solution to problem set2 - Eco 572 Research methods in...

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Eco 572: Research methods in Demography Solutions to Problem Set 2 [1] U.S. Mortality (a) Calculate age-specific period mortality rates for 1992 for blacks and whites . insheet using (5 vars, 19 obs) . gen mw = deathsw/expow . gen mb = deathsb/expob . list age mw mb +--------------------------+ | age mw mb | |--------------------------| 1. | 0 .007809 .019579 | 2. | 1 .000426 .000776 | 3. | 5 .000213 .000375 | 4. | 10 .000282 .000449 | 5. | 15 .00106 .002184 | |--------------------------| 6. | 20 .001354 .00321 | 7. | 25 .001533 .003617 | 8. | 30 .001958 .004644 | 9. | 35 .002455 .006096 | 10. | 40 .003122 .008032 | |--------------------------| 11. | 45 .004325 .010657 | 12. | 50 .006634 .014193 | 13. | 55 .010715 .021036 | 14. | 60 .017297 .029243 | 15. | 65 .026885 .040291 | |--------------------------| 16. | 70 .040124 .0572491 | 17. | 75 .061488 .07502 | 18. | 80 .097005 .1096976 | 19. | 85 .179562 .1671715 | +--------------------------+ (b) Plot the rates against age on a log scale. Exclude the last group, which is actually 85+. . gen agem = (age+age[_n+1])/2 // excludes 85+ (1 missing value generated) . line mw mb agem, title(U.S. Males 1992) /// > yscale(log) ytitle(asmr) xtitle(age) /// > legend(order(1 "White" 2 "Black") ring(0) pos(5) cols(1)) (1 of 14) [2/11/2008 2:16:28 PM]
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Eco 572: Research methods in Demography . graph export ps2fig1.png, replace (file ps2fig1.png written in PNG format) (c) Fit a Gompertz model over an appropriate age range. Seems clear that black mortality rates are Gompertz over a wider range than those for whites, which seeem to show a change of slope around age 50. To avoid biases I decided to start at 40-44. To simplify interpretation of the intercept I subtract 40 from age. (You could plot the fits but this was not required.) . gen lmw=log(mw) . gen lmb=log(mb) . gen lb40 = agem-40 (1 missing value generated) . reg lmw lb40 if age >= 40 Source | SS df MS Number of obs = 9 -------------+------------------------------ F( 1, 7) = 6841.82 Model | 11.4650463 1 11.4650463 Prob > F = 0.0000 Residual | .011730118 7 .001675731 R-squared = 0.9990 -------------+------------------------------ Adj R-squared = 0.9988 Total | 11.4767764 8 1.43459705 Root MSE = .04094 (2 of 14) [2/11/2008 2:16:28 PM]
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Eco 572: Research methods in Demography ------------------------------------------------------------------------------ lmw | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lb40 | .0874263 .001057 82.72 0.000 .084927 .0899256 _cons | -6.053246 .0274181 -220.78 0.000 -6.118079 -5.988412 ------------------------------------------------------------------------------ . di exp(_b[_cons]) ", " exp(_b[lb40]) .00235022, 1.0913619 . reg lmb lb40 if age >= 40 Source | SS df MS Number of obs = 9 -------------+------------------------------ F( 1, 7) = 6962.81 Model | 6.50170578 1 6.50170578 Prob > F = 0.0000 Residual | .006536431 7 .000933776 R-squared = 0.9990 -------------+------------------------------ Adj R-squared = 0.9989 Total | 6.50824221 8 .813530276 Root MSE = .03056 ------------------------------------------------------------------------------ lmb | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lb40 | .0658367 .000789 83.44 0.000 .063971 .0677024 _cons | -5.02427 .0204671 -245.48 0.000 -5.072667 -4.975873 ------------------------------------------------------------------------------ . di exp(_b[_cons]) ", " exp(_b[lb40]) .00657639, 1.0680523 (d) Interpret the resulting parameters in terms of the underlying hazard rate and the rate of aging. The hazard for blacks is almost triple that of whites at age 40 (.00658 versus .00235, from exponentiating the intercepts), but blacks "age" more slowly as the hazard increases only 6.8% per year, compared to 9.1% for whites (from exponentiating the slopes). (e) Could one include the 85+ group? What assumption(s) might be needed? Yes, one could, but assumptions are required. We attributed other hazards to the midpoint of the age group. The question is what to do with 85+. (1) Use 87.5, as if the group was 85-90, or maybe 90. (2) Assume constant hazard after 85. Then expectation of life at 85 is the reciprocal of the death rate, and adding 85 gives us midpoints of 90.6 for whites and 91 for blacks. (3) Assume Gompertz hazards. Use the model fitted to ages below 85 to see what age would correspond to the observed mortality rate. This gives 102 for whites and 95.3 for blacks. Given the uncertainty, the choice of omitting 85+ seems prudent. (Also, there is some evidence that after age 80 the hazard increases more slowly than in a Gompertz model, but that's a different issue.)
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