Effect size converter

Inverse-variance pooling only works when every study speaks the same language. A meta-analysis cannot average an odds ratio from one trial against a standardized mean difference from another and a correlation from a third, because the three numbers live on incompatible scales. An effect size converter rewrites each result as its equivalent on one chosen metric, so the whole evidence base can enter a single analysis. It is the bridge between the messy reality of how studies report and the tidy uniformity that pooling demands, and it sits inside the broader workflow described in our guide to working with effect sizes across study designs.

This tool uses Cohen’s d as a hub. Whatever you enter is first reduced to d, and the log odds ratio, the odds ratio, and the correlation are then derived from it using the standard Borenstein conversion formulas. Routing everything through a single hub keeps the conversions internally consistent, so a round trip from one scale to another and back returns the value you started with.

The conversions and their assumptions

The link between the standardized mean difference and the log odds ratio is the Cox logistic approximation: it treats a binary outcome as a coarse view of an underlying continuous trait with a logistic distribution. The link between d and the correlation assumes the two groups are roughly equal in size and the relationship is approximately linear. Neither is exact, and both degrade when their assumptions are stretched, so a converted value is an estimate rather than a measurement. When some studies report binary outcomes you may want to build those from raw counts first using the odds ratio and risk ratio calculator before converting.

Choosing the target scale

Pool on whatever metric most of your studies report naturally, then convert the minority into it. If the bulk of the evidence is binary, the log odds ratio is the sensible hub; if most outcomes are continuous, standardise to a mean difference. The aim is to minimise the number of conversions, because every conversion injects a little approximation error. When you have a mix of continuous scales to standardise, the standardized mean difference calculator produces the per-study inputs you will then keep on one scale.

A worked example

Imagine a review where most trials report a standardized mean difference but one large trial reports only an odds ratio of 1.85. Take the natural log to get a log odds ratio of about 0.615. Multiply by the square root of three divided by pi, roughly 0.5513, and you arrive at a Cohen’s d near 0.34. That converted d, with its standard error transformed the same way, now joins the other standardized mean differences in one pooled analysis. As a sanity check, converting that d to a correlation gives d divided by the square root of d squared plus four, around 0.17, a small positive association consistent with the modest odds ratio you began with. Once every study is on one scale, feed them into the pooled meta-analysis calculator to produce the combined estimate.

Common mistakes

• Converting the estimate but not its uncertainty. Reusing the original standard error after rescaling the point estimate corrupts the weighting and the pooled interval.
• Treating an approximation as exact. The Cox and correlation conversions rest on assumptions; report that conversions were used and probe their influence.
• Chaining unnecessary conversions. Each step adds error, so convert directly to the target scale rather than hopping through several.
• Ignoring the sign. A favourable direction on one scale can flip when the outcome is framed as benefit versus harm; check that every converted effect points the same way before pooling.
• Forgetting to test robustness. Where converted studies carry real weight, confirm the result holds with a sensitivity analysis that drops the converted studies.