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 = $$\frac{\mbox{difference in means}}{\mbox{sd of (population) difference in means}}$$ where the population sd is $$\frac{1}{n-1}$$ (sum of squared deviations from the average difference).
d, above, can also be expressed as
d = t $$\sqrt{\frac{1}{n}} \sqrt{\frac{n}{n-1}}$$ = $$ ( \frac{\mbox{difference in means}}{\mbox{sd of sample difference}} ) \sqrt{\frac{n}{n-1}}$$ (see p.248 of Baguley (2012)) where the sd of the sample 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:
$$\mbox{Cohen's d} = \sqrt{\mbox{2}} \frac{\sqrt{\mbox{F}}}{\sqrt{\mbox{n}}}
where n is the assumed equal number of observations for each combination of the two factors. If these are unequal then we use the harmonic mean of the sample sizes.
The two sample t-test with equal sample sizes is a special case since t equals $$\sqrt{F}$$ and df is made equal to 2n.
Pearson Correlation
Rosenthal (1994) also gives a conversion formula to turn a t-statistic into a correlation
Correlation = square root of $$[\mbox{t}2$$
divided by
$$(\mbox{t}2 + df)]$$
General Conversions
Jamie DeCoster has written a 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.