Effects Size Estimates

Cohen’s D

Cohen’s D is the number of standard deviations an effect is shifted above or below the populations mean stated by the null hypothesis.

z test

When we have known population parameters, we use them to calculate \[ d = \frac{\bar{X} - \mu}{\sigma} \] Note that \(\sigma\) is the standard deviation and NOT the standard error of the mean.

One-sample t test

\[ d = \frac{\bar{X} - \mu}{s} \]

Independent sample t test

\[ d = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{s^2_P}} \] Consult a introductory statistics textbook for the calculation fo \(s^2_p\).

Cohen’s Effect Size Guidelines

d description
0.2 Small
0.2 - 0.8 Medium
0.8 Large

But don’t take these guidelines too seriously. It’s more important to compare the effect size to thos expected or typical within the field of study.

Proportion of Variance Effect Size Measures

\[ \text{Proportion of Variance} = \frac{\text{variability explained}}{\text{total variability}} \]

Eta-Squared (\(\eta^2\))

\[ \eta^2 = \frac{t^2}{t^2 + df}. \]

Omega-Squared (\(\omega^2\))

While very popular, \(\eta^2\) is biased and tends to overestimate effect size, particularly for studies with small sample size. Therefore, with small samples it is recommended that \(\omega^2\) be used instead, which is a modified vesion of \(\eta^2\). Note that \(\omega^2\) is very similar to \(\eta^2\), the only difference is that 1 is substraced from the numerator. This has the effect of making \(\omega^2\) more conservative than \(\eta^2\).

\[ \omega^2 = \frac{t^2 - 1}{t^2 + df}. \]

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