Many similar questions here, but this question combines two threads: relationship between correlation and regression, and correlation between a binary and a continuous variable.

  1. Since relationship between a binary variable and a continuous variable is not linear, is Pearson correlation ever appropriate measure of whether these two variables move together? Several sources mention that Point Biserial is equivalent to Pearson correlation in case one variable is binary and the other continuous. So is point Biserial correlation identical to Pearson correlation in this case? Is Point Biserial more appropriate/the most appropriate?
  2. Is it still the case that R-squared from OLS equals the square of Pearson correlation, if dependent variable is binary and independent is continuous? Why?
  3. Is there some type of regression between these variables for which R-squared is square of Point Biserial correlation?
  • 1
    $\begingroup$ One crucial difference is that correlation is 'symmetrical': $Cor(X,Y) = Cor(Y,X).$ However, regression is not: Regression line of $Y$ on $X$ is not at all the same as regression line of $X$ on $Y.$ $\endgroup$ – BruceET Jan 24 '19 at 22:56

When you have Y = 0, 1 and X is a continuous, you must check the dependence by means of a t test (special case of a ANOVA) (if X is normal or if you have a significative sample size)

  • $\begingroup$ I believe that would tell me if the means are statistically different. I am interested in how the two variables co-vary about their means. $\endgroup$ – user497996 Jan 24 '19 at 22:27
  • $\begingroup$ How which two variables co-vary, the predictor and the response? $\endgroup$ – Dave Jun 18 '20 at 11:06

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