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 unpaired t

Rosenthal (1994) states for sample sizes using an unpaired t-test $$\mbox{n}_text{1}$$, $$\mbox{n}_text{2}$$,

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 = $$\frac{\mbox{2t}}{\sqrt{\mbox{df}}}$$ (See also Howell (2013), p.649)

Paired t

Baguley (2012, p.271) gives a formula, amongst other conversion formulae, for converting a paired t to d using a joint group size equal to n:

d = $$\frac{\mbox{difference in means}{sd of (population) difference}}$$ where the population sd is $$\frac{1}{n-1}$$ (sum of square deviations from average difference).

d, above, can also be expressed as

d = t \sqrt{\frac{1}{n}} \sqrt{\frac{n}{n-1}}$$ = $$ ( \frac{\mbox{difference in means}}{\mbox{sd of difference}} ) \sqrt{\frac{n}{n-1}}$$ (see p.248 of Baguley (2012)) where the sd of the difference is the square root of 1/n (sum of the squared deviations of each difference from the overall sample mean difference) as defined on page 23 of Baguley (2012) and $$\sqrt{\frac{n}{n-1}}$$ is the correction factor for estimating a population sd from a sample sd (pages 26-27 of Baguley).

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 = $$\sqrt{\frac{\mbox{t}^{text{2}}{\mbox{t}}text{2} + \mbox{df}}$$

General Conversions

Jamie DeCoster has written a [attachment:effconv.xls spreadsheet] to convert a range of commonly used effect sizes such as Cohen's d, Pearson's r and odds ratios.

References

Abelson, R. P. and Prentice, D. A. (1997) Contrast tests of interaction hypotheses. Psychological Methods 2(4) 315-328.

Baguley, T. (2012) Serious Stats. A guide to advanced statistics for the behavioral sciences. Palgrave Macmillan:New York. In addition to those mentioned above, Chapter 7 gives some conversion formulae including converting from r to g, where g is an effect size estimator which is very closely related to d.

Howell, D. C. (2013) Statistical methods for psychologists. 8th Edition. International Edition. Wadsworth:Belmont, CA.

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.