Directly standardised rates (DSRs)

An introduction to standardisation, and an introduction to confidence intervals are available elsewhere in this guidance.


Calculation of directly standardised rates

Directly age-standardised rates express an indicator in terms of the overall rate that would occur in a standard population age structure if it experienced the age-specific rates of the observed population.

The directly standardised rate \((DSR)\) is given by:

\[ DSR = \frac{1}{\sum_{i}w_{i}}\sum_i\frac{w_{i}O_{i}}{n_{i}} \]

where:

  • \(O_{i}\) is the observed number of events in the local or subject population in age group \(i\)
  • \(n_{i}\) is the number of individuals in the local or subject denominator population in age group \(i\), or the population multiplied by the period at risk (‘person-years’, for example)
  • \(w_{i}\) is the number (or proportion) of individuals in the reference or standard population in age group \(i\)

In 2017 Public Health England commissioned research to investigate the validity of directly standardised rates when the number of events \(O\) is small (Morris et al., 2018). The results of this research, provides the following guidance:

  • Directly standardised rates should not be calculated when the total count across all age groups is less than 10
  • The number of age or age and sex strata does not matter
  • Cells with zero counts do not matter, however many of them there are


Calculation of confidence intervals for directly standardised rates

The directly standardised rate is a weighted sum of the independent age-specific rates. This means that its variance is a weighted sum of the variances of each of those age-specific rates.

While there are a number of methods of calculating confidence intervals for directly standardised rates (Morris et al., 2018), Dobson’s method (Dobson et al., 1991) with Byar’s method (Breslow and Day, 1987) is recommended.

In this method the exact interval is found for the crude count and then weighted and scaled to give the interval for the directly standardised rate. The weight used is the ratio of the standard error of the DSR to the standard error of the crude count.

For rates, which assume the Poisson distribution, the \(100(1 − \alpha)\%\) confidence limits for the DSR are given by:

\[ DSR_{lower} = DSR + \sqrt{\frac{\operatorname{Var}(DSR)}{\operatorname{Var}(O)}}(O_{lower}-O) \]

\[ DSR_{upper} = DSR + \sqrt{\frac{\operatorname{Var}(DSR)}{\operatorname{Var}(O)}}(O_{upper}-O) \]

where:

  • \(O\) is the total observed count of events in the local or subject population
  • \(O_{\text{lower}}\) and \(O_{\text{upper}}\) are the lower and upper confidence limits for the observed count of events, determined using the Byar’s approximation (see below)
  • \(\operatorname{Var}(O)\) is the variance of the total observed count \(O\)
  • \(\operatorname{Var}(DSR)\) is the variance of the directly standardised rate

\(O_{\text{lower}}\) and \(O_{\text{upper}}\), calculated using Byar’s method, 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

The variances of the observed count and DSR \(\operatorname{Var}(O)\) and \(\operatorname{Var}(DSR)\) are estimated by:

\[ \operatorname{Var}(O) = \sum_iO_{i} \]

\[ \operatorname{Var}(DSR) = \frac{1}{\left(\sum_{i}w_{i}\right)^2}\sum_i\frac{{w_{i}}^2O_{i}}{{n_{i}}^2} \]

where:

  • \(O_{i}\) is the observed number of events in the local or subject population in age group \(i\)
  • \(n_{i}\) is the number of individuals in the local or subject denominator population in age group \(i\), or the population multiplied by the period at risk (‘person-years’, for example)
  • \(w_{i}\) is the number (or proportion) of individuals in the reference or standard population in age group \(i\)

Calculation of confidence intervals when observed events are not independent

The Dobson and Byar’s approach described above assumes that events are unique and not interrelated. However, in some circumstances this is not the case. For example, many long term health conditions require a number of admissions to hospital. In this instance, it may, therefore, be inappropriate to treat the observed admissions as independent events when calculating directly standardised hospital admission rates.

Instead an alternative approach to calculating confidence intervals is needed. We can assume that individuals are independent of each other, and base the confidence interval calculations on the number of distinct people, while still presenting the rate in terms of numbers of admissions.The overall directly standardised rate is calculated in the same way, based on the number of events (such as admissions).

However, the variance of the DSR needs to be calculated by separating out individuals into those having 1, 2, 3, up to the maximum number \(n\) events.

\[ \operatorname{Var}(DSR) = \frac{1} { \left( \sum_i w_{i} \right) ^2} \sum_j \left( \sum_i \frac{{w_{i}}^2O_{ij}}{{n_{i}}^2}j^2 \right) \] where:

  • \(O_{ij}\) is the observed number of people in age group \(i\) who had \(j\) events
  • \(n_{i}\) is the number of individuals in the local or subject denominator population in age group \(i\), or the population multiplied by the period at risk (‘person-years’, for example)
  • \(w_{i}\) is the number (or proportion) of individuals in the reference or standard population in age group \(i\)
  • \(j\) is the number of events (people with one visit, 2 visits, 3 visits, and so on)

This formula can then replace \(\operatorname{Var}(DSR)\) in the standard equation (Calculation of confidence intervals for directly standardised rates) to give upper and lower confidence limits.

Tools

The following tools are available to calculate directly standardised rates:


Page last updated: August 2024