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Quantifying the regression to the mean effect in Poisson processes.

Manzoor Khan1,2, Jake Olivier2

  • 1Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan.

Statistics in Medicine
|June 27, 2018
PubMed
Summary
This summary is machine-generated.

Regression to the mean (RTM) can be mistaken for real change. This study quantifies RTM for bivariate Poisson distributions, offering new methods for analyzing count data in various real-world scenarios.

Keywords:
Poisson processesregression to the mean

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Area of Science:

  • Statistics
  • Probability Theory

Background:

  • Regression to the mean (RTM) describes the statistical tendency for extreme measurements to be followed by more average ones.
  • Existing RTM quantification methods are limited to bivariate normal distributions.
  • Many real-world phenomena, such as incident counts, follow Poisson processes.

Purpose of the Study:

  • To derive and quantify regression to the mean (RTM) effects for bivariate Poisson distributions.
  • To extend RTM analysis to count data, accommodating both homogeneous and inhomogeneous Poisson processes.
  • To provide a statistical framework for understanding natural variability versus real change in count data.

Main Methods:

  • Derivation of analytical expressions for RTM in bivariate Poisson distributions (homogeneous and inhomogeneous cases).
  • Evaluation of statistical properties through simulation studies.
  • Derivation of asymptotic distributions for RTM estimators.
  • Application of maximum likelihood estimation to real-world road accident data.

Main Results:

  • Novel expressions for quantifying RTM in bivariate Poisson settings are established.
  • Simulation studies confirm the statistical properties and validity of the derived estimators.
  • Asymptotic distributions for RTM estimators are theoretically derived.
  • The RTM effect is estimated for road accident fatalities in New South Wales.

Conclusions:

  • The derived methods effectively quantify regression to the mean (RTM) for bivariate Poisson distributions.
  • This work extends RTM analysis beyond normal distributions, enabling its application to count data.
  • The findings have implications for accurately interpreting variability in diverse real-world count processes.