Health and Care of People with Learning Disabilities Experimental Statistics 2020 to 2021

Indicator IDs and descriptions can be found in the Business Rules.

Details of the indicators relating to each age band are provided as part of the publication.

The proportion p is given by \(p = \frac{O}{n}\)

where: 

O is the numerator observed number of individuals in the sample/population having the specified characteristics.

n is the denominator total number of individuals in the sample/population.

The 95% confidence intervals are given by

\(p_{lower}=\frac{(2O+z^2-z\sqrt{z^2+4Oq})}{2(n+z^2)}\)

\(p_{upper}=\frac{(2O+z^2+z\sqrt{z^2+4Oq})}{2(n+z^2)}\)

where: q is 1-p

z is the 97.5th percentile of the Standard Normal Distribution (1.96)

The rate of events r is given by: \(r=\frac{O}{n}\)

where

O is the numerator observed number of individuals in the sample/population having the specified characteristics.

n is the denominator population-years at risk.

The 95% confidence intervals are given by

\(r_{lower}=\frac{\text{O }_{lower}}{n}\)

\(r_{upper}=\frac{\text{O }_{upper}}{n}\)

where O_lower and O_upper 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 X² distributions, the equations for O_lower and O_upper above can be replaced by

\(O_{lower}=\frac{\chi^2\text{ }_{ lower}}{2}\)

\(O_{upper}=\frac{\chi^2\text{ }_{ upper}}{2}\)

where

χ²_lower is the 100(1–0.05/2)th percentile value from the χ² distribution with 2O degrees of freedom

χ²_upper is the 100(0.05/2)th percentile value from the χ² distribution with 2O+2 degrees of freedom

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

O_ij is the observed number of events in the subject population for each combination of age group i and sex j

E_ij is the expected number of events in the subject population for each combination of age group i and sex j

n_ij is the number of individuals in the subject population for each combination of age group i and sex j

λ_ij 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

O_lower and O_upper 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 O_lower and O_upper above can be replaced by

\(O_{lower}=\frac{\chi^2\text{ }_{ lower}}{2}\)

\(O_{upper}=\frac{\chi^2\text{ }_{ upper}}{2}\)

where

χ²_lower is the 100(1–0.05/2)th percentile value from the χ² distribution with 2O degrees of freedom

χ²_upper is the 100(0.05/2)th percentile value from the χ² distribution with 2O+2 degrees of freedom