ADDITIONAL POPULATION EXPLANATION
For your use, two additional population tables have been added as WK1 files
in the POP directory on this CD-ROM. These files: 1) populations by marital
status, race, sex, and age, 1991, and 2) populations by marital status,
Hispanic origin, race, sex, and age, 1991 (for 48 states and the District of
Columbia). These populations can be used to calculate death rates by marital
status.
The SETS software cannot calculate these rates. The user will have to extract
the number of deaths from the mortality file and calculate the rates using
these populations. The explanation of these populations and the reliability
of the death rates follows. Please read the following text before calculating
any death rates by marital status.
Reliability measures for estimated death rates for marital status by race and
Hispanic origin for data year 1991
Computation of death rates
Death rates are computed by dividing the number of deaths in a group by the
population of a similarly defined group, and then scaling the rate per 100,000
estimated population. For marital status by race group the population base is
the entire United States, while for marital status by Hispanic origin group,
the population is the area covered by the reporting area of 48 States and the
District of Columbia (excluding New Hampshire and Oklahoma).
For death rates for marital status by Hispanic origin groups, deaths of
unknown origin may be proportionately allocated for rate computation. For
death rates for marital status by race and/or Hispanic origin, it is not
necessary for deaths for the "not stated age" group to be allocated since the
percentage of all deaths that have not stated age is very small.
Random variation for estimated death rates
Both the number of deaths and population total that define a death rate may be
subject to random variation. The number of deaths is considered as a Poisson
process subject to random variation, and the population total may be an
estimate based upon survey data subject to sampling error. Typically, the
reliability of an estimated death rate decreases as either or both the
numerator or denominator decrease. In addition, both mortality data and
survey data are subject to non-sampling errors.
Denominators for computing rates
The population estimates (furnished by the U.S. Bureau of the Census) used for
computing death rates by marital status represent the populations residing in
the specified areas for 1991 (1). The population estimates provided for
specified marital status groups and the specified Hispanic origin groups are
based on The Current Population Survey (CPS), and thus subject to sampling
error. The aggregated population estimates, all marital status groups
combined and all Hispanic origin groups combined, however, are considered
Census control totals and not based upon the CPS. These estimates are treated
as not having random variation.
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Relative Standard Errors, 95% Confidence Intervals and Rate Comparisons
Formulas for computing approximate relative standard errors (RSE) and
confidence intervals (CI) for crude and age-specific death rates are shown in
Table 1 and for age-adjusted death rates in Table 2. When presenting rates,
NCHS replaces the rate with an asterisk whenever the RSE is greater than or
equal to 0.23; these rates are considered to be statistically unreliable.
For testing the equality of two death rates, R1 and R2, the z-test may be used
(when both rates are based on 50 deaths or more) or the overlap of 95% CIs of
the rates may be used (when either or both of the rates are based on less than
50 deaths).
The z-test is determined as follows:
z = (R1-R2)/SQRT[R1^2 RSE(R1)^2 + R2^2 RSE(R2)^2]
to define a significance test statistic. If z is between -1.96 and 1.96 the
difference would be considered non-significant at the 0.05 level. If z is
less than or equal to -1.96 or greater than or equal 1.96, then the difference
would be considered statistically significant at the 0.05 level.
As a hypothetical example, if the death rate, R1, for never married Hispanics
is 38.7 (based on N=60 deaths and T=155,000 population) and R2 for married
Hispanics is 13.8 (based on N=180 deaths and T=1,300,000 population), then the
RSEs and z-test are computed using information from Table 1 as follows:
RSE(R1) = SQRT[1/60 + .67*(-.000017 + 4,786/155,000)] = 0.1932
RSE(R1) = SQRT[1/180 + .67*(-.000017 + 4,786/1,300,000] = 0.0895
and
z = 38.7 - 13.8/SQRT[(38.7)^2*(.1932)^2 + (13.8)^2*(.0895)^2] = 3.29
Since z is greater than 1.96, the two rates are statistically significantly
different from one another at the 0.05 level of significance.
If either of two rates is based on less than 50 deaths, then one may determine
if the 95% CI overlap as an indication of a statistically significant or
non-significant difference.
As a hypothetical example, if the death rate, R3, for divorced Mexicans is
66.7 (based on N=40 deaths and T=60,000 population) and R4 for divorced
non-Hispanic whites is 92.3 (based on N=600 deaths and T=650,000 population),
then the 95% CIs are computed using information from Tables 1 and A as
follows:
95%CI for R3
Lower = 66.7* .70266*(1-2.576*SQRT[.67*(-.000297 + 6,865/60,000)] = 13.5
Upper = 66.7*1.37991*(1+2.576*SQRT[.67*(-.000297 + 6,865/60,000)] = 157.6
95%CI for R4
RSE(R4) = SQRT[1/600 + .67*(-.000017 + 4,786/650,000)] = .0812
Lower = 92.3 - 1.96*92.3*.0812 = 77.6
Upper = 92.3 + 1.96*92.3*.0812 = 107.0
Since the CIs overlap, the difference between R3 and R4 is not statistically
significant.
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Reference
1. Bureau of the Census. Population estimates based on unpublished tabulations
prepared by the Housing and Household Economic Statistics Division, Bureau of
the Census.
Table 1. Formulas for computing approximate relative standard errors and 95%
confidence intervals for CRUDE AND AGE-SPECIFIC DEATH RATES for marital status
by race and Hispanic origin: 1991
Marital status, race, origin, and statistical measure
All marital status groups combined
Relative standard error [RSE(R)]: SQRT[1/N ]
95% Confidence Interval: 1-49 deaths
Lower: R*L(alpha =.95,N)
Upper: R*U(alpha =.95,N)
95% Confidence Interval: 50+ deaths
Lower: R - 1.96*S(R)
Upper: R + 1.96*S(R)
where
N = number of observed deaths or deaths after allocation/imputation of
unknown origin or unknown Hispanic
R = rate (deaths per 100,000 population)
S(R) = R*RSE(R)
L(alpha =.95,N) and U(alpha =.95,N) are shown in Table A
Formulas applicable to:
For all races, white, black, American Indian, Asian and Pacific Islander,
All origins, total Hispanic, total non-Hispanic, non-Hispanic white,
non-Hispanic black
Never married, married, widowed, divorced
Relative standard error [RSE(R)]: SQRT[1/N + 0.67*(a + (b/T))]
95% Confidence Interval: 1-49 deaths
Lower: R*L(alpha =.96,N)*(1-2.576*SQRT[0.67*(a + (b/T))])
Upper: R*U(alpha =.96,N)*(1+2.576*SQRT[0.67*(a + (b/T))])
95% Confidence Interval: 50+ deaths
Lower: R-1.96*S(R)
Upper: R+1.96*S(R)
where
N = number of observed deaths or deaths after allocation/imputation of
unknown origin or unknown Hispanic
R = rate (deaths per 100,000 population)
T = estimated population
S(R) = R*RSE(R)
L(alpha =.96,N) and U(alpha =.96,N) are shown in Table A
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Formulas applicable to:
All races, white, American Indian, All origins, total Hispanic, total
non-Hispanic, non-Hispanic white:
a = -.000017 and b = 4,786
Black, non-Hispanic black:
a = -.000204 and b = 6,865
Asian and Pacific Islander:
a = -.000719 and b = 6,865
All marital status groups combined and never married, married, widowed,
divorced
Mexican, Puerto Rican, Cuban, Other Hispanic:
a = -.000297 and b = 6,865
Table 2. Formulas for computing approximate relative standard errors and 95%
confidence intervals for AGE-ADJUSTED DEATH RATES for marital status by race
and Hispanic origin: 1991
Marital status, race, origin, and statistical measure
All marital status groups combined
Relative standard error [RSE(R")]: SQRT[SUM{wi^2 * Ri^2 *{1/Ni}] / R"
95% Confidence Interval: 1-49 deaths
Lower: R"*L(alpha =.95,Nnew)
Upper: R"*U(alpha =.95,Nnew)
95% Confidence Interval: 50+ deaths
Lower: R" - 1.96*S(R")
Upper: R" + 1.96*S(R")
where
R" = age-adjusted rate (per 100,000 population)
Ri = age-specific rate (per 100,000) for the ith age group
wi = ith age-specific Standard Population such that SUM(wi)=1.000000
Ni = number of observed deaths or deaths after allocation/imputation of
unknown origin or unknown Hispanic for the ith age group
Nnew = 1/[RSE(R")]^2 rounded to nearest integer
S(R") = R"*RSE(R")
L(alpha =.95,Nnew) and U(alpha =.95,Nnew) are shown in Table A
Formulas applicable to:
All races, white, black, American Indian, Asian and Pacific Islander,
All origins, total Hispanic, total non-Hispanic, non-Hispanic white,
non-Hispanic black
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Never married, married, widowed, divorced
Relative standard error [RSE(R")]:
SQRT[SUM{wi^2 * Ri^2 * {(1/Ni)+ 0.67*(a + (b/Ti)}}] /R"
95% Confidence Interval: 1-49 deaths
Lower: R"*L(alpha =.96,Nnew)*(1-2.576*RSE(Tnew))
Upper: R"*U(alpha =.96,Nnew)*(1+2.576*RSE(Tnew))
95% Confidence Interval: 50+ deaths
Lower: R"-1.96*S(R")
Upper: R"+1.96*S(R")
where
R"= age-adjusted rate (per 100,000 population)
Ri = age-specific rate (per 100,000)for ith age group
wi = ith age-specific Standard Population
such that SUM(wi)=1.000000
Ti = estimated population for the ith age group
Tnew = SUM(wi*Ti)
RSE(Tnew) = SQRT[SUM{wi^2*Ti^2*0.67*(a + b/Ti)}] /Tnew
Ni = number of observed deaths or deaths after
allocation/imputation of unknown origin or unknown
Hispanic for the ith age group
Nnew = smaller of SUM(Ni)or 1/[{RSE(R")}^2 - {RSE(Tnew)}^2]
If negative, set to SUM(Ni)
S(R") = R"*RSE(R")
L(alpha =.96,Nnew) and U(alpha =.96,Nnew) are shown in Table A
Formulas applicable to:
All races, white, American Indian, All origins, total Hispanic, total
non-Hispanic, non-Hispanic white:
a = -.000017 and b = 4,786
Black, non-Hispanic black:
a = -.000204 and b = 6,865
Asian and Pacific Islander:
a = -.000719 and b = 6,865
All marital status groups combined and never married, married, widowed,
divorced
Mexican, Puerto Rican, Cuban, Other Hispanic:
a = -.000297 and b = 6,865
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Table A. Lower and upper 95% and 96% confidence limit factors for a death rate
based on a Poisson variable of 1 through 49 deaths, N or Nnew
L(alpha=.96,N)or U(alpha=.96,N) or
N or Nnew L(alpha=.95,N) U(alpha=.95,N) L(alpha=.96,Nnew) U(alpha=.96,Nnew)
1 0.02532 5.57164 0.02020 5.83392
2 0.12110 3.61234 0.10735 3.75830
3 0.20622 2.92242 0.18907 3.02804
4 0.27247 2.56040 0.25406 2.64510
5 0.32470 2.33367 0.30591 2.40540
6 0.36698 2.17658 0.34819 2.23940
7 0.40205 2.06038 0.38344 2.11666
8 0.43173 1.97040 0.41339 2.02164
9 0.45726 1.89831 0.43923 1.94553
10 0.47954 1.83904 0.46183 1.88297
11 0.49920 1.78928 0.48182 1.83047
12 0.51671 1.74680 0.49966 1.78566
13 0.53246 1.71003 0.51571 1.74688
14 0.54671 1.67783 0.53027 1.71292
15 0.55969 1.64935 0.54354 1.68289
16 0.57159 1.62394 0.55571 1.65610
17 0.58254 1.60110 0.56692 1.63203
18 0.59266 1.58043 0.57730 1.61024
19 0.60207 1.56162 0.58695 1.59042
20 0.61083 1.54442 0.59594 1.57230
21 0.61902 1.52861 0.60435 1.55563
22 0.62669 1.51401 0.61224 1.54026
23 0.63391 1.50049 0.61966 1.52602
24 0.64072 1.48792 0.62666 1.51278
25 0.64715 1.47620 0.63328 1.50043
26 0.65323 1.46523 0.63954 1.48888
27 0.65901 1.45495 0.64549 1.47805
28 0.66449 1.44528 0.65114 1.46787
29 0.66972 1.43617 0.65652 1.45827
30 0.67470 1.42756 0.66166 1.44922
31 0.67945 1.41942 0.66656 1.44064
32 0.68400 1.41170 0.67125 1.43252
33 0.68835 1.40437 0.67575 1.42480
34 0.69253 1.39740 0.68005 1.41746
35 0.69654 1.39076 0.68419 1.41047
36 0.70039 1.38442 0.68817 1.40380
37 0.70409 1.37837 0.69199 1.39743
38 0.70766 1.37258 0.69568 1.39134
39 0.71110 1.36703 0.69923 1.38550
40 0.71441 1.36172 0.70266 1.37991
41 0.71762 1.35661 0.70597 1.37454
42 0.72071 1.35171 0.70917 1.36938
43 0.72370 1.34699 0.71227 1.36442
44 0.72660 1.34245 0.71526 1.35964
45 0.72941 1.33808 0.71816 1.35504
46 0.73213 1.33386 0.72098 1.35060
47 0.73476 1.32979 0.72370 1.34632
48 0.73732 1.32585 0.72635 1.34218
49 0.73981 1.32205 0.72892 1.33818
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