Both correlation and linear regression explain the linearity in data but to get a high correlation coefficient the data must be linear with a slope close to 1. In some cases you can have linear data that can be fit on a regression line with a slope less than one, in which case the correlation coefficient will be low. My question is, should not we consider linear regression rather than correlation?

  • $\begingroup$ When we understand "slope" to mean the standardized coefficient (obtained by first standardizing both variables to have unit variances), your initial assertion is correct. But the mathematical relationship between the results of two different statistical procedures, having different assumptions and different aims, doesn't imply those procedures are interchangeable! $\endgroup$
    – whuber
    Jun 24 '20 at 13:28

Your conjecture that the correlation is only one for slope one is wrong, as you can easily test with data on a line with slope 0.5:

cor(c(2,4,6), c(1,2,3))

This returns 1 because the correlation is 1 whenever all data points lie exactly on a line with slope greater than zero.

  • $\begingroup$ The qualifier "iff" is quite correct but may not be familiar to all readers who would be interested in this thread. I have used a simple English word. $\endgroup$
    – Nick Cox
    Jun 20 '20 at 9:51
  • $\begingroup$ @Nich Cox: Thanks. I should have written it out: "if and only if", but non-matematicians presumably would not understand that either. "Whenever" only implies a sufficient condition, but maybe colloquially it can mean "iff", so this improves the understandability. $\endgroup$
    – cdalitz
    Jun 20 '20 at 10:02
  • $\begingroup$ I am a non-mathematician -- no formal study of the subject beyond age 17 -- and understood your post fine.... I was taught the jargon and the ideas of necessary and sufficient conditions at age 13. But we agree: making the answer widely understandable is the key. $\endgroup$
    – Nick Cox
    Jun 20 '20 at 10:07
  • $\begingroup$ @NickCox Thank you. I don't know where did get the idea that correlation coefficient of 1 means a slope one. $\endgroup$ Jun 20 '20 at 19:22

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