J. R. Statist. Soc. A (2005)
168, Part 1, pp. 233–243
A comparison study of realtime fatality rates:
severe acute respiratory syndrome in Hong Kong,
Singapore, Taiwan, Toronto and Beijing, China

Paul S. F. Yip, K. F. Lam, Eric H. Y. Lau, Pui-Hing Chau and Kenneth W. Tsang University of Hong Kong, People's Republic of China National Tsing Hua University, Hsinchu, Taiwan [Received August 2003. Final revision June 2004] Summary. In an outbreak of a completely new infectious disease like severe acute respiratory
syndrome (SARS), estimation of the fatality rate over the course of the epidemic is of clinical
and epidemiological importance. In contrast with the constant case fatality rate, a new measure,
termed the ‘realtime' fatality rate, is proposed for monitoring the new emerging epidemic at a
population level. A competing risk model implemented via a counting process is used to esti-
mate the realtime fatality rate in an epidemic of SARS. It can capture and reflect the time-varying
nature of the fatality rate over the course of the outbreak in a timely and accurate manner. More
importantly, it can provide information on the efficacy of a certain treatment and management
policy for the disease. The method has been applied to the SARS data from the regions affected,
namely Hong Kong, Singapore, Toronto, Taiwan and Beijing. The magnitudes and patterns of
the estimated fatalities are virtually the same except in Beijing, which has a lower rate. It is
speculated that the effect is linked to the different treatment protocols that were used. The stan-
dard estimate of the case fatality rate that was used by the World Health Organization has been
shown to be unable to provide useful information to monitor the time-varying fatalities that are
caused by the epidemic.
Keywords: Competing risk; Counting process; Severe acute respiratory syndrome; Time-varying fatality rate Severe acute respiratory syndrome (SARS) is a highly contagious and severe atypical pneumoniawhich is caused by a new coronavirus and is predominantly transmitted by droplets and contactwith contaminated fomites (Tsang, Ho, Ooi et al., 2003; Rota et al., 2003; Seto et al., 2003;Riley et al., 2003). SARS has rapidly spread worldwide and there were in total 8098 reportedcases with 774 fatalities at July 31st, 2003 (World Health Organization, 2003a). An on-goingcontroversial topic is the estimation of the fatality rate of SARS. The usual definition of thefatality rate that has been adopted by the World Health Organization (WHO) is the ratio of thecumulative number of deaths to the cumulative number of infected people in the course of theepidemic (World Health Organization, 2003b). This is often called the case fatality rate and isassumed to be a constant.
Address for correspondence: Paul S. F. Yip, Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, People's Republic of China.
E-mail:  2005 Royal Statistical Society P. S. F. Yip, K. F. Lam, E. H. Y. Lau, P.-H. Chau, K. W. Tsang and A. Chao While the outbreak was on going and there were patients still in hospitals over the course of the epidemic, the WHO estimate assumed implicitly that all remaining SARS in-patientswould eventually recover. It therefore led to an underestimation of the true case fatality rate.
For example, in the midst of the SARS outbreak at April 15th, 2003, the fatality rate in HongKong was 4.5% according to the WHO estimate, but it hit a record high of 17.0% at the endof the epidemic. Furthermore, the WHO estimate cannot reflect the changing fatality rate overthe course of the epidemic in a timely manner owing to the use of cumulative data in the cal-culation. The WHO estimates of the fatality rates of Hong Kong, Singapore, Taiwan, Beijingand Toronto were 17.0%, 13.9%, 10.7%, 6.5% and 15.8% at July 31st, 2003, respectively. HongKong appeared to have the highest fatality rate.
Donnelly et al. (2003) were the first to apply a survival analysis framework to analyse the SARS data and to estimate the time delay distributions of the times since infection to onset,onset to admission, admission to death and admission to discharge. They assumed a fixed prob-ability π (the case fatality rate) and 1 − π for the respective outcomes of death and recovery foreach in-patient on admission during the whole course of the epidemic. Their approach projectedan estimate of the case fatality rate, and the estimate depends on whether the age of the patient isover 60 years or not. Their formulation can also be easily extended to a mixture of regression orthreshold models to incorporate exogenous variables. In this setting, the constant case fatalityrate is often interpreted as a measure of the severity or virulence of an infectious disease.
For a completely new disease like SARS, it would be worthwhile and informative to inves- tigate how the rate of death (given that a patient either dies or recovers) varies with time asclinical experience on diagnosis and management improve over time. When modelling this rate,there is no need to model the infectious process as our interest is in measuring the fatality thatis inflicted by the epidemic. Also, the duration of hospitalization of individual patients is nottaken into account since the initiation of hospitalization of an SARS patient is dependent onseveral factors, including the very early screening of hospital staff who are exposed, leading todetection and hospitalization of very early and asymptomatic disease, which is in contrast withthe reluctance of some cases to seek medical attention, thereby delaying their hospitalizationin the process. In this paper, we call this rate the realtime fatality rate. The rapid accumula-tion of experience and knowledge on the diagnosis, investigation and management, along withdifferent age groups being affected at different times, compounded by differences in viral loadsand other confounding factors, do not permit the assumption of a constant rate throughout theperiod of the outbreak . To monitor the fatality of an emerging epidemic, an estimator for sucha time-specific rate at the population level should be used to assess the efficacy of treatment inrealtime, especially when the disease is rapidly evolving. For instance, there is also considerablecontroversy on the treatment of SARS (Wenzel and Edmond, 2003). The efficacy of ribavirinand corticosteroids, which were commonly used in Hong Kong for an empirical treatment ofSARS, is not proven by controlled trial studies that were impossible to organize during theinitial outbreak. In the absence of large scale clinical trials to assess the efficacy of various treat-ments, investigators are interested in the estimation of the realtime fatality rate over the courseof the epidemic with a broad assessment of the effect of other factors, including variation inclinical management and other exogenous (including unidentified or unspecified) factors overthe course of the outbreak.
In this paper, a counting process in a competing risk set-up is used to model the transitions from in-patient to death and from in-patient to recovery (Andersen et al., 1993; Yip and Lam,1992, 1993). The in-patients are subject to death, or recovery or to remain in the hospital overthe course of the epidemic. Here, the timescale of interest is the calendar time, rather thanthe usual time to event from admission in standard survival analysis studies as in Donnelly Severe Acute Respiratory Syndrome et al. (2003). A new measure termed the realtime fatality rate is proposed in this paper and isdefined as the probability of death conditioned on a transition to death or recovery at time t.
This formulation can therefore model the fatality rate in realtime for the infected communityin a macro perspective. Since rate estimates from survival models are usually estimated as stepfunctions, a kernel method is used to smooth the estimates of the transition rates for recoveryand death at time t, and for the realtime fatality rate. This realtime fatality rate is also usefulin providing information on the efficacy of certain treatments and on the management of thedisease during the outbreak of the disease. The approximate pointwise confidence limits of therespective transition and the realtime fatality rates are readily available. The method is appliedto the SARS data from the regions affected, namely Hong Kong, Singapore, Toronto, Taiwanand Beijing.
Clinical data collection The daily data on admission, discharge and death for all SARS patients in Hong Kong,Singapore, Taiwan, Beijing and Toronto were obtained from the Web sites of the respectiveGovernment agencies and authorities: Hong Kong,; Singapore,;Taiwan,; Beijing,; Toronto, All patients with probable SARS were diagnosed according to thecriteria that were established by the WHO (World Health Organization, 2003b). Briefly,patients with probable SARS presented with high fever (temperature above 38 ◦C), cough-ing or breathing difficulty, contact history with SARS patients and pulmonary infiltrates thatwere consistent with pneumonia on chest radiographs or high resolution computed tomog-raphy of the thorax that could not be explained by an alternative diagnosis (Tsang, Mok,Wong and Ooi, 2003). The criteria from the WHO and Centers for Disease Control and Preven-tion (2003) do not require microbiological confirmation although most patients with SARS inHong Kong have positive serological evidence of SARS coronavirus infection, i.e. a significantrise in immunoglobulin G after a period of 21 days (Peiris et al., 2003).
Statistical model For each in-patient, the final outcome must be either death or recovery. The primary target ofmost medical treatments and policies for lethal epidemics such as SARS is to lower the fatalityrate during the outbreak of the epidemic, i.e. to lower the force of death relative to recovery,rather than necessarily to shorten the period of hospitalization. Hence a measure of treatmentefficacy should be time sensitive (calendar time but not chronological time) and focus on theireffects on death and recovery. By considering these two outcomes as competing ‘risks' as inYip and Lam (1992), we can obtain a realtime fatality rate which serves well for treatment orpolicy evaluation during the epidemic. The two risks, which are not necessarily independent,are assumed to operate simultaneously on the in-patients. Since the times of the outbreak inSARS-affected areas are different, we have different time origins (different calendar times) fordifferent regions depending on the time of onset of the SARS epidemic and the availability ofthe data, with all areas followed longitudinally up to June 30th, 2003. The method proposedmakes use of the information only on the cumulative number of deaths, cumulative number ofdischarges and SARS patients who are still in hospital on day t (for the Hong Kong region, P. S. F. Yip, K. F. Lam, E. H. Y. Lau, P.-H. Chau, K. W. Tsang and A. Chao t = 0 for March 1st, 2003, t = 1 for March 2nd, 2003 and so on), denoted by N1.t/, N2.t/ and I.t/ respectively. Suppose that N1.t/ and N2.t/ are counting processes with intensity processesor cause-specific instantaneous transition rates for a transition to death and to recovery on thetth day, denoted by γ1.t/ and γ2.t/ respectively, that satisfy the multiplicative intensity model P{dNi.t/ = 1 Ft} = γi.t/ I.t/ dt where Ft = {I.u/, Ni.u/, for i = 1, 2; 0  u < t}, the σ-field representing the history of the epi-demic process up to but not including time t. By the Doob–Meyer decomposition (see Yip andLam (1992) and Andersen et al. (1993), we have that Mi.t/ = Ni.t/ − γi.u/ I.u/ du are zero-mean martingales with respect to the increasing family of σ-fields {Ft, t  0}.
On the basis of standard martingale theories, the cumulative cause-specific transition rates can be estimated easily by the usual Nelson–Aalen estimator. A smoothed version of the cause-specific transition rates can be estimated by the Ramlau-Hansen (1983) estimator given by where ti1 < ti2 < . . denote the ordered exit times due to cause i, b is the window width of akernel function, which determines the degree of smoothing, and B is a kernel function. Onefrequently used kernel function is the Epanechnikov kernel function B.x/ = 0:75.1 − x2/ Details on the estimation of γ1.t/ and γ2.t/ can be found in Yip and Lam (1992, 1993). Sincethe time unit is days, there were situations where several people recovered or died on the sameday. For simplicity we assume that all the deaths occurred before the discharges on each day.
This assumption has very little effect on the estimates. It should be noted that only those exittimes til for which t − b<til <t + b would contribute to the estimation of γi.t/,whereas the infor-mation on those days that are outside the range would have no contribution. Under some mildregularity conditions, estimator (1), suitably normalized, is distributed according to a normaldistribution asymptotically (see Andersen et al. (1993), pages 241–243) and the variance of theestimator is consistently estimated by var{ ˆγi.t/} = Hence approximate 100.1 − α/% pointwise confidence limits can be constructed for the γi easily.
The choice of the bandwidth b for the kernel function in equation (1) leads to a trade-off betweenthe variance and bias of ˆγi.t/: The optimal bandwidth for γ can be obtained by minimizing theintegrated mean-square error (Andersen et al., 1993) in theory. However, a subjective value ofb by visually examining the fitted γ would be sufficient for our purpose. After many trials ofdifferent bandwidths, window widths of 5, 7, 6, 3 and 12 days were chosen for Hong Kong,Singapore, Taiwan, Beijing and Toronto respectively, depending on the volatility and sparse-ness of the data. A natural estimator for the ratio of the transition rates θ.t/ = γ2.t/=γ1.t/ Severe Acute Respiratory Syndrome θ.t/ = ˆγ2.t/=ˆγ1.t/: By the martingale theories, it can be shown that the estimates of the cumulative intensities i.u/ du, for i = 1, 2, are nearly orthogonal and hence the covariance is 0. The delta method can be used to estimate the variance of ˆ θ and is thus given by var{ ˆθ.t/} = var{ ˆγ Finally, we propose a new measure of the fatality rate π.t/ defined as the probability of deathconditioned on a transition to death or recovery at time t. Mathematically π.t/ = P{ dN1.t/ = 1 dN1.t/ + dN2.t/ = 1} = γ1.t/ + γ2.t/ 1 + θ.t/ A natural estimator for π.t/ is thus 1 + ˆθ.t/ and, again using the delta method, the variance estimator for ˆ π.t/ is given by var{ ˆπ.t/} = 1 + ˆθ.t/ Approximate 100.1 − α/% confidence limits for π.t/ can also be obtained in a similar mannerto that for γ.t/: Also, it is important to note that, as mentioned earlier, there is no need tomodel the complex hospitalization process I (u) in our formulation as far as the estimation ofthe realtime fatality rate is concerned.
For the entire Hong Kong region up to July 31st, 2003, altogether 1755 cases were hospitalized.
Of these, 1456 and 299 cases were discharged and died from SARS respectively. Figs 1(a) and1(b) depict the estimates of γ1.t/ and γ2.t/ respectively, together with the associated 95% confi-dence bands. Both transition rates increased with time since the beginning of the epidemic. Theestimated intensity of transition to death, ˆγ1, peaked around March 20th, decreased and thenincreased again around May 15th, 2003. The estimated intensity of recovery, ˆγ2, peaked aroundApril 1st, where most of the SARS patients who were medical health workers from a regionalhospital (Prince of Wales Hospital) were discharged, and another two peaks were also foundaround the end of April and on May 19th. Fig. 1(c) depicts the ratio estimate of the two rates, θ.t/, while Fig. 1(d) shows the plot of the estimated realtime fatality rate at time t, ˆπ.t/, over thecourse of the epidemic together with the WHO estimate. The WHO methodology estimated thecase fatality rates of SARS in Hong Kong at March 30th, April 15th and May 25th, 2003,as 2.5%, 9.9% and 15.8% respectively and started to stabilize thereafter, with a final estimate of17% at June 30th. In contrast, the realtime fatality rate was chaotically high at the beginning ofthe outbreak, since few patients were discharged in the early stages of the outbreak. The real-time fatality rate seemed to have stabilized from early April, fluctuating between 15% and 20%.
Similar figures for the estimated realtime fatality rates (with different origins) for Singapore,Taiwan, Beijing and Toronto are given in Fig. 2.
Kernel estimates of the SARS outbreak in Hong Kong: (a) instantaneous death-rate γ1.t/; (b) instantaneous recovery rate γ2.t/; (c) ratio of instan- taneous recovery and death-rates θ.t/; (d) realtime fatality rate π.t/ and WHO estimate of case fatality rate Fatality rate 0.3
Realtime fatality rate and WHO estimate of the case fatality rate for the regions (a) Singapore, (b) Taiwan, (c) Beijing and (d) Toronto P. S. F. Yip, K. F. Lam, E. H. Y. Lau, P.-H. Chau, K. W. Tsang and A. Chao The number of SARS cases in Singapore was smaller than that of Hong Kong probably because of the confinement of outbreaks within hospitals and the lack of major communityoutbreaks, such as that occurring in Amoy Gardens in Hong Kong. There were altogether 238cases with 33 deaths with a WHO-estimated case fatality rate of 13.9%. Fig. 2(a) shows thatthe realtime fatality rate appeared to stabilize approximately 4 weeks after the outbreak, whichwas similar to what happened in Hong Kong. The fatality rates fluctuated between 15% and19% after the middle of April.
Data from Taiwan became available from April 4th, 2003. Altogether 37 deaths were found out of the 346 cases with a WHO case fatality rate of 10.7%. However, the time-varying ratesuggested an unusually high rate in the early part of the outbreak, which subsequently improved.
Most of the major outbreaks were found within hospitals in Taiwan.
For Toronto the data were available from March 18th, 2003, with a total of 39 deaths from 247 infected cases giving a WHO case fatality rate of 15.8%. SARS had made a come-back in thecommunity (
html). The most recent date of onset of illness and isolation among probable cases was June12th. Most of the SARS cases in Canada were found in Toronto.
For Beijing, a special hospital was built within two weeks to treat the SARS patients in May 2003. The Government implemented very effective measures to prevent infection in the commu-nity and in hospital, including the removal of the Minister of Health and the Mayor of BeijingCity, who failed to deal with the epidemic control. A very high level working group was set upand headed by a Vice-Premier. The realtime fatality rate was estimated to be high in the earlystage of the epidemic and continuously improved over the course of the epidemic such that therate was less than 5% by June 10th. However, the WHO case fatality rate was much higher thanthis June estimate of the time-varying rate, since the former was very much affected by the earlydata.
An estimation of the fatality rate in the situation of an emerging outbreak or epidemic is neverstraightforward. This is particularly true for SARS, when clinicians all over the world had todeal with this highly contagious and frightening pneumonic illness with no idea of its aetiology,pathogenesis, clinical features, management issues or prognosis, other than anecdotal infor-mation from colleagues from mainland China (Tsang and Lam, 2003; Lee et al., 2003). Theoriginal estimate of the fatality rate for SARS was only about 4.5%, by both the WHO andthe Hong Kong Health Authority (Department of Health of Hong Kong Government, 2003),even as late as the beginning of April 2003. This was later adjusted to about 10% in May 2003.
This underestimation of the fatality rate from SARS must have undoubtedly affected the healthcare planning measures that were taken to contain SARS, and thus the outcome of the SARSoutbreak at the community level.
Our comparison of the estimated realtime fatality rates among the SARS-affected areas shows that the SARS fatality rate appears to be changing with time in all regions. Besides Beijing, therealtime fatality rates of all other areas seem to fluctuate around 12–20%. The constant fatal-ity assumption, which was adopted by Donnelly et al. (2003), does not reflect the situation atthe population level. Indeed the approach of Donnelly et al. (2003) can be extended easily toregression models set up to allow for differences in the diagnosis and management of patientsby allowing the (time constant) case fatality rate to depend on relevant (possibly time-varying)explanatory variables. However, a time-varying fatality rate at a population level is alwayshelpful and meaningful to medical researchers.
Severe Acute Respiratory Syndrome In addition, this changing fatality pattern is not reflected by the definition of case fatality rate that has been adopted by the WHO that derives from the ratio of cumulative deaths tothe total number of cases diagnosed. Any variation in clinical outcome measures, e.g. as aresult of improved treatment modalities, would not be depicted by such a definition. By usingthe actual calendar timescale, rather than duration of hospitalization, the method proposedprovides a visual assessment of the effect of other factors, including variation in clinical man-agement and other exogenous (including unidentified) factors over the course of the outbreak.
A stable estimate of the fatality rate could reflect a more mature approach to the diagnosis andtreatment of SARS, although the efficacy of using high dose corticosteroid and ribavirin, todeal with the lung parenchymal inflammation and coronavirus itself, are largely controversial(Booth et al., 2003; Wenzel and Edmond, 2003; Cyranoski, 2003). The constant case fatalityassumption is more suitable for some well-known infectious diseases such as rabies or small-pox because the aetiology, pathogenesis and treatment have been well studied with a standardtreatment protocol. It is recommended that the realtime fatality rate be investigated at the out-break of an emerging epidemic. It can serve as an indicator of the change in the fatality rateover time and can provide information on the effectiveness of the policy and treatment that areadopted.
Our data show that SARS fatality rates among Hong Kong, Singapore and Toronto were similar, whereas Beijing appeared to have a lower rate after the first month of the epidemic.
In the absence of data from controlled clinical trials for the treatment of SARS, it would beinteresting to speculate whether the time-varying fatality rates that are calculated by our modelcould be related to the different clinical treatment protocols that were used by these areas. Forinstance, the use of ribavirin was essentially forbidden in Canada by late April 2003, whereasit has consistently been the standard empirical practice in Hong Kong. The use of continuouspositive airway pressure, delivered by nasal masks, to patients with hypoxia has been partlyattributed as a cause of the low case fatality and low intensive care unit admission in mainlandChina (personal communication from Professor N. S. Zhong of Guangdong). The use of tradi-tional Chinese medicine has also been claimed to be the cause of low case fatalities in Beijing,and glycyrrhizin, an active component of liquorice roots, has been shown to inhibit the repli-cation of SARS-associated coronavirus (Cinatl et al., 2003). The Singaporian practice of usingribavirin and corticosteroid in the treatment of SARS is not uniform, and the standard proto-col that is practised in Taiwan has not been fully published (Twu et al., 2003; Hsu et al., 2003).
Although these differences in clinical practices could contribute to a difference in case fatali-ties, it should be cautioned that they do not necessarily equate to conclusions that are drawnfrom controlled clinical trials. Not withstanding these limitations, especially acknowledging theinsensitivity of the case fatality rate as a clinical outcome and the possible discrepancy in thediagnostic stringency, it is possible that ribivarin treatment could have no effects in the clinicaloutcome of SARS patients.
There is increasing concern about emerging infections all over the world, and SARS is only an example of how ignorant we are in dealing with the statistical–epidemiological and clinicalmanagement of this condition. Our model of using the realtime fatality rate should allow epi-demiologists to monitor not only SARS but also other emerging infections and conditions inthe future.
The authors pay tribute to all the front line medical health workers in those affected areas whohave contributed to the control of SARS. The authors also thank the reviewers, an Associate P. S. F. Yip, K. F. Lam, E. H. Y. Lau, P.-H. Chau, K. W. Tsang and A. Chao Editor and the Joint Editor for their helpful comments and suggestions that improved the pres-entation of the paper.
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