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I was curious as to the potential pitfalls of a Pearson correlation under these circumstances:

For example, let's say I wanted to test the correlation between an independent continuous variable (e.g. years of doing task A) and another dependent variable in two groups (Group 1 - trained in task A; Group 2 - not trained in task A), where the two groups significantly differ on the independent variable.

I ultimately want to investigate if "years of doing task A is correlated with the dependent variable". However, in such a case, the data for the non-experts (by the very nature of the group characteristic) may be clumped all around the value of 0 years, whereas the data for the experts may be clumped around a specific range (e.g. 7 to 10 years). I was wondering with such a distribution of the data (with one group almost all clustered around the 0 mark for the independent variable), if a Pearson correlation is still "ok", or is there specific criteria that this situation violates? Thank you.

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The issue here is one of conditioning; the correlation given group membership may be vastly different (even opposite in sign) from one that ignores it. Which is to say, The correlation between the two variables for experts might be negative, say, and the correlation between the two variables for non-experts might also be negative, but because the two groups have different means on the variables, if you ignore the grouping you can get a strongly positive correlation.

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The question is, which do you want? Most people will want the conditional (i.e. within-group) correlation, rather than the marginal (ignoring-group) one.

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  • $\begingroup$ Hi Glen_b, thanks for the great response. I just wanted to ask that when you refer to within-group correlation, it refers to for example just doing correlation analysis within the expert group, and not including the control group, or vice versa? And if you were to just do a correlation analysis with the expert group, could you still use that to infer a relationship between the independent and dependent (i.e. years of doing task A is correlated with the dependent variable) even with its distribution not spanning the years (i.e. clumped around 7 to 10)? Thanks again. $\endgroup$
    – pdhami
    May 7, 2015 at 12:48
  • $\begingroup$ Within-group correlation might be assumed to be the same within each group, or it might differ across groups. I'll show an example of what I was talking about where the two are the same; you can imagine a similar plot where they're different. $\endgroup$
    – Glen_b
    May 7, 2015 at 13:45
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Pearson correlation is used in a linear context. See http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient If you only use two clusters, then there is an assumption that the empty space between the clusters somehow could be filled linear. In practice the real relationship could turn out non-linear.

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