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If $r$ is the effect size for the correlation between $A$ and $B$, then $r^2$ is the amount of variance in $B$ that can be attributed to variable $A$.

  1. Is it important to report both indexes in a report, or just one or the other?
  2. How do you explain them in plain English (for a non-statistical audience)?
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General points on the term 'effect size'

The term 'effect size' can have both narrow and specific meanings.

  • narrowest meaning: Some authors use the term 'effect size' almost exclusively within the context of standardised group mean differences (i.e., $d$).
  • narrow meaning: Any of a set of standardised statistics that quantify relationships
  • broad meaning: Any value that quantifies the degree of effect, including unstandardised measures of relationship.

Just to be clear, $r^2$ is a measure of effect size, just as $r$ is a measure of effect size. $r$ is just a more commonly used effect size measure used in meta-analyses and the like to summarise strength of bivariate relationship.

When to report $r$ versus $r^2$

  • A convention in psychology and probably other areas is that correlations (i.e., $r$) are typically reported when summarising one or often a matrix of bivariate relationships and that $r^2$ is reported in the context of models predicting a variable (e.g., multiple regression). This makes sense for several rasons. First, correlation communicates the direction of the relationship whereas $r^2$ does not; however, directional information is communicated in predictive models by interpreting model coefficients. Second, where correlations are typically ranging between .1 and .3, then the correlation seems to be a bit more nuanced than $r^2$, and thus, fewer decimal places are required to be displayed.

Explaining $r$ and $r^2$ in plain English

  • $r$ is a standardised measure of the strength and direction of linear relationship between two variables ranging from -1 for a perfect negative relationship and 1 for a perfect positive relationship.
  • You may want to give your non-statistical audience a sense of some rules of thumb set out by Cohen and others (something like r = .1 = small; r = .3 = medium; r = .5 = large), while at the same time telling them not to take such presciptions too literally. You might also present some scatterplots of various correlations and some examples of typical correlation sizes in their field of interest.
  • One somewhat intuitive interpretation of $r$ is that it is equivalent to a standardised regression coefficient.
  • I think that the interpretation of $r^2$ as the percentage of variance explained by the linear relationship between two variables is relatively intuitive.
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  • $\begingroup$ Thanks! I have learnt so much from your detailed responses. $\endgroup$ – Adhesh Josh Sep 27 '11 at 6:52
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If you refer to the term "effect size", there are some standards on how to report them (Cohen, 1992). The most common is Cohen's $d$, which can be directly transformed into a correlation-based measure of effect-size, $r_{ES}$:

$r_{ES} = \frac{d}{\sqrt{(d2 + 4)}}$

For ANOVAs, you usually report $\eta^2$, which directly refers to "variance explained".

If the original statistics was a correlation, just report the correlation. It already is a measure of effect size.

To explain them in plain English, I would refer to Cohen's table of effect size magnitudes. For correlations, it says:

  • <.10: trivial
  • .10 - .30: small to medium
  • .30 - .50: medium to large
  • >.50: large to very large

Cohen, J. (1992). A power primer. Psychological Bulletin, 112, 155-159. doi:10.1037/0033-2909.112.1.155

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  • $\begingroup$ Many thanks but how does this relate to vaiance. (Yes, I am interested in the correlation test) $\endgroup$ – Adhesh Josh Sep 19 '11 at 11:57
  • $\begingroup$ You can convert any effect size measure into r_ES (I added the formula from d to r into my answer). Than you can square r to obtain the variance explained. $\endgroup$ – Felix S Sep 19 '11 at 13:00
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    $\begingroup$ I believe the formula as written works only for equal sample sizes. Also, it assumes a certain form of Cohen's d. I think in this case it is Cohen's d where n is used in the denominator for pooled standard deviation, not n - 2. $\endgroup$ – Sal Mangiafico Feb 3 '19 at 18:02

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