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From equation 13, it is clear that the shape of the distribution of deaths depends only on the number of deaths.

For the death rate, R , E ( R ) = R and Var ( R ) = D/P 2 . It follows, in this case, that y = D and z = P 1 and thus,

14. R ~ ( D ,P 1).

A useful property of the gamma distribution is that for X ~ ( y ,Z ), one

can divide

X

by

z such

that

X

/

z ~

(y ,1). This

converts

the

gamma

distribution into a simplified, standard form dependent only on parameter y . Expressing equation 14 in its simplified form gives

R 15.

P 1 = D ~ ( D ,1)

From equation 15, it is clear that the shape of the distribution of the death rate is also dependent solely on the number of deaths.

Using the results of equations 13 and 15, one can use the inverse gamma distribution to calculate upper and lower confidence limits. Lower and upper 100 (1 ) percent confidence limits for the number of deaths, L(D) and U(D) , are estimated as

16. L( D ) = 1 ( D ,1)( / 2) and U ( D ) = 1 ( D +1,1) (1 / 2)

where 1 represents the inverse of the gamma distribution and D+1 in the formula for U(D) reflects a continuity correction made neces- sary by the fact that D is a discrete random variable and the gamma distribution is a continuous distribution. For a 95 percent confidence interval, =.05. For the death rate, it can be shown that

L( D ) U ( D )17. L( R ) = P and

U ( R ) = P

For more detail regarding the derivation of the gamma method and its application to age adjusted death rates and other mortality statistics, see references 7, 79, and 84.

National Vital Statistics Reports, Vol. 54, No. 13, April 19, 2006 119

Availability of mortality data

Mortality data are available in publications, unpublished tables, and electronic products as described on the mortality Web site available from: http://www.cdc.gov/nchs/deaths.htm . More detailed analysis than provided in this report is possible by using the Mortality public use data set issued each data year. Since 1991, the data set is available through NCHS in CD ROM format. Data are also available in the Vital Statistics of the United States , Mortality, and Vital and Health Statistics , Series 20 reports, and the National Vital Statistics Reports through NCHS.

Definitions of terms

Infant deaths Deaths of infants aged under 1 year. Neonatal deaths Deaths of infants aged 027 days. Postneonatal deaths Deaths of infants aged 28 days1 year. Crude death rate Total deaths per 100,000 population for a

specified period. The crude death rate represents the average chance of dying during a specified period for persons in the entire population.

Age-specific death rate Deaths per 100,000 population in a specified age group, such as 14 years or 59 years for a specified period.

Age-adjusted death rate The death rate used to make compari- sons of relative mortality risks across groups and over time. This rate should be viewed as a construct or an index rather than as direct or actual measure of mortality risk. Statistically, it is a weighted average of the age specific death rates, where the weights represent the fixed population proportions by age (86).

http://www.cdc.gov/nchs/deaths.htmhttp://www.cdc.gov/nchs/deaths.htm

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