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| {{Expert-subject|statistics|date=May 2011}}
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| In [[statistics]], an '''effect size''' is a measure of the strength of a phenomenon<ref name="Kelley2012">{{cite journal | last=Kelley | first=Ken |coauthors=Preacher, Kristopher J. | title=On Effect Size | year=2012 | journal=Psychological Methods | volume=17 | pages=137–152 | doi=10.1037/a0028086 | issue=2}}</ref> (for example, the change in an outcome after experimental intervention). An effect size calculated from [[data]] is a [[descriptive statistics|descriptive statistic]] that conveys the estimated magnitude of a relationship without making any statement about whether the apparent relationship in the data reflects a true relationship in the [[statistical population|population]]. In that way, effect sizes complement [[inferential statistics]] such as [[p-value|''p''-values]]. Among other uses, effect size measures play an important role in [[meta-analysis]] studies that summarize findings from a specific area of research, and in [[statistical power]] analyses.
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| The concept of effect size already appears in everyday language. For example, a weight loss program may boast that it leads to an average weight loss of 30 pounds. In this case, 30 pounds is the claimed effect size. Another example is that a tutoring program may claim that it raises school performance by one letter grade. This grade increase is the claimed effect size of the program. These are both examples of "absolute effect sizes", meaning that they convey the average difference between two groups without any discussion of the [[variance|variability]] within the groups. For example, if the weight loss program results in an average loss of 30 pounds, it is possible that every participant loses exactly 30 pounds, or half the participants lose 60 pounds and half lose no weight at all.
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| Reporting effect sizes is considered good practice when presenting empirical research findings in many fields.<ref name="Wilkinson1999">{{cite journal | last=Wilkinson | first=Leland |coauthors=APA Task Force on Statistical Inference | title=Statistical methods in psychology journals: Guidelines and explanations | year=1999 | journal=American Psychologist | volume=54 | pages=594–604 | doi=10.1037/0003-066X.54.8.594 | issue=8}}</ref><ref name="Nakagawa2007">{{cite journal | last=Nakagawa | first=Shinichi | coauthors=Cuthill, Innes C | year=2007 | title=Effect size, confidence interval and statistical significance: a practical guide for biologists | journal = Biological Reviews Cambridge Philosophical Society | volume=82 | pages=591–605 | doi=10.1111/j.1469-185X.2007.00027.x | pmid=17944619 | issue=4}}</ref> The reporting of effect sizes facilitates the interpretation of the substantive, as opposed to the statistical, significance of a research result.<ref name="Ellis2010">{{Cite book
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| | author = Ellis, Paul D.
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| | title = The Essential Guide to Effect Sizes: An Introduction to Statistical Power, Meta-Analysis and the Interpretation of Research Results
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| | publisher = Cambridge University Press
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| | location = United Kingdom
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| | year = 2010
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| }}</ref>
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| Effect sizes are particularly prominent in social and medical research. Relative and absolute measures of effect size convey different information, and can be used complementarily. A prominent task force in the psychology research community expressed the following recommendation:
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| {{Quotation|Always present effect sizes for primary outcomes...If the units of measurement are meaningful on a practical level (e.g., number of cigarettes smoked per day), then we usually prefer an unstandardized measure (regression coefficient or mean difference) to a standardized measure (''r'' or ''d''). | L. Wilkinson and APA Task Force on Statistical Inference (1999, p. 599)}}
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| ==Overview==
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| ===Population and sample effect sizes===
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| The term ''effect size'' can refer to a statistic calculated from a sample of [[data]], or to a parameter of a hypothetical statistical population. Conventions for distinguishing sample from population effect sizes follow standard statistical practices — one common approach is to use Greek letters like ρ to denote population parameters and Latin letters like ''r'' to denote the corresponding statistic; alternatively, a "hat" can be placed over the population parameter to denote the statistic, e.g. with <math>\hat\rho</math> being the estimate of the parameter <math>\rho</math>.
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| As in any statistical setting, effect sizes are estimated with error, and may be biased unless the effect size estimator that is used is appropriate for the manner in which the data were [[sampling (statistics)|sampled]] and the manner in which the measurements were made. An example of this is [[publication bias]], which occurs when scientists only report results when the estimated effect sizes are large or are statistically significant. As a result, if many researchers are carrying out studies under low statistical power, the reported results are biased to be stronger than true effects, if any.<ref name="Brand2008">{{Cite journal | author = Brand A, Bradley MT, Best LA, Stoica G | year = 2008 | title = Accuracy of effect size estimates from published psychological research | journal = [[Perceptual and Motor Skills]] | volume = 106 | issue = 2 | pages = 645–649 | doi=10.2466/PMS.106.2.645-649 | url = http://mtbradley.com/brandbradelybeststoicapdf.pdf | pmid=18556917}}</ref> Another example where effect sizes may be distorted is in a multiple trial experiment, where the effect size calculation is based on the averaged or aggregated response across the trials.<ref name="Brand2011">{{Cite journal | author = Brand A, Bradley MT, Best LA, Stoica G | year = 2011 | title = Multiple trials may yield exaggerated effect size estimates | journal = [[The Journal of General Psychology]] | volume = 138 | issue = 1 | pages = 1–11 | doi=10.1080/00221309.2010.520360 | url = http://www.ipsychexpts.com/brand_et_al_(2011).pdf}}</ref>
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| ===Relationship to test statistics===
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| Sample-based effect sizes are distinguished from [[test statistic]]s used in hypothesis testing, in that they estimate the strength of an apparent relationship, rather than assigning a [[statistical significance|significance]] level reflecting whether the relationship could be due to chance. The effect size does not determine the significance level, or vice-versa. Given a sufficiently large sample size, a statistical comparison will always show a significant difference unless the population effect size is exactly zero. For example, a sample [[Pearson correlation]] coefficient of 0.1 is strongly statistically significant if the sample size is 1000. Reporting only the significant ''p''-value from this analysis could be misleading if a correlation of 0.1 is too small to be of interest in a particular application.
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| ===Standardized and unstandardized effect sizes===
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| The term ''effect size'' can refer to a standardized measures of effect (such as ''r'', [[Cohen's d|Cohen's ''d'']], and odds ratio), or to an unstandardized measure (e.g., the raw difference between group means and unstandardized regression coefficients). Standardized effect size measures are typically used when the metrics of variables being studied do not have intrinsic meaning (e.g., a score on a personality test on an arbitrary scale), when results from multiple studies are being combined, when some or all of the studies use different scales, or when it is desired to convey the size of an effect relative to the variability in the population. In meta-analysis, standardized effect sizes are used as a common measure that can be calculated for different studies and then combined into an overall summary.
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| ==Types==
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| === Effect sizes based on "variance explained" ===
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| These effect sizes estimate the amount of the variance within an experiment that is "explained" or "accounted for" by the experiment's model.
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| ==== Pearson ''r'' (correlation) ====
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| [[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. Cohen gives the following guidelines for the social sciences:<ref name="CohenJ1988Statistical"/><ref name="CohenJ1992">{{cite journal | last=Cohen | first=J | year=1992 | title=A power primer | journal=Psychological Bulletin | volume=112 | pages=155–159 | doi=10.1037/0033-2909.112.1.155 | pmid=19565683 | issue=1}}</ref>
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| {| class="wikitable"
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| !| Effect size
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| !|''r''
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| |-
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| | Small
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| | 0.10
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| |-
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| | Medium
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| | 0.30
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| |-
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| | Large
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| | 0.50
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| |}
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| ===== Coefficient of determination =====
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| A related ''effect size'' is ''r²'', the [[coefficient of determination]] (also referred to as "''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 ''r²'' is always positive, so does not convey the direction of the correlation between the two variables.
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| ====== Eta-squared, η<sup>2</sup> ======
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| 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.
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| : <math> \eta ^2 = \frac{SS_\text{Treatment}}{SS_\text{Total}} .</math>
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| ====== Omega-squared, ω<sup>2</sup> ======
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| {{see also subsection|Adjusted R2|Coefficient of determination}}
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| A less biased estimator of the variance explained in the population is ω<sup>2</sup><ref>Bortz, 1999{{full|date=November 2012}}, p. 269f.;</ref><ref>Bühner & Ziegler{{full|date=November 2012}} (2009, p. 413f)</ref><ref name="Tabachnick 2007, p. 55">Tabachnick & Fidell (2007, p. 55)</ref>
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| :<math>\omega^2 = \frac{SS_\text{treatment}-df_\text{treatment} * MS_\text{error}}{SS_\text{total} + MS_\text{error}} .</math>
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| 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>Olejnik, S. & Algina, J. 2003. Generalized Eta and Omega Squared Statistics: Measures of Effect Size for Some Common Research Designs ''Psychological Methods''. 8:(4)434-447. http://cps.nova.edu/marker/olejnik2003.pdf</ref> In addition, methods to calculate partial Omega<sup>2</sup> for individual factors and combined factors in designs with up to three independent variables have been published.<ref name=OlejnikAlgina/>
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| ==== Cohen's ''ƒ''<sup>2</sup> ====
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| Cohen's ''ƒ''<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>).
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| The ''ƒ''<sup>2</sup> effect size measure for multiple regression is defined as:
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| :<math>f^2 = {R^2 \over 1 - R^2}</math>
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| :where ''R''<sup>2</sup> is the [[squared multiple correlation]].
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| Likewise, ''ƒ''<sup>2</sup> can be defined as:
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| :<math>f^2 = {\eta^2 \over 1 - \eta^2}</math> or <math>f^2 = {\omega^2 \over 1 - \omega^2}</math>
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| :for models described by those effect size measures.<ref name=Steiger2004>Steiger, J. H. 2004. Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis. ''Psychological Methods'' 9:(2) 164-182. http://www.statpower.net/Steiger%20Biblio/Steiger04.pdf</ref>
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| The <math>f^{2}</math> effect size measure for hierarchical multiple regression is defined as:
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| :<math>f^2 = {R^2_{AB} - R^2_A \over 1 - R^2_{AB}}</math>
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| :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 ''B''. By convention, ''ƒ''<sup>2</sup><sub>''A''</sub> effect sizes of 0.02, 0.15, and 0.35 are termed ''small'', ''medium'', and ''large'', respectively.<ref name="CohenJ1988Statistical"/>
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| Cohen's <math>\hat{f}</math> can also be found for factorial analysis of variance (ANOVA, aka the F-test) working backwards using :
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| : <math>\hat{f}_\text{effect} = {\sqrt{(df_\text{effect}/N) (F_\text{effect}-1)}}.</math>
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| In a balanced design (equivalent sample sizes across groups) of ANOVA, the corresponding population parameter of <math>f^2</math> is
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| : <math>{SS(\mu_1,\mu_2,\dots,\mu_K)}\over{K \times \sigma^2},</math>
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| 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]] manipulation in ANOVA.
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| === Effect sizes based on means or distances between/among means ===
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| {{Expert-subject|Statistics|date=March 2011}}
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| [[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 between two populations<ref name="HedgesL1985Statistical">{{Cite book
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| | author = [[Larry V. Hedges]] & [[Ingram Olkin]]
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| | title = Statistical Methods for Meta-Analysis
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| | publisher = [[Academic Press]]
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| | year = 1985
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| | location = Orlando
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| | isbn = 0-12-336380-2
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| }}</ref>{{Rp|78|date=November 2012}}
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| : <math>\theta = \frac{\mu_1 - \mu_2}{\sigma},</math>
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| 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.
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| 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.
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| 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]], so is not affected by the sample size.
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| ==== Cohen's ''d'' ====
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| Cohen's ''d'' is defined as the difference between two means divided by a standard deviation for the data, i.e.,
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| : <math>d = \frac{\bar{x}_1 - \bar{x}_2}{s}.</math>
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| Cohen's ''d'' is frequently used in [[estimating sample sizes]]. 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>[http://davidakenny.net/doc/statbook/chapter_13.pdf Chapter 13], page 215, in: {{cite book |author=Kenny, David A. |title=Statistics for the social and behavioral sciences |publisher=Little, Brown |location=Boston |year=1987 |pages= |isbn=0-316-48915-8 |oclc= |doi= |accessdate=}}</ref>
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| What precisely the standard deviation ''s'' is was not originally made explicit by [[Jacob Cohen (statistician)|Jacob Cohen]] because he defined it (using the symbol "σ") as "the standard deviation of either population (since they are assumed equal)".<ref name="CohenJ1988Statistical">{{Cite book
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| | author = [[Jacob Cohen (statistician)|Jacob Cohen]]
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| | title = Statistical Power Analysis for the Behavioral Sciences
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| | year = 1988
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| | edition = second
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| | publisher = [[Lawrence Erlbaum Associates]]
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| }}</ref>{{Rp|20|date=November 2012}}
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| Other authors make the computation of the standard deviation more explicit with the following definition for a [[pooled standard deviation]]<ref>{{Cite book
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| | author = Joachim Hartung, Guido Knapp & Bimal K. Sinha
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| | title = Statistical Meta-Analysis with Application
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| | publisher = [[John Wiley & Sons|Wiley]]
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| | location = Hoboken, New Jersey
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| | year = 2008
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| }}</ref>{{Rp|14|date=November 2012}} with two independent samples.
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| : <math>s = \sqrt{\frac{(n_1-1)s^2_1 + (n_2-1)s^2_2}{n_1+n_2-2}}</math>
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| : <math>s_1^2 = \frac{1}{n_1-1} \sum_{i=1}^{n_1} (x_{1,i} - \bar{x}_1)^2</math>
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| This definition of "Cohen's ''d''" is termed the [[maximum likelihood]] estimator by Hedges and Olkin,<ref name="HedgesL1985Statistical" />
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| and it is related to Hedges's ''g'' by a scaling factor (see below).
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| So, in the example above of visiting England and observing men's and women's heights, the data ([http://www.ic.nhs.uk/pubs/hlthsvyeng2004upd Aaron,Kromrey,& Ferron, 1998, November]; from a 2004 UK representative sample of 2436 men and 3311 women) are:
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| * '''Men''': mean height = 1750 mm; standard deviation = 89.93 mm
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| * '''Women''': mean height = 1612 mm; standard deviation = 69.05 mm
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| The effect size (using Cohen's ''d'') would equal 1.72 (95% [[confidence interval]]s: 1.66 – 1.78). This is very large and you should have no problem in detecting that there is a consistent height difference, on average, between men and women.
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| ==== Glass's Δ ====
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| 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|78|date=November 2012}}
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| : <math>\Delta = \frac{\bar{x}_1 - \bar{x}_2}{s_2}</math>
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| 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.
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| Under a correct assumption of equal population variances a pooled estimate for σ is more precise.
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| ==== Hedges's ''g'' ====
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| Hedges's ''g'', suggested by [[Larry Hedges]] in 1981,<ref>{{Cite journal
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| | author = [[Larry V. Hedges]]
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| | title = Distribution theory for Glass's estimator of effect size and related estimators
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| | journal = [[Journal of Educational Statistics]]
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| | volume = 6
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| | issue = 2
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| | pages = 107–128
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| | year = 1981
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| | doi = 10.3102/10769986006002107
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| }}</ref>
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| is like the other measures based on a standardized difference<ref name="HedgesL1985Statistical"/>{{Rp|79|date=November 2012}}
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| : <math>g = \frac{\bar{x}_1 - \bar{x}_2}{s^*}</math>
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| but its pooled standard deviation <math>s^*</math> is computed slightly differently from Cohen's ''d''. Initially, one can calculate the pooled standard deviation as if doing so for Cohen's ''d'': | |
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| : <math>s^* = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}.</math>
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| However, as an [[estimator]] for the population effect size ''θ'' it is [[Bias of an estimator|bias]]ed.
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| Nevertheless, this bias can be approximately corrected through multiplication by a factor
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| : <math>g^* = J(n_1+n_2-2) \,\, g \, \approx \, \left(1-\frac{3}{4(n_1+n_2)-9}\right) \,\, g</math>
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| 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''.
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| The exact form for the correction factor ''J()'' involves the [[gamma function]]<ref name="HedgesL1985Statistical"/>{{Rp|104|date=November 2012}}
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| : <math>J(a) = \frac{\Gamma(a/2)}{\sqrt{a/2 \,}\,\Gamma((a-1)/2)}.</math>
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| <!--
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| In the above 'height' example, Hedges's ''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's ''ĝ'' provides a more conservative estimate of effect size: 1.70 (with larger 95% confidence intervals: 1.35 – 2.05).
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| -->
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| ====Ψ, Root-Mean-Square Standardized Effect====
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| A similar effect size estimator for multiple comparisons (e.g., [[ANOVA]]) is the Ψ root-mean-square standardized effect.<ref name="Steiger2004"/> This essentially presents the omnibus difference of the entire model adjusted by the root mean square, analogous to ''d'' or ''g''. The simplest formula for Ψ, suitable for one-way ANOVA, is
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| : <math>\Psi = \sqrt{\left(\frac{1}{k-1}\right)\frac{\Sigma(\bar{x}_j-\bar{X})^2}{MS_{error}}}</math>
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| In addition, a generalization for multi-factorial designs has been provided.<ref name="Steiger2004"/>
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| ==== Distribution of effect sizes based on means ====
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| Provided that the data is [[Gaussian]] distributed a scaled Hedges's ''g'', <math>\sqrt{n_1 n_2/(n_1+n_2)}\,g</math>, follows a [[noncentral t-distribution|noncentral ''t''-distribution]] with the [[noncentrality parameter]] <math>\sqrt{n_1 n_2/(n_1+n_2)}\theta</math> and (''n''<sub>1</sub> + ''n''<sub>2</sub> − 2) degrees of freedom. Likewise, the scaled Glass's Δ is distributed with ''n''<sub>2</sub> − 1 degrees of freedom.
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| From the distribution it is possible to compute the [[Expected value|expectation]] and variance of the effect sizes.
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| In some cases large sample approximations for the variance are used.
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| One suggestion for the variance of Hedges's unbiased estimator is<ref name="HedgesL1985Statistical"/>{{Rp|86|date=November 2012}}
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| : <math>\hat{\sigma}^2(g^*) = \frac{n_1+n_2}{n_1 n_2} + \frac{(g^*)^2}{2(n_1 + n_2)}.</math>
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| ===Effect sizes for associations among categorical variables===
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| {| class="wikitable" align="right" valign
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| |-
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| | align="center" |
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| <math>\phi = \sqrt{ \frac{\chi^2}{N}}</math>
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| | align="center" |
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| <math>\phi_c = \sqrt{ \frac{\chi^2}{N(k - 1)}}</math>
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| |-
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| ! Phi (φ)
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| ! Cramér's V (φ<sub>c</sub>)
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| |}
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| 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 x 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.
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| Phi can be computed by finding the square root of the chi-squared statistic divided by the sample size.
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| 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'').
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| φ<sub>''c''</sub> is the intercorrelation of the two discrete variables<ref name="Ref_a">Sheskin, David J. (1997). Handbook of Parametric and Nonparametric Statistical Procedures. Boca Raton, Fl: CRC Press.</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.
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| Cramér's V may also be applied to 'goodness of fit' chi-squared models (i.e. those where ''c''=1). In this case it functions as a measure of tendency towards a single outcome (i.e. out of ''k'' outcomes). In such a case one must use ''r'' for ''k'', in order to preserve the 0 to 1 range of V. Otherwise, using ''c'' would reduce the equation to that for Phi.
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| === Odds ratio ===
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| 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 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.
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| === Relative risk ===
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| 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.
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| 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}}</ref> Relative risk is commonly used in [[randomized controlled trial]]s and [[cohort study|cohort studies]].<ref name="Ref_b">Medical University of South Carolina. [http://www.musc.edu/dc/icrebm/oddsratio.html Odds ratio versus relative risk]. Accessed on: September 8, 2005.</ref> When the incidence of outcomes are rare in the study population (generally interpreted to mean less than 10%), the odds ratio is considered a good estimate of the risk ratio. However, as outcomes become more common, the odds ratio and risk ratio diverge, with the odds ratio overestimating or underestimating the risk ratio when the estimates are greater than or less than 1, respectively. When estimates of the incidence of outcomes are available, methods exist to convert odds ratios to risk ratios.<ref>{{cite pmid|9832001 }}</ref><ref>{{cite pmid|15286014}}</ref>
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| ===Common language effect size===
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| As the name implies, the common language effect size is designed to communicate the meaning of an effect size in plain English, so that those with little statistics background can grasp the meaning. This effect size was proposed and named by Kenneth McGraw and S. P. Wong (1992). <ref>{{Cite journal | author = 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| url = | pmid = }}</ref>
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| The core concept of the common language effect size is the notion of ''a pair'', defined as a score in group one paired with a score in group two. For example, if a study has ten people in a treatment group and ten people in a control group, then there are 100 pairs. The common language effect size ranks all the scores, compares the pairs, and reports the results in the common language of the percent of pairs that support the hypothesis.
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| As an example, consider a treatment for a chronic disease such as arthritis, with the outcome a scale that rates mobility and pain; further consider that there are ten people in the treatment group and ten people in the control group, for a total of 100 pairs. The sample results may be reported as follows: "When a patient in the treatment group was compared with a patient in the control group, in 80 of 100 pairs the treated patient showed a better treatment outcome."
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| This sample value is an unbiased estimator of the population value. <ref>{{Cite journal | author = Grissom RJ | year = 1994| title = Statistical analysis of ordinal categorical status after therapies | journal = [[Journal of Consulting and Clinical Psychology]] | volume = 62| issue = 2| pages = 281-284| doi= 10.1037/0022-006X.62.2.281 | url = | pmid = }}</ref> The population value for the common language effect size can be reported in terms of pairs randomly chosen from the population. McGraw and Wong <ref>{{Cite journal | author = 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| url = | pmid = }}</ref> use the example of heights between men and women, and they describe the population value of the common language effect size as follows: "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" (p. 381).
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| == Confidence intervals by means of noncentrality parameters ==
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| Confidence intervals of standardized effect sizes, especially Cohen's <math>{d}</math> and <math>{f}^2</math>, rely on the calculation of confidence intervals of [[noncentrality parameter]]s (''ncp''). A common approach to construct the confidence interval of ''ncp'' is to find the critical ''ncp'' values to fit the observed statistic to tail [[quantile]]s ''α''/2 and (1 − ''α''/2). The SAS and R-package MBESS provides functions to find critical values of ''ncp''.
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| === ''t''-test for mean difference of single group or two related groups ===
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| For a single group, ''M'' denotes the sample mean, ''μ'' the population mean, ''SD'' the sample's standard deviation, ''σ'' the population's standard deviation, and ''n'' is the sample size of the group. The ''t'' value is used to test the hypothesis on the difference between the mean and a baseline ''μ''<sub>baseline</sub>. Usually, ''μ''<sub>baseline</sub> is zero. In the case of two related groups, the single group is constructed by the differences in pair of samples, while ''SD'' and ''σ'' denote the sample's and population's standard deviations of differences rather than within original two groups.
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| :<math>t:=\frac{M}{SD/\sqrt{n}}=\frac{\sqrt{n}\frac{M-\mu}{\sigma} + \sqrt{n}\frac{\mu-\mu_\text{baseline}}{\sigma}}{\frac{SD}{\sigma}}</math>
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| :<math>ncp=\sqrt{n}\frac{\mu-\mu_\text{baseline}}{\sigma}</math>
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| and Cohen's
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| <math>d:=\frac{M-\mu_\text{baseline}}{SD}</math>
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| is the point estimate of
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| : <math>\frac{\mu-\mu_\text{baseline}}{\sigma}.</math>
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| So,
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| :<math>\tilde{d}=\frac{ncp}{\sqrt{n}}.</math>
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| === ''t''-test for mean difference between two independent groups ===
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| ''n''<sub>1</sub> or ''n''<sub>2</sub> are the respective sample sizes.
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| :<math>t:=\frac{M_1-M_2}{SD_\text{within}/\sqrt{\frac{n_1 n_2}{n_1+n_2}}},</math>
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| wherein
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| : <math>SD_\text{within}:=\sqrt{\frac{SS_\text{within}}{df_\text{within}}}=\sqrt{\frac{(n_1-1)SD_1^2+(n_2-1)SD_2^2}{n_1+n_2-2}}.</math>
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| :<math>ncp=\sqrt{\frac{n_1 n_2}{n_1+n_2}}\frac{\mu_1-\mu_2}{\sigma}</math>
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| and Cohen's
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| : <math>d:=\frac{M_1-M_2}{SD_\text{within}}</math> is the point estimate of <math>\frac{\mu_1-\mu_2}{\sigma}.</math>
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| So,
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| :<math>\tilde{d}=\frac{ncp}{\sqrt{\frac{n_1 n_2}{n_1+n_2}}}.</math>
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| === One-way ANOVA test for mean difference across multiple independent groups ===
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| One-way ANOVA test applies [[noncentral F distribution]]. While with a given population standard deviation <math>\sigma</math>, the same test question applies [[noncentral chi-squared distribution]].
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| :<math>F:=\frac{\frac{SS_\text{between}}{\sigma^2}/df_\text{between}}{\frac{SS_\text{within}}{\sigma^2}/df_\text{within}}</math>
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| For each ''j''-th sample within ''i''-th group ''X''<sub>''i'',''j''</sub>, denote
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| : <math>M_i \left(X_{i,j}\right) := \frac{\sum_{w=1}^{n_{i}}X_{i,w}}{n_{i}};\; \mu_i \left(X_{i,j}\right) := \mu_i.</math>
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| While,
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| :<math>\begin{array}{ll}
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| SS_\text{between}/\sigma^{2}
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| & = \frac{SS\left(M_{i}\left(X_{i,j}\right);i=1,2,\dots,K,\; j=1,2,\dots,n_{i}\right)}{\sigma^{2}}\\
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| & = SS\left(\frac{M_{i}\left(X_{i,j}-\mu_{i}\right)}{\sigma}+\frac{\mu_{i}}{\sigma};i=1,2,\dots,K,\; j=1,2,\dots,n_{i}\right)\\
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| & \sim \chi^{2}\left(df=K-1,\; ncp=SS\left(\frac{\mu_i\left(X_{i,j}\right)}{\sigma};i=1,2,\dots,K,\; j=1,2,\dots,n_{i}\right)\right)\end{array}</math>
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| So, both ''ncp''(''s'') of ''F'' and <math>\chi^2</math> equate
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| :<math>SS\left(\mu_i(X_{i,j})/\sigma;i=1,2,\dots,K,\; j=1,2,\dots,n_i \right).</math>
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| In case of <math>n:=n_1=n_2=\cdots=n_K</math> for ''K'' independent groups of same size, the total sample size is ''N'' := ''n''·''K''.
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| :<math>\text{Cohens }\tilde{f}^2 := \frac{SS(\mu_1,\mu_2, \dots ,\mu_K)}{K\cdot\sigma^{2}} = \frac{SS\left(\mu_i\left(X_{i,j}\right)/\sigma;i=1,2,\dots,K,\; j=1,2,\dots,n_i \right)}{n\cdot K} = \frac{ncp}{n\cdot K}=\frac{ncp}N.</math>
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| The ''t''-test for a pair of independent groups is a special case of one-way ANOVA. Note that the noncentrality parameter <math>ncp_F</math> of ''F'' is not comparable to the noncentrality parameter <math>ncp_t</math> of the corresponding ''t''. Actually, <math>ncp_F=ncp_t^2</math>, and <math>\tilde{f}=\left|\frac{\tilde{d}}{2}\right|</math>.
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| == "Small", "medium", "large" effect sizes==
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| Some fields using effect sizes apply words such as "small", "medium" and "large" to the size of the effect.
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| Whether an effect size should be interpreted small, medium, or large depends on its substantive context and its operational definition. Cohen's conventional criteria ''small'', ''medium'', or ''big''<ref name="CohenJ1988Statistical"/> are near ubiquitous across many fields. Power analysis or [[Sample_size#Required_sample_sizes_for_hypothesis_tests|sample size planning]] requires an assumed population parameter of effect sizes. Many researchers adopt Cohen's standards as default [[Alternative hypothesis|alternative hypotheses]]. Russell Lenth{{Sic}} criticized them as ''T-shirt effect sizes''.<ref>{{Cite web
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| | author = Russell V. Lenth
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| | title = Java applets for power and sample size
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| | url = http://www.stat.uiowa.edu/~rlenth/Power/
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| | publisher = Division of Mathematical Sciences, the College of Liberal Arts or The University of Iowa
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| | accessdate = 2008-10-08
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| }}</ref>
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| <blockquote>''This is an elaborate way to arrive at the same sample size that has been used in past social science studies of large, medium, and small size (respectively). The method uses a standardized effect size as the goal. Think about it: for a "medium" effect size, you'll choose the same n regardless of the accuracy or reliability of your instrument, or the narrowness or diversity of your subjects. Clearly, important considerations are being ignored here. "Medium" is definitely not the message!''</blockquote>
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| For Cohen's ''d'' an effect size of 0.2 to 0.3 might be a "small" effect, around 0.5 a "medium" effect and 0.8 to infinity, a "large" effect.<ref name="CohenJ1988Statistical"/>{{Rp|25|date=November 2012}}
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| (But the ''d'' might be larger than one.)
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| Cohen's text<ref name="CohenJ1988Statistical"/> anticipates Lenth's concerns:
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| <blockquote>"The terms 'small,' 'medium,' and 'large' are relative, not only to each other, but to the area of behavioral science or even more particularly to the specific content and research method being employed in any given investigation....In the face of this relativity, there is a certain risk inherent in offering conventional operational definitions for these terms for use in power analysis in as diverse a field of inquiry as behavioral science. This risk is nevertheless accepted in the belief that more is to be gained than lost by supplying a common conventional frame of reference which is recommended for use only when no better basis for estimating the ES index is available." (p. 25)</blockquote>
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| In an ideal world, researchers would interpret the substantive significance of their results by grounding them in a meaningful context or by quantifying their contribution to knowledge. Where this is problematic, Cohen's effect size criteria may serve as a last resort.<ref name="Ellis2010"/>
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| ==See also==
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| *[[Estimation statistics]]
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| *[[Statistical significance]]
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| *[[Z-factor]], an alternative measure of effect size
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| ==References==
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| {{Reflist}}
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| === Further reading ===
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| *Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). 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. [http://www.eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?accno=ED433353 (ERIC Document Reproduction Service No. ED433353)]
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| *Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means, ''Psychological Methods'', 13, 99–109.
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| *Bonett, D. G. (2009). Estimating standardized linear contrasts of means with desired precision, ''Psychological Methods'', 14, 1–5.
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| *Cumming, G. and Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions. ''Educational and Psychological Measurement, 61'', 530–572.
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| *Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation.'' Journal of Statistical Software'', ''20''(''8''), 1–24. [http://www.jstatsoft.org/v20/i08/paper]
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| *Lipsey, M. W., & Wilson, D. B. (2001). ''Practical meta-analysis''. Sage: Thousand Oaks, CA.
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| ==External links==
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| {{wikiversity}}
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| '''Online applications'''
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| *[http://www.danielsoper.com/statcalc/calc05.aspx Free Effect Size Calculator for Multiple Regression]
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| *[http://www.danielsoper.com/statcalc/calc13.aspx Free Effect Size Calculator for Hierarchical Multiple Regression]
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| *[http://xiaoxu.lxxm.com/ncp Copylefted Effect Size Confidence Interval R Code with RWeb service for t-test, ANOVA, regression, and RMSEA]
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| '''Software'''
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| *[http://cran.r-project.org/web/packages/compute.es/index.html compute.es: Compute Effect Sizes] ([[R (programming language)|R package]])
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| *[http://cran.r-project.org/web/packages/MBESS/index.html MBESS] – One of [[R (programming language)|R's packages]] providing confidence intervals of effect sizes based non-central parameters
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| *[http://www.meta-analysis-made-easy.com MIX 2.0] Software for professional meta-analysis in Excel. Many effect sizes available.
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| *[http://myweb.polyu.edu.hk/~mspaul/calculator/calculator.html Effect Size Calculators] Calculate ''d'' and ''r'' from a variety of statistics.
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| *[http://www.clintools.com/victims/resources/software/effectsize/effect_size_generator.html Free Effect Size Generator] – PC & Mac Software
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| *[http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/ Free GPower Software] – PC & Mac Software
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| *[http://www.mdp.edu.ar/psicologia/vista/vista.htm ES-Calc: a free add-on for Effect Size Calculation in ViSta 'The Visual Statistics System']. Computes Cohen's d, Glass's Delta, Hedges's g, CLES, Non-Parametric Cliff's Delta, d-to-r Conversion, etc.
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| '''Further explanations'''
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| *[http://www.uccs.edu/~faculty/lbecker/es.htm Effect Size (ES)]
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| *[http://effectsizefaq.com/ EffectSizeFAQ.com]
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| * [http://davidmlane.com/hyperstat/effect_size.html Measuring Effect Size]
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| * [http://web.uccs.edu/lbecker/Psy590/es.htm#II.%20independent Effect size for two independent groups]
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| * [http://web.uccs.edu/lbecker/Psy590/es.htm#III.%20Effect%20size%20measures%20for%20two%20dependent Effect size for two dependent groups]
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| *[http://www.tqmp.org/doc/vol5-1/p25-34.pdf Computing and Interpreting Effect size Measures with ViSta]
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| {{statistics|collection|state=collapsed}}
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| {{Experimental design}}
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| {{DEFAULTSORT:Effect Size}}
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| [[Category:Effect size| ]]
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| [[Category:Clinical research]]
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| [[Category:Educational psychology research methods]]
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| [[Category:Hypothesis testing]]
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| [[Category:Pharmaceutical industry]]
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| [[Category:Meta-analysis]]
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| [[Category:Medical statistics]]
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