A seeker of knowledge inquires:

Q: Can you give me a good layman's term "working" definition of R squared? Questions that pop up are why is it important? What does it reflect? Is it predictive?

A: R Squared is simply the square of the correlation coefficient. As most will recall the correlation coefficient can range from +1 for a strong positive correlation to -1 for a strongly negative relationship between two variables. Two variables which are unrelated will usually have a correlation around zero.

So when we square the correlation coefficient we get a number between 0 and +1. Remember that the negative correlations become positive so there are no more negative numbers. We also almost always get a smaller absolute number because multiplying two numbers less than 1 always gives a small number (except for zero and 1).

There is another interpretation. The R^2 also generally associated with a regression model (but need not be). The R^2 can be thought of as representing the percent of variance which is explained by the model. Mathematically things are linear in the variance but not in the square roots such as the standard deviation and correlation. So we can decompose the total variance of a regression into the part that is explained by the model and the part that is unexplained (the error or residual variance). The three relationships are:

Total variance  =  Explained Variance  +  Unexplained variance
    100%       =        R^2           +       (1 - R^2)
Total SSq       =  Explained SSq       +  Unexplained SSq

In the last line the term SSq means the Sum of Squares. The sum of squares relationship simply comes about because the variance is simply the average sum of squares. The bottom line is that if you have an R^2 of 25% you know that it explains 25% of the variance in the variable you wish to predict. You also know that the correlation coefficient is +/-50% because .5 * .5 = .25. Conversely if you know that a correlation coefficient is 90% then you know the R Squared will be 81%.

From just the R Squared you do not know if the correlation is positive or negative however. For that you have to look at the beta coefficient of the regression which tells you which sign to choose for the correlation coefficient. 

Yishen Kuik adds:

I've found an intuitive interpretation of correlation coefficient (R) to be a measure of how in phase two datastreams are.

For two dataseries X and Y:

R = (sum of Zx * Zy)/(N-1), where Zx is the z-normalized series X and Zy is same for Y

Hence, the more the below-mean datapoints in X and Y coincide, the greater the value in the numerator, since the product of two negative Z scores is positive. The corollary is that above-mean datapoints will also coincide, and since the sumproduct of two positive numbers is also positive, also contributes to a larger numerator.

Hence I think of this coincidence of above/below mean datapoints as in phase.





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