Data I've recently had from a Masters student shows an odd-looking property of
the odds ratio as a measure of effect size. I'd be very interested to know
whether this has ever been commented on before, and if so, any reference.
Please don't reply suggesting to use the relative risk i.e. ratio of proportions
instead - which I realise obviously gets around the problem. Nevertheless it
remains an interesting issue that use of the odds ratio in this way may lead to
a dampened assessment of a risk that is the product of two identifiably
The data comes from a study on year 11 school pupils. The main interest is in
sexual behaviour, with 2 questions
"Have you ever had sex?"
and if they reply positively,
"Last time you had sex, was a condom used?".
Many explanatory variables were also elicited, including self-assessed school
performance. Comparing lowest and highest school performance groups yielded the
Total respondents 212
Never sex 168
Ever sex 44
Latest sex was protected 29
Latest sex was unprotected 15
Total respondents 65
Never sex 23
Ever sex 42
Latest sex was protected 21
Latest sex was unprotected 21
If we use the relative risk to compare the risk in the lowest performance group
relative to the highest performance group:
RR for any sex (42/65)/(44/212) = 3.11
RR for latest sex unprotected, given any sex (21/42)/(15/44) = 1.47
RR for latest sex unprotected as proportion of total respondents
(21/65)/(15/212) = 4.57.
Obviously 4.57 is the same as 3.11*1.47, and appropriately indicates a stronger
effect than either of the component RRs 3.11 and 1.47.
BUT if we calculate the corresponding odds ratios:
OR for any sex (42/23)/(44/168) = 6.97
OR for latest sex unprotected, given any sex (21/21)/(15/29) = 1.93
OR for latest sex unprotected as proportion of total respondents
(21/44)/(15/197) = 6.26.
Here, 6.26 isn't the same as 6.97*1.93 - as is obvious from how these figures
are derived. What is paradoxical is that 6.26 is actually *lower* than 6.97,
even though 1.93 is greater than 1. So the message that in both respects the
high performers engage in less risky behaviour than the low performers is
summarised very neatly by the relative risks, but the odds ratios present a less
Has anyone come across this phenomenon before?
Robert G. Newcombe PhD CStat FFPH
Reader in Medical Statistics
Wales College of Medicine
Cardiff CF14 4XN
Phone 029 2074 2329
Fax 029 2074 2898