4.3 Small number problems and smoothing

One problem in identifying areas of high morbidity or mortality occurs in areas where the number of people diagnosed with a disease is low. In such areas, the estimated disease rate is more uncertain because of the small size of the population. In a district with a large population, an incidence rate of 5% might mean that there are 100 actual cases of disease among a population of 2,000. In a small district, the incidence rate of 5% might be derived from 3 cases of the disease among a population of 60. Clearly, the relative risk in the small district is much more uncertain because the population is low. These ‘small number’ problems are more pronounced for diseases that are rare.

In mapping diseases, a trade-off exists between using geographic units that are large enough to give stable morbidity and mortality rates, but small enough to enable the reader to discern geographical patterns. Sometimes, disease data are available for geographic units with small populations such as unit postcodes in the UK . In such circumstances, instead of mapping the raw disease rates directly, ‘smoothing’ may be used to reduce small number problems. Two types of ‘smoothing’ are often used. Global smoothing techniques (such as empirical Bayes smoothing) modify sub-district disease rates so that they are closer to an overall mean rate for the whole study area. In Local smoothing, a local average disease rate is calculated, based on the rate for a given sub-district and its neighbours. This object considers the latter technique – local smoothing.

In local smoothing, a ‘window’ of constant area or population size is moved across the map and the number of disease cases lying within the window is calculated for successive locations. The number of healthy individuals lying within the window is also identified and then a local disease rate is calculated. This process is also known as spatial filtering. Examples of this process may be seen in Ali et al (2002) and Rushton (2003).

How large (in terms of distance or population) does a ‘window’ need to be to prevent small numbers problems? There is no straightforward answer to this problem – if the ‘window’ is too small, small numbers problems will persist but if the ‘window’ is too large, real variation in disease rates may be obscured. However, Talbot et al (2000) provide some guidance on using local smoothing and found that a ‘window’ with a population of 5,000 smoothed out real variation in birthweight data. A ‘window’ with a population in the range 250 – 2,500 maintained real variation in diseases but helped reduce small numbers problems.

Smoothing can also be employed on point data sets of cases and controls as well as areal data. An alternative to displaying disease cases as a dot map is to produce a map of disease intensity in different areas or grid squares. This can prevent disclosure of confidential material about individuals and aid visual interpretation of very large point data sets.