for it in the trunsored model differs from
that in the truncated model.
In the truncated model, the confidence interval for the fragile samples
is affected by the fluctuation effect due to the censoring time and
the parameters in the underlying probability distribution. In the
trunsored model, however, the confidence interval is affected by
two kinds of fluctuation effect: one is the same as in the truncated
model, and the other is the extra parameter which corresponds to
the ratio of the number of fragile samples to the total number of
samples.
When the censoring time becomes large, the width of the confidence
interval in the truncated model tends to zero, whereas the confidence
interval in the trunsored model tends to a positive constant value,
which is corresponding to the binomial case.
A typical example of the method applied to the case fatality ratio for the infectious
diseases such as SARS shows different confidence intervals between the trunsored
model and the truncated model. Using the truncated model we may have the paradoxical
case fatality ratio; using the trunsored model, however, we can obtain the reasonable
estimate for it. This indicates that we have to be cautious in selecting the
appropriate model when we deal with the incomplete data models.
|
|
|
|
|
|
trunsored data, truncated data, fragile,
durable, confidence interval, case fatality ratio, SARS.
|
|
|