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I'm looking at some NHS healthcare data on the number of deaths in England

The measure i'm looking at is called the SHMI - it's simply:

The number of observed deaths at a hospital / The expected number of deaths

To get the expected number of deaths, 140 statistical models are used, and there is overdispersion naturally, since deaths can be influenced by factors not included in the models for expected deaths - road accidents etc. So we have greater variability in the expected deaths than would be accounted for in the models

I understand this concept

The hospitals are examined as whether their SHMI scores are within the expected control limits:

"The 95 per cent control limits are calculated using an additive random-effects model with a 10 per cent trim for over-dispersion"

additive affects model is used because the models use groups of patients to get expected deaths, but these groups' composition may skew the expected results, so we want to account for the variability

Following on we get to this, which is where i'm lost:

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Why are they taking the natural log of the SHMI score? Why do you need to know how multiples of e gets you the SHMI score?

Can someone say in plain English what they're ranking and excluding the top and bottom 10% of

Thanks in advance

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I can't explain everything in your quotations, but in general logs are used to turn additive effects into multiplicative effects. See lognormal.

Lognormal data has a long right tail -- i.e. most values are lower but some values are very high. Think contract values, incomes, house prices, etc. So if the log'd data is normalish (unimodal, not too peaked, symmetrical, etc, etc), it's easier to work in log space. For example, a z-score is the centered data (mean subtracted) over the standard deviation. Both of these concepts make sense with normal data and not so much with non-normal data. (OLS regression requires normally-distributed error terms.)

The log'd data is thus hopefully normal-shaped and thus symmetrical, so the control limits -- and chopping 10% symmetrically off of both tails -- all make sense, along with mean, standard deviation, etc. They exponentiate (reversing the log) back into the data's actual space, and the control limits (calculated in log space) will no longer be symmetrical. The upper bound will be further right.

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  • $\begingroup$ Thanks, I think I understand what they're doing now. Does this explanation i've written seem logical to you: "Overdispersion is tackled through taking the log of the SHMI. We take the log so that we can get a normal(ish) distribution in order to remove the top 10% and bottom 10% percent. Then the exponentiate is taken to put the points back into their original units"? $\endgroup$ Oct 24, 2018 at 14:02
  • $\begingroup$ @MassiveOwl I wouldn't say the log is meant to deal with overdispersion. The whole procedure is done to deal with overdispersion, but the logs are taken because of the lognormal distribution of the data on which the procedure is being done. Once log'd, the data should be normal and concepts like symmetrical tails, mean, standard deviation, etc, become applicable. To actually use the results, you exponentiate to return your bounds to the original data's units. $\endgroup$
    – Wayne
    Oct 24, 2018 at 20:41

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