Evaluation of stability of directly standardized rates for sparse data using simulation methods

Abstract

Abstract Background Directly standardized rates (DSRs) adjust for different age distributions in different populations and enable, say, the rates of disease between the populations to be directly compared. They are routinely published but there is concern that a DSR is not valid when it is based on a “small” number of events. The aim of this study was to determine the value at which a DSR should not be published when analyzing real data in England. Methods Standard Monte Carlo simulation techniques were used assuming the number of events in 19 age groups (i.e., 0–4, 5–9, ... 90+ years) follow independent Poisson distributions. The total number of events, age specific risks, and the population sizes in each age group were varied. For each of 10,000 simulations the DSR (using the 2013 European Standard Population weights), together with the coverage of three different methods (normal approximation, Dobson, and Tiwari modified gamma) of estimating the 95% confidence intervals (CIs), were calculated. Results The normal approximation was, as expected, not suitable for use when fewer than 100 events occurred. The Tiwari method and the Dobson method of calculating confidence intervals produced similar estimates and either was suitable when the expected or observed numbers of events were 10 or greater. The accuracy of the CIs was not influenced by the distribution of the events across categories (i.e., the degree of clustering, the age distributions of the sampling populations, and the number of categories with no events occurring in them). Conclusions DSRs should not be given when the total observed number of events is less than 10. The Dobson method might be considered the preferred method due to the formulae being simpler than that of the Tiwari method and the coverage being slightly more accurate.

Keywords

• Direct standardization
• Monte Carlo simulation
• Confidence interval coverage
• Tiwari
• Dobson