The reporting of effect sizes in social-scientific articles is becoming increasingly widespread and encouraged, particularly when research and experimental designs are involved. Two widely used experimental designs where the uniqueness of estimation can be guaranteed, the cell means and treatment effect models, are first introduced. Then, under those two experimental designs, it is proposed to explore the distributions of the effect sizes such as eta-squared (\(\eta ^2\)), omega-squared (\(\omega ^2\)) and Cohen’s \(f^2\). For each effect size in every experimental design, it is found that the distribution or transformation of distribution belongs to the non-central Beta family. Confidence intervals for effect size in the corresponding hypothesis are obtained by applying the results from the distributions combined with the probability limits. Based on the first two moments of distributions, which lead to the mean and standard deviation, a simulation study is given to help better understand the behaviour of \(\eta ^2\) at different sample sizes and group numbers. This provides a reference for choosing sample and group sizes in experimental design. An application is reported for a psychological data set in order to illustrate how effect sizes perform in practice.

中文翻译：

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一些效应大小的统计推断

社会科学文章中效应大小的报告变得越来越普遍，并受到鼓励，特别是在涉及研究和实验设计时。首先介绍两种可以保证估计唯一性的实验设计，即细胞均值和治疗效果模型。然后，在这两个实验设计下，建议探索效应大小的分布，例如eta平方（\（\ eta ^ 2 \）），omega平方（\（\ omega ^ 2 \））和Cohen's \（f ^ 2 \）。对于每个实验设计中的每个效应大小，发现分布的分布或变换属于非中央Beta族。通过将分布结果与概率极限相结合，可以得出相应假设中效应量的置信区间。基于导致均值和标准差的分布的前两个时刻，进行了仿真研究，以帮助您更好地了解\（\ eta ^ 2 \）在不同样本量和组数下的行为。这为在实验设计中选择样本和组的大小提供了参考。报告了一个心理数据集的应用程序，以说明效果大小在实践中的表现。