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One-Way ANOVA: Unequal Sample Sizes

One-Way ANOVA: Unequal Sample Sizes

One-way ANOVA can be performed on three or more samples of unequal sizes. However, calculations get complicated when sample sizes are not always the same. So, while performing ANOVA with unequal samples size, the following equation is used:

One-Way ANOVA: Equal Sample Sizes

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.
Different sample means can result in different values for the variance estimate: variance between samples. This is because the variance between samples is calculated as the product of the sample size and the variance between the...

Cluster Sampling Method

Cluster Sampling Method

Appropriate sampling methods ensure that samples are drawn without bias and accurately represent the population. Because measuring the entire population in a study is not practical, researchers use samples to represent the population of interest.
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Test for Homogeneity

Test for Homogeneity

The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to conclude whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence. The hypotheses for the test for homogeneity can be stated as...

Sampling Plans

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Sample Size Calculation

Sample Size Calculation

Knowledge of the sample size is the first requirement to conduct random sampling or an experiment. The sample size is the total number of units, observations, or groups (in some cases) used to get the data to estimate a population parameter. As the name suggests, the sample size is that of the sample drawn from the population and differs from the population size.
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Related Experiment Video

Updated: Jun 5, 2026

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Sample size determination for testing equality in a cluster randomized trial with noncompliance.

Kung-Jong Lui¹, Kuang-Chao Chang

¹Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, USA. kjl@rohan.sdsu.edu

Journal of Biopharmaceutical Statistics

|December 31, 2010

Summary

This study introduces a new sample size calculation for cluster randomized trials (CRTs) that accounts for patient noncompliance. This method ensures accurate statistical power in trials where participants may not follow assigned treatments.

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Area of Science:

Biostatistics
Clinical Trials Methodology
Epidemiology

Background:

Cluster randomized trials (CRTs) are frequently used for administrative or cost efficiencies.
Patient noncompliance with assigned treatments is a common issue in clinical research due to ethical or personal reasons.
A practical sample size calculation for CRTs with noncompliance is needed.

Purpose of the Study:

To develop a sample size calculation procedure for cluster randomized trials (CRTs) that incorporates noncompliance.
To provide a method for testing treatment equality among compliers in CRTs with noncompliance.
To address the practical challenges of designing effective CRTs with imperfect adherence.

Main Methods:

Developed an asymptotic test procedure using a tanh(-1)(x) transformation under the exclusion restriction model.
Derived a sample size formula considering noncompliance and intraclass correlation for a specified power (1 - β) and significance level (α).
Utilized Monte Carlo simulations to assess the performance of the test procedure and the accuracy of the sample size formula.

Main Results:

The proposed test procedure demonstrates reliable type I error rates in simulations.
The derived sample size formula accurately estimates the required sample size to achieve desired statistical power.
The methodology was illustrated using data from a CRT on vitamin A supplementation and child mortality.

Conclusions:

The developed sample size calculation method is practical and accounts for key factors in CRTs with noncompliance.
This approach enhances the design and efficiency of cluster randomized trials.
Accurate sample size determination is crucial for the validity and power of CRTs with imperfect adherence.