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Related Concept Videos

One-Way ANOVA: Equal Sample Sizes01:15

One-Way ANOVA: Equal Sample Sizes

One-Way ANOVA can be performed on three or more samples with equal or unequal sample sizes. When one-way ANOVA is performed on two datasets with samples of equal sizes, it can be easily observed that the computed F statistic is highly sensitive to the sample mean.
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Factorial Design

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One-Way ANOVA: Unequal Sample Sizes01:15

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Sampling Plans01:23

Sampling Plans

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Cluster Sampling Method

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Published on: September 11, 2021

Addressing Cluster-Level Treatment Effect Heterogeneity in Sample Size Determination for Hierarchical 2 × 2 Factorial

Jiaqi Tong1,2, Fan Li1,2,3,4, Guangyu Tong1,4,5

  • 1Department of Biostatistics, Yale School of Public Health, New Haven, Connecticut, USA.

Biometrical Journal. Biometrische Zeitschrift
|June 15, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces new sample size formulas for hierarchical 2 × 2 $2\times 2$ factorial trials, accounting for varying treatment effects across clusters. These methods improve the design of complex intervention studies.

Keywords:
intracluster correlation coefficientlinear mixed modelspower analysisrandom slope modelsample size estimationsplit‐plot design

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Sampling Soils in a Heterogeneous Research Plot
07:11

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Last Updated: Jun 16, 2026

Problem-Solving Before Instruction (PS-I): A Protocol for Assessment and Intervention in Students with Different Abilities
10:26

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Sampling Soils in a Heterogeneous Research Plot
07:11

Sampling Soils in a Heterogeneous Research Plot

Published on: January 7, 2019

Area of Science:

  • Biostatistics
  • Clinical Trial Design
  • Public Health Interventions

Background:

  • Hierarchical 2 × 2 $2\times 2$ factorial designs involve cluster-level and individual-level randomization.
  • Standard sample size calculations assume uniform treatment effects, which may not hold in cluster randomized trials.
  • Heterogeneous treatment effects across clusters introduce additional variability.

Purpose of the Study:

  • To extend sample size calculation methods for hierarchical 2 × 2 $2\times 2$ factorial trials.
  • To address the challenge of heterogeneous treatment effects across clusters.
  • To provide formulas for testing controlled, marginal, and interaction effects.

Main Methods:

  • Utilized a generalized least squares framework.
  • Developed sample size formulas for models with and without intervention interactions.
  • Employed simulation studies to validate the derived formulas.

Main Results:

  • Derived novel sample size formulas for hierarchical 2 × 2 $2\times 2$ factorial trials.
  • Formulas account for potential heterogeneity in treatment effects at the cluster level.
  • Simulation studies confirmed the accuracy of the proposed sample size calculations.

Conclusions:

  • The proposed methods offer a robust approach to sample size determination for complex factorial trials.
  • Accurate sample size calculations are crucial for detecting treatment effects in the presence of cluster heterogeneity.
  • The methods are illustrated using a suicide prevention trial example.