To assess sample size for an expected Pearsons correlation ρ₀ between two continuous variables

n = ([z_{1 – α} + z_{1 − β}]²/z₀²) + 3

Where z₀ = 0.5 (ln [1 + ρ₀] − ln [1 − ρ₀])

This tests the hypothesis, H₀: ρ = 0 versus H₁: ρ = ρ₀ > 0

Estimation in R

library(pwrss)
pwrss.z.corr(r = 0.50, 
             r0 = 0, 
             power = 0.95, 
             alpha = 0.05, 
             alternative = "not equal")
Code language: R (r)

Output

 A Correlation against a Constant (z Test) 
 H0: r = r0 
 HA: r != r0 
 ------------------------------ 
  Statistical power = 0.95 
  n = 47 
 ------------------------------ 
 Alternative = “not equal” 
 Non-centrality parameter = 3.605 
 Type I error rate = 0.05 
 Type II error rate = 0.05 

https://cran.r-project.org/web/packages/pwrss/vignettes/examples.html#:~:text=10.1%20One-,Correlation

https://cran.r-project.org/web/packages/pwrss/pwrss.pdf

Useful R Packages

• pwrss
• pwr
• presize – requires shiny, ggplot2, shinydashboard – load and use launch_presize_app()
  • Also see: https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-022-01694-7 Sample size calculation for prevalence studies using Scalex and ScalaR calculators
• epiR
  • https://cran.r-project.org/web/packages/epiR/vignettes/epiR_sample_size.html

REFERENCES

• https://journals.lww.com/crst/fulltext/2020/03040/basics_of_statistics___4__sample_size_calculation.29.aspx
• Hulley SB, Cummings SR, Browner WS, Grady D, Newman TB. Designing clinical research : an epidemiologic approach. 4th ed. Philadelphia, PA: Lippincott Williams & Wilkins; 2013. Appendix 6C, page 79.
• https://www2.ccrb.cuhk.edu.hk/stat/other/correlation.htm#:~:text=Suppose%20one%20wishes%20to%20detect,47%20(n%3D47).
• Cleveland Calculators – https://riskcalc.org/samplesize/
• https://cran.r-project.org/web/packages/pwrss/vignettes/examples.html
• https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-022-01694-7