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I'm aware that it isn't possible to calculate a correlation coefficient from the means of two data sets.

However, I'm wondering if there is a way to get a correlation coefficient (with confidence intervals) in the following scenario:

We are planning on running a meta-analysis with the main question of looking at the association between two variables (A and B). However, nearly all studies report the means and standard deviations (or 95% confidence intervals etc) of the two variables individually; they do not report correlation coefficients.

As such, is there a way of taking these means from each study (i.e. each study will have a paired mean data set), and then calculating the correlation coefficient for this meta-data set.

My thinking is this would require knowledge of the mean, spread of the data (i.e. SD) and sample size. Combined this would then give a "best guess" at the correlation coefficient and then a confidence boundary based on the individual data sets spread etc.

Thank you for any thoughts/guidance on this.

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    $\begingroup$ Why do you need a correlation coefficient? What is your objective? $\endgroup$ – user2974951 Feb 11 at 12:43
  • $\begingroup$ We are looking at if there is an association between the two variables. This is a hypothesis of ours and are looking at the literature. Although they collect and report the variables (as means and SD) for the same sample, no one has investigated the association. As such, I'm wondering if there is a way of taking these data and run a meta-analytical approach to observing a potential association. We could take the means of each variable from each study and then calculate the correlation coefficient, but this would not take into account the variability and sample size of the different studies. $\endgroup$ – StuartMiller Feb 11 at 16:30
  • $\begingroup$ Can you not get the raw data from the authors? Do you know the sample size? $\endgroup$ – user2974951 Feb 12 at 7:40
  • $\begingroup$ Sample sizes known from the papers. Unfortunately not able to get raw data on enough etc. All things we've tried before. $\endgroup$ – StuartMiller Feb 12 at 9:13
  • $\begingroup$ Interesting question. You may approach it by focusing at case reports rather than studies. Or focus only on very small studies (eg <30 cases), where regression to the mean and ecological fallacy may play a lower role. $\endgroup$ – Joe_74 Feb 12 at 9:48
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I do not see a way to make this work. For a correlation coefficient order is fundamental. Take for ex. two samples, one with values from 1 to 5, the other from 5 to 1. Both samples have mean 3 and the correlation coefficient is equal to -1. Reverse the order of one sample and the correlation coefficient is equal to +1.

So just by using a mean you do not have enoguh information to infer about the correlations.

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  • $\begingroup$ Yep, I'm aware of that if using an individual mean. But I'm wondering if by taking paired means from multiple studies, can we fit a correlation coefficient to these? If the between-group variability is greater than the within-group variability (we could include this into the modelling through the spread measures for each individual mean), could this then lead to a more "population"-specific correlation? Obviously if the within-group variability is much greater than the between-group variability, this would lead to no confidence in any coefficient produced. $\endgroup$ – StuartMiller Feb 12 at 11:14
  • $\begingroup$ @StuartMiller This looks very wishy washy, I would not trust such an estimate. $\endgroup$ – user2974951 Feb 13 at 10:18

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