Discussion Paper No. 501

A NEWLY DEVELOPED JAPANESE PNEUMONIAAND INFLUENZA MORTALITY MODEL AND
STATISTICAL ANALYSIS OF EXCESS MORTALITYBY STOCHASTIC FRONTIER ESTIMATION

Nahoko Sindo
Masako Ii
Yasushi Ohkusa
andKiyosu Taniguchi

February 2000

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

A Newly Developed Japanese Pneumonia andInfluenza Mortality Model and Statistical Analysis
of Excess Mortality by Stochastic Frontier
Estimation
Nahoko Sindo,Jikei University School of Medicine and National Institute for Infectious
Diseases

Masako Ii,Yokohama National University
Yasushi Ohkusa,Osaka UniversityandKiyosu Taniguchi,National Institute for Infectious Diseases

Abstract
Models which estimate inﬂuenza mortality probably overestimate this mortality because
their baseline estimates include only nonepidemic weeks. Because deaths due to noninﬂuenza illnesses rises during weeks in which inﬂuenza does, the models incorrectly conclude
that all deaths during the inﬂuenza season are due to inﬂuenza. We present a stochastic
frontier estimation model which better reﬂects inﬂuenza mortality.

Key Words : pneumonia, inﬂuenza, excess mortality, models, statistical.

In Japan, the inﬂuenza epidemic of the 1997-98 season was the most severe of the decade.
There were more then 500,000 inﬂuenza-like illness cases and 50,000 pneumonia and inﬂuenza deaths. Like other industrialized countries, there is a growing elderly population in
Japan and they are the ones most aﬀected by inﬂuenza epidemics. Over 90% of inﬂuenza
deaths occur among those aged 65 or older(1,2) . Accurately determining the magnitude ofmortality caused by inﬂuenza highlights the impact of inﬂuenza on the aged population,and may increase the commitment of resources to lessen this morbidity.
To assess the severity of inﬂuenza epidemics, ”excess mortality”, the number of deaths
actually recorded in excess of the number expected on the basis of past seasonal experience, has been used as a major index. This is because the number of deaths caused byacute respiratory diseases sharply rises during inﬂuenza epidemic, although most of them
are without pathological diagnosis and few are virologicaly conﬁrmed. In 1963,Dr.Robert
E.Serﬂing graphically presented expected numbers of seasonal pneumonia-inﬂuenza mortality with a regression method. He estimated inﬂuenza and pneumonia deaths using
Fourier Series with Linear Trend(3) . Subsequently, Choi and Thacker applied a seasonal
ARIMA (autoregressive integrated moving average) model to estimate numbers of deaths
attributable to inﬂuenza and pneumonia in the absence of an epidemic(4) . They stated that
the Serﬂing method tends to underestimate the expected numbers of deaths since data fromepidemic weeks are omitted in deriving the regression equation of expected death. Their
model included data for epidemic weeks, but used data from past nonepidemic weeks toreplace them. They arbitrarily deﬁned epidemic weeks as ones in which there were rises
in mortality beyond what was expected, along with widespread reporting of inﬂuenza-like
illness and laboratory evidence of infection. The Serﬂing method is currently used withsome modiﬁcations by the Centers for Disease Control and Prevention and also has been
applied to several inﬂuenza studies by Who and other countries(5) .
We believe it is conceptually incorrect to include only nonepidemic weeks in these models.
Because deaths due to noninﬂuenza illnesses rise during weeks in which inﬂuenza does, the1models will overestimate mortality because they incorrectly assume that all deaths during
the inﬂuenza season are due to inﬂuenza. We present a new pneumonia and inﬂuenza
mortality model, based on stochastic frontier estimation. This model draws the base line
curve of expected deaths without excluding any data during the year. In addition, by
doing this, there is no need to make an arbitrary decision about epidemic periods. Thismodel provides numerical information about how rarely, from a statistical point of view,
epidemics of inﬂuenza mortality occur.
The estimation equation for pneumonia and inﬂuence death at year t, Pt , islog Pt = α + γ1 D95 + γ2 Tt + γ3 D95 Tt + γ4 Tt2 + γ5 D95 Tt2 +βi Mit + εt(1)where D95 is a dummy variable which takes one after January 1995 and zero otherwise,
because ICD-10 (International Statistical Classiﬁcation of Diseases and Related Health
Problem s, Tenth Revision) was used since January 1995 on Japanese medical certiﬁcatesof death. The number of pneumonia deaths in (10,995) was 77.8% of those in 1994 (14,129).
This decrease is explained as an artifactual change due to mortality and morbidity coding
rules stated in ICD-10 to chose underlying cause of death(6) . For example, a patient with acerebral hemorrhage followed by pneumonia would be coded as a pneumonia death in 1994
but a cerebral hemorrhage death in 1995. Pneumonia and inﬂuenza mortality includes
ICD-9 code 480-487 or ICD-10 code J10-J18 in this study. A variable Tt is linear timetrend. The third degree polynomials are not signiﬁcant (0.74 and 0.95), therefore timetrend variables for the third and higher degree polynomial are not included in the model.
A disturbance term is deﬁned asεt = υt + |ωt |(2)

where υt and ωt are mutually independent random variables, and υt ∼ N(0, µ2 ) and ωt ∼N(0, ξ 2 ). A random variable υt is purely stochastic disturbance and ωt represents the2magnitude of the epidemic which is a non-negative deviation term from expected numberof deaths. The equation form (1) is selected by statistical inference.
The probability density function of this model isf (εt ) =2(µ2 + ξ 2 )0.5φ(ξ/µεt)Φ(εt 2)
220.5(µ + ξ )(µ + ξ 2 )0.5

(3)where εt are disturbance terms in equation (1), and deﬁned by the equation (2). φ isprobability density function for standard normal distribution and Φ is its cumulative distribution function. The likelihood function is deﬁned as the production of probability
density function over time : L(ε1 , ε2 , · · · , εt ) =f (εt ). The estimators for α, γ’s, β’s, ξ/µand (µ2 + ξ 2)0.5 are chosen so as to maximize the likelihood function. The obtained estimators are maximum likelihood estimators. This model is known in economics as stochastic
frontier estimation (7,8) . In economics, ωt is often deﬁned as an ineﬃciency, and stochastic
frontier estimation is used to measure this ineﬃciency. For example, in estimating production function the deviation from the most eﬃcient production is deﬁned as ineﬃcient, andin estimating cost function the deviation from the least cost is deﬁned as ineﬃcient. In thispaper, we deﬁned ωt as the magnitude of epidemic. In this way we can estimate the excess mortality without making an arbitrary assumption of what epidemic or non-epidemic
weeks are. For example, the referred models arbitrarily deﬁne non-epidemic weeks and estimate the excess mortality during the epidemic weeks using these

non-epidemic weeks

as baseline.
For the years 1998 - February 1999, the largest monthly excess mortality from pneumonia
and inﬂuenza occurred in January, 16.8 % more than that in December, while in September
it was the smallest ( 26.1 % less ) ( Table 1 (a) ). Excess mortality from pneumonia andinﬂuenza grew at the rate of 1.1 % per year until 1995 ( Linear Time Trend in Table 1 (a)). The number of total death in January was also the largest, 7.2 % more than that inDecember ( Table 1 (b) ).

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The epidemic threshold is usually deﬁned as an upper limit of 95 % conﬁdence interval forstochastic disturbance terms, υt . However, υt are originated from measurement errors andnot from the magnitude of epidemics. The magnitude of an epidemic should be measuredby ωt . Therefore, in this paper, threshold is deﬁned as a summation of estimated numberof death and, ω which is deﬁned in the following equation : ω ∞
0−∞fˆ(υ + |ω|)dυdω = 0.95

(4)where, fˆ is evaluated using µ and ξ estimated in equation (3).
Thus, in our model the actual number of deaths from pneumonia and inﬂuenza was above
the threshold in 1990, 1993, 1995, 1997, 1998 and 1999(Figure 1 (a) ). Excess mortality
was 1, 951 in January 1990, 2,191 in February 1993, 2, 641 February 1995, 5, 032 inJanuary 1997, 2, 820 in February 1998, and 6, 798 in January 1999. For these months theprobability values of excess mortality occurrence are less than 0.00001, which is calculatedby the following equation : ∞  ∞−∞ε̂−υfˆ(υ + |ω|)d|ω|dυ(5)

In the ﬁgure 1 (b), the threshold indicates that actual number of total deaths is above thethreshold in 1990, 1993,1995,1997,1998 and 1999. Excess mortality was 7,465 on January
1990, 7,192 in January 1993, 18,184 in January 1995, 12,405 in January 1997, 6,569 in
February 1998, and 22,503 in January 1999. For these months the probability values ofexcess mortality occurrence are less than 0.00001 when calculated by equation (5).
We believe that , compared to the two referred models, stochastic frontier estimation
has at least four advantages in estimating excess mortality. One is that the model does
not overestimate mortality caused by inﬂuenza. Indeed, by using in its base line all datafrom epidemic weeks, it is likely, if anything, to underestimate mortality. Ideally, we would
like to know the proportion of all deaths reported as ICD-10 J10-18 actually caused by4

inﬂuenza. Unfortunately, neither we norm to our knowledge, other countries have thisinformation. Until such information is available, models will need to be used. A second
advantage is that stochastic frontier estimation does not depend on arbitrarily deﬁned
epidemic weeks. Third, this estimation model gives numerical information that shows howrare epidemics that cause greatly increased mortality are. Fourth is utilizing long term
past data for making base line curve while the current CDC model is looking back data for
only ﬁve-year period.
This models should be applicable for weekly analysis and useful as a prediction tool. We
will be to get national weekly mortality data in the near future, and hope to promptly
predict outbreaks of inﬂuenza by making use of this mortality model.

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Acknowledgments

The authors thank Dr. Mark white and Dr. George Schmid for their comments.

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References[1] Statistics and Information Department, Minister’s Secretariat, Ministry of Health andwelfare. Causes of death-Table 3. Death by Causes ( the condensed list of causes ofdeath for Japan ), sex and month of occurrence : Japan, 1998-1997. ( Vital Statistics
Mortality Data, Total death, inﬂuenza death, pneumonia death ), VITAL STATISTICS OF JAPAN, vol.3, 1997-1999.
[2] Statistics and Information Department, Minister’s Secretariat, Ministry of Health andwelfare. Monthly Vital Statistics Report ( preliminary), Total death, inﬂuenza deaths,
pneumonia death,1998 Jan.-1999 Feb.
[3] Serﬂing RE.,” Methods for current statistical analysis of excess pneumonia-inﬂuenza
deaths,” Public Health Rep 1963; 78 : 494-506.
[4] Choi K and Thacker SB.,”An evaluation of inﬂuenza mortality surveillance, 19621979”, I. Am J Epidemiology 1981; 113 :215-226.
[5] Assad F, Cockburn WC, Sundaresan TK.,”Use of excess mortality from respiratory
diseases in the study of inﬂuenza,” Bull WHO 1973; 49 : 219-233.
[6] Statistics and Information Department, Minister’s Secretariat, Ministry of Health andwelfare. Reference table. VITAL STATISTICS OF JAPAN, vol.1, 1995; p456.
[7] Aigner AD, Lovell K and Schmidt P.,”Formulation and estimation of stochastic frontier
production function models,” Journal of Econometrics 1977; pp.21-37.
[8] Jondrow J, Lovell K, Materov S and Schmidt P.,”On estimation of technical ineﬃciency
in the stochastic frontier production function model,” Journal of Econometrics 1982 ;pp.233-239.

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Table 1: Stochastic Frontier Estimation for Mortality, January
1988 - February 1999(a) Pneumonia and Inﬂuenza
Estimated
t-statistics
Coeﬃcient
Constant
8.20787
107.946
Linear Time Trend
0.010526
4.25308
Dummy for after 1995
-0.154352-2.39021
Linear Time Trend
× Dummy for after 1995
-0.078245-5.0944
Linear Time Trend squared -0.000043-1.89429
Linear Time Trend squared
× Dummy for after 1995
0.000329
4.98485
Monthly Dummies
January0.167703
3.27628
February
0.096200
1.90556
March
0.092937
1.77739
April-0.026695-0.419345
May-0.071048-1.09857
June-0.199961-3.29451
July-0.156862-2.32106
August-0.170525-2.70645
September-0.260593-3.98653
October-0.183964-2.69883
November-0.109000-1.48611(µ2 + ξ 2 )0.5
0.148406
14.6737
ξ/µ4.16483
2.61146
Number of observations
134
Log likelihood-166.329
Variable8

Variable(b) Total Death
Estimated
Coeﬃcient11.13350.001473

Constant
Linear Time Trend
Monthly Dummies
January
February
MarchApril
May
June
July
AugustSeptember
OctoberNovember(µ2 + ξ 2 )0.5
ξ/µNumber of observations
Log likelihood

0.072121-0.038273
0.003039-0.088863-0.109695-0.193285-0.162045-0.166225-0.206316-0.111206-0.084749
0.056938
3.31355134-290.511

9t-statistics
691.067
14.0684
4.25377-2.26390.174893-4.47695-5.23517-9.25166-8.86165-9.18318-9.58921-4.96559-4.031713.3600
2.93399

Figure 1(a): Number of Monthly Death from Pneumonia & Influenza
16000

14000

# of Death

12000

10000

8000

6000

4000

2000
88

9092

94
96
−:data,− −:predict,...:upper

98

100

412

Figure 1(b): Number of Monthly Death, Total

x 1011# of Death

10

987

6

588

90

92

94
96
−:data,− −:predict,...:upper

98100