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I have read this link.

Pearson's or Spearman's correlation with non-normal data

I cannot comment on the original post. Therefore I ask here:

I have two variables are not normally distributed. I implement different methods to calculate the first variables in different formats, and calculate the Pearson's correlation between the first and the second variables.

  1. If the Pearson's correlation calculated by Method A, is much higher than Method B. Can I say Method A is better than Method B?

  2. If the Pearson correlation is very high between two not normally distributed variables, 0.8, for example. Can I say these two variables are linearly related?

06-12-14:50

More details

This is my project for measuring the sense relatedness between synonyms

I extract sentences containing those synonyms, transfer the neighbouring words appearing in those sentences to vectors by different methods (e.g. TF-IDF, using PMI to do feature selection), calculate the cosine distance between different vectors. These are the first variables.

The second variables are the sense relatedness from this paper (Page 628)

Rubenstein, Herbert, and John B. Goodenough. "Contextual correlates of synonymy." Communications of the ACM 8.10 (1965): 627-633.

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  • $\begingroup$ This is all rather cryptic. What methods? What variables? In what way are they not normal? $\endgroup$ – Peter Flom Jun 12 '14 at 13:46
  • $\begingroup$ More details are included $\endgroup$ – Ming Li Jun 12 '14 at 13:53
  • $\begingroup$ What I understood is: you have different methods to compute sample values for variable X. We denote with X_A the variable produced by method A, and so on. We have also sample values for variable Y which are taken from a paper. And you use corr(X_A,Y) and corr(X_B,Y) to infer results? $\endgroup$ – rapaio Jun 12 '14 at 14:26
  • $\begingroup$ @rapaio Yes, it is. $\endgroup$ – Ming Li Jun 12 '14 at 16:32
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If the Pearson's correlation calculated by Method A, is much higher than Method B. Can I say Method A is better than Method B?

Strictly speaking no. For the purpose of prediction, perhaps. For the purpose of measurement, then we'd need to know if you have calibrated the tool.

For example, a person is using a measurement tape to measure the length of different sticks. However, instead of starting at 0", the person started at 5", so all the measurements were boosted up by 5 inches. In this case, the measured length and the actual length might be nearly perfectly linear, but all the measurements are biased.

In a nut shell, your choice of word "better" can be of so many connotations that it's better (no pun intended) to explicitly describe what property of the model is being compared.

If the Pearson correlation is very high between two not normally distributed variables, 0.8, for example. Can I say these two variables are linearly related?

No, Pearson correlation is estimated by first dividing the data into four parts by mean of x and mean of y. The data in the top right and lower left quadrants contribute to the positivity of the coefficient side and the data in the top left and lower right contribute to the negativity of the coefficient. Since means are sensitive to extreme values, if your distribution is far from normal, the result will not be trust worthy because a handful of outliers can change your correlation magnitude or even sign. See Anscombe's quartet for a demonstration.

Linear relationship cannot be supported by Pearson's correlation alone, it's important to also visually check the data and have a good theorized reason behind to support the linear relationship.

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