# What is the distribution of changes in adjusted R squared when adding a random normal variable?

Suppose we have a model with $n$ data points and $p$ predictors. The model has an $R^2_{\text{adjusted}}=q$.

What sort of distribution does the change in $R^2_{\text{adjusted}}$ have when a random Gaussian predictor is added to the model?

I am assuming that the expected value is zero, since $R^2_{\text{adjusted}}$ is supposed to adjust for predictors that have no explanatory power. But I can't find the answer anywhere, and I'm having trouble working it out myself. ...

• Wikipedia confirms your supposition that the expectation will be zero. The distribution, as a linear transformation of a ratio of sums of squares, tends to be strongly skewed. You might consider beginning with the case of no predictors (except a constant) and consider what happens when you include one independent predictor. – whuber Jun 26 '13 at 4:15

Here's a quick simulation in R that gives you a feel for the distribution of $R^2_{adj}$ when adding a single random normal variable.
set.seed(1234)
summary(fit)$adj.r.squared } # Simulate adjusted r-square values simulations <- 1000 n <- 100 x <- sapply(seq(simulations), function(X) adj_r_simulation(n=n)) # Display results mean(x) sd(x) plot(density(x), main=paste0("n=",n, "; simulations=", simulations), xlab='Adjusted r-squared') abline(v=mean(x))  The simulation above simulates two independent normal variables, x and y of sample size n=100. It then fits a linear model predicting y from x and returns$R^2_{adj}$. This process is repeated 1000 times and the values are saved. The sample mean of simulated$R^2_{adj}$values is close to zero: > mean(x) [1] 0.0002181863 > sd(x) [1] 0.01425647  Below you can see the density plot of the simulated values of$R^2_{adj}\$. This illustrates whuber's point about the distribution being highly skewed.