####
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.

Back to SMR exercise

*An Introduction to Medical Statistics
*contents

Reports on deaths
associated with volatile substance abuse

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