Effect Size Analysis

The choice depends on your design. Cohen's d is standard for two-group mean comparisons. Pearson's r or r-squared suits correlations and simple regressions. Eta-squared or omega-squared fits ANOVA designs. Cramer's V or phi is appropriate for chi-squared tests on categorical data. Hedges' g is preferred over Cohen's d when sample sizes differ across groups.

Omega-squared is preferred for inferential reporting because eta-squared is positively biased — it overestimates the population variance explained, especially in small samples. Eta-squared equals the sample proportion of variance explained and is useful for descriptive purposes, but omega-squared or intraclass correlation provides a less biased population estimate.

Yes. Effect sizes and p-values answer different questions. The p-value addresses whether the observed result is consistent with the null hypothesis given sampling variability. The effect size addresses how large the result is. Both are needed: a significant p with a tiny d suggests a real but negligible effect; a large d with a non-significant p suggests the study was underpowered.

Yes. Cohen's d can be recovered from a t statistic and sample sizes; r from t or F; eta-squared directly from the ANOVA table. Many meta-analytic calculators (e.g., the compute.es R package or the online effect size calculators) automate these conversions.

Cohen's small/medium/large benchmarks were calibrated on social-science data and should be treated as rough defaults, not rigid rules. In high-stakes clinical or safety contexts, an effect of d = 0.2 may be very important. In cognitive psychology where effects are typically large, d = 0.5 may be unremarkable. Always contextualise against prior literature in your specific field.