Scottish Indices of Deprivation 2003
Appendix 1: The 'Shrinkage' Technique
The 'shrunken' estimate of a ward-level proportion (or ratio) is a weighted average of the two 'raw' proportions for the ward and for the corresponding local authority. 9 The weights used are determined by the relative magnitudes of within-ward and between-ward variability.
If the rate for a particular indicator in ward j is r j events out of a population of n j , the empirical logit for each ward is:
| [1] |
whose estimated standard error (s j ) is the square root of:
| [2] |
The corresponding counts r out of n at local authority level give the local authority-level logit:
| [3] |
The 'shrunken' ward-level logit is then the weighted average:
| [4] |
where w j is the weight given to the 'raw' ward- j data and (1-w j) the weight given to the overall rate for the local authority. The formula used to determine w j is:
| [5] |
where t 2 is the inter-ward variance for the k wards in the local authority, calculated as:
| [6] |
Thus large wards, where precision 1/s 2j is relatively large, have weight w j close to 1 and so shrinkage has little effect. The shrinkage effect is greatest for small wards in relatively homogeneous local authorities.
The final step is to back-transform the shrunken logit z j* using the 'anti-logit', to obtain the shrunken ward level proportion:
| [7] |
for each ward.