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Subject:

Odds ratios - an odd-looking lack of transitivity

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Mon, 13 Dec 2004 12:46:48 +0000

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 ```Dear Colleagues, 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 meaningful proportions. 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 following. Highest performance: Total respondents 212 Never sex 168 Ever sex 44 Latest sex was protected 29 Latest sex was unprotected 15 Lowest performance: 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 cogent message. Has anyone come across this phenomenon before? Robert G. Newcombe PhD CStat FFPH Reader in Medical Statistics Wales College of Medicine Cardiff University Heath Park Cardiff CF14 4XN Phone 029 2074 2329 Fax 029 2074 2898 http://www.cardiff.ac.uk/medicine/epidemiology_statistics/research/statistics/newcombe.htm ```