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Area of Science:

  • Statistics
  • Econometrics
  • Psychometrics

Background:

  • Existing textbooks often provide incorrect formulas for standard error estimates of standardized regression coefficients.
  • Current analytic methods like the delta method and covariance structure modeling are insufficient for unstandardized regression coefficients involving products of Z-scores.
  • Nonparametric bootstrap procedures are currently recommended for testing the significance of unstandardized regression coefficients with interaction terms.

Purpose of the Study:

  • To propose a simple analytic approach for calculating standard error estimates.
  • To extend applicability to both standardized regression coefficients (without interaction terms) and unstandardized regression coefficients (with interaction terms of Z-scores).
  • To compare the performance of the proposed analytic approach against existing methods and simulation studies.

Main Methods:

  • Development of a novel analytic approach for standard error estimation.
  • Comparison with the delta method, covariance structure modeling, and nonparametric bootstrap procedures.
  • Evaluation through numerical examples and simulation studies at finite sample sizes.

Main Results:

  • Regular regression performs adequately only with small predictor variable variances.
  • The proposed analytic approach demonstrates good performance for sample sizes of 200 and above.
  • The nonparametric bootstrap procedure exhibits near-perfect performance across all tested conditions.

Conclusions:

  • The proposed analytic approach provides a viable method for estimating standard errors, particularly for regression models with interaction terms.
  • While effective, the analytic approach has limitations at smaller sample sizes compared to the nonparametric bootstrap.
  • The nonparametric bootstrap remains a robust and highly accurate method for significance testing in these complex regression scenarios.