Our primary analytic approach for describing socioeconomic gradients by areabased socioeconomic measures has been to use geocodes to append areabased socioeconomic data to case records, to stratify these records into discrete categories based on ABSM, and to aggregate numerators and denominators over areas, within levels defined by ABSM. This method avoids the problem of unstable rates arising from small areas by assuming that cases and population denominators from areas with similar socioeconomic characteristics can be legitimately combined into the same strata. An alternative approach, which preserves the spatial information of the geocodes, is discussed in the section on multilevel analyses.
The following steps are used to generate agestandardized disease rates stratified by areabased socioeconomic measures, once the case data have been geocoded and appropriate ABSMs have been generated from census data.
• Aggregate the case data into numerators (age cells within areas/geocodes).
• Aggregate population denominator data into age cells within areas/geocodes.
• Merge the numerators and denominators with ABSMs, by area/geocode.
• Aggregate over areas into strata defined by categorical ABSM and age category.
• Generate agestandardized rates and other summary measures.
[See the ‘Tools” section for a step by step comparison of the analytic methods, the relevant task of the Case Example, and sample SAS code.]
Aggregating Numerator Data
Data from public health databases are typically formatted such that each record represents one person (or case report). Once these data have been geocoded, they need to be aggregated before linking to denominator and ABSM data. Before aggregating, however, one should exclude all records that are not geocoded, do not meet the case definition, or are missing data on the important covariates (e.g. age, in the case of simple agestandardized analyses; age, sex, and race/ethnicity in the case of more complex stratified analyses).
One can think of the basic unit of aggregation as a cell, defined by age and other covariates, within an area/geocode. Once aggregated, this cell within an area can be linked to a relevant population denominator. The cell contains a count of all cases within that area that meet the specified age and other covariate criteria. Since our goal is eventually to create rates, we call this count of cases the “numerator.”
Example:
We intend to agestandardize in 5 broad age categories, 014, 1524, 2544, 4564, 65+. Therefore, we need to aggregate the records in each census tract into cells defined by the corresponding ages. As an example, consider the following 23 records from census tracts 25009250500 and 25009250800.
Before aggregating:  
Record #  Geocode  Age of death 
1  25009250500  <1 
2  25009250500  <1 
3  25009250500  <1 
4  25009250500  17 
5  25009250500  19 
6  25009250500  27 
7  25009250500  38 
8  25009250500  40 
9  25009250500  40 
10  25009250500  44 
11  25009250800  <1 
12  25009250800  <1 
13  25009250800  5 
14  25009250800  22 
15  25009250800  24 
16  25009250800  26 
17  25009250800  31 
18  25009250800  36 
19  25009250800  36 
20  25009250800  40 
21  25009250800  43 
22  25009250800  43 
23  25009250800  43 
After aggregating:  
Geocode  Age category  Number of deaths (numerator) 
25009250500  014  3 
25009250500  1524  2 
25009250500  2544  5 
25009250800  014  3 
25009250800  1524  2 
25009250800  2544  8 
Aggregating Denominator Data
Denominator data at the census tract level typically come from the decennial census. In 1990, the US Census reported population counts by age in 31 categories (<1, 12, 34, 5, 6, 79, 1011, 1213, 14, 15, 16, 17, 18, 19, 20, 21, 2224, 2529, 3034, 3539, 4044, 4549, 5054, 5559, 6061, 6264, 6569, 7074, 7579, 8084, 85+). In the 1990 US Census STF3, age specific population counts were reported in table P013. Variable P0130001 gave the count of residents <1 year old, P0130002 gave the count of residents 12 years old, etc.
For the purposes of age standardization, these age categories need to be reaggregated to match the age categories used for categorizing case data (numerators, above) and the age categories from the standard million reference population. Additionally, when using case data from multiple years, in order to calculate an average annual incidence rate, one needs to use a persontime denominator (population count multiplied by number of years of case data). For example, in the case of the Massachusetts allcause mortality data, we have three years worth of cases (19891991). Therefore, we multiply the population count in each age category by 3.
Example:
For census tract 25009250800 in 1990, we wish to age standardize using the same five broad age categories as in the numerator example above (014, 1524, 2544, 4564, 65+):
Before:  
Census variable  Ages (years)  Population count 
P0130001  <1  115 
P0130002  12  243 
P0130003  34  197 
P0130004  5  92 
P0130005  6  59 
P0130006  79  237 
P0130007  1011  160 
P0130008  1213  141 
P0130009  14  77 
P0130010  15  62 
P0130011  16  54 
P0130012  17  94 
P0130013  18  65 
P0130014  19  89 
P0130015  20  101 
P0130016  21  128 
P0130017  2224  387 
P0130018  2529  571 
P0130019  3034  746 
P0130020  3539  422 
P0130021  4044  354 
P0130022  4549  317 
P0130023  5054  176 
P0130024  5559  174 
P0130025  6061  65 
P0130026  6264  214 
P0130027  6569  158 
P0130028  7074  316 
P0130029  7579  178 
P0130030  8084  112 
P0130031  85+  69 
In order to collapse these variables into the five broad age categories, we have to sum up census variables as follows:
After:  
Age category  Population count  Persontime denominator (x 3 years of case data) 

P0130001  <1  1321  115 
P0130002  12  980  243 
P0130003  34  2093  197 
P0130004  5  946  92 
P0130005  6  833  59 
Merging numerators with denominators and ABSM
Once the numerators and denominators have the same structure (AREAKEY x AGECAT), they can be merged together, along with the ABSM data (by AREAKEY). For age cells within areas where no cases were reported, we set the numerator to zero.
Example:
Before merging with ABSM:  
Numerator dataset:  
Geocode/Areakey  Age category  Number of deaths (numerator) 
25009250500  014  3 
25009250500  1524  2 
25009250500  2544  5 
25009250500  4564  7 
25009250500  65+  26 
25009250800  014  4 
25009250800  1524  3 
25009250800  2544  8 
25009250800  4564  13 
25009250800  65+  132 
Denominator dataset:  
Geocode/Areakey  Age category  Persontime denominator (x 3 years of case data) 
25009250500  014  4152 
25009250500  1524  1953 
25009250500  2544  3489 
25009250500  4564  1233 
25009250500  65+  1212 
25009250800  014  3963 
25009250800  1524  2940 
25009250800  2544  6279 
25009250800  4564  2838 
25009250800  65+  2499 
After merging with ABSM:  
Geocode  Age category  Poverty  Numerator  Denominator 
25009250500  1  4  3  4152 
25009250500  2  4  2  1953 
25009250500  3  4  5  3489 
25009250500  4  4  7  1233 
25009250500  5  4  26  1212 
25009250800  1  3  4  3963 
25009250800  2  3  3  2940 
25009250800  3  3  8  6279 
25009250800  4  3  13  2838 
25009250800  5  3  132  2499 
Aggregating OVER areas in ABSM strata
Next, in order to generate rates for categories of a specific ABSM, it is necessary to aggregate OVER areas into strata defined by AGECAT and ABSM. Numerators and denominators from census tracts with missing ABSM data for a particular ABSM are typically excluded from that analysis.
Example: In Suffolk County, Massachusetts, there are a total of 189 census tracts. We wish to examine all cause mortality rates by poverty, with poverty categorized into 4 strata (04.9%, 59.9%, 1019.9%, and 20100%).
ABSM: CT Poverty  Number of census tracts 
0.04.9%  10 
5.09.9%  37 
10.019.9%  56 
20.0100.0%  83 
Missing poverty data  3 
Thus, to obtain the mortality rates in the least impoverished stratum (0.04.9% below poverty), we need to aggregate the cases and the population at risk OVER the ten census tracts in that stratum (preserving the age structure WITHIN each poverty stratum so that we can age standardize in the following step, below). For the next poverty stratum (5.09.9%) we need to aggregate the cases and the population denominator over 37 census tracts, and so on. Cases and population at risk in the three census tracts with missing poverty data are excluded from the analysis.
This yields the following table:
ABSM: CT poverty  Age category  Numerator  Denominator 
0.04.9%  014  1  10,608 
0.04.9%  1524  5  9,984 
0.04.9%  2544  54  29,190 
0.04.9%  4564  106  16,710 
0.04.9%  65+  657  15,825 
5.09.9%  014  40  69,939 
5.09.9%  1524  39  64,065 
5.09.9%  2544  252  179,595 
5.09.9%  4564  792  90,042 
5.09.9%  65+  4,535  80,916 
10.019.9%  014  101  88,989 
10.019.9%  1524  93  93,147 
10.019.9%  2544  531  224,793 
10.019.9%  4564  962  100,479 
10.019.9%  65+  3,944  71,955 
20.0100.0%  014  182  155,193 
20.0100.0%  1524  170  217,593 
20.0100.0%  2544  831  288,882 
20.0100.0%  4564  1,291  108,588 
20.0100.0%  65+  3,645  72,720 
Generating Rates and Other Summary Measures/Measures of Effect
1. Agestandardized incidence rates
The standard practice of public health departments in reporting population rates of mortality and disease incidence is to calculate agestandardized rates, which facilitates comparisons between regions or subgroups of interest. The agestandardized rate is interpretable as the rate that would be observed in a population if that population had the same age distribution as a given reference population. Standardization by the direct method involves taking a weighted average of the age specific incidence rates observed in the area or subgroup of interest, where the weights come from a standard age distribution, such as the year 2000 standard million $latex (1)&s=4$.
“Standard million” reference populations are available based on the US population age distribution for 1940, 1970, 1980, 1990, and 2000. Here we present the standard million in 11 age categories.
Age (years)  Standard million reference population  
Year 1940  Year 1970  Year 1980  Year 1990  Year 2000  
<1  15,343  17,150  15,598  12,936  13,818 
14  64,718  67,265  56,565  60,863  55,317 
514  170,355  200,511  154,238  141,584  145,565 
1524  181,677  174,405  187,542  147,860  138,646 
2534  162,066  122,567  163,683  173,600  135,573 
3544  139,237  113,616  113,155  151,095  162,613 
4554  117,811  114,265  100,641  101,416  134,834 
5564  80,294  91,481  95,799  85,030  87,247 
6574  48,426  61,192  68,775  72,802  66,037 
7584  17,303  30,112  34,116  40,429  44,842 
85+  2,770  7,436  9,888  12,385  15,508 
For our project, we used five broad age categories to age standardize, in order to obtain more stable rates in each age stratum, particularly for outcomes with sparse data. The relationship between our five categories and the standard eleven categories is illustrated in the table below.
Age in 11 categories  Year 2000 standard million  Age in 5 categories  Year 2000 standard million 
<1  13,818 
<15 
214,700 
14  55,317  
514  145,565  
1524  138,646  1524  138,646 
2534  135,573  2544  298,186 
3544  162,613  
4554  134,834  4564  222,081 
5564  87,247  
6574  66,037  65+  126,387 
7584  44,842  
85+  15,508 
If $latex cases_j$ represents the number of cases in age group j of the group or region of interest and $latex pop_j$ represents the population associated with that age group, then the standardized rate $latex IR_{st}$ for the group or region is
$latex \displaystyle IR_{st} =\frac{\sum_j w_j \left(\frac{\mbox{cases}_j}{\mbox{pop}_j}\right)}{\sum_j w_j} = \frac{\sum_j w_j IR_j}{\sum_j w_j}$
where $latex w_j$ is the weight associated with category j in the reference (standardizing) population (e.g. the population size or the proportion of the total population). The estimated variance of the standardized rate is given by:
$latex \displaystyle Var(IR_{st})=\frac {{\sum_j {w^2}_j} {\left(\frac{\mbox{cases}_j}{\mbox{pop}^2_j}\right)}} {\left({\sum_j w_j}\right)^2} $
(When the $latex w_j$s are proportions, then $latex \displaystyle IR_{st} ={\sum_j w_j IR_j} $ and $latex \displaystyle Var(IR_{st})={\sum_j {w^2}_j} \left(\frac{\mbox{cases}_j}{\mbox{pop}^2_j}\right) $).
Example:
To calculate the agestandardized all cause mortality rates in each of the four poverty strata in Suffolk County, we start with the agespecific mortality data. In each poverty stratum, the age standardized mortality rate is calculated as a weighted sum of the agespecific mortality rates, with the weights for each age stratum defined by the Year 2000 standard million.
ABSM: CT poverty  Age category  Numerator  Denominator  Year 2000 standard million  $latex w_ j$ (weight)  $latex IR_j$ (incidence rate per 100,000)  $latex IR_{st}$ (age standardized rate per 100,000) 
0.04.9%  014  1  10,608  214,700  0.215  9.4 
729.7 
0.04.9%  1524  5  9,984  138,646  0.139  50.1  
0.04.9%  2544  54  29,190  298,186  0.298  185  
0.04.9%  4564  106  16,710  222,081  0.222  634.4  
0.04.9%  65+  657  15,825  126,387  0.126  4,151.70  
5.09.9%  014  40  69,939  214,700  0.215  57.2 
966.2 
5.09.9%  1524  39  64,065  138,646  0.139  60.9  
5.09.9%  2544  252  179,595  298,186  0.298  140.3  
5.09.9%  4564  792  90,042  222,081  0.222  879.6  
5.09.9%  65+  4,535  80,916  126,387  0.126  5,604.60  
10.019.9%  014  101  88,989  214,700  0.215  113.5 
1,014.0 
10.019.9%  1524  93  93,147  138,646  0.139  99.8  
10.019.9%  2544  531  224,793  298,186  0.298  236.2  
10.019.9%  4564  962  100,479  222,081  0.222  957.4  
10.019.9%  65+  3,944  71,955  126,387  0.126  5,481.20  
20.0100.0%  014  182  155,193  214,700  0.215  117.3  1,019.30 
20.0100.0%  1524  170  217,593  138,646  0.139  78.1  
20.0100.0%  2544  831  288,882  298,186  0.298  287.7  
20.0100.0%  4564  1,291  108,588  222,081  0.222  1,188.90  
20.0100.0%  65+  3,645  72,720  126,387  0.126  5,012.40 
2. Confidence intervals for directly standardized rates
Traditional confidence limits for the direct standardized rates are based on the normal distribution and require large cell counts. In our analyses, we found that they can also occasionally result in “impossible” lower limits that are less than zero. Because of this, we adopted an alternate method for calculating the confidence limits based on the inverse gamma function $latex (2)&s=4$. This method assumes that the direct standardized rate is a linear combination of independent Poisson random variables. Assuming that this linear combination also follows a Poisson distribution, the agestandardized rate E(X) = x follows a gamma distribution Γ(a,b) as follows:
$latex \displaystyle X \backsim \Gamma \left(\frac{x^2}{v}{,} \frac{v}{x} \right) $
where x is the agestandardized rate ($latex IR_{st}$ as estimated above) and v is its variance, as described above. Converting this to the gamma distribution in its standard form, i.e. where b=1, this yields
$latex \displaystyle \frac{X}{b} \backsim \Gamma \left(\frac{x^2}{v}{,}1 \right) $
which greatly simplifies calculations. Then the lower 100(1α) confidence limit for $latex \displaystyle \frac{x^2}{v} $ is given by
L$latex \displaystyle \left( \frac{x^2}{v} \right)= \Gamma^{1} \left( \frac{x^2}{v}{,}1 \right) \left( \alpha \diagup {2} \right) $
and the upper 100(1α) confidence limit for $latex \displaystyle \frac{x^2}{v} $ is given by
U$latex \displaystyle \left( \frac{x^2}{v} \right) = \Gamma^{1} \left(\frac {(x+{k_M})^2}{(v+k_M)}{,}1 \right) \left(1\frac {\alpha} {2}\right)$
where $latex k=k_M=\mbox{max}_{je\left(1,…,j\right)} \left(k_j\right)$ is a continuity correction necessitated by using a continuous distribution to estimate confidence limits for a discrete random variable.
Increasing the number of events by 1 in an age stratum i results in a $latex \displaystyle k_j=\frac{w_j}{pop_j} $ increase in the agestandardized rate. If $latex k_j$ is constant for all age intervals, then $latex k_j = k$. However, since the $latex w_j$ and $latex pop_j$ typically vary across age strata, it is unclear what value of k to use. A very conservative upper limit can be obtained by using the maximum value of $latex k_j = k_M$. However, following the recommendation of the NCHS, we used a close approximation that alleviates the need to calculate $latex k_M$:
U$latex \displaystyle \left( \frac{x^2}{v} \right) = \Gamma^{1} \left(\frac{x^2}{v}+{1,1}\right)\left(1\frac {\alpha} {2}\right) $
To transform these intervals to obtain the desired confidence limits for X, we use L(X) = $latex \displaystyle \frac {L\left(x^2 \diagup v \right)}{x \diagup v} $ and U(X) = $latex \displaystyle \frac {U\left(x^2 \diagup v \right)}{x \diagup v} $.
Example:
In the following analysis of mortality due to homicide and legal intervention among hispanic women in Massachusetts, the lower confidence limits on the rate in the 5.09.9% poverty stratum is negative, using the traditional normal approximation method. In contrast, the lower confidence limit based on the gamma distribution yields a more reasonable confidence limit.
ABSM: CT poverty  Rate per 100,000  Confidence Limits  Deaths  Persontime at risk  
Normal approximation  “Gamma” interval  
Lower  Upper  Lower  Upper  
0.04.9%  0  (0.0  ,0.0)  (0.0  ,9.2)  0  40,182 
5.09.9%  3.5  (0.5  ,7.5)  (0.7  ,10.3)  3  67,458 
10.019.9%  3.8  (0.1  ,7.5)  (1.0  ,9.7)  4  87,336 
20.0100.0%  4.2  (1.4  ,7.0)  (1.9  ,8.0)  11  228,288 
3. Confidence intervals for IRst
When the observed rate is zero (i.e. there were zero cases), the gamma method is unable to produce confidence limits for the direct standardized rates. In this situation, we adopt the following convention for the confidence limit. The lower limit is simply set to zero. For the upper limit, we assume that the number of cases (i.e. the count) follows a Poisson distribution, and use the formula for the “exact” upper confidence limit of a Poisson random variable $latex (3)&s=4$:
U(Y) = $latex \displaystyle \frac{1}{2}\chi^{1}_{2\left(y+1\right)df} \left(1\frac{\alpha}{2} \right)$
where y is the count, i.e. zero. When α = 0.05 (i.e. for a 95% confidence limit) this simplifies to U(Y) = $latex \displaystyle \frac{\chi^{1}_{2df}\left({1\alpha \diagup 2}\right)}{2} $ = 3.689.
We can then divide this upper limit on the count by the population denominator to give the upper limit on the rate.
Example:
In the analysis of mortality due to homicide and legal intervention among Hispanic women in Massachusetts, the estimated rate in the least impoverished group is zero, since there were no deaths reported in census tracts with 04.9% below poverty. In the table below, the normal approximation method yields a confidence interval of (0,0) for the rate in the least impoverished group, as well (as “impossible” negative lower limits on the rates in the 5.09.9% poverty stratum, as we saw above). The gamma method also yields a (0,0) interval for the rate in the least impoverished group, so we have corrected the entry for the upper confidence limit as described above. Using the “exact” upper limit on the count of 3.689, we divide this by the denominator (40,182) to give an upper limit of 9.2 per 100,000.
ABSM: CT poverty  $latex IR_{st}$ (age standardized rate per 100,000)  Confidence Limits  Deaths  Persontime at risk  
Normal approximation  “Gamma” interval  
Lower  Upper  Lower  Upper  
0.04.9%  0  (0.0  ,0.0)  (0.0  ,9.2)  0  40182 
5.09.9%  3.5  (0.5  ,7.5)  (0.7  ,10.3)  3  67458 
10.019.9%  3.8  (0.1  ,7.5)  (1.0  ,9.7)  4  87336 
20.0100.0%  4.2  (1.4  ,7.0)  (1.9  ,8.0)  11  228288 
4. Agestandardized incidence rate difference and rate ratio
Two commonly used measures for comparing incidence rates from two different groups are the incidence rate difference (IRD) and the incidence rate ratio (IRR). The incidence rate difference compares the rates on the absolute scale, and summarizes the excess rate comparing the larger to the smaller rate. The incidence rate ratio compares the rates on a relative scale, summarizing the size of one rate relative to the other rate.
To compare two agestandardized incidence rates on the absolute scale, the agestandardized incidence rate difference ($latex IRD_{st}$) is the rate in one group minus the rate in the other, i.e. $latex IR_{st1} – IR_{st0}$. The variance of this agestandardized incidence rate difference is simply the sum of the estimated variance of the two agestandardized rates $latex (4)&s=4$,
$latex \displaystyle Var \left({IRD} \right) = Var \left(IR_{st1} \right) + Var \left(IR_{st0} \right) $
To compare agestandardized rates from two different groups or regions on the relative scale, the agestandardized incidence rate ratio ($latex \mbox{IRR}_{st}$) is simply $latex \mbox{IR}_{st1}$/$latex IR_{st0}$. Confidence intervals can be calculated using the variance estimator $latex (4)&s=4$:
$latex \displaystyle Var \left[log \left(IRR_{st} \right) \right] = \frac {Var \left (IR_{st1} \right)}{{IR_{st1}}^2} + \frac {Var \left(IR_{st0} \right)}{{IR_{st0}}^2}$
Example:
To compare the agestandardized incidence rates in the most and least impoverished census tracts in Suffolk County, we start with the agespecific data for these two strata (note: for ease of presentation, we present variances in scientific notation in the table below):
ABSM: CT Poverty  Age category  Numerator  Denominator  $latex \mbox{w}_j$ (weight)  $latex \mbox{ IR}_j$ (age specific rate)  Var($latex \mbox{IR}_j$) (variance of the age specific rate)  $latex \mbox{IR}_{st}$ (age standardized rate)  Var($latex \mbox{IR}_{st}$) (variance of the age standardized rate) 
0.04.9%  014  1  10,608  0.2147  0.000094  8.89E09  0.007297  6.76E08 
0.04.9%  1524  5  9,984  0.1386  0.000501  5.02E08  
0.04.9%  2544  54  29,190  0.2982  0.00185  6.34E08  
0.04.9%  4564  106  16,710  0.2221  0.006344  3.80E07  
0.04.9%  65+  657  15,825  0.1264  0.041517  2.62E06  
20.0100.0%  014  182  155,193  0.2147  0.001173  7.56E09  0.010193  1.77E08 
20.0100.0%  1524  170  217,593  0.1386  0.000781  3.59E09  
20.0100.0%  2544  831  288,882  0.2982  0.002877  9.96E09  
20.0100.0%  4564  1,291  108,588  0.2221  0.011889  1.10E07  
20.0100.0%  65+  3,645  72,720  0.1264  0.050124  6.89E07 
The agestandardized rate difference is simply 1,019.3 per 100,000 – 729.7 per 100,000 = 289.6 per 100,000 (or, in scientific notation, 2.896 x 103).
Using the formula above, we calculate the variance of $latex \mbox{IRD}_{st}$.
$latex \displaystyle Var \left({IRD}_{st} \right) = 6.76 \times 10^{8} + 1.77 \times 10^{8} = 8.54 \times 10^{8} $
Then the lower and upper confidence limits are derived as follows:
$latex \displaystyle L_{IRDst} = 2.896 \times 10^{3} – \left(1.96 * \sqrt{8.54 \times 10^{8}} \right) = 0.002323 $
$latex \displaystyle U_{IRDst} = 2.896 \times 10^{3} – \left(1.96 * \sqrt{8.54 \times 10^{8}} \right) = 0.003469 $
or, expressed per 100,000, 232.2 to 346.9 per 100,000.
The agestandardized rate ratio is simply 1,019.3 per 100,000/729.7 per 100,000 = 1.40.
Using the formula above, we calculate the variance of log$latex \left(IRR_{st}\right)$:
$latex \displaystyle Var \left[\mbox{log} \left(IRR_{st}\right) \right] = \frac {6.76 \times 10^{8}}{0.007297^2} + \frac{1.77 \times 10^{8}}{0.010193^2} = 0.001441 $
Then the lower and upper confidence limits are derived as follows:
$latex \displaystyle L_{IRR_{st}} = \mbox{exp} \left[ \mbox{log} \left(1.40 \right) – 1.96\sqrt{0.001441 }\right] = 1.30 $
$latex \displaystyle U_{IRR_{st}} = \mbox{exp} \left[ \mbox{log} \left(1.40 \right) – 1.96\sqrt{0.001441 }\right] = 1.50 $
5. Relative Index of Inequality (RII)
Comparisons of socioeconomic gradients based on categorical ABSM may be complicated by differences in the population distributions of areabased socioeconomic measures. For example, it may be expected that the classifications producing smaller groups at the margins would lead to larger incidence rate ratios, comparing the most deprived to the most affluent, because finer discrimination of extremes of socioeconomic position is achieved. The relative index of inequality (RII) has been proposed as a measure which explicitly addresses this problem $latex (57)&s=4$. Assuming ordinality of the ABSM categories, the RII is calculated by regressing the incidence rate in each ABSM category on the total proportion of the population that is more deprived in the socioeconomic hierarchy. Because the RII combines information about the magnitude of the socioeconomic gradient with information about the distribution of the socioeconomic variable in the population, it can be conceptualized as a measure of “total population input”.
In practice, this latter quantity is represented by the cumulative distribution function (cdf). We approximate the cdf for the jth level of a given ABSM by summing the proportion of the population represented by the categories $latex \mbox{ABSM}_1$,…, $latex \mbox{ABSM}_{j1}$, and adding onehalf the proportion of the population represented by the category $latex \mbox{ABSM}_j$.
Example:
In order to calculate the RII for poverty and all cause mortality in Massachusetts, we begin by calculating the approximate cumulative distribution function as follows:
ABSM: CT poverty  Population denominator  Proportion  Formula  Approximate cdf 
0.04.9%  7,626,117  0.423  0.2115  0.211 
5.09.9%  5,508,912  0.305  0.5755  0.576 
10.019.9%  2,782,194  0.154  0.805  0.805 
20.0100.0%  2,120,208  0.118  0.941  0.941 
In order to compare RII meaningfully across groups with differing age composition, we developed an agestandardized RII, standardized to the year 2000 standard million, as follows. Let observed$latex ij&s=4$ be the observed number of cases in the ith age group and the jth category of ABSM, and pop$latex ij&s=4$ be the population at risk in the corresponding category. First, we calculate the agestandardized rate IR$latex st&s=4$ in each stratum j defined by ABSM, as described above. For each stratum j, we estimate the expected number of cases in stratum j, $latex \mbox{expected}_{j}$, by multiplying the agestandardized rate IR$latex st&s=4$ by the population denominator, pop$latex j&s=4$ = Σ$latex i&s=4$pop$latex ij&s=4$. We determine the “marginal” cumulative distribution function, cdf(ABSM$latex j&s=4$), of the ABSM over the entire population, as noted above.
Example:
The column of red numbers shows the expected number of cases in each poverty stratum.
ABSM: CT Poverty  IRst (age standardized rate per 100,000)  Observed deaths  Population denominator  Expected deaths  Approximate cdf 
04.9%  757  57,256  7,626,117  57,731.70  0.211 
59.9%  840.3  52,583  5,508,912  46,291.70  0.576 
1019.9%  915.9  27,730  2,782,194  25,482.00  0.805 
20100%  1,035.30  17,842  2,120,208  21,950.70  0.941 
To calculate the agestandardized RIIst, we fit the following Poisson model for the expected cases:
$latex \displaystyle expected_{ij} \sim Poisson \left(\lambda_{ij} \right)$
$latex \displaystyle log \left(\lambda_{ij} \right) = log\left(pop_{ij}\right) + \beta_0 + \beta_1 * cdf\left(ABSM_j\right)$
Exponentiation of the $latex \beta_1$ yields the RII, which is interpretable as an incidence rate ratio comparing the rates in the bottom to the top of the socioeconomic hierarchy. A larger RII indicates a greater the degree of inequality across a socioeconomic hierarchy, which may be due to a steep socioeconomic gradient or large inequalities in the distribution of the ABSM itself.
Example:
Fitting this model to the data presented above yields a $latex \beta_1$ of 0.379. Exponentiating this, we obtain an RII of 1.46. In the figures below, we can see how the RII for poverty is obtained. In the left figure, the height of light blue bars represents the all cause mortality rate per 100,000 in each of the four poverty strata (04.9, 59.9, 1019.9, 20100%), with width of bars proportional to population size of poverty stratum (in order from least to most impoverished). Open circles are plotted along the xaxis at the interpolated midpoints of each bar, approximating the cumulative distribution function of CT level poverty. The solid line represents fitted RII line. In the left figure, this line is not a straight line since the fitted line comes from a Poisson model. The right figure shows the plotted points and fitted RII line on the log scale, where the line is truly straight.
6. Population Attributable Fraction
The population attributable fraction (PAF) is a useful summary measure for characterizing the public health impact of an exposure on population patterns of health and disease. It is defined as “the fraction of all cases (exposed and unexposed) that would not have occurred if exposure had not occurred.” $latex (8)&s=4$ For a polytymous exposure, the population attributable fraction is a weighted sum of the attributable fractions for each level of the exposure, with the weights defined by the case fractions (number of exposed cases divided by overall number of cases):
$latex \displaystyle PAF = CF_1 \times \frac{RR_1 – 1}{RR_1} + CF_2 \times \frac{RR_2 1}{RR_2} + … + CF_j \times \frac{RR_j – 1}{RR_j} $
In order to aggregate multiple PAFs over several age strata i=1,…,I, note that
$latex \displaystyle PAF_{agg} = \frac{\sum_i \mbox{excess number of cases}}{\mbox{number of cases}}$
$latex \displaystyle = \frac {\sum_i \mbox{number of cases} \times \frac {\mbox{excess number of cases}}{\mbox{number of cases}}}{\sum_i\mbox{number of cases}} $
$latex \displaystyle =\frac {\sum_i \mbox{number of cases} \times PAF_1}{\sum_i\mbox{number of cases}}$
that is, a weighted average of stratum specific PAFs, with the number of cases in each age stratum as weights.
Example:
To calculate the population attributable fraction of all cause mortality due to poverty, we begin by tabulating the cases and population persontime at risk in each poverty stratum j within each age group i. Within each age group, the case fraction $latex \mbox{CF}_{ij}$ is the number of cases in that poverty stratum, divided by the total number of cases within the age group. The incidence rate ratio $latex \mbox{IRR}_j$ for a particular poverty stratum, relative to the reference category of the least impoverished group, is calculated by dividing the rate in that poverty stratum by the rate in the least impoverished group. For each age stratum, we calculate a separate agespecific PAF, as seen in the column of red numbers in the table below. These agespecific PAFs range from 5% to 23%.
Age category (i)  ABSM: CT poverty (j)  Cases  Persontime denominator  Rate per 100,000  Case Fraction ($latex \mbox{CF}_{ij}$)  Incidence rate ratio ($latex \mbox{IRR}_{ij}$)  Population attributable fraction ($latex \mbox{PAF}_i$) 
014  04.9% (reference)  303  727,947  41.6  40.7%  1.00 
0.1626 
5.09.9%  253  461,958  54.8  34.0%  1.32  
10.019.9%  113  206,214  54.8  15.2%  1.32  
20.0100.0%  75  100,716  74.5  10.1%  1.79  
Total cases:  744  
1524  04.9% (reference)  377  510,645  73.8  40.6%  1.00 
0.0506 
5.09.9%  323  349,518  92.4  34.8%  1.25  
10.019.9%  152  179,928  84.5  16.4%  1.14  
20.0100.0%  76  153,273  49.6  8.2%  0.67  
Total cases:  928  
2544  04.9% (reference)  1,569  1,201,002  130.6  34.7%  1.00 
0.2266 
5.09.9%  1,392  873,072  159.4  30.7%  1.22  
10.019.9%  933  405,366  230.2  20.6%  1.76  
20.0100.0%  633  200,457  315.8  14.0%  2.42  
Total cases:  4,527  
4564  04.9% (reference)  5,314  763,464  696.0  39.7%  1.00 
0.2210 
5.09.9%  4,429  461,451  959.8  33.1%  1.38  
10.019.9%  2,287  191,934  1,191.6  17.1%  1.71  
20.0100.0%  1,369  82,674  1,655.9  10.2%  2.38  
Total cases:  13,399  
65+  04.9% (reference)  19,470  376,002  5,178.2  38.8%  1.00 
0.0725 
5.09.9%  17,784  314,181  5,660.4  35.4%  1.09  
10.019.9%  8,734  146,091  5,978.5  17.4%  1.15  
20.0100.0%  4,248  63,594  6,679.9  8.5%  1.29  
Total cases:  50,236 
To aggregate these PAFs across age strata, we weight the contribution of each age stratum by the proportion of cases in that age stratum. As seen in the table below, this results in an aggregated population attributable fraction $latex \mbox {PAF}_{agg}$ of 11%.
Age category (i)  Cases  Population attributable fraction ($latex \mbox{PAF}_i$)  $latex \rightarrow$  Aggregated population attributable fraction ($latex \mbox{PAF}_{agg}$) 
014  744  0.1626  (744*0.1626 + 928*0.0506 + 4527*0.2266 + 13399*0.2210 + 50236*0.0725)/ 69834 
=0.1116 
1524  928  0.0506  
2544  4,527  0.2266  
4564  13,399  0.221  
65+  50,236  0.0725  
Total cases:  69,834 
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