# How do R-squared and p(F-statistic) impact the fitness of a linear regression model?

Given below the attached summary statistics of a linear regression model.

Noting that Adjusted R-square is only 15% but p(F-statistic) is very low (almost equal to zero):

• Can it be said that it is a significant fit?
• Can it be said that the relationship is linear?

$$R^2=0.15$$ might not sound so great, but it might be quite acceptable for what you're doing. If that value is too low to be interesting, then you have a practically insignificant result resulting in a statistically significant p-value due to a large sample size that detects subtle differences. This means that there probably is a difference (statistical significance) but not enough of a difference for the scientist to care (practical significance).

You're not a lazy or negligent scientist to say, "Sure, $$p<\alpha$$, but it's not enough for me to be interested. They're basically the same."

1) Yes, your fit is significantly better than just estimating every value as the mean of all response variables pooled together. This is what the p-value tells you.

2) No, you definitely cannot conclude that. Imagine fitting a line to the right half of a parabola. You will get some kind of fit, but the relationship between the predictor and response certainly isn't linear. Let's look at some R code.

set.seed(2020)
x <- seq(0,3,0.01)
err <- rnorm(length(x),0,1)
y <- x^2 + err
plot(x,y)
L <- lm(y~x^2)
lines(x,predict.lm(L),col='red')
summary(L)


I get $$R^2 = 0.933$$ and $$p<2.2\times 10^{-16}\approx 0$$, which sound pretty good, but the plot shows a clearly nonlinear relationship.