KACD
Articles
- Page Path
- HOME > Restor Dent Endod > Volume 41(3); 2016 > Article
- Open Lecture on Statistics Statistical notes for clinical researchers: Sample size calculation 3. Comparison of several means using one-way ANOVA
- Hae-Young Kim
-
2016;41(3):-234.
DOI: https://doi.org/10.5395/rde.2016.41.3.231
Published online: July 26, 2016
Department of Health Policy and Management, College of Health Science, and Department of Public Health Sciences, Graduate School, Korea University, Seoul, Korea.
- Correspondence to Hae-Young Kim, DDS, PhD. Associate Professor, Department of Health Policy and Management, College of Health Science, and Department of Public Health Sciences, Graduate School, Korea University, 145 Anam-ro, Seongbukgu, Seoul, Korea 02841. TEL, +82-2-3290-5667; FAX, +82-2-940-2879; kimhaey@korea.ac.kr
©Copyrights 2016. The Korean Academy of Conservative Dentistry.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
- 1,526 Views
- 20 Download
- 25 Crossref
In this third article about sample size determination, we will discuss sample size determination procedure for comparison of several means. Usually analysis of such data is performed using the analysis of variance (ANOVA) procedure. Because of the complex nature that more than two group means are compared, various types of effect sizes have been suggested including Cohen's f, Eta squared (η2), Partial Eta squared (η2p
), and Omega squared (ω2). Therefore, we will discuss sample size determination procedure using Cohen's f and then will explore various types of effect sizes for ANOVA and their interchangeability.
), and Omega squared (ω2). Therefore, we will discuss sample size determination procedure using Cohen's f and then will explore various types of effect sizes for ANOVA and their interchangeability.Sample size determination using Cohen's f measure
Cohen's f measure is an extended version of Cohen's d which is defined as a standardized difference, difference divided by standard deviation (d=µ1−µ2σ
) in comparison of two sample means. Cohen's f is expressed as a square root of mean squared difference divided by variance, whose numerator represents an average difference of group means from the grand mean and denominator represents the common standard deviation.
) in comparison of two sample means. Cohen's f is expressed as a square root of mean squared difference divided by variance, whose numerator represents an average difference of group means from the grand mean and denominator represents the common standard deviation.
Cohen's f=√∑pj=1(µj−µ)2/pσ2
, where p is the number of groups. Based on the data in Table 1, Cohen's f is calculated as follows: √100.6/48.62=√25.273.9=0.58
, where p is the number of groups. Based on the data in Table 1, Cohen's f is calculated as follows: √100.6/48.62=√25.273.9=0.58
Table 1
Download Table 
Descriptive statistics of a variable from four groups
| Group | Mean | SD | (µj - µ)2 |
|---|---|---|---|
| 1 | 18.4 | 8.6 | 36.6 |
| 2 | 22.2 | 8.6 | 5.1 |
| 3 | 25.1 | 8.6 | 0.4 |
| 4 | 32.1 | 8.6 | 58.5 |
| Total | 24.5 | 8.6 |
∑pj=1(µj−µ)2=100.6
|
Cohen (1988) suggested that interpretation of Cohen's f was such as f = 0.1, small effect, f = 0.25, medium effect, and f = 0.40, large effect for the behavioral science.1 Therefore, the calculated Cohen's f value 0.58 can be interpreted as a large effect. If we anticipate outcome values for four comparative groups as appeared in Table 1, we can calculate the required sample size to keep small Type 1 error and large power. Let's set the condition, as α error level = 0.05 and power level = 0.8. Also four group means are assumed as 18.4, 22.2, 25.1, and 32.1, respectively, and a common standard deviation 8.6 is provided. The free software G*Power gives the result of an appropriate sample size as ten per group (total sample size = 40) with the practical power value of 0.85. The steps to perform the sample size calculation are displayed as follows:
Step 1: Select statistical test types
Menu: Tests - Means - Many Groups ANOVA: One-way (one independent variable)
Step 2: Calculation of Cohen's f measure
Menu: Determine - select the procedure as Effect size from means - Number of groups: 4 - Provide means and common SD - Calculate - Calculate and transfer to main window
Step 3: Set α error probability = 0.05 and β error probability = 0.2 - Calculate (a total sample size of 40 and actual power level of 0.847 are obtained.)
Additionally, we can check the changes of power level (1 - β error probability) as the total sample size changes from 12 to 80, given the prior conditions are fixed except β error probability as shown in the figure below.
Other types of effect sizes for ANOVA
Several types of effect sizes for ANOVA are more commonly reported compared to Cohen's f because most statistical software programs provide statistics such as total sum of squares (SStotal), sum of squares of effects (SSeffect) or sum of squares of error (SSerror), which are related to them. If previous studies report only other types of effect sizes different from Cohen's f and if group means and variances are not available, researchers should convert those effect sizes into Cohen's f to calculate an adequate sample size.
Eta squared is expressed as sum of squares between groups (SSeffect) divided by the total sum of squares of the dependent variable (SStotal), η2=SSeffectSStotal
. The quantity is the same with the usual r squared (R2) which we use as a measure of degree that a model explains the data. The estimate of η2 value was calculated as 0.341(=SSeffectSStotal=1996.9985863.715)
, which means the ANOVA model using the MATERIAL independent variable explained 34.1% of variability in the dependent variable (Table 2). Eta squared can be converted into Cohen's f and vice versa as follows: f=√η2/(1−η2)
or η2 = f
2 / (1 + f
2).
. The quantity is the same with the usual r squared (R2) which we use as a measure of degree that a model explains the data. The estimate of η2 value was calculated as 0.341(=SSeffectSStotal=1996.9985863.715), which means the ANOVA model using the MATERIAL independent variable explained 34.1% of variability in the dependent variable (Table 2). Eta squared can be converted into Cohen's f and vice versa as follows: f=√η2/(1−η2) or η2 = f
2 / (1 + f
2).
Table 2
Download Table
An exemplary ANOVA table
| Sum of Squares | df | Mean Square | F | |
|---|---|---|---|---|
| Between Groups | 1996.998 | 3 | 665.666 | 13.084 |
| Within Groups | 3866.717 | 76 | 50.878 | |
| Total | 5863.715 | 79 |
)The effect size of partial Eta square measure is preferred to Eta squared in a two-way factorial design. The main reason is that when other independent variables are included in the model, η2 value becomes smaller compared to the original value, therefore it cannot represent an effect size in multivariate situation. The partial Eta squared is expressed as SSeffect divided by the sum of SSeffect and SSerror, η2p=SSeffectSSeffect+SSerror
. Partial Eta squared measure can be obtained by selecting 'estimates of effect size' on the option window during performing two-way ANOVA by selecting successive procedures of Analysis - General Linear Model - Univariate in IBM SPSS statistical package version 23.0 (IBM Corp., Armonk, NY, USA). As appeared in Table 3, the estimates of η2p values for MATERIAL, LIGHT, and the interaction term were calculated as 0.469, 0.015, and 0.410, respectively. Partial Eta squared can be converted into Cohen's f for a specific term using the formula, f=√η2p/(1−η2p)
. Partial Eta squared measure can be obtained by selecting 'estimates of effect size' on the option window during performing two-way ANOVA by selecting successive procedures of Analysis - General Linear Model - Univariate in IBM SPSS statistical package version 23.0 (IBM Corp., Armonk, NY, USA). As appeared in Table 3, the estimates of η2p values for MATERIAL, LIGHT, and the interaction term were calculated as 0.469, 0.015, and 0.410, respectively. Partial Eta squared can be converted into Cohen's f for a specific term using the formula, f=√η2p/(1−η2p)
Table 3
Partial Eta squared measures from two-way ANOVA with an interaction term
Download Table
The Omega squared measure was suggested to correct the biasedness of Eta squared measure. The Eta squared was slightly biased because the calculation procedure was made purely based on statistics from the sample without any adjustment considering population measure. Omega-squared is calculated as ω2=SSeffect−dfeffect*MSerrorSStotal−MSerror
, where SStotal and MSresidual represent total sum of squares and mean square of error, respectively, and dfeffect is degrees of freedom of the effect. An Omega square has slightly lower value and generally is considered more accurate compared to an Eta squared (Table 4). Omega squared can be approximately converted into Cohen's f using the formula, f≈√ω2/(1−ω2)
.
, where SStotal and MSresidual represent total sum of squares and mean square of error, respectively, and dfeffect is degrees of freedom of the effect. An Omega square has slightly lower value and generally is considered more accurate compared to an Eta squared (Table 4). Omega squared can be approximately converted into Cohen's f using the formula, f≈√ω2/(1−ω2).
Table 4
Download Table
Omega squared and Eta squared calculated in a two way factorial design
| Effect | SSeffect* | dfeffect* | MSerror* | ω2 | |
|---|---|---|---|---|---|
| MATERIAL | 1997 | 3 | 31.41 | 1902.77/5832.31 | 0.326 |
| LIGHT | 34.72 | 1 | 31.41 | 3.31/5832.31 | 0.001 |
| MATERIAL * LIGHT | 1570.39 | 3 | 31.41 | 1476.16/5832.31 | 0.253 |
*Figures were from Table 3.
Cohen (1988) suggested interpretation of effect sizes expressed as η2 or ω2: small effect, η2 or ω2 = 0.01; medium effect, η2 or ω2 = 0.06; large effect, η2 or ω2 = 0.14 for the behavioral science.1
- 1. Cohen J. Statistical power analysis for the behavioral science. 2nd ed. Hillsdale: Lawrence Erlbaum Associates; 1988. p. 284-288.
REFERENCES
Tables & Figures
Table 1
Download Table
Descriptive statistics of a variable from four groups
| Group | Mean | SD | (µj - µ)2 |
|---|---|---|---|
| 1 | 18.4 | 8.6 | 36.6 |
| 2 | 22.2 | 8.6 | 5.1 |
| 3 | 25.1 | 8.6 | 0.4 |
| 4 | 32.1 | 8.6 | 58.5 |
| Total | 24.5 | 8.6 |
∑pj=1(µj−µ)2=100.6
|
Table 2
Download Table
An exemplary ANOVA table
| Sum of Squares | df | Mean Square | F | |
|---|---|---|---|---|
| Between Groups | 1996.998 | 3 | 665.666 | 13.084 |
| Within Groups | 3866.717 | 76 | 50.878 | |
| Total | 5863.715 | 79 |
Table 3
Partial Eta squared measures from two-way ANOVA with an interaction term
Download Table
Table 4
Download Table
Omega squared and Eta squared calculated in a two way factorial design
| Effect | SSeffect* | dfeffect* | MSerror* | ω2 | |
|---|---|---|---|---|---|
| MATERIAL | 1997 | 3 | 31.41 | 1902.77/5832.31 | 0.326 |
| LIGHT | 34.72 | 1 | 31.41 | 3.31/5832.31 | 0.001 |
| MATERIAL * LIGHT | 1570.39 | 3 | 31.41 | 1476.16/5832.31 | 0.253 |
*Figures were from Table 3.
REFERENCES
- 1. Cohen J. Statistical power analysis for the behavioral science. 2nd ed. Hillsdale: Lawrence Erlbaum Associates; 1988. p. 284-288.
Citations
Citations to this article as recorded by 
- Maternal dietary folate imbalance alters cerebellar astrocyte morphology and density in offspring
Philip Maseghe Mwachaka, Peter Gichangi, Adel Abdelmalek, Paul Odula, Julius Ogeng’o
IBRO Neuroscience Reports.2025; 18: 78. CrossRef - Fc gamma receptors activate different protein kinase C isoforms in human neutrophils
Omar Rafael Alemán, Carlos Blanco-Camarillo, Nathalia Naranjo-Pinto, Nancy Mora, Carlos Rosales
Journal of Leukocyte Biology.2025;[Epub] CrossRef - The Effects of Concurrent Training on Molecular, Functional, and Clinical Outcomes in Breast Cancer Survivors: A Pilot Study
Celia García-Chico, Susana López-Ortiz, Salvador Santiago-Pescador, Paloma Guillén-Rogel, Saúl Peñín-Grandes, Lisa Musso-Daury, Francisco Javier Iruzubieta-Barragán, José Pinto-Fraga, Sergio Maroto-Izquierdo, Lourdes del Río Solá, Alejandro Santos-Lozano
Cancers.2025; 17(12): 1967. CrossRef - Effect of first epinephrine administration time on cerebral perfusion pressure and cortical cerebral blood flow in a porcine cardiac arrest model
Dong Hyun Choi, Ki Jeong Hong, Ki Hong Kim, Sang Do Shin, Kyoung Jun Song, Yoonjic Kim, Yoon Ha Joo, Jeong Ho Park, Young Sun Ro, Hyun Jeong Kang
Resuscitation.2024; 195: 109969. CrossRef - Okul Öncesi Öğretmenlerin Erken Okuryazarlık Bilgileri ile Okuma Yazmaya Hazırlık Çalışmalarına İlişkin Yeterlikleri Arasındaki İlişkinin İncelenmesi
Halime Kübra Türköz, Yakup Burak, Emine Ahmetoğlu
Okuma Yazma Eğitimi Araştırmaları.2024; 12(1): 27. CrossRef - Dimensional accuracy and surface characteristics of complete-arch cast manufactured by six 3D printers
Mi-Young Sim, June-Beom Park, Deok-Yeoung Kim, Hae-Young Kim, Ji-Man Park
Heliyon.2024; 10(10): e30996. CrossRef - Promoting learners' self-regulated textbook use for overcoming impasses in solving mathematics exercises
Mari Fukuda, Emmanuel Manalo
Research in Mathematics Education.2024; 26(1): 133. CrossRef - The development of resin-coating materials for enhancing elemental release of coated glass ionomer cements
Piyaphong Panpisut, Arnit Toneluck, Chutikarn Khamsuk, Somruethai Channasanon, Siriporn Tanodekaew, Naruporn Monmaturapoj, Parichart Naruphontjirakul
Heliyon.2024; 10(14): e34512. CrossRef - Behavioral signatures of the rapid recruitment of long-term memory to overcome working memory capacity limits
Kirsten C. S. Adam, Chong Zhao, Edward K. Vogel
Memory & Cognition.2024; 52(8): 1816. CrossRef - Viscoelastic properties of the equine hoof wall
Christian Bonney, Siyuan Pang, Marc A. Meyers, Iwona Jasiuk
Acta Biomaterialia.2024; 184: 264. CrossRef - Assessment of physical/mechanical properties and cytotoxicity of dual-cured resin cements containing Sr-bioactive glass nanoparticles and calcium phosphate
Sasipin THANYASIRI, Parichart NARUPHONTJIRAKUL, Chawal PADUNGLAPPISIT, Bharat MIRCHANDANI, Anne M. YOUNG, Piyaphong PANPISUT
Dental Materials Journal.2023; 42(6): 806. CrossRef - Maxillary arch dimensions, occlusion assessment and space conditions in patients with cleft palate in the period of deciduous dentition—A retrospective study
Jakob Sajovic, Anina Setnikar Lesjak, Alja Plut, Andreja Eberlinc, Jasmina Primožič, Eva Drevenšek, Martina Drevenšek
Journal of Stomatology, Oral and Maxillofacial Surgery.2023; 124(1): 101356. CrossRef - Okul Psikolojik Danışmanlarının Yaşam Kalitesi ve Psikolojik Danışma Öz-Yeterliği Düzeylerinin İncelenmesi
Büşra YILDIRIM, Hakan SARIÇAM, Rabia Şeyda AYDIN
Bayburt Eğitim Fakültesi Dergisi.2023; 18(40): 1189. CrossRef - Depression, anxiety, and stress in pregnancy and postpartum: A longitudinal study during the COVID-19 pandemic
Emily P. Rabinowitz, Lindsay A. Kutash, Alexis L. Richeson, MacKenzie A. Sayer, Marielle R. Samii, Douglas L. Delahanty
Midwifery.2023; 121: 103655. CrossRef - A Meta-Analytic Examination of the Cost and Benefits of Overlay versus Separate Displays
Amelia C. Warden, Christopher D. Wickens, Benjamin A. Clegg, Francisco R. Ortega
Proceedings of the Human Factors and Ergonomics Society Annual Meeting.2023; 67(1): 1501. CrossRef - ÖĞRETMEN, REHBER ÖĞRETMENİ, VE OKUL YÖNETİCİLERİNİN KAYNAŞTIRMA/BÜTÜNLEŞTİRME UYGULAMALARININ BAŞARISINI ETKİLEYEN ETMENLERİ DEĞERLENDİRMESİ
Yakup BURAK, Emine AHMETOĞLU
Atatürk Üniversitesi Kazım Karabekir Eğitim Fakültesi Dergisi.2022; (44): 54. CrossRef - The age differences and effect of mild cognitive impairment on perceptual-motor and executive functions
Yupaporn Rattanavichit, Nithinun Chaikeeree, Rumpa Boonsinsukh, Kasima Kitiyanant
Frontiers in Psychology.2022;[Epub] CrossRef - Monomer conversion, biaxial flexural strength, apatite forming ability of experimental dual-cured and self-adhesive dental composites containing calcium phosphate and nisin
Piyaphong PANPISUT, Thunchanok SUPPAPATPONG, Athiporn RATTANAPAN, Patamawadee WONGWARAWUT
Dental Materials Journal.2021; 40(2): 399. CrossRef - Children Emotion Regulation: Development of Neural Marker by Investigating Human Brain Signals
Raja Majid Mehmood, Hyung-Jeong Yang, Sun-Hee Kim
IEEE Transactions on Instrumentation and Measurement.2021; 70: 1. CrossRef - Attitudes of Future Doctors Towards LGBT Patients in Conservative Malaysian Society
Andrew L. S. Foong, Jun Wei Liow, Sivalingam Nalliah, Wah Yun Low, Alexander Lourdes Samy, Zahra Fazli Khalaf
Sexuality & Culture.2020; 24(5): 1358. CrossRef - Overexpression of miR-124 Protects Against Neurological Dysfunction Induced by Neonatal Hypoxic–Ischemic Brain Injury
Liulin Xiong, Haoli Zhou, Qiong Zhao, Lulu Xue, Mohammed Al-Hawwas, Jingyuan He, Maxiu Wu, Yu Zou, Mingan Yang, Jing Dai, Manxi He, Tinghua Wang
Cellular and Molecular Neurobiology.2020; 40(5): 737. CrossRef - COX5A over-expression protects cortical neurons from hypoxic ischemic injury in neonatal rats associated with TPI up-regulation
Ya Jiang, Xue Bai, Ting-Ting Li, Mohammed AL-Hawwas, Yuan Jin, Yu Zou, Yue Hu, Lin-Yi Liu, Ying Zhang, Qing Liu, Hao Yang, Jun Ma, Ting-Hua Wang, Jia Liu, Liu-Lin Xiong
BMC Neuroscience.2020;[Epub] CrossRef - Mood impairment is stronger in young than in older adults after sleep deprivation
Johanna Schwarz, John Axelsson, Andreas Gerhardsson, Sandra Tamm, Håkan Fischer, Göran Kecklund, Torbjörn Åkerstedt
Journal of Sleep Research.2019;[Epub] CrossRef - Different strength declines in leg primary movers versus stabilizers across age—Implications for the risk of falls in older adults?
Franziska Daun, Armin Kibele, Riccardo Di Giminiani
PLOS ONE.2019; 14(3): e0213361. CrossRef - Heart Rate Variability following Combined Aerobic and Resistance Training in Sedentary Hypertensive Women: A Randomised Control Trial
Sidra Masroor, Pooja Bhati, Shalini Verma, Monis Khan, M. Ejaz Hussain
Indian Heart Journal.2018; 70: S28. CrossRef
ePub Link
Cite
XML DownloadStatistical notes for clinical researchers: Sample size calculation 3. Comparison of several means using one-way ANOVA
Statistical notes for clinical researchers: Sample size calculation 3. Comparison of several means using one-way ANOVA
Descriptive statistics of a variable from four groups
| Group | Mean | SD | (µj - µ)2 |
|---|---|---|---|
| 1 | 18.4 | 8.6 | 36.6 |
| 2 | 22.2 | 8.6 | 5.1 |
| 3 | 25.1 | 8.6 | 0.4 |
| 4 | 32.1 | 8.6 | 58.5 |
| Total | 24.5 | 8.6 |
An exemplary ANOVA table
| Sum of Squares | df | Mean Square | F | |
|---|---|---|---|---|
| Between Groups | 1996.998 | 3 | 665.666 | 13.084 |
| Within Groups | 3866.717 | 76 | 50.878 | |
| Total | 5863.715 | 79 |
Partial Eta squared measures from two-way ANOVA with an interaction term
| Source | Type III sum of Squares | df | Mean Square | Partial Eta Squared |
|---|---|---|---|---|
| Corrected model | 3602.103a | 7 | 514.586 | 0.614 = 3602.1 / (3602.1 + 2261.6) |
| Intercept | 47894.642 | 1 | 47894.642 | 0.955 = 47894.6 / (47894.6 + 2261.6) |
| LIGHT | 34.716 | 1 | 34.716 | 0.015 = 34.7 / (34.7 + 2261.6) |
| MATERIAL | 1996.998 | 3 | 665.666 | 0.469 = 1997 / (1997 + 2261.6) |
| LIGHT * MATERIAL | 1570.389 | 3 | 523.463 | 0.410 = 1570.4 / (1570.4 + 2261.6) |
| Error | 2261.612 | 72 | 31.411 | |
| Total | 53758.357 | 80 | ||
| Corrected total | 5863.715 | 79 |
Omega squared and Eta squared calculated in a two way factorial design
| Effect | SSeffect* | dfeffect* | MSerror* | ω2 | |
|---|---|---|---|---|---|
| MATERIAL | 1997 | 3 | 31.41 | 1902.77/5832.31 | 0.326 |
| LIGHT | 34.72 | 1 | 31.41 | 3.31/5832.31 | 0.001 |
| MATERIAL * LIGHT | 1570.39 | 3 | 31.41 | 1476.16/5832.31 | 0.253 |
*Figures were from
Table 1 Descriptive statistics of a variable from four groups
Table 2 An exemplary ANOVA table
Table 3 Partial Eta squared measures from two-way ANOVA with an interaction term
Table 4 Omega squared and Eta squared calculated in a two way factorial design
*Figures were from
