January
5, 2006
Standardized Mortality Ratio
and Confidence Interval
Kevin M. Sullivan, PhD, MPH, MHA cdckms@sph.emory.edu
This module tests for statistical significance and calculates various confidence intervals for SMR, based on a number of different methods. First, the user is prompted to enter observed and expected number of deaths in the respective data entry cells. Please note that the observed number of cases must be an integer as they are assumed to be Poisson variates (random variables with a Poisson distribution). The user can change the confidence interval settings as seen in the data entry dialog box (or) at ‘Options/Setting’ at the main menu screen.
The output from the example above is as follows:
P-values are calculated under the assumption that the observed deaths
are Poisson variates (random variables with a Poisson distribution) and the
expected deaths are invariate. Exact confidence intervals and p-values should
be used when the number of observed deaths is less than or equal to five. For
greater numbers of observed deaths, approximation methods are as nearly
accurate as exact tests.
In the output window, the statistical significance test between observed and expected number of deaths by Mid-P exact method shows p=0.6571.
The point estimate of SMR is 1.212, and six different methods are used to calculate the confidence interval around this estimate: Mid-P exact test, Fisher’s exact test, normal approximation, Byar approximation, Rothman/Greenland method, Vandenbroucke method and Ury & Wiggins method. Of these methods, the Mid-P exact test is generally the preferred method.
Based on p-values and confidence intervals that include null value ‘1’ in the output table, the interpretation is that there is no significant excess or deficit of mortality rate in the study population compared to that of general population.
For confidence limit estimates < 0.0, the value 0.0 is shown. All confidence intervals calculated are two-sided and depend on the setting of user’s choice (90%, 95%, 99%, 99.9% or 99.99%). Formulas for the methods are provided in the following section.
Formulae
The notation for the formulae is:
a = the observed number of deaths
λ = the expected number of deaths
SMR= a/b;
= the two-sided Z
value
Significance Tests
(two-tailed P-value)
Mid-P exact test (see Rothman and Boice):
If a>λ:
If a<λ:
Exact Test based on
Poisson distribution (see Rosner):
If a<λ:
If a>λ:
Byar approximation (see Rothman and Boice):
If a>λ then a=a; If a<λ then a=a+1;
Calculation of
Confidence Intervals
Exact Tests (Mid-P
and Fisher)
Exact confidence limits for an SMR can be derived by setting
limits for the numerator and assuming the expected number in the denominator to
be a constant. The limits for ‘a’ with 100(1-α) percent confidence are the
iterative solutions and
.
Computing iterative solutions and
is below……..
A. Mid-P exact
test (see Rothman and Boice):
Lower bound:
Upper bound:
B. Fisher’s exact
test (see Rothman and Boice):
Lower bound:
Upper bound:
Therefore, the exact
lower and upper limits for SMR equal to “a/λ” would be
, respectively.
Byar Approximation: (see Rothman and Boice):
Lower bound:
Upper bound:
Rothman Greenland Method:
Lower bound:
Upper bound:
Ury & Wiggins Method: (only 90%, 95% and 99%CI available)
For 90%CI
Lower bound:
Upper bound:
For 95%CI
Lower bound:
Upper bound:
For 99%CI
Lower bound:
Upper bound:
Vandenbroucke Method: (only 95%CI available)
References
Rosner B. Fundamentals of Biostatistics, 5th Edition. Duxbury Press, 2000.
Rothman KJ, Boice JD Jr: Epidemiologic analysis with a programmable calculator. NIH Pub No. 79-1649. Bethesda , MD : National Institutes of Health, 1979;31-32.
Rothman KJ, Greenland S. Modern Epidemiology, 2nd Edition. Lippincott-Raven Publishers, Philadelphia , 1998.
Ury HK, Wiggins AD. Another shortcut method for calculating the confidence interval of a poisson variable (or of a standardized mortality ratio). Am J Epidemiol 1985; 122:197-8.
Vandenbroucke JP. A shortcut method for calculating the 95 percent confidence interval of the standardized mortality ratio. (Letter). Am J Epidemiol 1982; 115:303-4.