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Jamie DeCoster has written a [effconv.xls spreadsheet] to convert effect sizes. | Jamie DeCoster has written a [effconv.xls: spreadsheet] to convert effect sizes. |
How do I convert a t-statistic into an effect size?
One sample t
Since $$\frac{\sqrt{\mbox{n}} \mbox{mean}}{\mbox{sd}}$$ = t
Cohen's d = $$\frac{t}{\sqrt{n}} = \frac{\mbox{mean-constant}}{\mbox{sd}}$$
This value of Cohen's d is used by Lenth (2006) in his one sample and (paired sample) t-test option.
Two sample t
Rosenthal (1994) states for sample sizes $$\mbox{n}_text{1}$$, $$\mbox{n}_text{2}$$,
Cohen's d = $$\frac{t(\mbox{n}_text{1}+\mbox{n}_text{2})}{\sqrt{\mbox{df}}\sqrt{\mbox{n}_text{1}}\sqrt{\mbox{n}_text{2}} }$$
When $$\mbox{n}_text{1}$$ = $$\mbox{n}_text{2}$$
Cohen's d = $$\frac{\mbox{2t}}{\sqrt{\mbox{df}}}
2 way interaction
Abelson and Prentice (1997) suggest a way of converting a F statistic from a two-way interaction into Cohen's d:
$$\mbox{Cohen's d} = \sqrt{\mbox{2}} \frac{\sqrt{\mbox{F}}}{\sqrt{\mbox{n}}}
where n is the assumed equal number of observations for each combination of the two factors. If these are unequal then we use the [http://en.wikipedia.org/wiki/Harmonic_mean harmonic mean] of the sample sizes.
The two sample t-test with equal sample sizes is a special case since t equals $$\sqrt{F}$$ and df is made equal to 2n.
Pearson Correlation
Rosenthal (1994) also gives a conversion formula to turn a t-statistic into a correlation
Correlation = $$\sqrt{\frac{\mbox{t}text{2}}{\mbox{t}text{2} + \mbox{df}}$$
Jamie DeCoster has written a [effconv.xls: spreadsheet] to convert effect sizes.
References
Abelson, RP and Prentice, DA (1997) Contrast tests of interaction hypotheses. Psychological Methods 2(4) 315-328.
Lenth, R. V. (2006). Java Applets for Power and Sample Size [Computer software]. Retrieved month day, year, from http://www.stat.uiowa.edu/~rlenth/Power.
Rosenthal, R. (1994) Parametric measures of effect size. In H. Cooper and L.V. Hedges (Eds.). The handbook of research synthesis. New York: Russell Sage Foundation.