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Various authors discuss the standardized mortality ratio (SMR) and provide exact and approximate confidence limits for the true SMR. These results are also applicable to the standardized incidence ratio (SIR). The following sections provide a brief outline of the results and give references to more detailed discussions.
Definition of the SIR
Suppose the person-time from the study group (i.e. cohort) is allocated among M cells defined by the cross-classification of various adjustment variables such as gender, race, attained age group, and attained calendar year group. Let tk represent the person-time and Dk represent the observed events that the cohort subjects contribute to the kth cell, and let Mathematical expression represent the standard rate for the kth cell, where k = 1, 2, ..., M. Given this notation, the SIR is defined as:
\( {\huge SIR = \frac{\sum_{k=1}^{M} D_k}{\sum_{k=1}^{M}t_k \lambda_k^\ast} = \frac{D}{E^\ast}} \)
where the total number of events observed in the cohort is \( D = \sum_{k=1}^{M} D_k\) Mathematical expression, and the total number of expected events is \( E^\ast = \sum_{k=1}^{M}\) \( E_k^\ast = \sum_{k=1}^{M}t_k \lambda_k^\ast \) Mathematical expression (Breslow and Day, 1987; Sahai and Khurshid, 1996).
Exact Confidence Limits for the True SIR
Sahai and Khurshid (1993, 1996) discuss the following method for obtaining the exact confidence limits, SIRL and SIRU, for the true SIR, φ. Assuming D is Poisson distributed with mean μ = E(D), confidence limits for μ are obtained using the relationship between the Poisson distribution and the chi-square distribution. Then these limits are divided by the total number of expected events, E*, to obtain the limits
\({\huge SIR_L = \frac{X_{2D,\alpha/2}^2}{2E^\ast} }\) and \( {\huge SIR_U = \frac{X_{2(D+1),1-\alpha/2}^2}{2E^\ast} }\)
where \( X_{v,\alpha}^2\) is the 100a percentile of the chi-square distribution with v degrees of freedom.
Approximate Confidence Limits for the True SIR
The exact limits, SIRL and SIRU, can be approximated by applying the Wilson and Hilferty (1931) approximation for chi-square percentiles, which is
\( {\huge X_{v,\alpha}^2 = v \left(1- \frac{2}{9v} + Z_\alpha\sqrt{\frac{2}{9v}}\right)^3 }\)
The resulting approximate limits for the true SIR, φ, are
\( {\huge SIR_{\widetilde{L}} = \frac{D}{E^\ast}\left(1 - \frac{1}{9D} + \frac{Z_{}\alpha/2}{3\sqrt{D}} \right)^3 }\) and
\( {\huge SIR_{\widetilde{U}} = \frac{D+1}{E^\ast}\left(1 - \frac{1}{9(D+1)} + \frac{Z_{}1-\alpha/2}{3\sqrt{D+1}} \right)^3 }\)
where Zα is the 100α percentile of the standard normal distribution (Rothman and Boice, 1979, 1982; Breslow and Day, 1987; Sahai and Khurshid, 1993, 1996). Rothman and Boice (1979, 1982) mention that these limits were first proposed by Byar (unpublished).
References
Breslow NE, Day NE (1987). Statistical Methods in Cancer Research. Vol. II, The Design and Analysis of Cohort Studies (IARC Scientific Publication No. 82). Lyon, France: International Agency for Research on Cancer.
Rothman KJ, Boice JD, Jr. (1979). Epidemiologic Analysis with a Programmable Calculator (NIH Publication 79-1649). Washington DC: US Government Printing Office.
Rothman KJ, Boice JD, Jr. (1982). Epidemiologic Analysis with a Programmable Calculator, New Edition. Boston, MA: Epidemiology Resources, Inc.
Sahai H, Khurshid A (1993). Confidence Intervals for the Mean of a Poisson Distribution: A Review. Biometrical J, 35: 857-67.
Sahai H, Khurshid A (1996). Statistics in Epidemiology: Methods, Techniques, and Applications. Boca Raton, FL: CRC Press, Inc.
Wilson EB, Hilferty MM (1931). The Distribution of Chi-Square. Proc Natl Acad Sci USA, 17: 684-8.
