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What are the different measures of effect size used for chi-squared tests / binomial data? I'm already at least somewhat familiar with the phi coefficient and odds ratio. I've heard of Cramer's V and relative risk as well. However, I'm still not sure how to decide which effect sizes to report in different circumstances. What are the arguments for / against the different measures of effect size and in what circumstances are each most informative?

Additionally, though I have resources for calculating confidence intervals for the odds ratio, I have struggled to find information regarding confidence intervals for the other effect size measures.

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I think relative risk is the best index of effect size when using binomial data. Odds ratios are somewhat difficult to interpret (you can read the article entitled "Down with odds ratios!" by Sackett and colleagues). In contrast, relative risk provides an intuitive and easily interpretable value (e.g., one group has 40% greater risk of belonging to group X). Cramer's V is potentially useful, but is also difficult to interpret. Specifically, the meaning of the Cramer's V value is difficult to ascertain without knowledge of how it is calculated (and it is hard to explain in plain English).

I suggest presenting relative risk and absolute risk for maximum interpretability and transparency.

As for confidence intervals, I don't know how to calculate them by hand. Most websites that calculate them for odds ratios also do so for relative risk.

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    $\begingroup$ +1 Just a quick note that odds ratio has the property of "invertibility" if you change the level of the outcome -- e.g. if smoking (compared to not smoking) has an OR of 2 for having disease X, then smoking has an OR of 1/2 for NOT having disease X. This is not true of relative risks (whether this is trivial or not depends on the application.) For a worked example see Wikipedia on odds ratios $\endgroup$ – James Stanley Sep 5 '13 at 5:34

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