American Journal of Epidemiology Advance Access originally published online on August 2, 2005
American Journal of Epidemiology 2005 162(6):604-605; doi:10.1093/aje/kwi239
American Journal of Epidemiology Copyright © 2005 by the Johns Hopkins Bloomberg School of Public Health All rights reserved
RE: "A CHAIN MULTINOMIAL MODEL FOR ESTIMATING THE REAL-TIME FATALITY RATE OF A DISEASE, WITH AN APPLICATION TO SEVERE ACUTE RESPIRATORY SYNDROME"
Eric H. Y. Lau and
Paul S. F. Yip
Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong, People's Republic of China
In a recent article (1
), we and our colleagues proposed a method of estimating a fatality rate in real time based on a chain multinomial model. We demonstrated that the proposed estimator is more sensitive than that of the World Health Organization. We proposed a kernel smoothing method that we applied to the crude estimator of the daily death and recovery probabilities, p1t and p2t, under the chain multinomial model. However, the kernel function we us,ed is a two-sided one with a bandwidth b, implying that when we estimate the fatality rate at time t, data within the period [t – b, t + b] are needed. In a retrospective study, this is obviously not a problem. However, when the method is applied during an epidemic, there is no observation for the period (t, t + b]. Here we suggest a one-sided kernel function method such that only information up to time t is needed to obtain a timely estimate of fatality up to time t during the course of the epidemic. More specifically, a one-sided Epanechnikov kernel function (2)
 |
can be used in calculating
the real-time fatality rate at time
t. Let
Kb(
x) =
b–1K(
x/
b);
then only data within the period [
t –
b,
t] are needed.
The estimated real-time fatality rate
t is
given by
where
i = 1, 2, are crude estimators of the death and
recovery probabilities at time
t. Let
N1t and
N2t be the numbers
of deaths and recoveries on day
t, while
Ht–1 is the number
of inpatients at the start of day
t. Then
Finally, the standard error of
is given by
and hence the variances
and covariances and the confidence interval can be found as
in our paper 1
. Since the bandwidth is chosen using the empirical
bias bandwidth selection (EBBS) method, in order to minimize
the empirical mean squared error, it will be larger when data
are more stable or sparse. For a one-sided kernel, we can expect
that the bandwidth will be comparably larger and hence more
data preceding the point of estimation are needed. This could
decrease the sensitivity of the estimator.
We applied the proposed one-sided kernel function to data on severe acute respiratory syndrome (SARS) from Hong Kong and Beijing, China. Results are shown in figure 1.
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FIGURE 1. Top left: estimated real-time fatality rate during the 2003 severe acute respiratory syndrome (SARS) epidemic obtained using a one-sided kernel (with its 95% confidence limits (CLs)) and the traditional fatality ratio for SARS in Hong Kong. Top right: estimated real-time fatality rate obtained using one-sided and two-sided kernels and the traditional fatality ratio in Hong Kong. Bottom left: estimated real-time fatality rate during the 2003 SARS epidemic obtained using a one-sided kernel (with its 95% confidence limits) and the traditional fatality ratio for SARS in Beijing. Bottom right: estimated real-time fatality rate obtained using one-sided and two-sided kernels and the traditional fatality ratio in Beijing.
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The top left panel in
figure 1 shows the SARS fatality rate
in Hong Kong estimated using a one-sided kernel, as well as
the traditional fatality ratio for SARS. The top right panel
compares the estimated fatality rates obtained using one-sided
and two-sided kernels. The bottom panels show the corresponding
estimates in Beijing. From the left panels, one can see that
the confidence intervals are wide at the beginning of the epidemic,
for approximately 2 weeks, and near the end of the epidemic
in Hong Kong. Otherwise, the confidence intervals are generally
narrow enough to provide sufficient information for estimating
the fatality level in both regions. The right panels show that
using a one-sided kernel causes instability of the estimated
fatality rate, especially at the beginning of the epidemic.
Nevertheless, both estimates converge to nearly the same trend
thereafter. In other words, the fatality rate estimated using
a one-sided kernel allowed us to practically provide an estimator
in real time, while the loss of accuracy in comparison with
the two-sided kernel is insignificant, especially in the midst
of an epidemic.
It is also important to note that the performance of the kernel-type estimator is highly dependent on the bandwidth chosen. We used the EBBS method to choose an optimal bandwidth for the estimator at each time point. In such cases, if the optimal b over the course of the epidemic is too large, the ability to reflect real-time information will also be weakened.
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References
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- Yip PSF, Lau EHY, Lam KF, et al. A chain multinomial model for estimating the real-time fatality rate of a disease, with an application to severe acute respiratory syndrome. Am J Epidemiol 2005;161:700–6.[Abstract/Free Full Text]
- Epanechnikov VA. Nonparametric estimation of a multidimensional probability density. Theory Probab Appl 1969;14:153–8.[CrossRef]
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