Editing Effect size (section)

==== 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>
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In the above 'height' example, Hedges' ''ĝ'' effect size equals 1.76 (95% confidence intervals: 1.70&nbsp;−&nbsp;1.82). Notice how the large sample size has increased the effect size from Cohen's&nbsp;''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&nbsp;–&nbsp;2.05).
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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.