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Manzoor Khan1,2, Jake Olivier2

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Regression to the mean (RTM) is quantified for bivariate binomial distributions. This method provides accurate analysis for pre-post studies involving success counts, unlike normal or Poisson approximations.

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Area of Science:

  • Biostatistics
  • Statistical Modeling
  • Quantitative Research Methods

Background:

  • Regression to the mean (RTM) is a statistical phenomenon where extreme measurements tend to move closer to the population mean upon re-measurement.
  • Pre-post study designs are susceptible to inaccurate conclusions due to RTM.
  • Existing methods for quantifying RTM are primarily for bivariate normal and Poisson distributions.

Purpose of the Study:

  • To derive expressions for quantifying RTM effects under the bivariate binomial distribution.
  • To investigate the impact of correlation (positive or negative) on RTM severity in bivariate binomial models.
  • To compare RTM quantification under the bivariate binomial distribution with normal and Poisson approximations.

Main Methods:

  • Derivation of analytical expressions for RTM under the bivariate binomial distribution.
  • Development of maximum likelihood estimation for RTM with its asymptotic distribution.
  • Conducting a simulation study to evaluate the RTM estimator's statistical properties.
  • Application of methods to real-world data examples (obesity counts, cardboard can defects).

Main Results:

  • Novel expressions for quantifying RTM in bivariate binomial distributions were successfully derived.
  • The bivariate binomial distribution allows for both positive and negative correlations, with negative correlation leading to greater RTM severity.
  • The percentage relative difference highlights significant discrepancies between bivariate binomial RTM and its normal/Poisson approximations.
  • The simulation study confirmed the statistical properties of the derived RTM estimator and its asymptotic distribution.

Conclusions:

  • The derived methods provide accurate quantification of RTM for bivariate binomial data, crucial for pre-post studies with success counts.
  • Ignoring the bivariate binomial nature and using approximations can lead to substantial errors in RTM assessment.
  • The findings are applicable to various fields dealing with count data, such as public health and quality control.