Indirectly standardised rate (ISR) is given by
\(ISR=\frac{O}{E}\times 100 = \frac{\sum O_{ij}\\_{ij}}{\sum E_{ij}\\_{ij}}\times 100 = \frac{\sum O_{ij}\\_{ij}}{\sum n_{ij} \lambda_{ij}\\_{ij}}\times 100\)
where
Oij is the observed number of events in the subject population for each combination of age group i and sex j
Eij is the expected number of events in the subject population for each combination of age group i and sex j
nij is the number of individuals in the subject population for each combination of age group i and sex j
λij is the crude age-specific rate in the standard population for each combination of age group i and sex j
The 95% confidence intervals are given by
\(ISR_{lower}=\frac{\text{O }_{ lower}}{E}\)
\(ISR_{upper}=\frac{\text{O }_{ upper}}{E}\)
where
Olower and Oupper are the lower and upper confidence limits for the observed number of events.
Using Byar's method, the 95% confidence limits for the observed number of events are given by
\(O_{lower}=O\times\begin{pmatrix}1-\frac{1}{9O}-\frac{z}{3\sqrt{O}}\end{pmatrix}^3\)
\(O_{upper}=(O+1)\times\begin{pmatrix}1-\frac{1}{9(O+1)}-\frac{z}{3\sqrt{(O+1)}}\end{pmatrix}^3\)
Where z is the 97.5th percentile of the standard Normal distribution (1.96)
For small numerators, Byar's method can be less accurate and an exact method based on the Poisson distribution is used for all numerators less than 389.
Using the link between the Poisson and χ2 distributions, the equations for Olower and Oupper above can be replaced by
\(O_{lower}=\frac{\chi^2\text{ }_{ lower}}{2}\)
\(O_{upper}=\frac{\chi^2\text{ }_{ upper}}{2}\)
where
χ2lower is the 100(1–0.05/2)th percentile value from the χ2 distribution with 2O degrees of freedom
χ2upper is the 100(0.05/2)th percentile value from the χ2 distribution with 2O+2 degrees of freedom