1. Direct age-adjustment, fictitious State

All rates are �per 100,000�

(A) ...crude mortality rate in fictitious State. 15,984 / 2,450,000 � 100,000 = 652 
(B) age-specific rates in fictitious State: 

Age	Rate (per 100,000)
0 - 4	238
5 -24	64
25 - 44	208
45 - 64	809
65 - 74	2221
75+	6887

Compare these to Florida's�.The age-specific rates are identical. 

(C) Explain... Florida's population is much older. For example, 8% of Florida's residents are over 75, compared to 4% of this State's. 

(D)� adjust the fictitious state's death rate using the direct method

Method A formula aR_direct = ∑(N_i � r_i) / ∑N_i

Age	Fictitious state	Standard Million (Ni)	Product  (Niri)
0 - 4	238	76158	18125604
5 -24	64	286501	18336064
25 - 44	208	325971	67801968
45 - 64	809	185402	149990218
65 - 74	2221	72494	161009174
75+	6887	53474	368275438
SUMS		1,000,000	783,538,466

aR_direct = Sum(N_ir_i) / Sum(N_i) = 783538466 / 1000000 = 784

Method B formula aR_direct = ∑ w_i�r_i where w_i = N_i / N

Age	Fictitious state	Standard Million (wi)	Product  (wiri)
0 - 4	238	0.076158	18.125604
5 -24	64	0.286501	18.336064
25 - 44	208	0.325971	67.801968
45 - 64	809	0.185402	149.990218
65 - 74	2221	0.072494	161.009174
75+	6887	0.053474	368.275438
		1.0000000	784

aR_direct = Sum(w_ir_i) = sum or last column = 784

How do the adjusted rates compare? The adjusted rate in the fictitious state and Florida are the same.

2. 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).

3. Ahlbom & Norell, 1990, p. 44, #10

� Use the formula �_i = n_i � R_i to calculate the expected number of cases within stratum i.
�₁ = (8000 � 0.5/1000) = 4
�₂ = (2000 � 4/1000) = 8
�₃ = (2000 � 9/1000) = 18

� Expected frequency (�) = �₁ + �₂ + �₃ = 4 + 8 + 18 = 30

� The observed number of cases was 40. Thus, SMR = observed / expected = 40 / 30 = 1.33

� Interpretation: the incidence was 33% higher than expected.

4. 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.

5. Ahlbom & Norell, 1990

	Age-specific rates
Age	Group A	Group B
Younger	4 / 2000 = 0.002	20 / 4000 = 0.005
Older	32 / 4000 = 0.008	15 / 1000 = 0.015

aR_{Group A} = (0.5)(0.002) + (0.5)(0.008) = 0.005

aR_{Group B} = (0.5)(0.005) + (0.5)(0.015) = 0.010

 The adjusted rate in Group B is half that of Group A.

6.  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.