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Could someone please explain the purpose of the odds ratio and how it could be interpreted (i.e., the origin of their usage is in primary interest for me)? Why don't people simply use the difference between two proportions instead of odds ratio? Reference to the proper mathematical book will be great.

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One reason odds ratios are used so frequently is that their logarithms have an unrestricted range. That allows one to model covariate effects using ordinary unconstrained maximum likelihood estimation. A second reason is that odds ratios can transport to settings where the base risk is drastically different from the original sample. An odds ratio can be applied to any baseline risk. A risk difference or risk ratio cannot, otherwise you would get probabilities outside [0,1].

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Odds ratio for a success rate of $p$ is defined as $\frac{p}{1-p}$. A nice feature of this ratio is the plain English explanation you can do with it in a betting scenario. As an example, assume $p = 0.8$. Your odds ratio is $\frac{0.8}{1 - 0.8} = 4$. So you could say you are four times more likely to win than loose or vice versa. Lots of horse race betting (and their pay off) also works in the same way.

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    $\begingroup$ This seems only to push the question back one inconsequential step: why, then, do people use odds in betting rather than (say) differences of chances? Moreover, you need to go one step further: once the use of odds to express chances is justified, why would one use odds ratios to compare chances rather than, say, odds differences or something else? $\endgroup$ – whuber Aug 14 '14 at 14:32

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