Statistical inference of some effect sizes
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Online ISSN 2005-2863 Print ISSN 1226-3192
RESEARCH ARTICLE
Statistical inference of some effect sizes Jun Zhao1 · Sung‑Chur Sim2,3 · Hyoung‑Moon Kim1 Received: 10 September 2019 / Accepted: 23 December 2019 © Korean Statistical Society 2020
Abstract 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 ( 𝜔2 ) 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. Keywords Experimental design · Effect size · Non-central beta distribution · Confidence interval
1 Introduction Effect size is a measure that quantifies the strength of the relationship between two variables or the magnitude of the difference between populations with a standard metric. A summary of various effect size measures is provided in Kirk (2013). In some research and experimental studies, it is required to report certain measures of
* Hyoung‑Moon Kim [email protected] 1
Department of Applied Statistics, Konkuk University, Seoul, Korea
2
Department of Bioresources Engineering, Sejong University, Seoul, Korea
3
Plant Engineering Research Institute, Sejong University, Seoul, Korea
13
Vol.:(0123456789)
Journal of the Korean Statistical Society
effect size along with tests for statistical significance. We focus on the effect sizes that belong to the correlation family based on “variance explained”, such as etasquared ( 𝜂 2 ), omega-squared ( 𝜔2 ), and Cohen’s f 2. 𝜂 2 is also known as the correlation ratio and is defined as the proportion of the total subpopulation variance made up by the population means, as depicted in the books by Cohen (1988), Hays (1994) and Kirk (2013). The occurrence of 𝜂 2 can be dated back at least to Pearson (1911), and some of its characteristics make it a useful estimate of effect size for research and experimental studies. First, the interpretation of 𝜂 2 in terms of the proportion of variance accounted for by a variable or model offers a useful way to understand the effect size. Se
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