Statistical inference of some effect sizes

• Published in: Journal of the Korean Statistical Society Vol. 49; no. 3; pp. 976 - 1007
• Main Authors: Zhao, Jun, Sim, Sung-Chur, Kim, Hyoung-Moon
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01.09.2020; 한국통계학회
• Subjects: Applied Statistics; Bayesian Inference; Mathematics and Statistics; Research Article; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; 통계학; Non-central beta distribution; Experimental design; Effect size; Confidence interval
• ISSN: 1226-3192; 2005-2863
• DOI: 10.1007/s42952-019-00046-4

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 ( η 2 ), omega-squared ( ω 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 η 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.