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Sample Size Calculation for Estimating or Testing a Nonzero Squared Multiple Correlation Coefficient.

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This study introduces precise methods for sample size calculation in hypothesis testing and interval estimation for the squared multiple correlation coefficient. These methods ensure accurate statistical power and confidence interval width for multivariate normal distributions.

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Area of Science:

  • Statistics
  • Multivariate Analysis
  • Psychometrics

Background:

  • Hypothesis testing and interval estimation are crucial for understanding relationships in multivariate data.
  • The squared multiple correlation coefficient (R²) is a key measure of effect size in regression models.
  • Accurate sample size determination is essential for reliable statistical inference.

Purpose of the Study:

  • To develop and present exact methods for sample size calculation for one-sided hypothesis tests concerning the squared multiple correlation coefficient (R²).
  • To provide a method for sample size calculation to achieve a specified expected width for confidence intervals of R².
  • To evaluate the performance of one-sided tests and confidence intervals for R² in multivariate normal distributions.

Main Methods:

  • Derivation of exact sample size formulas for one-sided tests of R² with specified power.
  • Development of sample size calculation methods for confidence intervals of R² with a target width.
  • Tabulation of sample sizes across various parameter configurations and dimensions.

Main Results:

  • One-sided tests for R² are uniformly most powerful.
  • One-sided confidence intervals for R² are uniformly most accurate.
  • Exact sample size calculation methods are provided for both hypothesis testing and interval estimation.

Conclusions:

  • The study offers practical tools for researchers to determine appropriate sample sizes for analyzing R² in multivariate settings.
  • The findings enhance the reliability of statistical inference when assessing the squared multiple correlation coefficient.
  • The presented methods and tables facilitate robust study design in multivariate statistical analysis.