A recent conversation about the membership of the ^{2}gether NHS Foundation Trust provided new insight into the membership figures that are published each year in the trust’s Annual Report, when a governor colleague made a clever suggestion that a simple experiment now shows is probably right.

Anyone in England who has an interest in the work of the trust can become a member. (If you are not already a member, you can join online here: Membership form.) However, most of the members live in Gloucestershire, the trust’s home county, as you would expect. Not all that many live in Herefordshire, the other county where the trust provides mental health services.

Mainly for the purposes of electing governors, members are allocated to constituencies. The constituencies match the six local government districts of Gloucestershire, plus a constituency called Greater England for everyone else (including Herefordshire).

This post is about the six Gloucestershire constituencies, or districts.

### The problem

The problem with the membership figures is that there seem to be far too many members in Gloucester (1,335), and far too few in Cotswold (381). If the Gloucestershire members were equally distributed, there would be 748 in each constituency.

Here’s a chart showing the difference between the actual number in each constituency and 748. Constituencies with “too many” members (like Gloucester) are shown in green, while constituencies with “too few” (like Cotswold) are shown in red:

A statistical measure of how badly this equal distribution model fits the data is given by its standard deviation, which is a whopping 40% (of the average membership of a constituency). A perfect fit would have a standard deviation of zero.

### The usual solution

The usual line of reasoning within the trust is that membership depends on the population. The Gloucester population (118,000) is much greater than the Cotswold population (84,000), so it’s said to be understandable that Gloucester has more members.

It’s easy to calculate how many members each constituency “should” have based on its population. Gloucester, for example, should have 891 (when it actually has 1,335), and Cotswold should have 634 (when it actually has 381). So there are still too many members in Gloucester and too few in Cotswold.

Here’s a chart showing the difference between the actual number in each constituency and the number it should have according to the population model. You can see that the deviations are not so extreme. Indeed, Tewkesbury is spot on with 618 members when the model says it should have 619 — therefore its bar is too small to be visible on the chart:

You can see from the chart that this model reduces the size of the bars somewhat. This model’s standard deviation is 29%, which is better but still not good.

### An improved solution

A better way of thinking about things, it turns out, is to relate membership to the amount of mental illness in the community, which differs between districts. This was my colleague’s clever suggestion.

It’s hard to find health statistics that tell you about differences between districts. Most only go to county level, if that. One exception is the health profiles that Public Health England published last year. They contain 32 health indicators at district level, and one of those indicators is a mental health indicator.

Using this indicator the picture is a lot different. It predicts that Gloucester should have 1,322 members (when it actually has 1,335) and Cotswold should have 410 members (when it actually has 381). The figures are pretty close.

This model makes the Forest of Dean look as if it has too many members, predicting 435 when it actually has 570, an excess of 135 members. And it makes Cheltenham look as if it has too few, predicting 960 when it actually has 806, a shortfall of 154 members. Here’s the chart:

Overall, the model’s standard deviation is only 12%, less than half that of the population-based model. This method of making comparisons between constituencies gets much closer to the actual membership numbers.

It looks as if the trust would be better to base its discussions about membership numbers on an estimate of the incidence of mental illness in each district, not on population. And perhaps recruitment efforts should focus on Cheltenham, not Cotswold.

### The proxy

The mental health indicator in Public Health England’s health profiles happens to be based on the number of hospital admissions for self harm in 2011-12. It might not be optimal to use this particular indicator as a proxy for the overall incidence of mental illness. I’ve only used it here because it was the thing that came most easily to hand. There might be data available to the trust that would provide an even better model of membership.

On the other hand, there will always be a need to use *some* proxy indicator or other for this kind of exercise. There’s no single perfect indicator for the overall incidence of mental illness, and no perfect way to predict membership.