Indirect Standardization and SMR
Menu location: Analysis_Rates_Indirect Standardization and SMR.
This function calculates standardized mortality ratios (SMR) with exact confidence intervals.
Indirect standardization is used to calculate the expected mortality rate for the index population, given age specific mortality rates from a reference population.
The method applies not only to mortality rates but also to any rates of uncommon events (i.e. the Poisson distribution can be applied). An example of a different application is calculation of standardized incidence rate ratio (SIR) for a disease in a population.
If you want to standardize by both age and sex then enter two sets of age groups (i.e. 10 rows instead of 5 for the example below) split into male and female consecutively. Provided that you have sex and age specific mortality rates for the reference population and age and sex specific population sizes for the index population then you can produce an age and sex standardized SMR.
SMRs from different index/study populations are not strictly comparable because they are calculated using different weighting schemes that depend upon the age structures of the index/study populations. SMRs can, however, be compared if you make the assumption that the ratio of rates between index and reference populations is constant; this is similar to the assumption of proportional hazards in Cox regression (Armitage and Berry, 1994). Direct standardization is an alternative to indirect standardization that does provide comparable measures. Direct standardization, however, has other weaknesses, such as greater susceptibility than the indirect method to error with small numbers, which make the choice of method a matter for careful statistical consideration. If you are in doubt, please consult with a Statistician and/or Epidemiologist.
Please note that methods of standardization can mask differences in rates between populations; in order to avoid this you should supplement SMR analysis with individual comparisons of group (e.g. age) specific rates.
Data input
- Groups/strata, e.g. age bands for indirect age standardization
- Rates for each group from a reference population
- Person-time for each group from the index/study population (e.g. size of each group if just one year and all subjects were followed up)
- Multiplier for reference rates, e.g. 10000 if mortality entered as deaths per 10000, 1 if mortality entered as a decimal fraction
The SMR is expressed in ratio and integer (ratio * 100) formats with a confidence interval.
A test based on the null hypothesis that the number of observed and expected deaths is equal is also given. This test uses a Poisson distribution to calculate probability (Armitage and Berry, 1994, Bland, 2000; Gardner and Altman, 1989).
Technical validation
The confidence intervals are calculated by the exact Poisson method of Owen, this gives better coverage than the frequently quoted Vandenbroucke approximation or other asymptotic methods (Ulm, 1990; Greenland, 1990).
- where SMR_{l} and SMR_{u} are lower and upper confidence limits respectively for the SMR, χ^{2}_{α, ν} is the (100*α)th chi-square centile with ν degrees of freedom, d is the number of observed deaths, e is the number of expected deaths, n_{i} is the person-time for the ith study group stratum and R_{i} is the reference population rate for the ith stratum.
Example
From Bland (2000).
Test workbook (Rates worksheet: Reference Mortality, Index Group Sizes, Age Groups).
The following data represent the age-specific mortality rates for liver cirrhosis in men and the number of male doctors in each age stratum:
Age group | Mortality per million men per year | Number of male doctors |
15-24 | 5.859 | 1080 |
25-34 | 13.050 | 12860 |
35-44 | 46.937 | 11510 |
45-54 | 161.503 | 10330 |
55-64 | 271.358 | 7790 |
To analyse these data in StatsDirect you must select indirect standardization and SMR from the rates section of the analysis menu. Enter the mortality rate and group size for each age group. Note that group size refers to the study sample of doctors and not to the male population used to derive mortality data. Enter the mortality denominator as 1000000. Enter the observed deaths as 14.
For this example:
Indirectly Standardized Rates and SMR
Reference rate | Observed person-time | Expected deaths |
0.000006 | 1080 | 0.006328 |
0.000013 | 12860 | 0.167823 |
0.000047 | 11510 | 0.540245 |
0.000162 | 10330 | 1.668326 |
0.000271 | 7790 | 2.113879 |
Total = 4.4966004 |
Standardized Mortality Ratio (SMR) = 3.113463
SMR (*100 as integer) = 311
Exact 95% confidence interval = 1.702159 to 5.223862 (170 to 522)
Probability of observing 14 or more deaths by chance P = .0002
Probability of observing 14 or fewer deaths by chance P > .9999
Here we can see that the total expected deaths from liver cirrhosis in male doctors is 4.5 per year. The observed number, 14, was statistically highly significantly greater than expected. With 95% confidence we can state that male doctors in this country exhibit between 1.7 and 5.2 times the number of deaths from liver cirrhosis than expected from the general male population of a similar age distribution.