Estimating confidence limits on a standardised mortality ratio when the expected number is not error free

Area of Science:

• Epidemiology
• Biostatistics
• Statistical Modeling

Background:

• Standardized Mortality Ratios (SMRs) are crucial in epidemiology for comparing observed deaths to expected deaths.
• Accurate confidence intervals for SMRs are essential for reliable interpretation, especially when expected event counts are uncertain.

Purpose of the Study:

• To demonstrate the application of the beta distribution for calculating confidence limits of SMRs with variable expected events.
• To compare these beta distribution-derived limits with traditional exact methods and Fieller-based intervals.

Main Methods:

• Explored the relationship between binomial and beta distributions.
• Developed a method for SMR confidence limits analogous to relative risk calculations in cohort studies.
• Utilized hypothetical data and a MINITAB macro for analysis.

Main Results:

• Exact confidence intervals incorporating expected number variation are substantially wider than standard exact intervals.
• Fieller intervals approximate the new exact method under conditions of large observed and expected numbers.
• The standard method may yield different conclusions than the new approach even with 100 expected events.

Conclusions:

• When expected deaths in SMR calculations are subject to sampling error, the proposed beta distribution method for exact confidence limits is recommended.
• Alternatively, approximate Fieller-based limits are suitable if sufficient events are observed and expected to ensure approximate normality.