Indirectly standardised ratios (ISRs)

Calculation of indirectly standardised ratios

The indirectly standardised ratio is the ratio of the observed number of events relative to the number of events that would be expected if standard age-specific rates were applied to the particular observed population’s age structure. A common example is the standardised mortality ratio (SMR).

The indirectly standardised ratio \((ISR)\) is given by:

\[ ISR = \frac{O}{E} = \frac{\sum_{i}O_{i}}{\sum_{i}E_{i}} = \frac{\sum_{i}O_{i}}{\sum_{i}n_{i}\lambda_{i}} \]

where:

• \(O_{i}\) is the observed number of events in the subject population in age group \(i\)
• \(E_{i}\) is the expected number of events in the standard population in age group \(i\) given the standard rates
• \(n_{i}\) is the number of individuals in the subject population in age group \(i\)
• \(\lambda_{i}\) is the crude age-specific rate in the standard population in age group \(i\)

For presentation purposes, the ratio is usually multiplied by 100. By definition, the standard population has a ratio of 100. Ratios above 100 indicate that the number of events observed is greater than that expected from the standard rates, and ratios below 100 that it is lower. The ratio can be multiplied by the overall crude rate in the standard population and presented as an indirectly age-standardised rate.

Calculation of confidence intervals for indirectly standardised ratios

For the purposes of calculating the confidence interval of the ratio, the expected count is considered to be precise (that is, it has no variance and contributes no uncertainty to the confidence interval. The imprecision in the ratio is therefore dependent only on the imprecision of the observed count.

The \(100(1 − \alpha)\)% confidence interval limits of the ratio are given by:

\[ ISR_{lower} = \frac{O_{lower}}{E} \] \[ ISR_{upper} = \frac{O_{upper}}{E} \]

where:

• \(O _{lower}\) and \(O _{upper}\) are the lower and upper confidence limits for the observed number of events.

\(O _{lower}\) and \(O _{upper}\) can be calculated using either Byar’s method or the \(\chi^2\) exact method, depending on the size of the observed number of events.

The confidence limits as calculated above should then be multiplied by any scaling factor that has been used in presenting the ratio itself.

Counts of 10 or greater

Where the observed number of events \(O\) is less than 10, the variability in \(O\) is described by the Poisson distribution. This can be calculated using Byar’s approximation as it is computationally simple and gives very accurate approximations to the exact Poisson probabilities (Breslow and Day, 1987).

Using Byar’s method, the lower and upper confidence limits of the observed number of events, are given by:

\[ O _{lower} = O \left(1-\frac{1}{9O}-\frac{z}{3\sqrt{O}}\right)^3 \]

\[ O _{upper} = (O+1) \left(1-\frac{1}{9(O+1)}+\frac{z}{3\sqrt{O+1}}\right)^3 \]

where:

• \(O\) is the total observed count of events in the local or subject population
• \(z\) is the \(100\left(1 - \frac{\alpha}{2}\right)\)th percentile value from the standard normal distribution

Counts of less than 10

Where the observed number of events \(O\) is less than 10, the \(\chi^2\) exact method should be used to calculate the lower and upper confidence limits of the observed number of events. Using the link between the Poisson and \({\chi}^2\) distributions (Armitage and Berry, 2002), these are given by:

\[ O _{lower} = \frac{{\chi}^2_{lower}}{2} \] \[ O _{upper} = \frac{{\chi}^2_{upper}}{2} \] where:

• \(O\) is the total observed count of events in the local or subject population

• \({\chi}^2_{lower}\) is the \(100\left(1 - \frac{\alpha}{2}\right)\)th percentile value from the \({\chi}^2\) distribution with \(2{O}\) degrees of freedom

• \({\chi}^2_{upper}\) is the \(100\left(1 - \frac{\alpha}{2}\right)\)th percentile value from the \({\chi}^2\) distribution with \(2{O}+2\) degrees of freedom

Tools

The following tools are available to calculate indirectly standardised ratios:

The PHEindicatormethods R package includes the function calculate_ISRatio(), which can be used to calculate ISRs.

The Excel tool for common public health statistics and their confidence intervals (xlsx) includes a template for calculating ISRs.

Page last updated: July 2025