Accounting for regression to the mean under the bivariate t-distribution.

Area of Science:

• Statistics
• Biostatistics
• Medical Research Methodology

Background:

• Regression to the mean describes the tendency for extreme observations to be followed by more typical ones.
• In intervention studies, this phenomenon can confound the assessment of treatment effectiveness.
• Ignoring regression to the mean may lead to incorrect conclusions about treatment efficacy.

Purpose of the Study:

• To derive formulae for quantifying regression to the mean.
• To develop a method for estimating regression to the mean using maximum likelihood estimation under a bivariate t-distribution.
• To compare the proposed method with existing approaches and evaluate its performance in real-world data.

Main Methods:

• Derivation of regression to the mean formulae.
• Application of maximum likelihood estimation for numerical estimation of regression to the mean.
• Simulation studies to assess estimator properties (unbiasedness, consistency, asymptotic normality).
• Comparison with methods assuming bivariate normality.
• Decomposition of total effect into regression to the mean and treatment effects.

Main Results:

• Formulae for regression to the mean are derived.
• Maximum likelihood estimation provides a robust method for estimating regression to the mean, even with unequal degrees of freedom.
• The proposed method demonstrates good statistical properties and outperforms existing methods under certain conditions.
• Accounting for regression to the mean significantly impacts the statistical significance of paired t-tests.

Conclusions:

• Accurate quantification of regression to the mean is crucial for valid intervention effect estimation.
• The proposed maximum likelihood estimation method offers a reliable approach to address regression to the mean.
• Application to schizophrenia patient data demonstrates the practical utility of the method in decomposing treatment effects.