Samuel Clark
Department of Sociology, University of WashingtonInstitute of Behavioral Science, University of Colorado at Boulder
Agincourt Health and Population Unit, University of the Witwatersrand
Age-Standardization & Decomposition
2
Period Age-Specific Death Rate
Death Rate for ages x to x+n during the period spanning 0 to T:
0,0,
0,n x
n xn x
D TM T
N T
M is the death rateD is the number of deathsN is the population
3
Lexis Diagram
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30 35 40
Time (years)
Ag
e (y
ears
)
C1 C2 C3
4
Components of the Crude Death Rate
Dropping the period notation:
00
0 0
n xn xn x
xx n x n x n xn x n x
x xn x
DND
N D NDCDR M C
N N N N N
nCx is the proportion of the population between ages x and x+n
0
; 1.0n xn x n x
x
NC C
N
5
Standardization
CDR is a function of the mortality schedule AND the age distribution
Changes in either or both affect the level of the CDR When comparing CDRs, it is important to isolate the
source of the differences:– Differences in age-specific mortality rates?– Differences in age distributions?
Age standardization holds the age structure constant so that the only source of difference is the mortality schedule
Same applies to any division of the population that produces differing rates (or proportions)
6
Age-Standardized CDR = ASCDR
i ii
CDR M C Replacing the n,x notation with i:
; 1.0s si i i
i i
ASCDR M C C The Age-Standardized Crude Death Rate is:
Where Cs is a standard age distribution
7
Selection of a Standard
There is no “correct” way to choose a standard As the covariance between the standard and the
schedule increases, so will the value of the standardized rate
The average of the proportionate distributions being compared is a good choice in general:
1
Nxi
xi
CC
N
Where there are N distributions indexed over x
8
Age Standardization: CDR
We want to compare the crude death rate from two populations
P1 has lower child and higher old-age mortality P2 has higher child and lower old-age mortality P1’s age distribution is almost constant, comparatively
unloaded on young ages and loaded on old ages P2’s age distribution is loaded on younger ages and
unloaded on older ages
9
Example Mortality Schedules
Mortality Rate
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0-4
5-9
10-1
4
15-1
9
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
9
80-8
485
+
Age
Mo
rtal
ity
Rat
e
P1 P2
10
Example Age Distributions
Proportionate Age Distribution
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0-4
5-9
10-1
4
15-1
9
20-2
4
25-2
9
30-3
4
35-3
9
40-4
4
45-4
9
50-5
4
55-5
9
60-6
4
65-6
9
70-7
4
75-7
9
80-8
485
+
Age
Pro
po
rtio
n o
f P
op
ula
tio
n
P1 P2 Average
11
Calculation of Standardized CDRs
AverageAge MR1 MR2 C1 C2 AC MR1*C1 MR2*C2 MR1*C2 MR2*C1 MR1*AC MR2*AC
0-4 0.027 0.079 0.060 0.114 0.087 0.0016 0.0090 0.0031 0.0048 0.0024 0.00695-9 0.011 0.030 0.060 0.103 0.081 0.0007 0.0031 0.0012 0.0018 0.0009 0.0025
10-14 0.003 0.005 0.059 0.093 0.076 0.0002 0.0005 0.0003 0.0003 0.0002 0.000415-19 0.003 0.005 0.059 0.084 0.072 0.0002 0.0004 0.0003 0.0003 0.0002 0.000420-24 0.005 0.006 0.058 0.076 0.067 0.0003 0.0004 0.0004 0.0003 0.0003 0.000425-29 0.008 0.007 0.057 0.069 0.063 0.0005 0.0005 0.0006 0.0004 0.0005 0.000430-34 0.012 0.008 0.057 0.063 0.060 0.0007 0.0005 0.0007 0.0005 0.0007 0.000535-39 0.015 0.010 0.056 0.057 0.056 0.0008 0.0006 0.0008 0.0006 0.0008 0.000640-44 0.018 0.012 0.056 0.051 0.053 0.0010 0.0006 0.0009 0.0007 0.0009 0.000745-49 0.020 0.016 0.055 0.046 0.051 0.0011 0.0007 0.0009 0.0009 0.0010 0.000850-54 0.021 0.020 0.055 0.042 0.048 0.0012 0.0008 0.0009 0.0011 0.0010 0.001055-59 0.030 0.026 0.054 0.038 0.046 0.0016 0.0010 0.0011 0.0014 0.0014 0.001260-64 0.041 0.038 0.054 0.034 0.044 0.0022 0.0013 0.0014 0.0020 0.0018 0.001765-69 0.067 0.058 0.053 0.031 0.042 0.0036 0.0018 0.0021 0.0031 0.0028 0.002470-74 0.097 0.084 0.053 0.028 0.040 0.0051 0.0024 0.0027 0.0044 0.0039 0.003475-79 0.159 0.130 0.052 0.025 0.039 0.0083 0.0033 0.0040 0.0068 0.0062 0.005080-84 0.271 0.194 0.051 0.023 0.037 0.0139 0.0045 0.0062 0.0100 0.0101 0.007285+ 0.436 0.276 0.051 0.021 0.036 0.0222 0.0058 0.0091 0.0141 0.0156 0.0099
Sum 1.000 1.000 0.0650 0.0372 0.0367 0.0533 0.0509 0.0452per 1,000 CDR = 65.05 CDR = 37.18 SCDR = 36.72 SCDR = 53.3 SCDR = 50.88 SCDR = 45.24
Age Standardized CDRMortality Rates Age Distributions Crude Death Rates Cross Standardized CDR
12
Comparison of CDRs
Crude Death Rate
P1 P2
P1 65.05 53.30
P2 36.72 37.18
Av
e. P
1-P
2
50.88 45.24
Mortality Rate
Ag
e D
istr
ibu
tio
n
13
Standardization: Income
We want to compare male and female average income distributions for the working population
The proportionate measure is the job category-specific average income, AIj, for the period 0 to T:
0,0,
0,j
jj
I TAI T
N T
14
Job Category Standardized Average Income
As with the CDR, AIj can be written as the product of two components: the job category-specific average income and the proportion of the population holding jobs of each category:
; 1.0s sj j j
j j
J CSAI AI C C
15
An Employment Distribution Effect - Chart
Same Average Income per Category: Different Employment Distributions
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Daily Wage Driver Professional Executive
Job Category
Pro
po
rtio
n o
f W
ork
ing
Peo
ple
R 0
R 10,000
R 20,000
R 30,000
R 40,000
R 50,000
R 60,000
Ave
rag
e In
com
e
Male Job Distribution Female Job Distribution Male Income Female Income
16
An Employment Distribution Effect
Job Category Men Women Men Women Average
Daily Wage R 1,000.00 R 1,000.00 0.40 0.84 0.62Driver R 2,500.00 R 2,500.00 0.30 0.04 0.17Professional R 6,000.00 R 6,000.00 0.20 0.10 0.15Executive R 50,000.00 R 50,000.00 0.10 0.02 0.06
AI: R 7,350.00 R 2,540.00 1.00 1.00 1.00JCSAI: R 4,945.00 R 4,945.00
Difference (M-F): R 4,810.00JCS Difference (M-F): R 0.00
Same Average Income per Category: Different Employment Distributions
Average Income Employment Distributions
17
Male-Biased Average Income per Category: Same Employment Distributions
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Daily Wage Driver Professional Executive
Job Category
Pro
po
rtio
n o
f W
ork
ing
Peo
ple
R 0
R 10,000
R 20,000
R 30,000
R 40,000
R 50,000
R 60,000
Ave
rag
e In
com
e
Male Job Distribution Female Job Distribution Male Income Female Income
An Average Income Distribution Effect - Chart
18
An Average Income Distribution Effect
Job Category Men Women Men Women Average
Daily Wage R 1,000.00 R 900.00 0.62 0.62 0.62Driver R 2,500.00 R 2,000.00 0.17 0.17 0.17Professional R 6,000.00 R 3,500.00 0.15 0.15 0.15Executive R 50,000.00 R 20,000.00 0.06 0.06 0.06
AI: R 4,945.00 R 2,623.00 1.00 1.00 1.00JCSAI: R 4,945.00 R 2,623.00
Difference (M-F): R 2,322.00JCS Difference (M-F): R 2,322.00
Male-Biased Average Income per Category: Same Employment Distributions
Average Income Employment Distributions
19
A Joint Effect - Chart
Female-Biased Average Income per Category: Different Employment Distributions
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Daily Wage Driver Professional Executive
Job Category
Pro
po
rtio
n o
f W
ork
ing
Peo
ple
R 0
R 10,000
R 20,000
R 30,000
R 40,000
R 50,000
R 60,000
R 70,000
R 80,000
Ave
rag
e In
com
e
Male Job Distribution Female Job Distribution Male Income Female Income
20
A Joint Effect
Job Category Men Women Men Women Average
Daily Wage R 1,000.00 R 1,200.00 0.40 0.84 0.62Driver R 2,500.00 R 3,000.00 0.30 0.04 0.17Professional R 6,000.00 R 8,000.00 0.20 0.10 0.15Executive R 50,000.00 R 75,000.00 0.10 0.02 0.06
AI: R 7,350.00 R 3,428.00 1.00 1.00 1.00JCSAI: R 4,945.00 R 6,954.00
Difference (M-F): R 3,922.00JCS Difference (M-F): -R 2,009.00
Female-Biased Average Income per Category: Different Employment Distributions
Average Income Employment Distributions
21
Decomposition
Decomposition refers to a technique that identifies the proportion of the difference between two crude death rates that results from the differences in the mortality schedules and the differences in the age distributions
As with the standardization technique described earlier, this is a general technique that can be used with any crude proportion formed as the sum of proportionate distribution and a proportional measure
22
Components of Difference in Crude Rates
1 2 1 21 2 1 2 1 2
2 2i i i i
i i i ii i
R R C CCR CR C C R R
ΔCR composition ave. rate weight rate ave. composition weight
ΔCR CC RC
23
Derivation of Decomposition
1 2
1 1 2 2
1 1 1 1 2 2 2 2
1 1 1 1 2 2 2 2 1 2 1 2
2 2 2 2
2 2 2 2 2 2
i i i ii i
i i i i i i i ii i i i
i i i i i i i i i i i ii i i i i i
CR CR
C R C R
C R C R C R C R
C R C R C R C R C R C R
2 1 2 1
1 2 1 2 1 2 1 21 2 1 2
1 2 1 21 2 1 2
2 2
2 2 2 2
2 2
i i i ii i
i i i i i i i ii i i i
i i i i
i i i ii i i i
i i
C R C R
R R R R C C C CC C R R
R R C CC C R R
24
Composition & Rate Contributions to Difference
CCComposition Contribution to Difference
ΔCR
RCRate Contribution to Difference
ΔCR
25
Decomposition Example: CDR
Average CC1 CC2 RC1 RC2Age MR1 MR2 C1 C2 AC MR1*C1 MR2*C2 C1-C2 0.5*(MR2+MR1) CC1*CC2 MR1-MR2 0.5*(C2+C1) RC1*RC2
0-4 0.027 0.079 0.060 0.114 0.087 0.0016 0.0090 -0.054 0.053 -0.003 -0.052 0.087 -0.0055-9 0.011 0.030 0.060 0.103 0.081 0.0007 0.0031 -0.043 0.021 -0.001 -0.019 0.081 -0.002
10-14 0.003 0.005 0.059 0.093 0.076 0.0002 0.0005 -0.034 0.004 0.000 -0.003 0.076 0.00015-19 0.003 0.005 0.059 0.084 0.072 0.0002 0.0004 -0.026 0.004 0.000 -0.002 0.072 0.00020-24 0.005 0.006 0.058 0.076 0.067 0.0003 0.0004 -0.018 0.005 0.000 -0.001 0.067 0.00025-29 0.008 0.007 0.057 0.069 0.063 0.0005 0.0005 -0.012 0.008 0.000 0.002 0.063 0.00030-34 0.012 0.008 0.057 0.063 0.060 0.0007 0.0005 -0.006 0.010 0.000 0.004 0.060 0.00035-39 0.015 0.010 0.056 0.057 0.056 0.0008 0.0006 0.000 0.012 0.000 0.005 0.056 0.00040-44 0.018 0.012 0.056 0.051 0.053 0.0010 0.0006 0.005 0.015 0.000 0.005 0.053 0.00045-49 0.020 0.016 0.055 0.046 0.051 0.0011 0.0007 0.009 0.018 0.000 0.004 0.051 0.00050-54 0.021 0.020 0.055 0.042 0.048 0.0012 0.0008 0.013 0.021 0.000 0.001 0.048 0.00055-59 0.030 0.026 0.054 0.038 0.046 0.0016 0.0010 0.016 0.028 0.000 0.005 0.046 0.00060-64 0.041 0.038 0.054 0.034 0.044 0.0022 0.0013 0.019 0.039 0.001 0.003 0.044 0.00065-69 0.067 0.058 0.053 0.031 0.042 0.0036 0.0018 0.022 0.063 0.001 0.010 0.042 0.00070-74 0.097 0.084 0.053 0.028 0.040 0.0051 0.0024 0.024 0.091 0.002 0.013 0.040 0.00175-79 0.159 0.130 0.052 0.025 0.039 0.0083 0.0033 0.027 0.145 0.004 0.029 0.039 0.00180-84 0.271 0.194 0.051 0.023 0.037 0.0139 0.0045 0.028 0.232 0.007 0.077 0.037 0.00385+ 0.436 0.276 0.051 0.021 0.036 0.0222 0.0058 0.030 0.356 0.011 0.160 0.036 0.006
Sum 1.000 1.000 0.0650 0.0372 Composition Component = 0.0222 Rate Component = 0.0056CDR = 65.05 CDR = 37.18 79.7% 20.3%
Decomposition of Difference in Crude Death RatesMortality Rates Age Distributions Crude Death Rates
26
Check Decomposition
Population CDR
A 65.05B 37.18
A-B 27.87
CC 22.22RC 5.65
CC+RC 27.87
%CC 79.74%%MC 20.26%
%CC+%MC 100.00%
