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. - 1 - 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. - 2 - 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 - 3 - 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 - 4 - 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 - 5 - 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 - 6 -