A nonsensical formula for the differential funding of police forces

9 Nov 15

The recent row over changes to the police funding formula has ignored the fact that the statistical methods underpinning it are illogical and flawed.

My response to the Home Office’s appeal for views about its new police funding formula is parked here ‘for the record’. It had to be polite, and it had to be academically watertight so that any professional statistician the Home Office might consult can approve it. It is therefore unreadable!

It nevertheless established, rather painfully, that the Home Office’s proposal falls at the first fence, breaking the first bit of the department’s first principle for the design of a funding formula – that it be ‘robust’ in the sense of being ‘analytically sound’. However you interpret that phrase, anything seriously lacking in logic cannot be ‘analytically sound’. The word ‘logic’ and the like is nowhere to be found in the Home Office’s consultation paper, in contrast to the Police Federation’s first benchmark for a high-quality funding formula – strong underlying logic.

The formula is like a pair of shoes made by two different cobblers with very different skills, sticking to their lasts in neighbouring villages. The cobbler making the left shoe claims to be skilled in the choice and cut of the constituent leathers. The one making the right shoe claims to know how to join them up. They do not communicate.      

The leathers are the five ‘indicators’ of police-force workload. These are: population; inverse of band-D property density; number of poor households; a proxy for ‘hard-pressed’ population; and bar density. Three of these workload indicators are genuinely volumes, but two are not.

The stitching corresponds to the deployment of ill-understood statistical methods. The crucial one is Principal Component Analysis (PCA).

The media noise about the formula has come from those who question the choice of indicators. There has been almost total silence about the statistical methods, and no one has yet explained in what way the proposal is ‘deeply flawed’. This is how the logic went AWOL. Five numbers (‘weights’) are to be extracted by PCA from the five by five correlation matrix of the indicators, and for no good reason adopted as the indicators’ shares of total funding. Each police force is to get a share of each of those shares, determined by its contribution to the total ‘volume’ of all 43 forces.   

The indicator variables are positively correlated, which the Home Office takes to mean that there is a proper measure of workload trying to get out, and that PCA will extract it as a linear combination of the indicator variables – after each variable has been rescaled to have unit variance. There is again no good reason for the rescaling other than that it conveniently disposes of a tricky question – whether, for example, to specify population with spurious precision or as the number of thousands. 

The rescaling is the first bit of AWOL. For no good reason, the weights in the linear combination are to be the weights for the division of total funding. The Home Office could have tested the logic of that, by asking what the weights would be with just two indicators. Mathematical statistics gives the answer – they are always 50%/50%, whatever the indicators! The Home Office could have been curious about why, for the only test it did make, the weights 24%, 25%, 25%, 16% and 10% turned out to be so close to equality. The fact is that the proposed weights are purely mathematical artefacts that should play no part in reforming the formula. If the Home Office had known that, would it have promised that “further work will … refine these weights before the model is introduced” ?

The second bit of AWOL is that indicator weights cannot be logical imperatives, but have to be value-judgements related to costs, complementing the judgements that need to be made in the choice of indicators. Agreement on value-judgements is, however, difficult to achieve, which is why formula makers may prefer the silence that mathematical complexity can induce.

The Home Office has mended its ways in some respects, but it has a long way to go. Chapter 6 and Annex B of the consultation document together reveal that the department has enough technical expertise to generate interesting errors of statistical logic. Annex E shows that the same expertise is unable to detect such errors in other funding formulas considered relevant to the police case. What is particularly disturbing, however, is that the report ignores the well-documented critiques of those formulas in the statistical and financial management journals of learned societies. It seems that genuinely independent external comment has failed to have any impact on the formula-making machinery of government departments.

The revenue support grant and clinical commissioning formulas have dominated allocations for decades. It is almost certain that hundreds of billions have been misallocated on both sides of postcode boundaries. The Home Office proposal may also have damaging and costly financial consequences if it is adopted. The reform the department should be pursuing is not the current formula but the machinery that generated it and its proposed replacement. I fear that implementing the Home Office proposal would simply trigger a fresh round of ultimately futile contestation.

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