= 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 [[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 = square root of $$[\mbox{t}^2^$$ 

divided by

$$(\mbox{t}^2^ + df)]$$

__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.