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In linear regression, we often get multiple R and R squared. What are the differences between them?

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Capital $R^2$ (as opposed to $r^2$) should generally be the multiple $R^2$ in a multiple regression model. In bivariate linear regression, there is no multiple $R$, and $R^2=r^2$. So one difference is applicability: "multiple $R$" implies multiple regressors, whereas "$R^2$" doesn't necessarily.

Another simple difference is interpretation. In multiple regression, the multiple $R$ is the coefficient of multiple correlation, whereas its square is the coefficient of determination. $R$ can be interpreted somewhat like a bivariate correlation coefficient, the main difference being that the multiple correlation is between the dependent variable and a linear combination of the predictors, not just any one of them, and not just the average of those bivariate correlations. $R^2$ can be interpreted as the percentage of variance in the dependent variable that can be explained by the predictors; as above, this is also true if there is only one predictor.

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    $\begingroup$ So if in a multiple regression R^2 is .76, then we can say the model explains 76% of the variance in the dependent variable, whereas if r^2 is .86, we can say that the model explains 86% of the variance in the dependent variable? What's the difference in their interpretation? $\endgroup$ – wizlog Jun 8 '16 at 14:50
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Multiple R actually can be viewed as the correlation between response and the fitted values. As such it is always positive. Multiple R-squared is its squared version.

Let me illustrate using a small example:

set.seed(32)
n <- 100
x1 <- runif(n)
x2 <- runif(n)
y <- 4 + x1 - 2*x2 + rnorm(n)

fit <- lm(y ~ x1 + x2)
summary(fit) # Multiple R-squared:  0.2347

(R <- cor(y, fitted(fit))) # 0.4845068
R^2                        # 0.2347469

There is no need to make a big fuss around "multiple" or not. This formula always applies, even in an Anova setting. In the case where there is only one covariable $X$, then R with the sign of the slope is the same as the correlation between $X$ and the response.

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I simply explain to my students that:

  1. the multiple R be thought of as the absolute value of the correlation coefficient (or the correlation coefficient without the negative sign)!

  2. The R-squared is simply the square of the multiple R. It can be through of as percentage of variation caused by the independent variable (s)

It is easy to grasp the concept and the difference this way.

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