Converting t to Cohen's d

[FRESCO, DAVID]

Colleagues,

I am trying to determine the appropriate equation to convert a paired sample t-test to Cohen’s d.

Rosenthal and Rosnow have published the equation of d = 2t/SQRT(df), which I thought was encompassing of all kinds of t tests.  

I recently read a paper that for a paired sample t-test another equation attributed to Rosenthal was as follows: d = t/SQRT(df).

Can some kind soul point me in the direction of sources and arguments for one or the other or perhaps something totally different.  CC’ing my email to replies would be appreciated as I only take the digest of this listserve.

Thanks.

David Fresco
------

David M. Fresco, Ph.D.               fresco@...
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[Meyer, Gregory J]

Hi David,

 

The first formula is technically to compute the d for independent samples (d_I) and it is the formula that assumes each group is the same size (i.e., the equal n formula). The more general formula is:

 

d_i = 2t_i / SQRT[df (p₁p₂)]

 

where p₁ is the proportion of total N in group 1 and p₂ is the remainder.

 

The second formula is to compute the d for correlated samples (d_C). It reflects the distance between the two means in SD units, but now the SD units are based on the SD of the difference scores, not the pooled SD of the original variable. The SD of the difference scores is a function of the correlation between the paired observations, with a stronger correlation generating a smaller SD of the differences.

 

Because the d for correlated observations is in part a function of power to detect the difference (i.e., the degree of correlation) rather than magnitude of the difference per se, Cohen noted that d_C cannot be compared to d_I. It is also for this reason that some have argued that d_C is typically misleadingly large and so only d_I should be reported. To do so, you would have to compute the independent samples t and convert it or compute d directly from the group Ms and SDs.

 

This issue is discussed at some length in the following citation:

 

Dunlap, W. P., Cortina, J. M., Vaslow, J. B., & Burke, M. J. (1996). Meta-analysis of experiments with matched groups or repeated measures designs. Psychological Methods, 1, 170-177

 

But note that the formulas in this article actually compute Hedge’s g rather than Cohen’s d. (For an explanation of the differences between d and g, as well as other symbols in the literature to represent standardized mean differences, see Table 1 in McGrath & Meyer, 2006, “When effect sizes disagree: The case of r and d”; Psych Methods).

 

Greg

 

From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of FRESCO, DAVID
Sent: Wednesday, July 08, 2009 12:32 PM
To: [hidden email]
Subject: Converting t to Cohen's d

 

Colleagues,

I am trying to determine the appropriate equation to convert a paired sample t-test to Cohen’s d.

Rosenthal and Rosnow have published the equation of d = 2t/SQRT(df), which I thought was encompassing of all kinds of t tests.  

I recently read a paper that for a paired sample t-test another equation attributed to Rosenthal was as follows: d = t/SQRT(df).

Can some kind soul point me in the direction of sources and arguments for one or the other or perhaps something totally different.  CC’ing my email to replies would be appreciated as I only take the digest of this listserve.

Thanks.

David Fresco
------

David M. Fresco, Ph.D.               fresco@...
Department of Psychology        Voice:  [phone]
Kent State University                   Fax:    [phone]
226 Kent Hall Annex             
P.O. Box 5190                              http://dept.kent.edu/psychology/
Kent, OH  44242-0001                http://www.personal.kent.edu/~dfresco/

ListOwner of Helplessness http://psych.upenn.edu/~fresco/helplessness.html
Listowner of Mindfulness http://listserv.kent.edu/archives/mindfulness.html