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Bootstrap standard error and confidence intervals for the correlation corrected for range restriction: a simulation

Wai Chan1, Daniel W-L Chan

  • 1Department of Psychology, The Chinese University of Hong Kong, Shatin, Hong Kong. wchan@psy.cuhk.edu.hk

Psychological Methods
|September 10, 2004
PubMed
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The bootstrap standard-error estimate improves accuracy for correlation corrected for range restriction, especially with small sample sizes and normal data. It offers a more reliable alternative to traditional methods for estimating population correlation (rho).

Area of Science:

  • Statistics
  • Psychometrics
  • Quantitative Psychology

Background:

  • The standard Pearson correlation coefficient is biased when predictor and criterion variables are range restricted.
  • The correlation corrected for range restriction (rc) is recommended, with available standard error estimation formulas.
  • Existing methods for estimating standard errors of rc may lack accuracy, particularly in certain conditions.

Purpose of the Study:

  • To propose and evaluate the bootstrap standard-error estimate as an alternative for the correlation corrected for range restriction (rc).
  • To compare the performance of the bootstrap estimate against traditional methods using Monte Carlo simulations.
  • To assess the accuracy of bootstrap confidence intervals for estimating the population correlation (rho).

Main Methods:

Related Experiment Videos

  • Monte Carlo simulation studies were employed.
  • Data included both normal and nonnormal distributions.
  • Simulations varied levels of population correlation (rho), selection ratio, sample size, and truncation types.

Main Results:

  • The bootstrap standard-error estimate demonstrated higher accuracy than the traditional estimate for normal data, especially with small sample sizes.
  • For nonnormal data, the performance of both estimates was highly dependent on the specific distribution type.
  • Bootstrap bias-corrected and accelerated (BCa) intervals consistently yielded the most accurate coverage probabilities for rho.

Conclusions:

  • The bootstrap method offers a more accurate standard-error estimate for correlation corrected for range restriction compared to traditional methods, particularly under normal data conditions and small sample sizes.
  • Careful consideration of data distribution is necessary when applying either bootstrap or traditional methods to nonnormal data.
  • Bootstrap bias-corrected and accelerated intervals are recommended for robust and accurate estimation of population correlation (rho) in range-restricted settings.