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In a psychological paper (Fujita et al., 2006), the authors perform a two sample t-test for independent means. They report the following values:

$M_1$ = 9.88, $M_2$ = 8.47,

t(66) = 2.25, p = 0.03, $p_{rep}$ = .94, r = .27

What does r mean in this context? Is it a correlation coefficient or is it an effect size? In the back of my head, I remember that r can be an effect size. I wish to know the effect size. What is the relation between the effect size and correlation? gives me some understanding, but how exactly can I interpret this r or convert it do Cohen's d?

For those who wonder what $p_{rep}$ is: wiki says it is a bad measure, nowadays its conception is proven to contain mathematical errors.


(The paper is behind a paywall)

Fujita, K., Henderson, M. D., Eng, J., Trope, Y., & Liberman, N. (2006). Spatial Distance and Mental Construal of Social Events. Psychological Science, 17(4), 278-282. https://doi.org/10.1111/j.1467-9280.2006.01698.x
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  • $\begingroup$ Please provide a full citation for (Fujita et al., 2006). Then -- if the paper is not behind a paywall -- the answers won't have to make guesses about the "usual" and "without further information". $\endgroup$
    – dipetkov
    Nov 23, 2023 at 12:34
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    $\begingroup$ @dipetkov I looked it up and the paper is behind a paywall, sorry. But actually the paper is not important though, because I included everything that is relevant in my question. It could be about any paper that reports $r$ along with a t-test. You have to take my word that there is nothing else that is useful in this paper to answer this question. $r$ is just dropped after a $t$ statistic and $p$ value, no explanations. $\endgroup$
    – uke
    Nov 23, 2023 at 21:41
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    $\begingroup$ i added the reference. And what you are saying, we won't even compute the correlation, is interesting. Why not? The important thing, that it is a two independent samples scenario was clearly stated in my question though. $\endgroup$
    – uke
    Nov 24, 2023 at 9:39
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    $\begingroup$ Thank you. Correlation is computed between two measurements taken from the same sample / person / object. Say we have two groups, men and women; we can't "align" those to compute a sample correlation coefficient; see wikipedia. Unless perhaps we have a sample of partnered couples; then we can pair those. $\endgroup$
    – dipetkov
    Nov 24, 2023 at 10:10
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    $\begingroup$ It's true, your question is clear & the thing about proper citing of sources is something of a pet peeve of mine. What I noticed is that the two answers are incompatible but I admit that's not due to the content of your question. $\endgroup$
    – dipetkov
    Nov 24, 2023 at 10:13

2 Answers 2

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The reported $r$ is usually referring to an effect size. It is calculated from the t-statistic and the degrees of freedom

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

where $t$ is the t-statistic and $df$ is the degrees of freedom ($N-2$) ​

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  • $\begingroup$ is this $r$ equivalent to calculating a certain correlation directly from the data? I wonder: is it possible to transform this formula to the formula of some correlation coefficient? $\endgroup$
    – uke
    Nov 24, 2023 at 9:51
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    $\begingroup$ As already commented, a correlation is only defined if the data are paired. $\endgroup$
    – Nick Cox
    Nov 24, 2023 at 10:06
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    $\begingroup$ Interesting, does this effect size have a name? Since usually effect sizes aim to provide a standardized measure of effect which is independent of sample size (in contrast to quantities such as the p-value), it seems a bit odd that $df$ is included in the formula. $\endgroup$
    – LuckyPal
    Nov 24, 2023 at 10:56
  • $\begingroup$ @LuckyPal I think it is about removing the influence of sample size ($df$) from the test statistic ($t$ in this case), that's why it is in the denominator. $\endgroup$
    – uke
    Feb 23 at 15:49
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Without any other information, I would say $r$ is correlation. And that is an effect size.

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