R-squared is a statistic that often accompanies regression output. It ranges in value from 0 to 1 and is usually interpreted as summarizing the percent of variation in the response that the regression model explains. So an R-squared of 0.65 might mean that the model explains about 65% of the variation in our dependent variable. Given this logic, we prefer our regression models have a high R-squared.

In R, we typically get R-squared by calling the summary function on a model object. Here’s a quick example using simulated data:

# independent variable
x <- 1:20 
# for reproducibility
set.seed(1) 
# dependent variable; function of x with random error
y <- 2 + 0.5*x + rnorm(20,0,3) 
# simple linear regression
mod <- lm(y~x)
# request just the r-squared value
summary(mod)$r.squared          
## [1] 0.6026682

One way to express R-squared is as the sum of squared fitted-value deviations divided by the sum of squared original-value deviations:

\[ R^{2} = \frac{\sum (\hat{y} – \bar{\hat{y}})^{2}}{\sum (y – \bar{y})^{2}} \]

We can calculate it directly using our model object like so:

# extract fitted (or predicted) values from model
f <- mod$fitted.values
# sum of squared fitted-value deviations
mss <- sum((f - mean(f))^2)
# sum of squared original-value deviations
tss <- sum((y - mean(y))^2)
# r-squared
mss/tss                      
## [1] 0.6026682

It is important to recognize that R-Squared is not a “good” measure of fit because of how often it is used in research. R-Squared is a very common measure used in understanding data, but it is very misleading, as seen in this example:

R-squared says nothing about prediction error, even with \(σ^2\) exactly the same, and no change in the coefficients. R-squared can be anywhere betweinen 0 and 1 just by changing the range of X. We’re better off using Mean Square Error (MSE) as a measure of prediction error.

MSE is basically the fitted y values minus the observed y values, squared, then summed, and then divided by the number of observations.

Let’s demonstrate this statement by first generating data that meets all simple linear regression assumptions and then regressing y on x to assess both R-squared and MSE.

x <- seq(1,10,length.out = 100)
set.seed(1)
y <- 2 + 1.2*x + rnorm(100,0,sd = 0.9)
mod1 <- lm(y ~ x)
summary(mod1)$r.squared
## [1] 0.9383379
# Mean squared error
sum((fitted(mod1) - y)^2)/100
## [1] 0.6468052

Now repeat the above code, but this time with a different range of x. Leave everything else the same:

 # new range of x
x <- seq(1,2,length.out = 100)      
set.seed(1)
y <- 2 + 1.2*x + rnorm(100,0,sd = 0.9)
mod1 <- lm(y ~ x)
summary(mod1)$r.squared
## [1] 0.1502448
# Mean squared error
sum((fitted(mod1) - y)^2)/100        
## [1] 0.6468052

The R-squared falls from 0.94 to 0.15 but the MSE remains the same. In other words the predictive ability is the same for both data sets, but the R-squared would lead you to believe the first example somehow had a model with more predictive power.

Instead of using R-Squared as a measure of fit, I would encourage you to use another method, such as MSE.

This content is from: University of Virginia Library