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How to interpret OR (odds ratio) decimals? Are they percentages?

OR with probabilities $p_1,p_2$ for two events is

$$\frac{\frac{p_1}{1-p_1}}{\frac{p_2}{1-p_2}}$$

Like if one gets an OR of, say, $2.25$, then does this mean that the nominator event is 225% more likely than the denominator event? What does "odds" mean?

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  • $\begingroup$ It might help to give you a concrete answer if you said what the events and groups are. As it stands all we can say is that the odds of something in one group are 2.25 times the odds of the something in some other group. $\endgroup$ – mdewey Nov 27 '16 at 12:05
  • $\begingroup$ @mdewey Well in this case I get 2.25 if I take the odds ratio of the chance of getting skin cancer when taking beta carotene ($p_1$) and not taking ($p_2$). $\endgroup$ – mavavilj Nov 27 '16 at 12:37
  • $\begingroup$ So whatever the odds were of getting skin cancer in the beta carotene group (ie the ratio of those who did to those who did not) it was 2.25 times the odds in the group not getting it. $\endgroup$ – mdewey Nov 27 '16 at 13:31
  • $\begingroup$ @mdewey But is odds the same as "2.25 more likely to"? $\endgroup$ – mavavilj Nov 27 '16 at 13:43
  • $\begingroup$ If you measure likeliness in terms of odds then yes, if you measure likeliness in terms of risk then no you need the risk ratio unless you are interested in absolute risk in which case you need the risk difference. $\endgroup$ – mdewey Nov 27 '16 at 13:47
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It's betting odds of 2.25 to 1 (same as 9 to 4). In contrast what you describe is the relative risk ratio.

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  • $\begingroup$ Yea so what does 2.25 to 1 mean? $\endgroup$ – mavavilj Nov 27 '16 at 10:16
  • $\begingroup$ In 9 out of 13 cases the first proportion is expected to be bigger in 4 out of 13 the second proportion. What that means in terms of relative risk ratios depends on the proportions we ate talking about. $\endgroup$ – Björn Nov 27 '16 at 18:24
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The odds of an event are the ratio of the number of times it occurred to the number of time it did not. We can distinguish this from the risk which is the number of times it occurred as a proportion of the total number of trials. In order to compare the likeliness of an event in two groups we can either take the ratio of the odds in those two groups (which is the formula you quote), take the ratio of the risks, or take the difference in risk. All of these are in wide use. Each of them defines a way of interpreting the English phrase 'more likely'.

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