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It seems clear to me odds ratio ($OR$) is not equivalent to difference in probability and it's not possible to derive the difference knowing nothing but the odds ratio.

As an example, say on a given day cat's have 49% chance of getting sun-burn while dogs have 52% chance. The difference between cats and dogs is 3 percentage points or $\frac{3}{49} \approx 6.12\%$ increase in probability for dogs over cats.

Meanwhile the odds ratio is $\dfrac{\frac{0.52}{1-0.52}}{\frac{0.49}{1-0.49}} \approx 1.128$

However I am finding a lot of social science literature intpreting $OR-1$ as equivalent to increase in probability. Example:

children whose parents were living together at the index observation were 5.4% more likely to be male than children whose parents were living apart (odds ratio of 1.054)

What am I missing?

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You are not missing anything, it is (some) social science literature which is wrong.

A bit more:

Saying that the odds ratio of 1.054 suggests a 5.4% increase of probability would imply that the odds ratio is in fact a probability ratio (which is it not).

Here insert some rant about statistics education in social science.

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    $\begingroup$ The mistake is common, but not nearly as common as you suggest. The confusion is one of terminology. In the social sciences it is pretty common to us "likely" as the generic term that could encompass both probability and odds. In that case the statement is correct, and one that often reads easier. So, in my subfield that is why that statement is pretty common, even though they are very aware of what odds and odds ratios are. $\endgroup$ – Maarten Buis Sep 1 '16 at 18:05
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    $\begingroup$ If the probability is very low, then it is almost correct, since the probability that is very close to 1 will not change very much. This can only really be the case when the intercept term is strongly negative to begin with. $\endgroup$ – Max Candocia Dec 23 '17 at 1:09

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