Sample Size for Test of One Correlation

Example clinical scenario

A researcher wants to test whether serum vitamin D concentration is linearly associated with bone mineral density among postmenopausal women. The null hypothesis specifies no population correlation, and the study is powered to detect a clinically plausible nonzero Pearson correlation.

Formula and methodological note

The calculator preserves the Fisher-z normal-approximation formula for testing one correlation. The null and expected correlations are transformed with Fisher’s transformation and combined with the selected Type I error and power critical values.

It assumes paired continuous measurements, independent participants, a linear association, and correlations strictly between −1 and 1. A one-sided alternative must be justified prospectively.

The method does not address partial correlations, covariate adjustment, clustering, non-linearity, measurement error, missing-data mechanisms, or multiplicity. The result is rounded upward before dropout/non-response inflation.

Method references

1. Fisher RA. On the probable error of a coefficient of correlation deduced from a small sample. Metron. 1921;1:3–32.
2. Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale: Lawrence Erlbaum Associates; 1988.