Population-Sample Regression in the Estimation of Population Proportions

Focusing on a single sample obtained randomly with replacement from a single population, this article examines the regression of population on sample proportions and develops an unbiased estimator of the square of the correlation between them. This estimator turns out to be the regression coefficient. Use of the squared-correlation estimator as a shrinkage coefficient applied to sample proportions in a Bayesian context results in credibility intervals that are narrower, sometimes considerably narrower, than conventional confidence intervals. On illustrative data involving 285 respondents who selected one of two optional responses, the 95% credibility interval was 33% narrower than the corresponding conventional confidence interval, while the regression or shrinkage effect, which decreases with increasing sample size, was about±0.02. In a review of prior work, this article also shows that shrinkage coefficients, applied to multisample means as well as single-sample proportions, correspond generally to estimators of squared population-sample correlations.