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====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 = 386β401 | 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=170β177 |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>
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