American Journal of Epidemiology 2005 161(7):700-706; doi:10.1093/aje/kwi088
American Journal of Epidemiology Copyright © 2005 by the Johns Hopkins Bloomberg School of Public Health All rights reserved
A Chain Multinomial Model for Estimating the Real-Time Fatality Rate of a Disease, with an Application to Severe Acute Respiratory Syndrome
Paul S. F. Yip1,
Eric H. Y. Lau1,
K. F. Lam1 and
Richard M. Huggins2
1 Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong, People's Republic of China
2 Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, Australia
Reprint requests to Dr. Paul S. F. Yip, Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam, Hong Kong, People's Republic of China (e-mail: sfpyip{at}hku.hk).
Received for publication June 25, 2004.
Accepted for publication November 1, 2004.
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ABSTRACT
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It is well known that statistics using cumulative data are insensitive
to changes. World Health Organization (WHO) estimates of fatality
rates are of the above type, which may not be able to reflect
the latest changes in fatality due to treatment or government
policy in a timely fashion. Here, the authors propose an estimate
of a real-time fatality rate based on a chain multinomial model
with a kernel function. It is more accurate than the WHO estimate
in describing fatality, especially earlier in the course of
an epidemic. The estimator provides useful information for public
health policy makers for understanding the severity of the disease
or evaluating the effects of treatments or policies within a
shorter time period, which is critical in disease control during
an outbreak. Simulation results showed that the performance
of the proposed estimator is superior to that of the WHO estimator
in terms of its sensitivity to changes and its timeliness in
reflecting the severity of the disease.
disease outbreaks; epidemiologic methods; fatality rate; models, statistical; multinomial model; severe acute respiratory syndrome
Abbreviations:
SARS, severe acute respiratory syndrome; WHO, World Health Organization
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INTRODUCTION
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The 2003 outbreak of severe acute respiratory syndrome (SARS)
affected more than 28 countries, with a total of 8,098 reported
cases and 774 reported fatalities as of July 31, 2003 (1
). The
areas worst hit by SARS included Hong Kong, China; Singapore;
Taiwan; Toronto, Canada; and Beijing, China. Estimation of fatality
rates for an outbreak or epidemic of an emerging disease is
not straightforward. This was particularly true for the SARS
outbreak, in which clinicians all over the world had to deal
with this highly contagious and frightening pneumonic illness
with almost no knowledge about its etiology, pathogenesis, clinical
features, management, or prognosis, other than anecdotal information
obtained from colleagues in China (2
–4
).
The World Health Organization (WHO) estimated SARS fatality rates in Hong Kong on March 30, April 30, May 25, and June 30, 2003, to be 2.5 percent, 9.9 percent, 15.8 percent, and 17.0 percent, respectively (5). These discrepancies do not reflect a change in the severity of the disease; rather, they are an artifact of WHO methodology. The WHO estimate of the fatality rate is the ratio of the cumulative number of deaths to the total number of cases diagnosed. In the early part of the epidemic, few patients died from SARS, since their infection had not run its course; hence, the estimated fatality rate was low. As the epidemic developed, more people reached the terminal stage of the disease and died, but more people were being diagnosed, so the estimated rate was still low. It was only towards the end of the epidemic, when there were relatively few new cases arising, that the WHO estimate could be expected to perform well. Apparently, this problem is known to public health professionals; for example, it was acknowledged in the cautious paragraph published along with the definition of "case fatality rate" in Last's Dictionary of Epidemiology (6). However, no viable alternative has been suggested in the literature. Underestimation of the fatality rate might affect how seriously an outbreak is taken and subsequent vaccination efforts aimed at controlling the epidemic.
This deficiency in the WHO estimate of the fatality rate of a disease motivated us to write this paper. We developed a new estimator using a chain multinomial model of an emerging epidemic. In this model, in time period t, hospital inpatients either die with probability p1t, recover with probability p2t, or remain in the hospital. The transition probabilities are assumed to depend on calendar time (days since the start of the epidemic) rather than chronologic time related to the onset of disease for each individual, since the latter times often are not known or cannot be obtained quickly, especially in the case of a new emerging disease. There is no need to model the number of infectious disease patients if our interest is in the fatality inflicted by the epidemic. The fatality rate 
t = p1t/(p1t + p2t) is the conditional probability of death given either recovery or death. Simple estimates of p1t and p2t may be obtained at each stage of the epidemic. However, these estimates may be based on limited numbers of cases and may be quite variable. We propose applying a kernel smoothing method to the simple estimates of the probabilities of recovery and death. Then, a simple transformation of the estimated transition probabilities yields an estimate of the fatality rate in real time. The variance of this estimator and associated pointwise confidence limits for the fatality rate are also given here.
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THE ESTIMATOR
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We take the time periods to be days. Time is measured in the
number of days since the first case of the disease was diagnosed
or reported. Suppose that we observe the numbers
N1t and
N2t of the
Ht–1 inpatients at the start of day
t who respectively
die from the disease or recover on day
t. Note that the number
of inpatients
Ht–1 includes new SARS admissions on day
t – 1 as well as existing inpatients who neither die nor
recover. On day
t, an inpatient dies or recovers with probabilities
p1t and
p2t, respectively. Using the information on the number
of inpatients at the end of day
t – 1,
N1t,
N2t|
Ht–1 
multinomial (
Ht–1,
p1t,
p2t) (see Becker (7
) and Yip
and Huggins (8
)). Consequently,
Nit and
Njs are conditionally
uncorrelated for
s 
t,
i,
j = 1, 2. Simple estimators for
p1t and
p2t are
N1t/
Ht–1 and
N2t/
Ht–1, respectively.
The estimated variances of
it are given by
it(1–
it)/
Ht–1 for
i = 1, 2, and the covariance between
1t and
2t is –
1t2t/
Ht–1. A simple estimator for the fatality
rate at time
t,
t, is
A plot of these unsmoothed
estimates for the Hong Kong data is given in
figure 1. These
daily estimates are highly variable, and there is no long-term
trend in the estimated fatality rates. The daily estimates are
generally larger than the WHO estimates, which increase throughout
the course of the epidemic. We propose to reduce the variability
and hence clarify the long-term trend by computing smooth estimates.
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FIGURE 1. Unsmoothed estimates of the death rate, 1t (top left), the recovery rate, 2t (top right), and the ratio of the death rate to the recovery rate, 1t/2t (bottom left), for the outbreak of severe acute respiratory syndrome in Hong Kong, China, 2003. Also shown are the estimated fatality rate, 1t/(1t+2t), and the World Health Organization estimate (thin line) of the fatality rate (bottom right).
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For computation of smooth estimates of the recovery and death
rates for SARS, a kernel method was applied to
1t and
2t. Let
Kb(
x) =
b–1K(
x/
b) be a
kernel function of bandwidth
b, with support in [–1.1].
The Nadaraya-Watson estimator (9
, 10
) for
pit is given by
where
T is the last day of the epidemic. Because
it and
is are conditionally
uncorrelated for
s t, the variances of the
it, i = 1, 2, are given by
and the covariance
of
1t and
2t is
An estimator for the ratio of the death and recovery transition probabilities is given by 
t = 1t/2t, and its variance can be obtained by the standard 
method (11, p. 388):
Then, we estimate
t by
The
method yields a variance estimator
for
t as
and approximate
(1 –
)
x 100 percent pointwise confidence limits for
t can be obtained as
t ±
z1–/2SE(
t), where
z1–/2 is the
normal quartile at the significance level
/2 and SE is the standard
error.
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APPLICATIONS
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Let
K be the Epanechnikov kernel function
K(
x) = (3/4)(1 –
x2)
I(|
x| < 1). To choose the bandwidth
b, we selected a local
bandwidth using the empirical bias bandwidth selection method
to minimize the empirical mean square error (12
). In our application,
we restricted the choice of the bandwidths within the set {3,
5, 7, 10, 15, 20, 30, 40}. In general, the bandwidth tends to
be larger when the data are more sparse or variable. It is well
known that the choice of kernel function in smoothing is less
important than the choice of the bandwidth.
Hong Kong
The first SARS case in Hong Kong was diagnosed on March 12, 2003, and 1,755 Hong Kong patients were hospitalized over the course of the epidemic (13). Of these cases, 1,456 patients were discharged and 299 patients died from SARS by July 31, 2003.
The top panels of figure 2 show real-time transition probabilities and associated 95 percent confidence intervals. The estimated death rate, 1t, peaked in mid-March, decreased for approximately 2 weeks, and then increased again until mid-May. The estimated recovery rate 2t, shown in the top right panel, rose in April, when the majority of SARS patients (mostly health-care workers) from a regional hospital (Prince of Wales Hospital) were discharged, and was then relatively steady until late June. The bottom left panel depicts the ratio of the two estimated transitional probabilities t, and the bottom right panel plots the estimated fatality rate at time t, t, along with the WHO fatality rate over the course of the epidemic for comparison.
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FIGURE 2. Kernel estimates of the death rate, 1t (top left), the recovery rate, 2t (top right), and the ratio of the death rate to the recovery rate, t (bottom left), for the outbreak of severe acute respiratory syndrome in Hong Kong, China, 2003. Also shown are the estimated fatality rate, t, and the World Health Organization estimate (thin line) of the fatality rate (bottom right). Dashed lines, 95% confidence interval.
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The proposed estimate of the fatality rates was high at the
beginning of the outbreak. This may reflect that the patients
who were initially admitted were quite ill but, as the epidemic
progressed, patients were admitted earlier in the progression
of their disease. The estimated fatality rate stabilized at
the beginning of April, and until mid-June it fluctuated between
15 percent and 25 percent. The stabilization of the estimated
fatality rate could reflect a more mature approach in the diagnosis
and treatment of SARS, although the efficacy of using high-dose
corticosteroid and ribavirin to deal with lung parenchymal inflammation
and the coronavirus itself is controversial (14
–16
). Alternatively,
it could reflect a stationary distribution of the chain multinomial
model. The WHO-adopted estimate increased gradually over the
course of the epidemic and converged to the final overall fatality
rate, which was approximately 17 percent.
Beijing
SARS data from Beijing have been available on the WHO website (http://www.who.int/csr) since April 21, 2003. By July 2, 2003, there had been 191 deaths among the 2,521 cases. Our estimates are plotted in the bottom right panel of figure 3. As in Hong Kong, the fatality rate was estimated to be high in the early stage of the epidemic. It then continuously improved over the course of the epidemic, with an overall fatality rate of approximately 8 percent. The reduction in the fatality rate is possibly related to the use of different treatments by the Chinese authorities, who also implemented effective measures to prevent infection in the community and in hospitals during the epidemic. In Beijing, a special hospital was built within 2 weeks of the start of the epidemic to treat SARS patients. Here, the WHO fatality rate overestimates the fatality rate in the latter part of the epidemic. The use of traditional Chinese medicine has also been suggested to have contributed to the low rate of fatality in Beijing, and glycyrrhizin, an active component of licorice root, has been shown to inhibit the replication of SARS-associated coronavirus (17).
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FIGURE 3. Kernel estimates of the death rate, 1t (top left), the recovery rate, 2t (top right), and the ratio of the death rate to the recovery rate,  t (bottom left), for the outbreak of severe acute respiratory syndrome in Beijing, China, 2003. Also shown are the estimated fatality rate,  t, and the World Health Organization estimate (thin line) of the fatality rate (bottom right). Dashed lines, 95% confidence interval.
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SIMULATED DATA
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We simulated data under various scenarios to compare the proposed
real-time fatality rate with the WHO fatality rate. In the simulation,
we used the daily number of inpatients in the Hong Kong SARS
epidemic as the basis for simulation for each day so that the
infection process was not simulated. The numbers of deaths and
recoveries for each day were then simulated according to the
assumed multinomial probabilities under different scenarios.
We chose bandwidths for the estimator by means of the empirical
bias bandwidth selection method.
Scenario I assumes constant death and recovery rates on each day, with values of 0.01 and 0.1, respectively. This can be the situation if no effective treatment is available and/or the lethality of the disease does not change over time. Scenario II is similar to scenario I except for the appearance of a sudden increase in the death rate from 0.01 to 0.03 at 30 days after the start of the epidemic. This mirrors the Amoy Gardens outbreak in Hong Kong, where the strain of the SARS virus was found to be more lethal. In this scenario, it is assumed that the death rate then drops back to its initial level after a period of time. Scenario III assumes a consistently decreasing death rate and an increasing recovery rate. In this scenario, the fatality rate would decrease monotonically.
The left panels in figure 4 show the true death and recovery rates under the different scenarios. The right panels show the corresponding true daily fatality rate p1t /(p1t + p2t), the average over 100 simulations of the proposed real-time fatality rate, and the WHO fatality rate. In all cases, the mean of the proposed fatality rate estimated the true fatality rate quite well and captured the trend or sudden changes in the true fatality for the first and third scenarios. In the second scenario, the abrupt change was not totally captured, but this is to be expected.
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FIGURE 4. Simulation results from estimations of the real-time fatality rate for severe acute respiratory syndrome in three different scenarios with constant (top panels), stepwise (middle panels), and decreasing (bottom panels) fatality rates, 2003. True death rates (thick lines) and recovery rates (thin lines) are shown in the left panels. Also shown are the average real-time fatality rates (thick solid lines) and average World Health Organization estimates (thin lines) of the case fatality rate in 100 simulations (right panels). The true fatality rates are shown by the thick dotted lines (right panels).
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In scenario I, the WHO fatality rate increased consistently
during the first 60 days and then stabilized, so it was underestimating
the true fatality rate in the earlier period of the epidemic.
In scenario II, the WHO fatality rate could only gradually reflect
the sudden increase in the true fatality rate, but it still
underestimated this rate. In addition, the WHO fatality rate
did not reflect the later decrease in the fatality rate. In
scenario III, after an initial increase, the WHO fatality rate
was relatively constant and underestimated the decrease in the
fatality rate.
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DISCUSSION
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We have shown that a real-time fatality rate estimated by means
of kernel smoothing methods has major advantages over the traditional
WHO estimate during the course of an epidemic. An iteratively
reweighted kernel estimator
where
is the weight to be iterated until convergence, can
also be used. This estimator should be more efficient, but the
improvement is expected to be marginal. In general, the weight
function, which puts emphasis on the information in different
periods or characteristics contributed to the estimator, does
not affect the estimator much in comparison with the choice
of bandwidth.
Our simulation studies show that variations in clinical outcome measures—resulting from improved treatment modalities, for example—are not accurately detected by the WHO estimator. On the other hand, our method reflects the possible impact of such factors, including variation in clinical management and other exogenous factors over the course of the outbreak. The spread of the SARS epidemic in Hong Kong was similar to that of scenario I, in which the recovery and mortality rates are nearly constant. In addition, the situation in Beijing was similar to that of scenario III, in which constant improvement in rates of recovery and death was observed. However, as was shown in the simulation, the WHO estimate fails to reflect the time-varying fatality rate in all regions.
The WHO estimates are simply not suitable for monitoring the fatality rate over the course of an emerging epidemic. The present formulation is also different from traditional survival analysis, which adopts chronologic time on the individual level rather than calendar time on the population level and thus requires more detailed data. The proposed chain multinomial model of using the real-time case fatality rate should allow epidemiologists to monitor not only SARS but also other emerging infections and conditions in the future. It can provide a timely and accurate estimate of the number of fatalities inflicted by an epidemic, especially in the earlier period when decisions on health policy and treatment are most critical. It works satisfactorily if the durations of hospital stays for recoveries and deaths are not that different. Yip et al. (18) adopted a competing risk model to provide an estimate of the real-time fatality for SARS using a counting process approach. In the absence of data from controlled clinical trials on SARS treatment, it is interesting to speculate as to whether the different estimated fatality rates between Hong Kong and Beijing could be related to the different clinical treatment protocols used.
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ACKNOWLEDGMENTS
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The authors would like to pay tribute to all of the frontline
health-care workers in the affected areas who have contributed
to the control of SARS.
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References
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E. H. Y. Lau and P. S. F. Yip
RE: "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.,
September 15, 2005;
162(6):
604 - 605.
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ABSTRACT
INTRODUCTION
