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Selecting Multiple Biomarker Subsets with Similarly Effective Binary Classification Performances
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Maximum likelihood estimation with binary-data regression models: small-sample and large-sample features.

Roland C Deutsch1, John M Grego, Brian Habing

  • 1Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27402.

Advances and Applications in Statistics
|July 20, 2010
PubMed
Summary
This summary is machine-generated.

This study verifies asymptotic normality for maximum likelihood estimators in generalized linear models with binomial responses, especially for less common link functions. It highlights potential issues with Wald confidence regions in small samples near parameter boundaries.

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Area of Science:

  • Statistics
  • Statistical Modeling

Background:

  • Generalized linear models (GLiMs) often depend on the asymptotic normality of maximum likelihood estimators (MLEs).
  • Existing research on MLE asymptotic normality in binomial response models is limited, particularly for non-standard link functions like complementary log-log and complementary log.
  • The conditions for asymptotic normality established by Fahrmeir & Kaufmann (1985) require further verification for a broader range of GLiM link functions.

Purpose of the Study:

  • To verify the asymptotic normality conditions for MLEs in binomial response GLiMs with various link functions.
  • To establish simple conditions for asymptotic normality applicable to any twice differentiable monotone link function.
  • To assess the accuracy of asymptotic Wald confidence regions, especially in challenging scenarios.

Main Methods:

  • Verification of asymptotic normality conditions for MLEs in binomial GLiMs.
  • Development of a generalized set of conditions for twice differentiable monotone link functions.
  • Analysis of the quality of asymptotic Wald confidence region approximations.

Main Results:

  • Asymptotic normality of MLEs is confirmed for binomial GLiMs with complementary log-log and complementary log links under fixed experimental groups.
  • A simplified set of conditions for asymptotic normality is presented for a broad class of link functions.
  • The approximation quality for Wald confidence regions can be problematic with small sample sizes and certain link functions, particularly near parameter space boundaries.

Conclusions:

  • The study extends the understanding of MLE asymptotic normality in GLiMs to less common binomial response link functions.
  • The findings provide practical guidance on the applicability and limitations of Wald confidence intervals in specific GLiM contexts.
  • Researchers should exercise caution when using asymptotic Wald confidence regions for small sample sizes or when parameters approach boundary values.