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```									It's not the figures themselves," she said finally, "it's
what you do with them that matters." Lamia Gurdleneck

SARS Survivor Function
Estimation from Cumulative
Reports (Draft March 14, 2006)

Larry George

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Vision Statement

   Cumulative case and death counts are
statistically sufficient to estimate
nonparametric survivor functions of
transient stochastic processes. Such
estimates would benefit biostatistics and
reliability statistics.

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Goal and Objective
   State the desired goal
   Help the WHO and biostatisticians with
epidemiological statistics
   State the desired objective
   Spread the use of survival analysis without life data, to
save computer storage requirements and money,
reduce errors, improve credibility, and obtain more
precise actuarial forecasts, without privacy violation
   Such estimates would benefit biostatistics with
statistical confidence limits on forecasts and regional
differences in survivor functions
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Today’s Situation
   Statisticians believe random samples of death
times are required for survival analysis
   “The WHO did not describe the estimation method and
just mentioned that this estimation requires detailed
individual patient data on the time from admission (or
illness onset) to death or full recovery.” [Yu et al]
   “…we also used a version of the Kaplan-Meier survival
curve, adapted to allow for two types of outcome
(death and discharge).” ”We thank David R. Cox for
developing a suitable nonparametric method for
estimation of the case fatality rate.” [Donnelly et al]

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Counterexample
   SARS data from
www.who.int/csr/sars/country/en/
   Daily cases, deaths, some recoveries
   Grouped by week and country
   Used the total for all countries
   Make nonparametric maximum likelihood and
least squares estimators of survivor functions
   Npmle [George and Agrawal, George 1999]
   Nplse [Oscarsson and Hallberg; Harris, Rattner, and
Sutton; George 1995]

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Typical report
(almost daily)

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Survivor function
SARS Survivor Function Estimates

1

0.98
npmle S(t) 17 weeks
npmle S(t) 14 weeks
0.96                                                    npmle S(t) 10 weeks
npmle S(t) 6 weeks
nplse S(t) 17 weeks
0.94                                                    nplse S(t) 14 weeks
nplse S(t) 10 weeks
nplse S(t) 6 weeks
0.92

0.9
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Time From Infection to Death, Weeks

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Weekly death rates
Weekly SARS Death Rate
as a Function of Weeks Since Case Report

0.05

0.04

0.03                                                     nplse 17 weeks
nplse 14 weeks
nplse 10 weeks
0.02
nplse 6 weeks

0.01

0
0        5          10           15     20
Weekly Death Rate

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Recovery (Survivor) function
SARS Recovery Time Survivor Function

1

0.9
0.8

0.7

0.6                                                    nplse S(t) 17 weeks
nplse S(t) 14 weeks
0.5
nplse S(t) 10 weeks
0.4                                                    nplse S(t) 6 weeks
0.3

0.2
0.1

0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Time from Infection to Recovery, Weeks

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Recommendations
   Use nonparametric estimates of age-or-time-
specific survivor and actuarial rate functions
from case, death, and recovery reports
   Make actuarial forecasts of deaths and
recoveries
   Estimate CFR, survival and recovery time
distributions, and estimate confidence limits
   Test hypotheses about country differences,
treatments, and so on

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References
    Donnelly, Christl A., et al., “Epidemiological determinants of spread of causal agent of
SARS in Hong Kong,” http://image, thelancet.com/extras/03art4453web.pdf
    George, L. L. and A. Agrawal, “Estimation of a Hidden Service Distribution of an M/G/
Service System,” Naval Research Logistics Quarterly, pp. 549-555, September, 1973,
Vol. 20, No. 3.
    Ibid., “Field Reliability Estimation Without Life Data,” ASA SPES Newsletter, Dec. 1999,
    Ibid., “Apply Field Reliability to Service and Spares,” QC95 Conference, ASQC Santa
Clara Valley, April 1995
    Harris, Carl M.; Rattner, Edward; Sutton, Clifton. Forecasting the extent of the HIV/AIDS
epidemic. Socio-Economic Planning Sciences, Vol. 26, No. 3, Jul 1992. 149-68 pp.
Elmsford, New York/Oxford, England.
    Oscarsson P and Hallberg Ö, “EriView 2000 -A Tool For The Analysis Of Field Statistics”,
Proc. ESREL 97, Lisbon, June 1997, ISBN 0-08-042835-5
Yu, Philip L. H. et al., “Statistical exploration from SARS,” Amer. Statistician, vol. 60, No.
11

1, pp 81-91, 2006
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