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If I run Pearson correlation between Variable A (buyer/nonbuyer of ice-cream) and Variable B (buyer/nonbuyer of yoghurt) and have:

  • Ho as "There is no relationship between...."
  • Ha as "There is a strong relationship..."

then what would "strong" mean in this context?

Is the meaning of "strong" dependent on the context of my study or are there some guidelines on what is strong (as opposed to - I presume - "weak" or "average").

Is this something to do with "one-tail" and "two-tail" testing?

rolando2
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Adhesh Josh
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  • This is probably described in the software you use. Which do you? – Nick Sabbe Sep 15 '11 at 14:18
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    This formulation is unusual. The alternative to $H_0$ almost always is either (i) there is *some* relationship (no matter how tiny) or (ii) there is *some* positive relationship (no matter how tiny). – whuber Sep 15 '11 at 16:34
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    Not a full answer, but re: "Is this something to do with 'one-tail' and 'two-tail' testing?" No. That involves the direction of the relationship, not its magnitude. – bnaul Sep 15 '11 at 20:54
  • I changed Parker to Pearson in title, as this seemed to be what you meant – Peter Flom Sep 23 '12 at 23:57

3 Answers3

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An r of 0.822 means that the linear relationship between x and y accounts for $0.822^2 = 0.68$ or 68% of the variance in y.

Is this strong? In some fields (e.g. psychology), it would be amazingly strong. In others (e.g. some aspects of physics) it would be disappointingly weak.

The p < .001 means that, if, in the population from which this sample was drawn the true r was 0, then it is very unlikely that, in a sample the size of yours, you would get an r this high or higher.

Peter Flom
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    You might want to read up on the difference between 1 and 2 tailed tests, if this topic is not familiar to you. – Joel W. Sep 24 '12 at 00:28
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It absolutely depends on context. With pharmaceuticals, if use/non-use of a heart medicine is correlated at -.1 with occurrence/non-occurence of heart failure, this could save many lives and earn a drug maker millions of dollars. In contrast, in educational testing, if SAT - Critical Reading score is not correlated at at least .65 with SAT - Math score in a large population, people will get very suspicious that something is wrong with the tests, which are important for many admissions and placement decisions and are huge money-makers in that industry.

Also, it is not uncommon to test an hypothesis that an r is at least, say, .4, that is, not necessarily a tiny value.

rolando2
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Three more thoughts to complement P.Flom's excellent answer:

You should make a scatter-plot of the data to see if there is any obvious indication of a non-linear relationship. The Pearson r is an index of the strength of a linear relationship. For the Pearson r, a non-linear relationship is considered noise.

A scatter-plot will also help you look for outliers. Sometimes outliers are due to data entry errors. Sometimes outliers can drive high correlations. If you have outliers, check your data.

If you are working in a field where a correlation this high is quite unusual, review your data carefully to be sure you have not made an error that spuriously contributes to the high correlation.

Joel W.
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  • Indeed. For a good example of how this can happen, see the [Anscombe Quartet](http://en.wikipedia.org/wiki/Anscombe%27s_quartet). Interestingly, the r for those data are about the same as in this question. – Peter Flom Sep 24 '12 at 00:50
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    A monotonic increasing function will have positive correlation even though it is nonlinear.But the correlation will be weaker than a linear function with the same residual noise variance. The Pearson correlation does measure degree of linearity. However the statement that for Pearson's r nonlinear functions are treated as noise is not a correct interpretation. Do look at the influence function for the correlation to find any specific points that have high influence on the correlation either for increasing or decreasing it. – Michael R. Chernick Sep 24 '12 at 01:42
  • See my answer in this very recent [post](http://stats.stackexchange.com/questions/37865/detecting-outliers) for more details. – Michael R. Chernick Sep 24 '12 at 01:42