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Scottish Indices of Deprivation 2003

Chapter 3: Combining the Domain Indices into an overall ward level Index of Multiple Deprivation

Standardising and Transforming the Domain Indices

Having obtained a set of Domain Indices these needed to be combined into an 'overall' Scottish Index of Multiple Deprivation. In order to combine Domain Indices which are each based on very different units of measurement there needed to be some way to 'standardise' the scores before any combination could take place. A form of standardisation and transformation was required that met the following criteria. First it must ensure that each domain has a common distribution; second, it must not be scale dependent (i.e. conflate size with level of deprivation); third, it must have an appropriate degree of 'cancellation' built into it (discussed below); and fourth, it must facilitate the easy identification of the most deprived wards. Having considered other options, the exponential transformation of the ranks best met these criteria.

Other procedures such as z-scores or untransformed ranks are alternative methods of standardisation. Using the ranks for each domain would solve some problems but would introduce others. Ranks would certainly put domains on to the same metric. The problem is that the distance between each of the scores underlying the ranks is not equal. Once ranked this 'distance' is made equal and some of the information of the data is lost. The symmetrical nature of ranks, and 'z scores' of normally distributed data, means that a 'good' score on one domain could fully cancel out a 'bad' score on another. This means that a relative lack of deprivation in one domain, would have had a major impact on a more severe deprivation in another domain, when combined into an overall deprivation result. The model of multiple deprivation proposed instead is cumulative and a 'good' score on one domain should not fully cancel out a 'bad' score on another domain. Z-scores of normally distributed data or untransformed ranks will not therefore serve as standardisation mechanisms.

The exponential distribution used here has a number of properties. First it transforms each domain so that all domains have a common distribution, the same range and identical maximum/ minimum value, so that when the domains are weighted and combined into a single Index of Multiple Deprivation, the impact of the weights is absolutely clear and explicit. Second, it is not affected by the size of the ward's population. Third, it effectively spreads out that part of distribution in which there is most interest - that is that part which contains the most deprived wards in each domain. Fourth, it enables one to determine the desired cancellation properties.

The exponential transformation involves ranking the scores in each domain. The ranking standardises the domain scores (between 1222 for the most deprived and 1 for the least deprived for the purposes of the calculation). These ranks are then transformed to an exponential distribution, using the formula presented in Appendix 5. This has the effect of transforming the ranked domain scores to a value between 0 (least deprived) and 100 (most deprived), on an exponential basis, that is larger (more deprived) scores are given greater emphasis.

The exponential transformation stretches out the distribution at the deprived end of the scale so that greater levels of deprivation score more highly. The most deprived 10% of wards have values between 50 and 100 after exponential transformation.

This issue of cancellation is clearly important for understanding the nature of multiple deprivation. As has been noted in Chapter 1, the approach in the Scottish Indices of Deprivation is to conceptualise the various deprivations as measured by each domain as separate and distinct, though they may have cumulative effects in an area (or for any individual). Thus to be poor and in ill-health is clearly a worse state than experiencing just one of these deprivations on their own. It would be conceptually inappropriate for someone who is poor but healthy to have their income deprivation ignored because they are fortunate enough to be in good health.

The significant advantage of the exponential transformation is that it gives control over the extent to which lack of deprivation in one domain cancels or compensates for deprivation in another domain. In particular, it allows precise regulation (though not the elimination) of these cancellation effects. The exponential transformation has been used in a way that reflects a level of cancellation appropriate to this approach to multiple deprivation.

The exponential transformation formula selected gives approximately 10% cancellation. This means that in the extreme case, a ward which was ranked top on one domain but bottom on another would overall be ranked at the 90th percentile in terms of deprivation (if the two domains were equally weighted). This compares with the 50 ^th percentile if the untransformed ranks or a normal distribution had been used instead. For example a ward that was the most deprived in terms of income deprivation but was least deprived on the Education Domain would still be at the 90 ^th percentile (top 10%) if these two domains were combined with equal weights. In fact income deprivation is weighted more highly, which would further reduce the impact of the non-deprived result for the Education Domain.

Weighting the domains

Weighting always takes place when elements are combined together. Thus if the domains are summed together to create an Index of Multiple Deprivation, this means they are given equal weight. It would be incorrect to assume that items can be combined without weighting.

How can one attach weights to the various aspects of deprivation? That is, how can one determine which aspects are more important than others? As has been shown, simply summing indicators can itself lead to weighting which may be driven more by the availability of indicators rather than from any conceptual model of multiple deprivation.

There are five possible approaches to weighting:

• driven by theoretical considerations
• empirically driven
• determined by policy relevance
• determined by consensus
• entirely arbitrary

Weights driven by theoretical considerations

In the theoretical approach, account is taken of the available research evidence which informs the theoretical model of multiple deprivation and weights are selected which reflect this theory.

Empirical approaches to weighting

There are two sorts of approaches that might be applicable here. First, a commissioned survey or re-analysis of an existing survey might generate weights. Here one might construct a proxy for multiple deprivation or exclusion - perhaps in terms of 'socially perceived necessities' and use multivariate predictive modelling to derive weights. A possible recent data set for re-analysis in this way is The Millennium Poverty and Social Exclusion Survey (Gordon et al, 2000). Second one might apply a technique such as factor analysis to extract a latent 'factor' called 'multiple deprivation' assuming, that is, that the analysis permitted a single factor solution (see Senior, 2002).

Weights determined by policy relevance

It might be that only the individual domain scores could be released and weighted for combination in accordance and (proportion) to the focus of particular policy initiatives or weighted in accordance with public expenditure on particular areas of policy.

Weights determined by consensus

Policy makers and other 'customers' or experts could simply be asked for their views and the results examined for consensus.

Weights that are entirely arbitrary

Simply choosing weights without reference to the above considerations, or even selecting equal weights in the absence of empirical evidence, would come into this category.

Weights for the SIMD

For the SIMD 2003, theoretical considerations prevailed. However - there was a modification to this. For domains with less robust indictors a decision was taken to reduce the weight. A different approach could have been that if the robustness of indicators didn't warrant inclusion with the full desired weight then the domain should be excluded.

The Income and Employment Domains were regarded as the most important contributors to the concept of multiple deprivation and the indicators comprising the domains were very robust. Hence it was decided that they should carry more weight than the other domains. The weightings of the domains is supported by the research team's work, the consultation process with the steering group and, where available, the wider academic literature.

On the second criterion it is important to stress that only indicators which are sufficiently robust have been included within the SIMD 2003. Nonetheless, some indicators are more robust than others, but only those which are sufficiently robust, as well as meeting the other criteria ('domain specific', measuring major features of that deprivation, up-to-date, capable of being updated on a regular basis, available across Scotland at a small area level) have been selected.

Based on these criteria the following weights have been used (weights must total 100%):

Each domain score is ranked and exponentially transformed, to standardise the distribution. The five transformed domain scores for each ward are then summed, using the weights in the table above. Thus, a ward's overall score is:

(0.3 x Income) + (0.3 x Employment) + (0.15 x Education) + (0.15 x Health) + (0.1 x Access)

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