Editing Effect size (section)

=== Correlation family: Effect sizes based on "variance explained" ===
These effect sizes estimate the amount of the variance within an experiment that is "explained" or "accounted for" by the experiment's model ([[Explained variation]]).

==== Pearson ''r'' or correlation coefficient ====
[[Pearson product-moment correlation coefficient|Pearson's correlation]], often denoted ''r'' and introduced by [[Karl Pearson]], is widely used as an ''effect size'' when paired quantitative data are available; for instance if one were studying the relationship between birth weight and longevity.  The correlation coefficient can also be used when the data are binary. Pearson's ''r'' can vary in magnitude from −1 to 1, with −1 indicating a perfect negative linear relation, 1 indicating a perfect positive linear relation, and 0 indicating no linear relation between two variables.

===== Coefficient of determination (''r''<sup>2</sup> or ''R''<sup>2</sup>) =====
A related ''effect size'' is ''r''<sup>2</sup>, the [[coefficient of determination]] (also referred to as ''R''<sup>2</sup> or "''r''-squared"), calculated as the square of the Pearson correlation ''r''.  In the case of paired data, this is a measure of the proportion of variance shared by the two variables, and varies from 0 to 1. For example, with an ''r'' of 0.21 the coefficient of determination is 0.0441, meaning that 4.4% of the variance of either variable is shared with the other variable.  The ''r''<sup>2</sup> is always positive, so does not convey the direction of the correlation between the two variables.

===== Eta-squared (''η''<sup>2</sup>) =====
Eta-squared describes the ratio of variance explained in the dependent variable by a predictor while controlling for other predictors, making it analogous to the ''r''<sup>2</sup>. Eta-squared is a biased estimator of the variance explained by the model in the population (it estimates only the effect size in the sample). This estimate shares the weakness with ''r''<sup>2</sup> that each additional variable will automatically increase the value of ''η''<sup>2</sup>. In addition, it measures the variance explained of the sample, not the population, meaning that it will always overestimate the effect size, although the bias grows smaller as the sample grows larger.
<math display="block"> \eta ^2 = \frac{SS_\text{Treatment}}{SS_\text{Total}} .</math>

===== Omega-squared (''ω''<sup>2</sup>) =====
{{see also|Coefficient of determination#Adjusted R2{{!}}Adjusted ''R''<sup>2</sup>}}
A less biased estimator of the variance explained in the population is ''ω''<sup>2</sup><ref name="Tabachnick 2007, p. 55">Tabachnick, B.G. & Fidell, L.S. (2007). Chapter 4: "Cleaning up your act. Screening data prior to analysis", p. 55 In B.G. Tabachnick & L.S. Fidell (Eds.), ''Using Multivariate Statistics'', Fifth Edition. Boston: Pearson Education, Inc. / Allyn and Bacon.</ref>
<math display="block">\omega^2 = \frac{\text{SS}_\text{treatment}-df_\text{treatment} \cdot \text{MS}_\text{error}}{\text{SS}_\text{total} + \text{MS}_\text{error}} .</math>

This form of the formula is limited to between-subjects analysis with equal sample sizes in all cells.<ref name="Tabachnick 2007, p. 55"/> Since it is less biased (although not ''un''biased), ''ω''<sup>2</sup> is preferable to η<sup>2</sup>; however, it can be more inconvenient to calculate for complex analyses. A generalized form of the estimator has been published for between-subjects and within-subjects analysis, repeated measure, mixed design, and randomized block design experiments.<ref name=OlejnikAlgina>{{cite journal | last1 = Olejnik | first1 = S. | last2 = Algina | first2 = J. | year = 2003 | title = Generalized Eta and Omega Squared Statistics: Measures of Effect Size for Some Common Research Designs | url = http://cps.nova.edu/marker/olejnik2003.pdf | journal = Psychological Methods | volume = 8 | issue = 4 | pages = 434–447 | doi = 10.1037/1082-989x.8.4.434 | pmid = 14664681 | s2cid = 6931663 | access-date = 2011-10-24 | archive-date = 2010-06-10 | archive-url = https://web.archive.org/web/20100610101507/http://cps.nova.edu/marker/olejnik2003.pdf | url-status = dead }}</ref> In addition, methods to calculate partial ''ω''<sup>2</sup> for individual factors and combined factors in designs with up to three independent variables have been published.<ref name=OlejnikAlgina/>

==== Cohen's ''f''<sup>2</sup> ====
Cohen's ''f''<sup>2</sup> is one of several effect size measures to use in the context of an [[F-test]] for [[ANOVA]] or [[multiple regression]]. Its amount of bias (overestimation of the effect size for the ANOVA) depends on the bias of its underlying measurement of variance explained (e.g., ''R''<sup>2</sup>, ''η''<sup>2</sup>, ''ω''<sup>2</sup>).

The ''f''<sup>2</sup> effect size measure for multiple regression is defined as:
<math display="block">f^2 = {R^2 \over 1 - R^2}.</math>

Likewise, ''f''<sup>2</sup> can be defined as:
<math display="block">f^2 = {\eta^2 \over 1 - \eta^2}</math> or <math display="block">f^2 = {\omega^2 \over 1 - \omega^2}</math>
for models described by those effect size measures.<ref name=Steiger2004>{{cite journal | last1 = Steiger | first1 = J. H. | year = 2004 | title = Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis | url = http://www.statpower.net/Steiger%20Biblio/Steiger04.pdf | journal = Psychological Methods | volume = 9 | issue = 2| pages = 164–182 | doi=10.1037/1082-989x.9.2.164| pmid = 15137887 }}</ref>

The <math>f^{2}</math> effect size measure for sequential multiple regression and also common for [[Partial least squares path modeling|PLS modeling]]<ref>Hair, J.; Hult, T. M.; Ringle, C. M. and Sarstedt, M. (2014) ''A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM)'', Sage, pp. 177–178. {{ISBN|1452217440}}</ref> is defined as:
<math display="block">f^2 = {R^2_{AB} - R^2_A \over 1 - R^2_{AB}}</math>
where ''R''<sup>2</sup><sub>''A''</sub> is the variance accounted for by a set of one or more independent variables ''A'', and ''R''<sup>2</sup><sub>''AB''</sub> is the combined variance accounted for by ''A'' and another set of one or more independent variables of interest ''B''. By convention, ''f''<sup>2</sup> effect sizes of <math>0.1^2</math>, <math>0.25^2</math>, and <math>0.4^2</math> are termed ''small'', ''medium'', and ''large'', respectively.<ref name="CohenJ1988Statistical"/>

Cohen's <math>\hat{f}</math> can also be found for factorial analysis of variance (ANOVA) working backwards, using:
<math display="block">\hat{f}_\text{effect} = {\sqrt{(F_\text{effect} df_\text{effect}/N)}}.</math>

In a balanced design (equivalent sample sizes across groups) of ANOVA, the corresponding population parameter of <math>f^2</math> is
<math display="block">{SS(\mu_1,\mu_2,\dots,\mu_K)}\over{K \times \sigma^2},</math>
wherein ''μ''<sub>''j''</sub> denotes the population mean within the ''j''<sup>th</sup> group of the total ''K'' groups, and ''σ'' the equivalent population standard deviations within each groups. ''SS'' is the [[Multivariate analysis of variance|sum of squares]] in ANOVA.

==== Cohen's ''q'' ====

Another measure that is used with correlation differences is Cohen's q. This is the difference between two Fisher transformed Pearson regression coefficients. In symbols this is
<math display="block"> q = \frac 1 2 \log \frac{ 1 + r_1 }{ 1 - r_1 } - \frac 1 2 \log \frac{1 + r_2}{1 - r_2} </math>

where ''r''<sub>1</sub> and ''r''<sub>2</sub> are the regressions being compared. The expected value of ''q'' is zero and its variance is
<math display="block"> \operatorname{var}(q) = \frac 1 {N_1 - 3} + \frac 1 {N_2 -3} </math>
where ''N''<sub>1</sub> and ''N''<sub>2</sub> are the number of data points in the first and second regression respectively.