Confidence Interval Width for Pearson’s Correlation Coefficient: A Gaussian‐Independent Estimator Based on Sample Size and Strength of Association

Abstract

Core Ideas The confidence interval (CI) of Pearson’s correlation coefficient (r) was investigated.Confidence interval width is inversely proportional to r and sample size (n).It is recommended to use 1000 or more bootstrap replicates in order to not underestimate CI width (CIw).A model to estimate CIw as a function of n and r is proposed. The nonparametric bootstrap percentile method has been widely used to estimate confidence intervals (CI) for Pearson’s product‐moment correlation coefficient (r). However, because most studies provide results for specific crops and pre‐stablished CIs, an innovative approach to CI estimation is needed. The aim of this study was to propose a model that predicts CI width (CIw) as a function of the sample size (n) and the strength of association among traits. Additionally, we also investigated the extent to which the number of bootstrap replicates (BRs) influences CI estimation. Seventy‐eight different r magnitudes from a maize field experiment were used. The 95% CI half‐width for each trait combination was estimated based on 991 different sample sizes and seven different numbers of BRs. A simple nonlinear model with n and r as predictors is proposed for estimating the CIw: , where δ, β0, and β1 are the model coefficients. Based on our data, the fitted model was: . This model exhibited excellent goodness of fit (R2 = 0.988; root mean square error [RMSE] = 0.011). Considering an assumed magnitude of association (r), the n for a desired CIw can then be calculated as: . We also recommend using ≥1000 BRs, to prevent underestimating CIw. Finally, we present an intuitive table that provides previously estimated n for 9 levels of half‐widths for 95% CIs (0.05, 0.1,... 0.45) and 19 magnitudes for r (0.05, 0.10,..., 0.95).