Chapter 7 Exercise (Key Odd)

7.1 Adjustment using the formula aR_direct = ∑ w_i�r_i where w_i = N_i / N

Rates are per 100,000 person-years.

Age	California Rates ri	Weights wi	Products wiri
0 - 4	207	0.076158	15.80
5 -24	65	0.286501	18.62
25 - 44	182	0.325971	59.32
45 - 64	720	0.185402	133.58
65 - 74	2418	0.072494	175.26
75+	7768	0.053474	415.41

aR_direct = ∑w_ir_i = 15.80 + 18.62 + 59.32 + 133.58 + 175.26 + 415.41 = 818

Comment 1: The crude mortality rate in California is lower than in Florida (708 vs. 1026). However, the population in California is substantially younger, e.g., 4% of California is over the age of 75 compared to 8% for Florida. After direct age-adjustment, the rate in California is 818, compared to 784 for Florida.

Comment 2: All adjustments should be viewed as potentially incomplete, i.e., showing incomplete control for confounding. For the current problem, for example, the broad age categories leave room for residual confounding.

7.3. The text includes an answer on p. 153, showing intermediate calculations in Table 7.17. To understand these calculations, recognize that the last column in table 7.17 (labeled "Product") represents the expected number of deaths within each age group. Sum the expected number of deaths within the age groups to derive the expected number of deaths in the study population. The SMR (“observed to expected ratio”) is 2.15. You may stop at this point because all the information is captured in the SMR.

7.5. Mortality in Latkaland

(A) Crude mortality rate, Latkaland = 300 / 55,163 = .005438 = 544 per 100,000

(B) Expected number of deaths in each age group in Latkaland: 

�₁ = .00229 � 7,909 = 18.11   

�₂ = .00062 � 24,560 = 15.23   

�₃ = .00180 � 13,764 = 24.78   

�₄ = .00789 � 6,921 = 54.61   

�₅ = .02618 � 1,485 = 38.88   

�₆ = .08046 � 524 = 42.16 
(C) Expected deaths in Latkaland (all ages combined) � = 18.11 + 15.23 + . . . + 42.16 = 193.77
(D) SMR = observed / expected =  300 / 193.77 = 1.55. 

Interpret this statistic. Mortality in Latkaland is 1.55 times that of the U. S. (after adjusting for age).

7.7 Ahlbom & Norell, 1990, p. 44, #11.

Observed = 8

Expected in stratum 1 = �₁ = (651 � 0/337000) = 0

Expected in stratum 2 = �₂ = (518 � 6/431000) = 0.007211

Expected in stratum 3 = �₃ = (500 � 90/522000) = 0.08621

Expected in stratum 4 = �₄ = (465 � 381/507000) = 0.34944

Expected in stratum 5 = �₅ = (211 � 626/367000) = 0.35991

Expected in all strata = ∑�_i = 0.8028

SMR = 8 / 0.8028 = 9.97. The observed rate is ~10 time the expected rate.

7.9 Ahlbom & Norell, 1990, p. 45, #12.

	Age-specific rates
Age	Population A	Population B
30 - 39	5 / 1000 = .0050	25 / 5000 = .0050
40 - 49	20 / 2000 = .0100	40 / 3000 = .0133
50 - 59	50 / 4000 = .0125	20 / 1000 = .0200
60 - 69	50 / 3000 = .0167	20 / 1000 = .0200

aR_{Population A} = (0.25)(.005) + (0.25)(0.010) + (0.25)(.0125) + (0.25)(0.01667) = .01104 = .011

aR_{Population B} = (0.25)(.005) + (0.25)(0.0133) + (0.25)(.02) + (0.25)(0.02) = .01459 = .015

After accounting for age, the rate in population B is a little higher than the rate in population A.