#### Extract from An Introduction to Medical Statistics by Martin Bland

These are the solutions to the exercise as given in An Introduction to Medical Statistics.

### Solution to Exercise 16E: SMRs

1. We obtain the rates for the whole period by dividing the number of deaths in an age group by the population size. Thus for ages 10-14 we have 44/4271 = 0.01030 cases per thousand population. This is for a 13 year period so the rate per year is 0.01030/13 = 0.00079 per 1000 per year, or 0.79 per million per year. The table shows the rates for each age group.
```     ---------------------------------
Age      Great Britain A.S.M.R.s
group   per million  per thousand
per year     per 13 years
---------------------------------
0-9      0.00         0.00000
10-14     0.79         0.01030
15-19     2.58         0.03358
20-24     0.87         0.01137
25-29     0.32         0.00415
30-39     0.08         0.00108
40-49     0.03         0.00033
50-59     0.09         0.00112
60+       0.03         0.00037
---------------------------------
Total
---------------------------------```
The rates are unusual because they are highest among the adolescent group, where mortality rates for most causes are low. Anderson et al. (1985) note that `... our results suggest that among adolescent males abuse of volatile substances currently account for 2% of deaths from all causes ...'. The rates are also unusual because we have not calculated them separately for each sex. This is partly for simplicity and partly because the number of cases in most age groups is small as it is.

2. The expected number of deaths by multiplying the number in the age group in Scotland by the death rate for the period, i.e. per 13 years, for Great Britain.

```     ----------------------------------------------
Great Britain
deaths per      Scotland      Scotland
Age     thousand        population    expected
group   per 13 years    (thousands)   deaths
----------------------------------------------
0-9      0.00000          653         0.00000
10-14     0.01030          425         4.37750
15-19     0.03358          447        15.01026
20-24     0.01137          394         4.47978
25-29     0.00415          342         1.41930
30-39     0.00108          659         0.71172
40-49     0.00033          574         0.18942
50-59     0.00112          579         0.64848
60+       0.00037          962         0.35594
----------------------------------------------
Total                                 27.19240
----------------------------------------------```
We then add these to get 27.19 deaths expected altogether. We observed 48, so the SMR is 48/27.19 = 1.77, or 177 with Great Britain as 100.

3. We find the standard error of the SMR by root(O)/E = root(48)/27.19 = 0.2548. The 95% confidence interval is then 1.77 - 1.96*0.2548 to 1.77 + 1.96*0.2548, or 1.27 to 2.27. Multiplying by 100 as usual, we get 127 to 227. The observed number is quite large enough for the Normal approximation to the Poisson distribution to be used.

4. Yes, the confidence interval is well away from zero. Other factors relate to the data collection, which was from newspapers, coroners, death registrations etc. Scotland has different newspapers and other news media and a different legal system to the rest of Great Britain. It may be that the association of deaths with VSA is more likely to be reported there than in England and Wales.

Reference
Anderson, H.R., MacNair, R.S., and Ramsey, J.D. (1985) Deaths from abuse of substances: a national epidemiological study. British Medical Journal 290, 304-7.