FAQ/td - CBU statistics Wiki

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How do I convert a t-statistic into an effect size?

One sample t

Since Sqrt(n(mean)) / sd = t

Cohen's d = t/Sqrt(n) = (mean-constant)/sd

This value of Cohen's d is used by Lenth (2006) in his one sample and (paired sample) t-test option.

Notice than as n goes up t will increase if the new observations are close to the mean which should be the case if, as assumed, the response is normally distributed.

Two sample unpaired t

Pustejovsky (2014) states for sample sizes using an unpaired t-test with sample sizes n1 and n2

Cohen's d = t [Sqrt(1/n1 + 1/n2)]

When n1 = n2

Cohen's d = 2t / df (See Rosenthal (1994) and Howell (2013), p.649)

As n1 and n2 increase the t value should also increase if the new observations are close to their group means (which will tend to be the case for the assumed Normal distributions within each group) since the group means should remain roughly constant and the variance of each mean is related to the spread about the group means and 1/group sizes.

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 = (difference in means) / (sd of (population) difference in means) where the population sd is 1/(n-1) (sum of squared deviations from the average difference).

d, above, can also be expressed as

d = t Sqrt(1/n) Sqrt(n/(n-1)) = difference in means / sd of sample difference [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[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:

Cohen's d = Sqrt(2) Sqrt(F)/Sqrt(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) and Field, Miles and Field (2012, p.581) also give a conversion formula to turn a t-statistic into a correlation

Correlation = square root [(t2 / (t2 + 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. Pustejovsky (2014) gives simple to use formulae for computing effect sizes from t and F statistics and converting between d, r and z.

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.

Field A, Miles J and Field Z (2012) Discovering statistics using R. Sage:London.

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.

Pustejovsky J. E. (2014) Converting from d to r to z when the design uses extreme groups, dichotomization, or experimental control. Psychological Methods 19(1) 92-112.

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.