Why does summary overestimate the R-squared with a "no-intercept" model formula

Question

I wanted to make a simple linear model (lm()) without intercept coefficient so I put -1 in my model formula as in the following example. The problem is that the R-squared return by summary(myModel) seems to be overestimated. lm(), summary() and -1 are among the very classic function/functionality in R. Hence I am a bit surprised and I wonder if this is a bug or if there is any reason for this behaviour.

Here is an example:

x <- rnorm(1000, 3, 1)
mydf <- data.frame(x=x, y=1+x+rnorm(1000, 0, 1))
plot(y ~ x, mydf, xlim=c(-2, 10), ylim=c(-2, 10))

mylm1 <- lm(y ~ x, mydf)
mylm2 <- lm(y ~ x - 1, mydf)

abline(mylm1, col="blue") ; abline(mylm2, col="red")
abline(h=0, lty=2) ; abline(v=0, lty=2)

r2.1 <- 1 - var(residuals(mylm1))/var(mydf$y)
r2.2 <- 1 - var(residuals(mylm2))/var(mydf$y)
r2 <- c(paste0("Intercept - r2: ", format(summary(mylm1)$r.squared, digits=4)),
        paste0("Intercept - manual r2: ", format(r2.1, digits=4)),
        paste0("No intercept - r2: ", format(summary(mylm2)$r.squared, digits=4)),
        paste0("No intercept - manual r2: ", format(r2.2, digits=4)))
legend('bottomright', legend=r2, col=c(4,4,2,2), lty=1, cex=0.6)

Answer 1

Your formula is wrong. Here is what I do in fastLm.R in RcppArmadillo when computing the summary:

## cf src/library/stats/R/lm.R and case with no weights and an intercept 
f <- object$fitted.values    
r <- object$residuals         
mss <- if (object$intercept) sum((f - mean(f))^2) else sum(f^2)     
rss <- sum(r^2)   

r.squared <- mss/(mss + rss) 
df.int <- if (object$intercept) 1L else 0L    

n <- length(f)     
rdf <- object$df     
adj.r.squared <- 1 - (1 - r.squared) * ((n - df.int)/rdf)      

There are two places where you need to keep track of whether there is an intercept or not.

Answer 2

Oh yeah, I fell into this trap too! Very good question!! It is because

and

• in case of model with intercept (your mylm1), the y̅ is mean(y_i) - this is what you expect, this is the SS_tot you basicly want for proper R²
• whereas in case of model without intercept, the y̅ is taken as 0 - so the SS_tot will be very high, so the R² will be very close to 1! SS_res will differ according to the worse fit (will be little higher without intercept), but not much.

Code:

attach(mylm1) # in general be careful with attach, here only for code clarity

y_fit <- mylm1$fitted.values
SSE <- sum((y_fit - y)^2)
SST <- sum((y - mean(y))^2)
1-SSE/SST  # R^2 with intercept

y_fit2 <- mylm2$fitted.values
SSE2 <- sum((y_fit2 - y)^2) # SSE2 only slightly higher than SSE..
SST2 <- sum((y - 0)^2)  # !!! the key difference is here !!!
1-SSE2/SST2 # R^2 without intercept

Note: It is not clear to me why in the model without intercept the y̅ is 0 and not mean(y_i), but that's how it is. I myself found out hard way by investigating and hacking with the above code..