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==Types== About 50 to 100 different measures of effect size are known. Many effect sizes of different types can be converted to other types, as many estimate the separation of two distributions, so are mathematically related. For example, a correlation coefficient can be converted to a Cohen's d and vice versa. === 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. === Difference family: Effect sizes based on differences between means === The raw effect size pertaining to a comparison of two groups is inherently calculated as the differences between the two means. However, to facilitate interpretation it is common to standardise the effect size; various conventions for statistical standardisation are presented below. ==== Standardized mean difference ==== [[File:Cohens d 4panel.svg|thumb|Plots of Gaussian densities illustrating various values of Cohen's d.]] A (population) effect size ''θ'' based on means usually considers the standardized mean difference (SMD) between two populations<ref name="HedgesL1985Statistical">{{Cite book | author = [[Larry V. Hedges]] & [[Ingram Olkin]] | title = Statistical Methods for Meta-Analysis | publisher = [[Academic Press]] | year = 1985 | location = Orlando | isbn = 978-0-12-336380-0 }}</ref>{{Rp|p=78|date=November 2012}} <math display="block">\theta = \frac{\mu_1 - \mu_2} \sigma,</math> where ''μ''<sub>1</sub> is the mean for one population, ''μ''<sub>2</sub> is the mean for the other population, and σ is a [[standard deviation]] based on either or both populations. In the practical setting the population values are typically not known and must be estimated from sample statistics. The several versions of effect sizes based on means differ with respect to which statistics are used. This form for the effect size resembles the computation for a [[t-test|''t''-test]] statistic, with the critical difference that the ''t''-test statistic includes a factor of <math>\sqrt{n}</math>. This means that for a given effect size, the significance level increases with the sample size. Unlike the ''t''-test statistic, the effect size aims to estimate a population [[parameter]] and is not affected by the sample size. SMD values of 0.2 to 0.5 are considered small, 0.5 to 0.8 are considered medium, and greater than 0.8 are considered large.<ref name="Andrade2020">{{cite journal | last1 = Andrade | first1 = Chittaranjan | title = Mean Difference, Standardized Mean Difference (SMD), and Their Use in Meta-Analysis | journal = The Journal of Clinical Psychiatry | date = 22 September 2020 | volume = 81 | issue = 5 | eissn = 1555-2101 | doi = 10.4088/JCP.20f13681 | pmid = 32965803 | s2cid = 221865130 | url = | quote = SMD values of 0.2-0.5 are considered small, values of 0.5-0.8 are considered medium, and values > 0.8 are considered large. In psychopharmacology studies that compare independent groups, SMDs that are statistically significant are almost always in the small to medium range. It is rare for large SMDs to be obtained.| doi-access = free }}</ref> ====Cohen's ''d'' {{anchor|Cohen's d}}==== Cohen's ''d'' is defined as the difference between two means divided by a standard deviation for the data, i.e. <math display="block">d = \frac{\bar{x}_1 - \bar{x}_2} s.</math> [[Jacob Cohen (statistician)|Jacob Cohen]] defined ''s'', the [[pooled standard deviation]], as (for two independent samples):<ref name="CohenJ1988Statistical">{{cite book |last=Cohen |first=Jacob |author-link=Jacob Cohen (statistician) |url=https://books.google.com/books?id=2v9zDAsLvA0C&pg=PP1 |title=Statistical Power Analysis for the Behavioral Sciences |publisher=Routledge |year=1988 |isbn=978-1-134-74270-7 |pages=}}</ref>{{Rp|p=67|date=July 2014|chapter-url = http://www.utstat.toronto.edu/~brunner/oldclass/378f16/readings/CohenPower.pdf#page=66}} <math display="block">s = \sqrt{\frac{(n_1-1)s^2_1 + (n_2-1)s^2_2}{n_1+n_2 - 2}}</math> where the variance for one of the groups is defined as <math display="block">s_1^2 = \frac 1 {n_1-1} \sum_{i=1}^{n_1} (x_{1,i} - \bar{x}_1)^2,</math> and similarly for the other group. Other authors choose a slightly different computation of the standard deviation when referring to "Cohen's ''d''" where the denominator is without "-2"<ref>{{Cite journal | author1 = Robert E. McGrath | author2 = Gregory J. Meyer | title = When Effect Sizes Disagree: The Case of r and d | journal = [[Psychological Methods]] | volume = 11 | issue = 4 | pages = 386–401 | year = 2006 | url = http://www.bobmcgrath.org/Pubs/When_effect_sizes_disagree.pdf | doi = 10.1037/1082-989x.11.4.386 | pmid = 17154753 | citeseerx = 10.1.1.503.754 | access-date = 2014-07-30 | archive-url = https://web.archive.org/web/20131008171400/http://www.bobmcgrath.org/Pubs/When_effect_sizes_disagree.pdf | archive-date = 2013-10-08 | url-status=dead }}</ref><ref>{{cite book | last1=Hartung|first1=Joachim | last2=Knapp|first2=Guido | last3=Sinha|first3=Bimal K. | title=Statistical Meta-Analysis with Applications | url=https://books.google.com/books?id=JEoNB_2NONQC&pg=PP1|year=2008|publisher=John Wiley & Sons | isbn=978-1-118-21096-3}}</ref>{{Rp|p=14|date=November 2012}} <math display="block">s = \sqrt{\frac{(n_1-1)s^2_1 + (n_2-1)s^2_2}{n_1+n_2}}</math> This definition of "Cohen's ''d''" is termed the [[maximum likelihood]] estimator by Hedges and Olkin,<ref name="HedgesL1985Statistical" /> and it is related to Hedges' ''g'' by a scaling factor (see below). With two paired samples, an approach is to look at the distribution of the difference scores. In that case, ''s'' is the standard deviation of this distribution of difference scores (of note, the standard deviation of difference scores is dependent on the correlation between paired samples). This creates the following relationship between the t-statistic to test for a difference in the means of the two paired groups and Cohen's ''d''' (computed with difference scores): <math display="block">t = \frac{\bar{X}_1 - \bar{X}_2}{\text{SE}_{diff}} = \frac{\bar{X}_1 - \bar{X}_2}{\frac{\text{SD}_{diff}}{\sqrt N}} = \frac{\sqrt{N} (\bar{X}_1 - \bar{X}_2)}{SD_{diff}}</math> and <math display="block">d' = \frac{\bar{X}_1 - \bar{X}_2}{\text{SD}_{diff}} = \frac t {\sqrt N}</math>However, for paired samples, Cohen states that d' does not provide the correct estimate to obtain the power of the test for d, and that before looking the values up in the tables provided for d, it should be corrected for r as in the following formula:{{sfn|Cohen|1988|p=49}} <math display="block">\frac{d'} {\sqrt{1 - r}}.</math>where r is the correlation between paired measurements. Given the same sample size, the higher r, the higher the power for a test of paired difference. Since d' depends on r, as a measure of effect size it is difficult to interpret; therefore, in the context of paired analyses, since it is possible to compute d' or d (estimated with a pooled standard deviation or that of a group or time-point), it is necessary to explicitly indicate which one is being reported. As a measure of effect size, d (estimated with a pooled standard deviation or that of a group or time-point) is more appropriate, for instance in meta-analysis.<ref name=":0">{{Cite journal |last=Dunlap |first=William P. |last2=Cortina |first2=Jose M. |last3=Vaslow |first3=Joel B. |last4=Burke |first4=Michael J. |date=1996 |title=Meta-analysis of experiments with matched groups or repeated measures designs. |url=http://doi.apa.org/getdoi.cfm?doi=10.1037/1082-989X.1.2.170 |journal=Psychological Methods |language=en |volume=1 |issue=2 |pages=170–177 |doi= |issn=1082-989X}} {{doi|10.1037//1082-989X.1.2.170}}</ref> Cohen's ''d'' is frequently used in [[estimating sample sizes]] for statistical testing. A lower Cohen's ''d'' indicates the necessity of larger sample sizes, and vice versa, as can subsequently be determined together with the additional parameters of desired [[significance level]] and [[statistical power]].<ref>{{cite book|last=Kenny|first=David A.|title=Statistics for the Social and Behavioral Sciences|url=https://books.google.com/books?id=EdqhQgAACAAJ&pg=PP1|year=1987|publisher=Little, Brown|isbn=978-0-316-48915-7|chapter=Chapter 13|chapter-url=http://davidakenny.net/doc/statbook/chapter_13.pdf}}</ref> ==== Glass' Δ ==== In 1976, [[Gene V. Glass]] proposed an estimator of the effect size that uses only the standard deviation of the second group<ref name="HedgesL1985Statistical"/>{{Rp|p=78|date=November 2012}} <math display="block">\Delta = \frac{\bar{x}_1 - \bar{x}_2}{s_2}</math> The second group may be regarded as a control group, and Glass argued that if several treatments were compared to the control group it would be better to use just the standard deviation computed from the control group, so that effect sizes would not differ under equal means and different variances. Under a correct assumption of equal population variances a pooled estimate for ''σ'' is more precise. ==== Hedges' ''g'' ==== Hedges' ''g'', suggested by [[Larry Hedges]] in 1981,<ref>{{Cite journal | author = Larry V. Hedges | title = Distribution theory for Glass' estimator of effect size and related estimators | journal = [[Journal of Educational Statistics]] | volume = 6 | issue = 2 | pages = 107–128 | year = 1981 | doi = 10.3102/10769986006002107 | s2cid = 121719955 | author-link = Larry V. Hedges }}</ref> is like the other measures based on a standardized difference<ref name="HedgesL1985Statistical"/>{{Rp|p=79|date=November 2012}} <math display="block">g = \frac{\bar{x}_1 - \bar{x}_2}{s^*}</math> where the pooled standard deviation <math>s^*</math> is computed as:<!---there is something missing here... otherwise it is identical with Cohen's d... --> <math display="block">s^* = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}.</math> However, as an [[estimator]] for the population effect size ''θ'' it is [[Bias of an estimator|bias]]ed. Nevertheless, this bias can be approximately corrected through multiplication by a factor <math display="block">g^* = J(n_1+n_2-2) \,\, g \, \approx \, \left(1-\frac{3}{4(n_1+n_2)-9}\right) \,\, g</math> Hedges and Olkin refer to this less-biased estimator <math>g^*</math> as ''d'',<ref name="HedgesL1985Statistical" /> but it is not the same as Cohen's ''d''. The exact form for the correction factor ''J''() involves the [[gamma function]]<ref name="HedgesL1985Statistical"/>{{Rp|p=104|date=November 2012}} <math display="block">J(a) = \frac{\Gamma(a/2)}{\sqrt{a/2 \,}\,\Gamma((a-1)/2)}.</math> <!-- In the above 'height' example, Hedges' ''ĝ'' effect size equals 1.76 (95% confidence intervals: 1.70 − 1.82). Notice how the large sample size has increased the effect size from Cohen's ''d''? If, instead, the available data were from only 90 men and 80 women Hedges' ''ĝ'' provides a more conservative estimate of effect size: 1.70 (with larger 95% confidence intervals: 1.35 – 2.05). --> There are also multilevel variants of Hedges' g, e.g., for use in cluster randomised controlled trials (CRTs).<ref>Hedges, L. V. (2011). Effect sizes in three-level cluster-randomized experiments. Journal of Educational and Behavioral Statistics, 36(3), 346-380. </ref> CRTs involve randomising clusters, such as schools or classrooms, to different conditions and are frequently used in education research. ====Ψ, root-mean-square standardized effect==== A similar effect size estimator for multiple comparisons (e.g., [[ANOVA]]) is the Ψ root-mean-square standardized effect:<ref name="Steiger2004"/> <math display="block">\Psi = \sqrt{ \frac{1}{k-1} \cdot \sum_{j=1}^k \left(\frac{\mu_j-\mu}{\sigma}\right)^2}</math> where ''k'' is the number of groups in the comparisons. This essentially presents the omnibus difference of the entire model adjusted by the root mean square, analogous to ''d'' or ''g''. In addition, a generalization for multi-factorial designs has been provided.<ref name="Steiger2004"/> ==== Distribution of effect sizes based on means ==== Provided that the data is [[Gaussian distribution|Gaussian]] distributed a scaled Hedges' ''g'', <math display="inline">\sqrt{n_1 n_2/(n_1+n_2)}\,g</math>, follows a [[noncentral t-distribution|noncentral ''t''-distribution]] with the [[noncentrality parameter]] <math display="inline">\sqrt{n_1 n_2/(n_1+n_2)}\theta</math> and {{math|(''n''<sub>1</sub> + ''n''<sub>2</sub> − 2)}} degrees of freedom. Likewise, the scaled Glass' Δ is distributed with {{math|''n''<sub>2</sub> − 1}} degrees of freedom. From the distribution it is possible to compute the [[Expected value|expectation]] and variance of the effect sizes. In some cases large sample approximations for the variance are used. One suggestion for the variance of Hedges' unbiased estimator is<ref name="HedgesL1985Statistical"/> {{Rp|p=86|date=November 2012}} <math display="block">\hat{\sigma}^2(g^*) = \frac{n_1+n_2}{n_1 n_2} + \frac{(g^*)^2}{2(n_1 + n_2)}.</math> ==== Strictly standardized mean difference (SSMD) ==== {{main|Strictly standardized mean difference}} As a statistical parameter, SSMD (denoted as <math>\beta</math>) is defined as the ratio of [[mean]] to [[standard deviation]] of the difference of two random values respectively from two groups. Assume that one group with random values has [[mean]] <math>\mu_1</math> and [[variance]] <math>\sigma_1^2</math> and another group has [[mean]] <math>\mu_2</math> and [[variance]] <math>\sigma_2^2</math>. The [[covariance]] between the two groups is <math>\sigma_{12}.</math> Then, the SSMD for the comparison of these two groups is defined as<ref name="ZhangGenomics2007">{{Cite journal |last=Zhang |first=XHD |year=2007 |title=A pair of new statistical parameters for quality control in RNA interference high-throughput screening assays |journal=Genomics |volume=89 |issue=4 |pages=552–61 |doi=10.1016/j.ygeno.2006.12.014 |pmid=17276655 |doi-access=}}</ref> :<math>\beta = \frac{\mu_1 - \mu_2}{\sqrt{\sigma_1^2 + \sigma_2^2 - 2\sigma_{12} }}.</math> If the two groups are independent, :<math>\beta = \frac{\mu_1 - \mu_2}{\sqrt{\sigma_1^2 + \sigma_2^2 }}.</math> If the two independent groups have equal [[variance]]s <math>\sigma^2</math>, :<math>\beta = \frac{\mu_1 - \mu_2}{\sqrt{2}\sigma}.</math> ==== Other metrics ==== [[Mahalanobis distance]] (D) is a multivariate generalization of Cohen's d, which takes into account the relationships between the variables.<ref>{{Cite journal | last=Del Giudice | first=Marco | date=2013-07-18|title=Multivariate Misgivings: Is D a Valid Measure of Group and Sex Differences? | journal=Evolutionary Psychology | language=en | volume=11 | issue=5 | pages=1067–1076 | doi=10.1177/147470491301100511 | doi-access=free| pmid=24333840 | pmc=10434404 }}</ref> ===Categorical family: Effect sizes for associations among categorical variables=== {| class="wikitable" align="right" valign |- | align="center" | <math>\varphi = \sqrt{ \frac{\chi^2}{N}}</math> | align="center" | <math>\varphi_c = \sqrt{ \frac{\chi^2}{N(k - 1)}}</math> |- ! Phi (''φ'') ! Cramér's ''V'' (''φ''<sub>''c''</sub>) |} Commonly used measures of association for the [[chi-squared test]] are the [[Phi coefficient]] and [[Harald Cramér|Cramér]]'s [[Cramér's V (statistics)|V]] (sometimes referred to as Cramér's phi and denoted as ''φ''<sub>''c''</sub>). Phi is related to the [[point-biserial correlation coefficient]] and Cohen's ''d'' and estimates the extent of the relationship between two variables (2 × 2).<ref name="Ref_">Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). [http://www.eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=ED433353&ERICExtSearch_SearchType_0=no&accno=ED433353 Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula.] Paper presented at the annual meeting of the Florida Educational Research Association, Orlando, FL. (ERIC Document Reproduction Service No. ED433353)</ref> Cramér's V may be used with variables having more than two levels. Phi can be computed by finding the square root of the chi-squared statistic divided by the sample size. Similarly, Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the length of the minimum dimension (''k'' is the smaller of the number of rows ''r'' or columns ''c''). φ<sub>''c''</sub> is the intercorrelation of the two discrete variables<ref name="Ref_a">{{cite book | last=Sheskin|first=David J. | title=Handbook of Parametric and Nonparametric Statistical Procedures | url=https://books.google.com/books?id=bmwhcJqq01cC&pg=PP1 | edition=Third | year=2003 | publisher=CRC Press | isbn=978-1-4200-3626-8}}</ref> and may be computed for any value of ''r'' or ''c''. However, as chi-squared values tend to increase with the number of cells, the greater the difference between ''r'' and ''c'', the more likely V will tend to 1 without strong evidence of a meaningful correlation. ==== Cohen's omega (''ω'') ==== Another measure of effect size used for chi-squared tests is Cohen's omega (<math> \omega</math>). This is defined as <math display="block"> \omega = \sqrt{ \sum_{i=1}^m \frac{ (p_{1i} - p_{0i})^2 }{p_{0i}} } </math> where ''p''<sub>0''i''</sub> is the proportion of the ''i''<sup>th</sup> cell under ''H''<sub>0</sub>, ''p''<sub>1''i''</sub> is the proportion of the ''i''<sup>th</sup> cell under ''H''<sub>1</sub> and ''m'' is the number of cells. ==== Odds ratio ==== The [[odds ratio]] (OR) is another useful effect size. It is appropriate when the research question focuses on the degree of association between two [[binary data|binary variables]]. For example, consider a study of spelling ability. In a control group, two students pass the class for every one who fails, so the odds of passing are two to one (or 2/1 = 2). In the treatment group, six students pass for every one who fails, so the odds of passing are six to one (or 6/1 = 6). The effect size can be computed by noting that the odds of passing in the treatment group are three times higher than in the control group (because 6 divided by 2 is 3). Therefore, the odds ratio is 3. Odds ratio statistics are on a different scale than Cohen's ''d'', so this '3' is not comparable to a Cohen's ''d'' of 3. ==== Relative risk ==== The [[relative risk]] (RR), also called '''risk ratio''', is simply the risk (probability) of an event relative to some independent variable. This measure of effect size differs from the odds ratio in that it compares ''probabilities'' instead of ''odds'', but asymptotically approaches the latter for small probabilities. Using the example above, the ''probabilities'' for those in the control group and treatment group passing is 2/3 (or 0.67) and 6/7 (or 0.86), respectively. The effect size can be computed the same as above, but using the probabilities instead. Therefore, the relative risk is 1.28. Since rather large probabilities of passing were used, there is a large difference between relative risk and odds ratio. Had ''failure'' (a smaller probability) been used as the event (rather than ''passing''), the difference between the two measures of effect size would not be so great. While both measures are useful, they have different statistical uses. In medical research, the [[odds ratio]] is commonly used for [[case-control study|case-control studies]], as odds, but not probabilities, are usually estimated.<ref>{{cite journal |author = Deeks J |year = 1998 |title = When can odds ratios mislead? : Odds ratios should be used only in case-control studies and logistic regression analyses |journal = BMJ |volume = 317 |issue = 7166 |pages = 1155–6 |pmid = 9784470 |pmc = 1114127|doi=10.1136/bmj.317.7166.1155a }}</ref> Relative risk is commonly used in [[randomized controlled trial]]s and [[cohort study|cohort studies]], but relative risk contributes to overestimations of the effectiveness of interventions.<ref name="Stegenga2015">{{Cite journal | last1 = Stegenga | first1 = J. | title = Measuring Effectiveness | journal = Studies in History and Philosophy of Biological and Biomedical Sciences | volume = 54 | pages = 62–71 | year = 2015 | url = https://www.academia.edu/16420844 | doi=10.1016/j.shpsc.2015.06.003| pmid = 26199055 }}</ref> ==== Risk difference ==== The [[risk difference]] (RD), sometimes called absolute risk reduction, is simply the difference in risk (probability) of an event between two groups. It is a useful measure in experimental research, since RD tells you the extent to which an experimental interventions changes the probability of an event or outcome. Using the example above, the probabilities for those in the control group and treatment group passing is 2/3 (or 0.67) and 6/7 (or 0.86), respectively, and so the RD effect size is 0.86 − 0.67 = 0.19 (or 19%). RD is the superior measure for assessing effectiveness of interventions.<ref name="Stegenga2015"/> ==== Cohen's ''h'' ==== {{main|Cohen's h}} One measure used in power analysis when comparing two independent proportions is Cohen's ''h''. This is defined as follows <math display="block"> h = 2 ( \arcsin \sqrt{p_1} - \arcsin \sqrt{p_2}) </math> where ''p''<sub>1</sub> and ''p''<sub>2</sub> are the proportions of the two samples being compared and arcsin is the arcsine transformation. ==== Probability of superiority ==== {{Main|Probability of superiority}} To more easily describe the meaning of an effect size to people outside statistics, the common language effect size, as the name implies, was designed to communicate it in plain English. It is used to describe a difference between two groups and was proposed, as well as named, by Kenneth McGraw and S. P. Wong in 1992.<ref name="McGraw KO, Wong SP 1992 361–365">{{Cite journal |vauthors=McGraw KO, Wong SP | year = 1992 | title = A common language effect size statistic | journal = [[Psychological Bulletin]] | volume = 111 | issue = 2 | pages = 361–365 | doi= 10.1037/0033-2909.111.2.361}}</ref> They used the following example (about heights of men and women): "in any random pairing of young adult males and females, the probability of the male being taller than the female is .92, or in simpler terms yet, in 92 out of 100 blind dates among young adults, the male will be taller than the female",<ref name="McGraw KO, Wong SP 1992 361–365"/> when describing the population value of the common language effect size. ==== Effect size for ordinal data ==== '''Cliff's delta''' or <math>d</math>, originally developed by [[Norman Cliff]] for use with ordinal data,<ref name="Cliff1993">{{cite journal | last=Cliff | first=Norman | title=Dominance statistics: Ordinal analyses to answer ordinal questions | year=1993 | journal=Psychological Bulletin | volume=114 | pages=494–509 | issue=3 | doi=10.1037/0033-2909.114.3.494}}</ref>{{Dubious|date=May 2024|reason=I'm at least 80% sure this is just a weird name for Kendall's tau.}} is a measure of how often the values in one distribution are larger than the values in a second distribution. Crucially, it does not require any assumptions about the shape or spread of the two distributions. The sample estimate <math>d</math> is given by: <math display="block">d = \frac{\sum_{i,j} [x_i > x_j] - [x_i < x_j]}{mn}</math> where the two distributions are of size <math>n</math> and <math>m</math> with items <math>x_i</math> and <math>x_j</math>, respectively, and <math>[\cdot]</math> is the [[Iverson bracket]], which is 1 when the contents are true and 0 when false. <math>d</math> is linearly related to the [[Mann–Whitney U test|Mann–Whitney U statistic]]; however, it captures the direction of the difference in its sign. Given the Mann–Whitney <math>U</math>, <math>d</math> is: <math display="block">d = \frac{2U}{mn} - 1</math> ==== Cohen's g ==== One of simplest effect sizes for measuring how much a proportion differs from 50% is Cohen's g.<ref name="CohenJ1988Statistical" />{{Rp|page=147}} It measures how much a proportion differs from 50%. For example, if 85.2% of arrests for car theft are males, then effect size of sex on arrest when measured with Cohen's g is <math>g = 0.852-0.5=0.352</math>. In general: <math>g = P - 0.50 \text{ or } 0.50 - P \quad (\text{directional}), </math> <math>g = |P - 0.50| \quad (\text{nondirectional}). </math> Units of Cohen's g are more intuitive (proportion) than in some other effect sizes. It is sometime used in combination with [[Binomial test]].
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