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Binomial Expansion Using Pascal's Triangle01:30

Binomial Expansion Using Pascal's Triangle

195
Expanding a binomial expression such as (a + b)n results in a predictable sequence of terms that can be systematically derived using Pascal’s Triangle. This triangular array of numbers plays a central role in understanding and computing the coefficients of binomial expansions.Pascal’s Triangle is constructed such that each row corresponds to the coefficients of a binomial raised to a power. The topmost row, known as the zeroth row, corresponds to (a + b)0, and each successive row...
195

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Augmented binary method for basket trials (ABBA).

Svetlana Cherlin1, James M S Wason1

  • 1Population Health Sciences Institute, Newcastle University, Newcastle upon Tyne, UK.

Statistical Methods in Medical Research
|December 5, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces an augmented binary method for basket trials, improving statistical power and efficiency in clinical research. The new approach enhances analysis by borrowing information across subtrials, particularly beneficial for responder outcomes.

Keywords:
Bayesian inferenceaugmented binary methodbasket clinical trialsimmune-mediated inflammatory diseaseslatent variable framework

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Area of Science:

  • Biostatistics
  • Clinical Trial Design
  • Pharmacological Research

Background:

  • Traditional clinical trials often use responder outcomes, which dichotomize continuous measures, potentially reducing statistical power.
  • Augmented binary methods have been proposed to leverage undichotomized data for improved power.
  • Basket trials are increasingly utilized in clinical research to evaluate treatments across multiple subtrials.

Purpose of the Study:

  • To extend an augmented binary method for use in basket trials, specifically for clinical areas employing responder outcomes.
  • To enhance the efficiency of basket trials by enabling information borrowing on treatment effects between subtrials.
  • To develop a novel statistical methodology within a latent variable framework using Bayesian hierarchical modeling.

Main Methods:

  • Proposed an augmented binary method tailored for basket trials with responder outcomes.
  • Employed a latent variable framework and Bayesian hierarchical modeling.
  • Investigated methodology performance through simulations, comparing point estimates and high-density intervals against standard binary outcome analyses for basket trials.

Main Results:

  • The proposed augmented binary method demonstrated a reduction in the 95% high-density interval of the posterior distribution for the log odds ratio.
  • The method showed an increase in statistical power when treatment effects were consistent across subtrials.
  • Simulations confirmed the enhanced efficiency and power compared to standard basket trial analyses.

Conclusions:

  • The novel augmented binary method offers a more efficient approach for analyzing basket trials with responder outcomes.
  • This methodology effectively borrows information across subtrials, leading to improved statistical power and precision.
  • The approach was successfully illustrated using real-world data from rheumatology clinical trials.