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  • 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.

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    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 (η2prde-41-231-i012.jpg), 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.
    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=µ1µ2σrde-41-231-i005.jpg) 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(µjµ)2/pσ2rde-41-231-i006.jpg, 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=0.58rde-41-231-i007.jpg
    Table 1

    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(µjµ)2=100.6rde-41-231-i008.jpg
    Download Table Download Table
    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.
    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.
    1. Eta squared (η2)
    Eta squared is expressed as sum of squares between groups (SSeffect) divided by the total sum of squares of the dependent variable (SStotal), η2=SSeffectSStotalrde-41-231-i009.jpg. 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)rde-41-231-i010.jpg, 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/(1η2)rde-41-231-i011.jpg or η2 = f 2 / (1 + f 2).
    Table 2

    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
    Download Table Download Table
    2. Partial Eta squared (η2prde-41-231-i012.jpg)
    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+SSerrorrde-41-231-i013.jpg. 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 η2prde-41-231-i012.jpg 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/(1η2p)rde-41-231-i014.jpg
    Table 3

    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
    Download Table Download Table
    3. Omega squared (ω2)
    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=SSeffectdfeffect*MSerrorSStotalMSerrorrde-41-231-i015.jpg, 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, fω2/(1ω2)rde-41-231-i016.jpg.
    Table 4

    Omega squared and Eta squared calculated in a two way factorial design

    Effect SSeffect* dfeffect* MSerror*
    SSeffectdfeffect*MSerrorSStotalMSerror
    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.

    Download Table Download Table
    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.

    Tables & Figures

    Table 1

    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(µjµ)2=100.6rde-41-231-i008.jpg
    Download Table Download Table
    Table 2

    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
    Download Table Download Table
    Table 3

    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
    Download Table Download Table
    Table 4

    Omega squared and Eta squared calculated in a two way factorial design

    Effect SSeffect* dfeffect* MSerror*
    SSeffectdfeffect*MSerrorSStotalMSerror
    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.

    Download Table Download Table

    REFERENCES

    • 1. Cohen J. Statistical power analysis for the behavioral science. 2nd ed. Hillsdale: Lawrence Erlbaum Associates; 1988. p. 284-288.

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      Statistical notes for clinical researchers: Sample size calculation 3. Comparison of several means using one-way ANOVA
      Restor Dent Endod. 2016;41(3):231-234.   Published online July 26, 2016
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    Statistical 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

    GroupMeanSD(µj - µ)2
    118.48.636.6
    222.28.65.1
    325.18.60.4
    432.18.658.5
    Total24.58.6pj=1(µjµ)2=100.6

    An exemplary ANOVA table

    Sum of SquaresdfMean SquareF
    Between Groups1996.9983665.66613.084
    Within Groups3866.7177650.878
    Total5863.71579

    Partial Eta squared measures from two-way ANOVA with an interaction term

    SourceType III sum of SquaresdfMean SquarePartial Eta Squared
    Corrected model3602.103a7514.5860.614 = 3602.1 / (3602.1 + 2261.6)
    Intercept47894.642147894.6420.955 = 47894.6 / (47894.6 + 2261.6)
    LIGHT34.716134.7160.015 = 34.7 / (34.7 + 2261.6)
    MATERIAL1996.9983665.6660.469 = 1997 / (1997 + 2261.6)
    LIGHT * MATERIAL1570.3893523.4630.410 = 1570.4 / (1570.4 + 2261.6)
    Error2261.6127231.411
    Total53758.35780
    Corrected total5863.71579

    Omega squared and Eta squared calculated in a two way factorial design

    EffectSSeffect*dfeffect*MSerror*SSeffectdfeffect*MSerrorSStotalMSerrorω2
    MATERIAL1997331.411902.77/5832.310.326
    LIGHT34.72131.413.31/5832.310.001
    MATERIAL * LIGHT1570.39331.411476.16/5832.310.253

    *Figures were from Table 3.

    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 Table 3.


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