Editing Effect size (section)

==== 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.