Sample size for a correlation study (Fisher z method)

The sampling distribution of a correlation coefficient is skewed, especially for large r, so the formula works on a transformed scale. Fisher’s transformation z′ = 0.5 ln((1 + r)/(1 − r)) (equal to arctanh r) is approximately normal with variance 1/(n − 3), whatever the true correlation. To test the null hypothesis of zero correlation against a true value r with two-sided alpha and power 1 − β, you need the standardised effect C√(n − 3) to equal z₁₋α/2 + z₁₋β. Rearranging gives n = ((z₁₋α/2 + z₁₋β) / C)² + 3 (Cohen; Chow, Shao and Wang). For alpha 0.05 two-sided and 80% power the two quantiles are 1.960 and 0.842, so z₁₋α/2 + z₁₋β = 2.802.

Say a study expects a correlation of 0.3 between a blood biomarker and six-minute walk distance. Then C = 0.5 × ln(1.3 / 0.7) = 0.3095, and n = (2.802 / 0.3095)² + 3 = 81.9 + 3 = 84.9, which rounds up to 85. A simulation of 20,000 studies with bivariate normal data and a Pearson t-test at n = 85 gave 80.3% power, with 84 giving 80.5% and 86 giving 81.0% (differences within simulation error), so the formula is accurate here. It also holds at the extremes: n = 29 for r = 0.5 (81.0% simulated) and n = 783 for r = 0.1 (80.8%). If one-sided testing is justified in the protocol, use z₁₋α instead of z₁₋α/2, which for r = 0.3 lowers n to 68; most protocols and journals expect two-sided.

Enrolment is a separate step. Divide the evaluable number by (1 − expected dropout): 85 / 0.9 = 94.4, so enrol 95. The dropout-adjusted sample size calculator explains why dividing is correct. If the real question is whether a correlation differs from a non-zero value r₀ (for instance, 0.4 against 0.2), replace C with C(r) − C(r₀): the result is 164 participants for that example.

Often the sample is fixed by cost or by the number of eligible people and the question is what it can find. Invert the formula: with 50 participants, 80% power and two-sided alpha 0.05, the smallest correlation you can reliably detect is about r = 0.39. With n = 60 the power to detect r = 0.3 is only 64.7%, and with n = 100 it is 86.2%; with n = 40 it falls to 46.9%. The power analysis calculator handles the same question for means and proportions, and the pilot study sample size calculator discusses studies whose purpose is estimation rather than a test.

Power to reject zero is not the same as precision. At n = 85 the 95% confidence interval around an observed r of 0.3 is 0.09 to 0.48 (back-transforming 0.3095 ± 1.96/√82). If the aim is to estimate the correlation well, not just to show it is nonzero, choose n to give an acceptable interval width and report the interval, not only the p-value. A correlation of 0.3 also explains only 9% of the variance, so say in the protocol why a correlation of that size would matter clinically.

The Fisher z method assumes pairs from independent participants, a roughly bivariate normal distribution, a linear relationship and one pair of values per person. Repeated measurements on the same people, or participants clustered within sites or families, break independence and need a design effect (the cluster-randomised sample size calculator shows one). A strong outlier can create or hide a correlation, so look at a scatterplot before and after.

For skewed or ordinal data a Spearman rank correlation is often planned. Bonett and Wright (Psychometrika 2000) give a simple adjustment: the variance of the transformed Spearman coefficient is about (1 + r²/2)/(n − 3), so the sample size is (1 + r²/2) × ((z₁₋α/2 + z₁₋β) / C)² + 3. For r = 0.3 this gives 88.6, so 89 participants rather than 85; for r = 0.5 it gives 33 rather than 30. A simulation using bivariate normal data with the matching Pearson correlation confirmed the adjusted n (81.6% at r = 0.3, n = 89, and 84.8% at r = 0.5, n = 33, so the adjustment is slightly conservative). If the data are strongly non-normal, run a simulation under the distribution you expect, since this formula no longer applies exactly.

Finally, a correlation is not a treatment effect. Observing r = 0.3 between a biomarker and an outcome does not show that the biomarker causes anything, and adjusting for confounders needs a regression, which has its own sample size rules (often a minimum number of participants per predictor). A statistician should confirm the design, the target correlation and the analysis.

ICH E9 asks the protocol to state the sample size, how it was determined and the assumed values. A sentence that does this reads: “A sample of 85 evaluable participants provides 80% power to detect a Pearson correlation of 0.30 between baseline biomarker concentration and six-minute walk distance at the two-sided 5% level (Fisher z method). Allowing for 10% dropout, 95 participants will be enrolled.” Put the same numbers in the statistical analysis plan, with the reason the target r is the smallest one that matters. The how to write a clinical trial protocol guide places the section in context, and the sample size calculation entry defines the terms.

In Capture, both variables become numeric eCRF fields with units and range edit checks, so an implausible biomarker value raises an auto-query while the participant is still at the site. Calculated fields can derive composite scores during entry, visit windows keep the two measurements close in time, the field-level audit trail records every change, and CSV or Excel exports arrive with a data dictionary, or as SDTM datasets in SAS XPT files with Define-XML, ready for the correlation analysis. The monitoring dashboard shows enrolment against the 95 you planned. For group comparisons see the two-means sample size calculator, and for repeated measures per participant the repeated measures sample size calculator.