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1) How do we statistically interpret a situation where the correlation coefficient r=0.87, denoting a strong relationship between the independent var and the response var, but p-value >.05 (not-significant)?

2) In a different scenario, if the p-value is significant (p<0.05), while r=0.286 (weak correlation), can we say that the independent var will influence the response variable (since p<0.05), but the relationship of the regression equation is weak. What else can be deduced?

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This is a classic confusion. An r-value indicates how accurately X (linearly) predicts Y in your particular data set. A P-value indicates how likely it is for an r-value of that magnitude to have occurred by chance in a world where X and Y are not actually (linearly) related.

If you have a small data set, it's easy to get a large r-value by chance: Suppose you observe two men with blue eyes and one woman with green eyes. That's an r=1 correlation between sex and eye color! Should you be highly confident that sex predicts eye color?

Contrapositively, if you have a large data set, it's quite possible to observe a very weak correlation, but still be very confident that it didn't occur by chance (i.e. have a low P-value): Suppose you measure the heights of a million men and women. The typical scatter in human heights is larger than the difference in male and female averages, so you would not expect to get a very high r-value. But you would get a fantastically small P-value, because even though the height difference between the sexes is relatively small, there is very definitely a real difference between those averages.

One pity formulation of this: "statistical significance is not the same a practical significance".

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  • $\begingroup$ Very interesting thoughts! How should I deduce my findings in each of the above cases? $\endgroup$
    – Vyas
    Mar 29, 2017 at 0:12
  • $\begingroup$ Do I need to first look at the p-value, if it is significant (p<0.05) then I can go ahead an check for r and deduce that there is statistical significance (i.e. a high probability that a relationship exist between X and Y, and write down whether it is weak, moderate or strong. (ii) By contrast, if P>0.05, then even if r=0.9 (relationship is strong between X and Y), do I ignore the strong relationship, because such an equation between X and Y will not exist (as it is not significant). Kindly clarify $\endgroup$
    – Vyas
    Mar 29, 2017 at 0:19
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    $\begingroup$ Basically, yes. The one thing I would add is that the P=.05 threshold isn't some sort of absolute border between truth and falsity. If I got a strong correlation with P=.06, I would be inclined to try to take more data to see if the P improves. Similiarly, if I got a weak correlation at the P=.04 level, I wouldn't be super confident that this wasn't one of those 5% of cases in which I should expect a false positive, and I would be inclined to try to take more data to make sure the effect didn't go away. $\endgroup$ Mar 29, 2017 at 5:47

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