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Health Sciences M.Sc. Programme
Applied Biostatistics
Week 6: Proportions, risk ratios and odds ratios
Risk ratio or relative risk
Chi-squared tests are tests of significance, they do not provide estimates of the strength of
relationships. There are different ways of doing this for different kinds of data and sizes of table,
but two are particularly important in health research: the risk ratio or relative risk and the odds
ratio. Both apply to tables with two rows and columns.
Table 1. Cough during the day or at night at age 14 and bronchitis before age 5
Bronchitis at age 5 Cough at
age 14 Yes No
Total
Yes 26
44
70
No 247
1002
1249
Total 273
1046
1319
For example, consider Table 1. We ask: do children with bronchitis in infancy get more respiratory
symptoms in later life than others? We have 273 children with a history of bronchitis before age 5
years, 26 of whom were reported to have day or night cough at the age 14. We have 1046 children
with no bronchitis before age 5 years, 44 of whom were reported to have day or night cough at age
14. We thus have two proportions, 26/273 = 0.095 = 9.5% and 44/1046 = 0.042 = 4.2%.
We can find the difference between them, 0.095 โ 0.042 = 0.053. We can find a standard error for
this, 0.019, and use it to calculate a 95% confidence interval for the difference:
0.053 โ 1.96 ร 0.019 to 0.053 + 1.96 ร 0.019 = 0.016 to 0.090. We could also write this difference
as 9.5% โ 4.2% = 5.3. We call this difference between two percentages 5.3 percentage points
rather than a percentage.
The proportion who cough is called the risk of cough for that population. Thus in Table 1 the risk
that a child with bronchitis before age 5 will cough at age 14 is 26/273 = 0.095, and the risk for a
child without bronchitis before age 5 is 44/1046 = 0.042. The difference is called the absolute risk
difference.
This difference in proportions may not very easy to interpret. The ratio of two proportions is often
more useful. The ratio of the proportion with cough at age 14 for bronchitis before 5 to the
proportion with cough at age 14 for those without bronchitis before 5 is 0.095/0.042 = 2.26.
Children with bronchitis before 5 are more than twice as likely to cough during the day or at night
at age 14 than children with no such history. We call this ratio the risk ratio or relative risk. Both
can be abbreviated to RR.
The standard error for this ratio is complex, and as it is a ratio rather than a difference it does not
approximate well to a Normal distribution. If we take the logarithm of the ratio, however, we get a
difference, because the logarithm of a ratio is the difference between the two logarithms. We can
find the standard error for the log ratio quite easily. For the example the log ratio = 0.817 and the
standard error = 0.238. The 95% confidence interval for the log ratio is therefore
0.817 โ 1.96 ร 0.238 to 0.817 + 1.96 ร 0.238 = 0.351 to 1.283. The 95% confidence interval for
the ratio of proportions itself is the antilog of this, 1.42 to 3.61. Thus we estimate that the
